Investigation of Photon Counting Pixel Detectors for X-Ray Spectroscopy and Imaging

Investigation of photon counting pixel detectors for X-ray spectroscopy and imaging

Der Naturwissenschaftlichen Fakult¨at der Friedrich-Alexander-Universit¨at Erlangen-Nurnberg¨ zur Erlangung des Doktorgrades Dr.rer.nat.

vorgelegt von Patrick Takoukam Talla aus Yaounde/Kamerun Als Dissertation genehmigt von der Naturwissen- schaftlichen Fakult¨at der Universit¨at Erlangen-Nurnberg¨

Tag der mundlichen¨ Prufung:¨ 07. April 2011 Vorsitzender der Promotionskommission: Prof. Dr. Rainer Fink Erstberichterstatter: Prof. Dr. Gisela Anton Zweitberichterstatter: Prof. Dr. Valeria Rosso Dedication

To my wife Sandrine Takoukam Talla To my parents Jean Baptiste Talla and Monique Talla To my late brother Guy Bertrand Wafeu Talla

Contents

1 Introduction 1

2 Interaction of X-rays with Matter 3 2.1 PhotoelectricEﬀect...... 3 2.2 ComptonScattering ...... 4 2.3 RayleighScattering...... 5 2.4 PairProduction ...... 5 2.5 Interaction of Electrons with Matter ...... 6 2.6 Conclusion ...... 7

3 The Medipix2 Detector 9 3.1 Description and operating Modes ...... 9 3.2 CountingPrinciple ...... 11 3.3 ChargeSharing...... 12 3.4 Conclusion ...... 15

4 The Medipix3 Detector 17 4.1 Motivation ...... 17 4.2 Description ...... 18 4.3 Functionalities and operating Modes ...... 20 4.4 ChargeSummingMode ...... 20 4.5 Conclusion ...... 21

5 Monte Carlo Simulations and Energy Responses of Medipix Detectors 23 5.1 MonteCarloToolROSI ...... 23 5.2 Energy Response of the Medipix2 Detector ...... 24 5.3 Energy Response of the Medipix3 Detector ...... 27 5.4 Impact of Noise Contributions on the Energy Resolution of Medipix Detectors 29 5.5 Conclusion ...... 31

6 Characterization of the Medipix3 Detector 33 6.1 Test Pulses Measurements to determine Optimal DAC Settings ...... 33 6.2 Equalisation Procedure of the Medipix3 Detector ...... 37 6.3 Energy Calibration of the Medipix3 Detector ...... 40 6.4 Energy Resolution of the Medipix3 Detector ...... 42 6.5 Parameters Extraction for Monte Carlo Simulations ...... 43 6.6 Energy Response of the Medipix3 Detector in Charge SummingMode . . 43 6.7 Count Rate Linearity of the Medipix3 Detector ...... 45 6.8 Conclusion ...... 47

iii Contents

7 Spectrum Reconstruction with hybrid Photon counting Detectors 49 7.1 Motivation ...... 49 7.2 Theory...... 50 7.3 Methods...... 51 7.4 Reconstruction with the Medipix2 Detector ...... 57 7.5 Reconstruction with the Medipix3 Detector in Charge Summing Mode . . 67 7.6 Conclusion ...... 71

8 Determination of the kVp with the Medipix2 Detector 73 8.1 Determination of the kVp with multiple Filter Combinations ...... 74 8.2 Determination of the kVp with only one Filter Combination...... 80 8.3 Conclusion ...... 81

9 Introduction to X-ray Imaging 83 9.1 ImageQualityMetrics ...... 83 9.2 Imaging with the Medipix3 Detector ...... 85 9.3 Conclusion ...... 89

10 Spatial Resolution of the Medipix3 Detector 91 10.1Methods ...... 91 10.2 Spatial Resolution of the Medipix3 Detector in Single Pixel Mode . . . . . 93 10.3 Spatial Resolution of the Medipix3 Detector in Charge SummingMode . 96 10.4Conclusion ...... 101

11 Material Reconstruction with Photon counting Detectors 103 11.1 Theory of Material Reconstruction ...... 103 11.2 CombinationMethod...... 105 11.3 Minimization Algorithms ...... 106 11.4 Reconstruction Results with Medipix2 and Medipix3 ...... 108 11.5 Material Reconstruction of the Solder Bumps of Medipix ...... 115 11.6Conclusion ...... 118

12 Redesign of Charge Summing Mode of Medipix3 119 12.1 New Architecture of Charge Summing Mode of Medipix3 ...... 119 12.2 Simulation of the proposed architecture ...... 120 12.3 Impact of the new Charge Summing Mode architecture on Imaging . . . . 120

13 Summary and Outlook 123

14 Zusammenfassung und Ausblick 125

Acknowledgements 127

Literaturverzeichnis 139

iv List of Abbreviations

ASIC Application Speciﬁc Integrated Circuit CMOS Complementary Metal Oxide Semiconductor MPX Medipix PCB Printed Circuit Board DAC Digital Analog Converter CSDA Continuous Slowing Down Approximation SPM Single Pixel Mode CSM ChargeSummingMode HGM High Gain Mode LGM Low Gain Mode CM Color Mode RW Read/Write Mode SNR Signal to Noise Ratio MTF Modulation Transfer Function PSF Point Spread Function LSF Line Spread Function ESF Edge Spread Function NPS Noise Power Spectrum DQE Detective Quantum Eﬃciency ROI Region of Interest

1 Introduction

X-rays and gamma-rays were discovered respectively by W. C. Röntgen in 1895 and by Paul Villard in 1900. Since then, they are used in a wide range of applications e.g. in medicine or in industry for Non Destructive Testing. Nowadays, seventy percent of all medical inspections in imaging are carried out using X-rays [1]. They are electromagnetic waves like radiowaves or light but have a much smaller wave length. In medical radiography the patient is irradiated using X-rays and the attenuated intensity is measured. This attenuation is material and energy dependent. This enables one to distinguish for example between soft tissues and bones. Films are usually used for the detection of X-rays in projective X-ray diagnostic. The rapid development in electronics in the last decades allows the digitization of the detection signals and therefore a jump from integrating to photon counting pixel detectors like the Medipix2 detector. With this detector, we can count the incoming photons and gain at the same time information about their energy. Those properties of the detector enable for example the reconstruction of a polychromatic spectrum impinging on to the detector. A major drawback of the Medipix2 detector is that it suffers from charge sharing: the charge carriers produced by one photon can be distributed over several pixels. Therefore, an incoming photon can be detected by more than one pixel. As a consequence, the incoming and the measured spectrum are different. In order to suppress the influence of charge sharing the Medipix3 detector was developed. With its Charge Summing Mode it is able to correct for charge sharing in real time. The aim of this thesis is a detailed characterization of the detectors of the Medipix family. The first three chapters are about the basic interactions of radiation with matter and the introduction to photon counting detectors of the Medipix family. Chapter 5 intro- duces the Monte Carlo Simulation Tool ROSI and explains how the response functions of the Medipix detectors to monochromatic radiation are modelled. Characterization of the brand new Medipix3 detector in measurements is the main focus of chapter 6. Chapter 7 investigates the spectroscopic properties of the Medipix detectors and chapter 8 is about using the Medipix2 detector for quality assurance and constancy checks. Chapter 9 intro- duces metric quantities used to characterize an imaging system. The spatial resolution of the Medipix3 detector is presented in chapter 10. Chapter 11 focuses on energy resolved material reconstruction with Medipix detectors. The last chapter is about the redesign of Charge Summing Mode of Medipix3.

2 Interaction of X-rays with Matter

Contents 2.1 PhotoelectricEﬀect ...... 3 2.2 ComptonScattering...... 4 2.3 RayleighScattering ...... 5 2.4 PairProduction ...... 5 2.5 InteractionofElectronswithMatter...... 6 2.6 Conclusion ...... 7

This chapter gives an overview of the interactions which occur when X-ray photons encounter matter. Photoelectric effect, Compton scattering, coherent scattering and pair production will be presented in the first part of the chapter. In fact these processes are the basis of all current photon detection devices and thus determine the sensitivity and the efficiency of a detector. The last part of the chapter focuses on the interaction of electrons with matter.

2.1 Photoelectric Eﬀect

In the photoelectric absorption process, a photon undergoes an interaction with an atom in which the photon completely disappears (see ﬁgure 2.1). An energetic photoelectron is released by the atom from one of its bound shells. For a photon of suﬃcient energy, the most probable origin of the photoelectron is the K-shell of the atoms since a free electron cannot absorb a photon and also conserve momentum. The recoil momentum is then absorbed by the nucleus. The energy of the ejected electron is given by [2]:

E − = hν Eb (2.1) e − where Eb is the binding energy of the photo electron in its original shell, h the Planck constant, ν the frequency of the incident photon. The ejection of the photoelectron leaves an ionised atom. The vacancy is ﬁlled through rearrangement of electrons from other shells of the atom. This leads to characteristic X-ray photons or to the ejection of auger electrons. In general the cross section of the photoelectric eﬀect increases with the atomic number Z and decreases with the energy E. For photon energies below 100 keV, a rough approximation of the cross section is given by [3]:

Z4 σpe (2.2) ∝ E3

3 2 Interaction of X-rays with Matter

Incoming photon Photoelectron from an inner shell + +++

Figure 2.1 Illustration of photoelectric eﬀect [4].

2.2 Compton Scattering

Compton scattering takes place between an incident photon and an electron of the outer atomic shell. In this process, the incoming photon is deﬂected through an angle θ with respect to its original direction (ﬁgure 2.2). The photon transfers a portion of its energy E to the electron. Using the equations for the conservation of energy and momentum, the energy E′ of the photon after the scattering is then [2]:

′ E E = E (2.3) 1+ 2 (1 cos θ) m·c · − θ is the scattering angle, c the speed of light, m the mass of the electron. The diﬀerential

Scattered electron Incoming photon from an outer shell

Scattered photon

+ +++

Figure 2.2 Illustration of Compton scattering [4].

dσ cross section dΩ for Compton scattering can be calculated with the Klein-Nishina formula and is given by [2]:

dσ 1 1 + cos2 θ α2(1 cos θ)2 = Zr2 1+ − dΩ 0 1+ α(1 cos θ) 2 (1 + cos2 θ)[1 + α(1 cos θ)] − −

4 2.3 Rayleigh Scattering

2 where α = hν/mc . r0 is the classical electron radius. The total cross section shows a weak energy dependence compared to the photoelectric eﬀect.

2.3 Rayleigh Scattering

Rayleigh scattering (ﬁgure 2.3) describes photon scattering by atoms as a whole. It is also called coherent scattering as the electrons of the atom contribute to the interaction in a coherent manner. No energy is transferred to the material. This elastic process changes only the direction of the incoming photon. The cross section σR for Rayleigh scattering decreases almost quadratically with the energy E of the incident photon and increases quadratically with the atomic number Z:

Z2 σR (2.4) ∝ E1.9

Incoming photon

+ +++ Elastic scattered photon

Figure 2.3 Illustration of Rayleigh scattering [4].

2.4 Pair Production

A gamma ray photon with an energy of at least 1.022 MeV can create an electron-positron pair when it is under the influence of the strong Coulomb field surrounding the nucleus (see figure 2.4). In this interaction the nucleus receives a very small amount of recoil energy to conserve momentum, but the nucleus is otherwise unchanged and the gamma ray photon disappears [2]. If the gamma ray energy exceeds 1.022 MeV, the excess energy is shared between the electron and positron as kinetic energy. Figure 2.5 depicts the dependence of photoelectric effect, Compton scattering and pair production upon photon energy for different atomic numbers Z. For silicon (dashed lines), photoelectric effect dominates for energies below 57 keV. Above this limit, Compton scattering is predominant. Coherent scattering is not taken into account since its cross section is negligible compared to the cross sections of the three other processes. In fact, only the photoelectric absorption and the Compton effect will be important in the energy range (10 keV - 160 keV) used in X-ray imaging for medical applications.

5 2 Interaction of X-rays with Matter

Positron Incoming photon

+ +++ Electron

Figure 2.4 Illustration of pair production [4].

Figure 2.5 Regions of predominance of photoelectric effect, Compton effect, and pair production as function of photon energy and for different atomic numbers Z. The region of predominance of photoelectric effect increases with the atomic number Z. As a result, high Z materials are indicated as sensor material for radiation detection [5].

2.5 Interaction of Electrons with Matter

The electrons that are released through interaction of primary photons with matter, can loose their energy through collisions with shell electrons. The transferred energy, when suﬃcient, can promote electrons from the valence band into the conduction band, resulting in electron-hole pairs in a semi conductor material. For electrons, due to their small mass, another energy loss mechanism comes into play: the emission of electromagnetic radiation (Bremsstrahlung) that may be understood as radiation arising from the acceleration of the electron by the electrical attraction of the nucleus [6]. The total energy loss of electrons Stot, therefore, is the sum of collision stopping power Scoll and radiative stopping power Srad [7]:

Stot = Scoll + Srad (2.5)

6 2.6 Conclusion with

dE Z m Scoll = ρ (2.6) dx ∝ · A · E !coll and

dE Z2 e 2 Srad = ρ E (2.7) dx ! ∝ · A · m · rad where ρ is the density, Z the charge number, A the atomic mass number of the medium. m is the electron mass, e the elementary charge and E the energy. The energy loss of electrons per collision at energies of a few MeV is comparatively small, resulting in a quasi-continuous energy loss of electrons on a track with relatively large scattering angles. They follow a zigzag course through the medium [8] (see section 3.3).

2.6 Conclusion

This chapter gave a brief overview of interactions of photons and electrons with matter. The two predominant eﬀects are Compton and photoelectric eﬀect in the energy range relevant in medical X-ray imaging.

3 The Medipix2 Detector

Contents 3.1 DescriptionandoperatingModes...... 9 3.2 CountingPrinciple...... 11 3.3 ChargeSharing...... 12 3.3.1 PathofanElectronintheSensorMaterial ...... 12 3.3.2 PhotonmultipleInteractions ...... 13 3.3.3 LateralChargeDiﬀusion...... 13 3.3.4 Repulsion ...... 14 3.4 Conclusion ...... 15

Nowadays, there are several types of devices to detect radiation. In X-ray radiography the film is used. With the rapid advances in microelectronics, the film is being progressively replaced by digital systems. We can distinguish between two classes of digital detectors. The first class is based on the integrating principle: charge carriers that are released in the detector are added up over the frame time and digitized through analog digital converters. The second category makes use of the photon counting principle. The first photon counting semiconductor detector of the Medipix family was developed in the nineties of the last century. It was based on the idea of the active pixel detector like the one in the LHC1/omega3 [9] experiment. The first generation was the Medipix1 detector, developed in the framework of the international Medipix collaboration [10]. After the success of the Medipix1 detector, the collaboration decided to develop the Medipix2 detector with higher performance and a smaller pixel pitch for a better spatial resolution for imaging purposes. After few years, other chips were designed: Medipix2- MXR, Timepix and recently Medipix3. The latter has more functionalities and operation modes than its predecessors. This chapter will focus on the Medipix2 detector.

3.1 Description and operating Modes

The Medipix2 detector is a pixelated, direct converting photon counting semiconductor ñ detector. It has 256 256 pixel with a pixel pitch of 55 ñm. The use of 0.25 m CMOS × (Complementary Metal Oxide Semiconductor) technology enables the implementation of more than 33 millions of transistors over the active area (1.982 cm2) of the chip. The detector is designed in a hybrid technology. Hybrid means that sensor material and

readout electronics (ASIC) are produced separately. An advantage is that diﬀerent sensor

ñ ñ materials (Si, GaAs, CdTe) and sensor thicknesses (300 ñm, 700 m, 1000 m) can be used depending on the speciﬁc application: for low-energy X-ray, silicon sensors are used whereas the other materials mentioned are appropriate for higher energies due to their

9 3 The Medipix2 Detector higher quantum efficiency. Sensor and ASIC are connected via solder bumps made of Ag/Sn or Pb/Sn as depicted in figure 3.1. Figure 3.2 shows a block diagram of a Medipix2 pixel cell. The diagram exhibits an analog and a digital part: the analog part comprises a charge sensitive preamplifier, two discriminators (a Low threshold THL (DiscL) and a high threshold THH (DiscH) ) and 2 Digital to Analog Converters (DAC) for threshold equalization. The digital part consists of a pulse processing circuitry (Double Disc Logic DDL) and a Shift Register that acts as a 13 bit counter during data acquisition. The digital output of the discriminators are given as input to the DDL that enables two operating modes: photons with an energy above a defined THL can be registered (Single Threshold Mode) or only photons with an energy in a defined THL-THH window (Energy Window Mode) are counted. Table 3.1 summerizes the characteristics of the Medipix2 detector.

Figure 3.1 Schematic view of a Medipix assembly [11]. The sensor is connected to the ASIC via bump bonds.

Table 3.1 Characteristics of the Medipix2 detector [5]. ñ Pixel size 55 ñm 55 m × Number of pixels 256 256 × Sensitive area 1.98 cm2 (87 % of the total area) ∼ ENC (Electronic Noise Charge) 140 e− ∼ Pixel count rate up to 1 MHz Threshold linearity < 3 % to 100k e− Threshold spread (not adjusted) 500 e− Threshold spread (adjusted) 100 e−

10 3.2 Counting Principle

Figure 3.2 Electronic circuitry of a Medipix2 pixel cell. The analog part consists of a preampliﬁer and two discriminators. The digital part receives the output of the discriminators and decides in the DDL (Double Disc Logic), if the counter will be incremented. This circuitry is also responsible for the readout of the data [12].

3.2 Counting Principle

An incoming X-ray photon impinging on the detector interacts with the sensor material through Compton or photoelectric effect. The electron that is released will loose its energy gradually in the sensor material. The energy loss, when sufficient, can promote electrons from the valence band into the conduction band: as a result electron-hole pairs are created. These charge carriers are separated and drifted towards the pixel electrodes using an external applied electrical potential difference. The voltage is typically 150 V

(for a 300 ñm Si sensor). On the electrodes’ side, mirror charges are created due to the motion of the carriers. As a consequence an electrical current is registered in the electrodes. The magnitude of the current can be determined using the so-called weighting potential (Ramo Theorem) [13]. The temporal integral over the current in a pixel electrode corresponds to the amount of charge collected by this electrode. For Medipix2, this current is preampliﬁed. After that the charge to voltage conversion is performed in the preampliﬁer integrating capacitance and compared directly with the energy threshold. If the pulse is above the threshold value, a counter is incremented. The Medipix2 detector

11 3 The Medipix2 Detector can therefore count photons with energies above a given threshold or within a deﬁned energy window.

3.3 Charge Sharing

Most of pixelated detectors based on planar technology suffer from the effect called charge sharing: this implies that a distribution of free charge carriers released by an electron in the sensor material can be collected by more than one pixel. Many effects can cause charge sharing: on one hand we can have multiple interactions of incoming photons, diffusion and repulsion of generated charge carriers, on the other hand the track of a primary electron can be extended over several pixels.

3.3.1 Path of an Electron in the Sensor Material

A released electron will deposit its energy progressively in the sensor material, since it is deviated (because of its charge) for example in the Coulomb ﬁeld of the nucleus of some atoms. As a consequence, the path of the electron is not a straight line and can be extended over several pixels. Depending on the threshold, the counter in more than one pixel will be incremented. Figure 3.3 shows the path of a 60 keV electron in silicon

calculated with the CSDA (Continuous Slowing down Approximation) [14]. It springs ñ from the ﬁgure that the track of the electron is around 20 ñm for a pixel pitch of 55 m. Thus electrons with higher energies will have their track extended over more than one pixel causing charge sharing. This fact can be deduced from ﬁgure 3.4 which depicts the path length as function of the electron energy in silicon.

6 m

µ 4 0

z in 2 −5 0 −10 −15 5 −20 µ 0 x in m y in µm

Figure 3.3 Illustration of the path of an electron in silicon. The extension of that path over several pixel leads to charge sharing.

12 3.3 Charge Sharing

160

140

120

100

CSDA range in µm 60

0 15 30 45 60 75 90 105 120 135 150 Energy in keV

Figure 3.4 CSDA range for electrons in silicon versus electron energy [15].

3.3.2 Photon multiple Interactions In some cases, an incoming ionizing particle can interact with the sensor material and release more than one electron. Figure 3.5 shows how a photon that impinges on the sensor at (0), interacts firstly through Compton effect (1) at a certain position in the sensor and then, the Compton scattered photon interacts through photoelectric effect (2) at another place. Each time a charge carrier distribution is created. The dashed circles represent the position of the collection of the charge carriers after the drift process. These diameters are bigger than the initial ones (solid lines) due to diffusion. In this example, depending on the threshold, the incoming photon will be detected by 0, 1, 2 or 3 pixel.

3.3.3 Lateral Charge Diffusion A released charge carrier distribution in the sensor material will experience a lateral diffusion while drifting towards the pixel electrodes. The solution of the diffusion equation for an initial point-like distribution is a Gaussian distribution as illustrated in figure 3.6.

The spread of this distribution in ñm is given by [17]:

σRadius = √2 Dt (3.1) where D is the diffusion coefficient of the charge carriers and t is the respective drift time. The diffusion coefficient is related to the temperature T and the mobility µ of the charge carriers through the following Einstein formula:

k T D = µ B (3.2) e An estimate of the upperlimit for the drift time of charge carriers is given by [18]:

d dsensor d tc(d)= · = (3.3) µ VBias µ E

13 3 The Medipix2 Detector

2 γein

γf

j-1

j+1

i-1 i i+1

Figure 3.5 Illustration of one effect leading to charge sharing [16]. An incoming photon (0) interacts with the sensor and creates at (1) through Compton and at (2) through photo effect a free electron. The bold dashed lines represent the path of the electron during the energy loss process. The dashed circles represent the end diameter of a charge cloud at the end of the drift process. The diameters due to diffusion are bigger than the initial diameters (solid circles) at the beginning of the drift process. dsensor and VBias denote the sensor thickness and the bias voltage, respectively. d is the drift distance. E is the electric field between the sensor and the electrodes and is given in V/cm. Taking into account equations 3.2 and 3.3, the spread due to diffusion (see equation 3.1) can be rewritten as follows:

kB T d σRadius = 2 r e · E

At room temperature, for drift distance in ñm and E in V/cm, the equation above becomes: d σRadius = 23 [µm] (3.4) rE Obviously, σRadius that denotes the spread of the charge cloud during the drift, increases with the sensor thickness. The thicker the sensor, the more probable is charge sharing at constant ﬁeld E.

3.3.4 Repulsion The released electron-hole pairs are separated through the applied electric ﬁeld. Electrons within a charge cloud push each other away (repulsion). The charge distribution due to

14 3.4 Conclusion d Sensor d Drift direction Sensor thickness Distance

σ Radius Pixel electrodes

Figure 3.6 Illustration of diﬀusion process for a punctual charge cloud while drifting towards the pixel electrodes adapted from M. Boehnel [19]. repulsion is considered to be cylindrical at the beginning and spherical at the end of the path [20]. The maximal repulsion radius is given in the cylindrical model by:

ρµt r (t)= (3.5) max 2πεε r 0 where ρ is the projected charge, t the driftime. ε and ε0 denote the permittivity in medium and vacuum, respectively. µ represents the mobility. In the spherical case, we have the following formula:

1 3Neµt 3 r (t)= (3.6) max,sph. 4πεε 0 N means the number of charge carriers within the cloud.

3.4 Conclusion

In this chapter the hybrid pixel detector Medipix2 was presented. Its adjustable threshold enables to gain information on energies of photons impinging on the detector. Medipix2 suffers from charge sharing. This effect is due to photon multiple interactions, extended paths of primary electrons over more than one pixel and to diffusion and repulsion effects. Charge sharing has an impact on the energy and spatial resolution of the detector. More details will be given in chapters 5 and 10. In order to correct for the charge sharing part due to diffusion, the Medipix3 detector was developed.

4 The Medipix3 Detector

Contents 4.1 Motivation ...... 17 4.2 Description ...... 18 4.3 FunctionalitiesandoperatingModes...... 20 4.4 ChargeSummingMode...... 20 4.5 Conclusion ...... 21

4.1 Motivation

The Medipix2 detector suffers from charge sharing mainly due to lateral diffusion of charge carriers released in the sensor material (see section 3.3). As a consequence, the energy information of incoming particles is distorted. Figure 4.1 shows a simulated response of the Medipix2 detector to 20 keV incoming photons. More details on the simulations of the energy response of photon counting detectors will be given in chapter 5. This response consists of 2 parts: The full energy peak at around 19.9 keV slightly shifted to lower energies and the tail due to charge sharing. This tail leads to a distortion in spectrum for reconstruction algorithms and to fixed pattern noise in the image. The Medipix3 detector was designed in order to correct for this distortion: With its Charge Summing Mode configuration the energy information should be properly restored.

17 4 The Medipix3 Detector

0.2 Intensity [a.u.] 0.1

0 0 5 10 15 20 25 30 35 Energy [keV]

Figure 4.1 Simulated energy response of Medipix2 to 20 keV incoming photons. The observed low energy tail is caused by the charge sharing eﬀect. The edge visible at 5 keV represents the discriminator threshold in the simulation.

4.2 Description

The Medipix3 detector is an improvement of the Medipix2 detector with additional fea- tures and operation modes. Figure 4.2 represents a Medipix3 chip connected to an USB Readout device via an extender board that is necessary in order to operate the chip. The

sensitive area is organised in a matrix of 256 256 pixels with a pitch of 55 ñm. The × electronics of Medipix3 is designed in a 0.13 ñm CMOS technology. The matrix contains approximately 105 Millions of transistors (three times more than in the Medipix2 detector). Each pixel can be divided into an analog and a digital part as depicted in figure 4.3: The analog part contains a charge sensitive preamplifier (CSA), a semi Gaussian pulse shaper that processes the output of the CSA and two discriminators. Each discriminator has a 5-bit Digital to Analog Converter to reduce the threshold dispersion caused by the mismatch in the transistors that provide the threshold and the summing currents. The digital part contains control logic, 13 configuration bits, arbitration circuits that decide to which pixel the charge is assigned (see figure 4.4). Moreover, each pixel contains two configurable depth registers which can function either as counters or as serial shift registers. The registers can be configured as two 1-bit, 4-bit or 12-bits registers, or a single 24-bit register [5]. Since the pixel has two registers, the read out can be programmed for Continuous Read-Write. In this mode of operation one register is configured as a counter while the other is shifting data out of the chip and vice versa. This makes simultaneous counting and read out possible, providing dead time free operation. The data can be read out e.g. via an USB2 device with a clock frequency of 10, 20, 40, 80 MHz. This enables to write out three frames per second.

18 4.2 Description

Figure 4.2 Medipix3 chip connected to an USB-Readout.

Figure 4.3 Medipix3 block diagram [5].

19 4 The Medipix3 Detector

4.3 Functionalities and operating Modes

Each pixel analog front-end can be conﬁgured in Fine Pitch Mode or in Spectroscopic

Mode. In Fine Pitch Mode, the pixel pitch is 55 ñm, so all the pixel are bump bonded to the ASIC as in a Medipix2 detector. In Spectroscopic Mode, only one pixel is bump

bonded in a cluster of 4 pixels. This enables a pixel pitch of 110 ñm and a binning of eight

energy thresholds is possible [5]. Only detectors with a pixel pitch of 55 ñm were produced so far. The Spectroscopic Mode will be available in the later assembly productions. For each of these two modes each pixel can work in Single Pixel Mode or in Charge Summing Mode. Single Pixel Mode is the same as the counting Mode of the Medipix2 where each pixel acts independently from the neighbours. In Charge Summing Mode the charge deposited in a cluster of 4 pixel is added and assigned to the summing circuit with the largest energy deposition. Table 4.1 shows a summary of the characteristics of the Medipix3 detector.

Table 4.1 Modes of operation of the Medipix3 detector [5].

System Conﬁguration Pixel Operating Modes # of Thresholds ñ Fine Pitch Mode 55 ñm 55 m Single Pixel Mode 2

→ × ñ Fine Pitch Mode 55 ñm 55 m ChargeSummingMode 2

→ × ñ Spectroscopic Mode 110 ñm 110 m Single Pixel Mode 8

→ × ñ Spectroscopic Mode 110 ñm 110 m ChargeSummingMode 8 → × Front-end Gain Modes Linearity # of Thresholds High Gain Mode 10 k e− 2 ∼ Low Gain Mode 20 k e− 2 ∼ Pixel Counter Modes Dynamic range # of Thresholds 1-bit 0 - 1 2 4-bit 0 - 15 2 12-bit 0 - 4095 2 24-bit 0 - 16777215 1 Pixel Read out Modes # of Active Counters Dead Time Sequential Count-Read 2 Yes Continuous Count-Read 1 No

4.4 Charge Summing Mode

The charge collected in a pixel is integrated by the preampliﬁer. Then, the output of the ampliﬁer is processed by the shaper and converted into a current with an amplitude proportional to the collected charge. The current is sent to node common to four adjacent pixels where summing is done. Figure 4.4 illustrates the summing principle: a charge is deposited in the corner of four adjacent pixels. Each of the four pixel detects a signal

20 4.5 Conclusion which is a fraction of the deposited energy. From each signal four copies1 are generated and send to the nodes located at the edge of the pixels. There, they are added and the reconstructed signal is compared with an energy threshold. The incoming quantum is attributed as a single hit to the summing node with the largest energy deposition. Multiple counts are therefore avoided, thus enabling a better recovery of the energy information. It is important to point out that ﬂuorescence photons are included in the charge sum (reconstructed charge) provided the ﬂuorescence photon deposits its energy in the neighbouring pixels with respect to the original impact point.

2 1 2 1 copy of signals

8 5 8 5

charge distribution signal value event sum of signals node registration

2 3 1 2 3 1

comparative logic 10 16 6 10 16 6 discrimination

8 13 5 8 13 5

Figure 4.4 Charge Summing Mode adapted from P. Bartl [21].

4.5 Conclusion

This chapter presents an overview of the Medipix3 chip. Its main motivation is to correct for charge sharing due to lateral diﬀusion of charge carriers in the sensor material in order to get a better energy resolution. The simulated energy responses of Medipix3 to monoenergetic photons in Single Pixel Mode and Charge Summing Mode will be presented in the next chapter.

1In fact eight current copies are generated since 2 thresholds are available per pixel.

5 Monte Carlo Simulations and Energy Responses of Medipix Detectors

Contents 5.1 MonteCarloToolROSI ...... 23 5.2 Energy Responseof the Medipix2 Detector...... 24 5.2.1 GeometryoftheMedipix2Detector ...... 24 5.2.2 PropagationofChargeCarriers ...... 24 5.2.3 EnergyResponseof the Medipix2 Detector ...... 27 5.3 Energy Responseof the Medipix3 Detector...... 27 5.4 Impact of Noise Contributions on the Energy Resolution of MedipixDetectors...... 29 5.5 Conclusion ...... 31

In order to understand physical processes occurring within the detector we make use of Monte Carlo simulations. Besides, simulations constitute a very important part of this thesis as the responses of the detector to monoenergetic photons are simulated and are used for the data analysis. This chapter will present the Monte Carlo tool ROSI and give a detailed description of the response functions of photon counting detectors.

5.1 Monte Carlo Tool ROSI

Simulations are performed using the Monte Carlo tool called ROSI (Roentgen Simula- tion) [22]. This tool was developed in our group and is being used extensively either for characterization of X-ray detectors or to model X-ray tube setups. It simulates the transport of charged and uncharged particles through matter on an event by event basis. The interactions of particles and their possible daughter particles are stored in a tree structure to provide easy access [23]. ROSI has been mainly developed using established program packages:

GISMO (Graphical Interface for Simulation and Monte Carlo Code with Objects): this class is used for geometry and photon transport.

LSCAT and EGS4: The EGS4 (Electron Gamma Shower) library describes particle interactions (photon, electron) with energies down to the keV. Additionally, the LSCAT (Low Energy Photon Scattering) expansion is needed for physical effects such as fluorescence photons in photoelectric effect or for scattering of polarized photons.

23 5 Monte Carlo Simulations and Energy Responses of Medipix Detectors

RAVAR (Random Variable): This library is used for the generation of random numbers. It is based on the random number generator algorithm after James [24] with a seed from the interval [0, 900 000 000].

LAM (Local Area Multicomputer Environment): This library is used for parallel computing.

The user interface consists of four functions:

InitialiseSetup: The user deﬁnes the properties and geometry of the detector, object or particle source.

InitialiseSimDataContainers: At this point, the user sets the format of histograms, in which the simulation results will be stored.

SimulatePackage: The simulation starts here. A photon is released and the desired interactions along its path are generated.

SaveSimData: Finally, histograms or simulation data are saved.

5.2 Energy Response of the Medipix2 Detector

The standard program of ROSI was extended to take into account the pixelated structure and motion of the electron hole pairs in the semiconductor sensor layer [23].

5.2.1 Geometry of the Medipix2 Detector

The geometry of the detector was implemented in the simulation as follows: ñ

55 m squared pixels. ñ 300 m thick silicon layer as sensor.

Bump bonds: These are made of Lead/Tin and are implemented as cubes with 16 ñ

ñm - 25 m sidelength. ñ

700 m silicon layer as ASIC (Application Speciﬁc Integrated Circuit). ñ 30 m silver containing glue. This glue connects the ASIC to the PCB (Printed Circuit Board). Under bump metallisations were neglected due to their small thicknesses.

5.2.2 Propagation of Charge Carriers The next step focuses on the generation and propagation of charge carriers in the sensor material. The energy deposited in the sensor layer is converted into electron-hole pairs taking into account the Fano factor [25]. This factor describes the statistical ﬂuctuation in the number of created charge carriers and is about 11.5 % for silicon. This means that the production process of the charge carriers has a statistical ﬂuctuation of approximately

24 5.2 Energy Response of the Medipix2 Detector

11.5 % of the root mean square of the corresponding Poisson distribution [23]. After, an electric field is applied to separate and drift the charge carriers towards the electrodes. The electric field map was determined using COMSOL Multiphysics [26]. In the next step, the drift time of an electron and a hole is calculated as function of their lifetime and drift distance. Taking into account diffusion and repulsion processes, the end position of the pair is computed and the corresponding induced signal from electrons is calculated. Electronic and threshold noise are added up to the induced charge in each pixel and the result is compared with a defined threshold. The counter in the corresponding pixel is then incremented if the collected charge is bigger. Figure 5.1 summarizes the steps of signal generation algorithm for photon counting detectors.

25 5 Monte Carlo Simulations and Energy Responses of Medipix Detectors

Emission of a photon from an X-ray tube

Photon propagation leads to a ﬁrst energy deposition in sensor material

calculate number of next energy de- electron hole pairs position step including fano noise

calculate drifttime (lifetime, driftdistance ) next electron-hole-pair of an electron and a hole

calculate end position taking into account diﬀusion, repulsion

calculate induced signal from electrons

more electron- yes hole- pairs?

more energy yes deposition steps?

no Addition of electronic noise, threshold noise and comparison of the charge collected in each single pixel with a deﬁned threshold

End of charge carrier transport and signal in- duction for the present photon

Figure 5.1 Flow chart of the signal generation algorithm for the simulation of a photon counting detector [16].

26 5.3 Energy Response of the Medipix3 Detector

5.2.3 Energy Response of the Medipix2 Detector Figure 5.2 depicts the energy responses of Medipix2 (simulation and measurement) to 59.3 keV incoming photons. The measurement was performed using a silicon monochro- mator [23]. These responses reveal a quite complex structure: We have the full energy peak at 59.3 keV and a low energy tail due to charge sharing. Moreover, it is noticeable, that the energy peak is slightly shifted to lower energies. This can lead to a systematic error in the energy calibration. Two peaks are clearly visible at 22 keV and 25 keV. They are the Kα-ﬂuorescence lines of silver and tin. Tin is part of the solder bumps and silver is contained in the glue used to connect the ASIC to the PCB. The steep edge around 11 keV represents the so called Compton edge which is the maximum amount of energy that a photon can transfer to an electron through Compton scattering. We can denote that simulation and measurement are in good agreement.

measurement simulation Intensity

10 20 30 40 50 60 energy deposition [keV]

Figure 5.2 Energy response of the Medipix2 detector to 59.3 keV incoming photons. Simu- lation and measurement are in good agreement [4].

5.3 Energy Response of the Medipix3 Detector

The Medipix3 detector geometry is almost the same in case of MPX2 but an organic glue was used for ASIC-PCB connection. The aim is to correct for distortion due to K-ﬂuorescence photons of silver. Based on our experience in modeling the response of Medipix2, Charge Summing Mode was implemented as follows [21]: the dynamics of charge carriers in the sensor material remains unchanged compared to Medipix2. Single Pixel Mode and Charge Summing Mode were implemented in the same code taking into account the behaviour of the detector electronics. For the simulation of Charge Summing Mode (CSM), it is necessary to consider the copy of the signals, the summing of these signals in a cluster of four neighbouring nodes and the comparative logic. In the implementation of CSM, four copies of the counts registered in a pixel are generated and sent to the nodes at the corner of the pixel. There, they are added up in a 2 2 cluster. One compares the resulting values with the neighbouring ones from a × cluster of 3 3 in order to ﬁnd out the node with the largest energy deposition. Its counter ×

27 5 Monte Carlo Simulations and Energy Responses of Medipix Detectors is incremented in the corresponding node if the collected charge is above threshold. For completeness in modeling of the energy response in CSM, following parameters were also included in the simulation code:

Gain variation: The gain factors of the pixels are slightly diﬀerent. This leads to a pixel-to-pixel gain variation. A look up table is generated containing a value for each pixel. The random values of the table are Gaussian distributed with a relative spread of 4.56% (design value).

Fluctuations during the copy procedure: In order to model ﬂuctuations produced during the copy of signals, a look up table with four values for each pixel was generated. These values also follow a Gaussian distribution with a relative spread of 1.47% (design value).

Other noise contributions: electronic noise and threshold dispersion are also taken into account and implemented as Gaussian distributions with the respective spread.

1e+05 Charge Summing Mode Single Pixel Mode

80000

60000 Counts 40000

20000

0 10 15 20 25 30 35 Energy deposition in keV

Figure 5.3 Simulated energy responses of the Medipix3 detector to 20 keV photons in CSM and SPM. The simulation was performed with 107 incident photons. The full energy peak in CSM is higher and symmetrical since charge sharing is almost completely suppressed (red curve). The plateau between 8 keV and 15 keV represents the L-ﬂuorescence lines of lead contained in the solder bumps. .

Figures 5.3 and 5.4 depict the energy responses of Medipix3 in SPM and CSM for 20 keV and 80 keV incoming photons. The simulation was performed with 107 incident photons. On the first figure one can denote that the full energy peak is higher in case of CSM and without any shift to lower energies. Moreover the tail due to charge sharing is suppressed. The plateau between 8 keV and 15 keV, that can be seen in the CSM curve, represents the L-fluorescence lines of lead contained in the solder bumps. This plateau is almost invisible in the SPM curve. The full energy peak at 80 keV (see figure 5.4) is clearly visible in CSM

28 5.4 Impact of Noise Contributions on the Energy Resolution of Medipix Detectors

Charge Summing Mode Single Pixel Mode

500 Counts

0 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 Energy deposition [keV]

Figure 5.4 Simulated energy responses of the Medipix3 detector to 80 keV incoming photons in CSM and SPM. The simulation was performed with 107 incident photons. Noise observed on the curves is quantum noise. The full energy peak in CSM in higher and symmetrical since charge sharing is almost completely suppressed (red curve). The peak at 25keV is due to the Kα-fluorescence line of tin contained in the solder bumps. This new architecture provides a better energy resolution. whereas it is almost invisible in SPM due to the long track length of the photo electron at that energy. However, charge sharing is not completely suppressed in this case: in fact at energies above 60 keV Compton scattering is the dominant process for photon interaction in silicon i.e. the charge sharing part due to Compton effect cannot be corrected when the charge carrier generation due to the scattered photon, takes place outside of the sensor volume of the pixels neighouring the initial interaction. Other structures can also be recognized in figure 5.4: the Kα-fluorescence line of tin at 25 keV is clearly visible in both modes. In addition to that, the Compton edge associated to the primary energy can be seen at 19 keV. Based on these simulated data, it can be concluded that Charge Summing Mode provides a better energy resolution. This remains to be confirmed experimentally.

5.4 Impact of Noise Contributions on the Energy Resolution of Medipix Detectors

In the previous sections, the energy responses of the photon counting detectors were presented. In case of Medipix2, the spread of the full energy peak is due to contributions of electronic noise, threshold dispersion and gain variation from pixel-to-pixel. In case of Medipix3, the width of the full energy peak is due to contributions originating from gain variation from pixel-to-pixel, electronic noise, current copy noise, and threshold dispersion. The simulation was performed using the design parameters listed in table 5.4. Since

29 5 Monte Carlo Simulations and Energy Responses of Medipix Detectors the electronic noise is energy independent, the contributions from the copy procedure and pixel-to-pixel gain variation should be the ones that have the biggest impact on the energy resolution of the detector [21]. The simulation was performed with 107 incident photons. Figure 5.5 illustrates the response of the Medipix3 detector to incoming photons for diﬀerent values of the gain variation: the full energy peak although at the right energy position, becomes broader with increasing gain variation. This behaviour is expected as strong variations of the gain factors implies a broader distribution. As a result the energy resolution of the detector decreases. For high variations in current copies (see ﬁgure 5.6), the peak is also broader but shifted to higher energies: these variations lead to energy deposition values above the primary photon energy. The shift is due to the fact that the result of the summing procedure is always allocated to the node with the largest energy deposition. It is therefore important to have small relative noise contributions for a better energy resolution.

1500 4 % 8 % 12 % 15 % 20 %

1000 Counts

500

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 Energy deposition [keV]

Figure 5.5 Energy responses of the Medipix3 detector to 60 keV photons in Charge Sum- ming Mode for diﬀerent values of gain variation. The simulation was performed with 107 incident photons. Although the position of the full energy peak remains unchanged, the energy resolution becomes poorer since the broadness of the peak increases with increasing gain variation.

30 5.5 Conclusion

1500 1.47 % 3 % 6.5 % 11.5 %

1000 Counts

500

0 10 20 30 40 50 60 70 80 90 100 Energy deposition [keV]

Figure 5.6 Energy responses of the Medipix3 detector to 60 keV photons in Charge Summing Mode for diﬀerent values of current copy variation. The simulation was performed with 107 incident photons. The full energy peak shifts to higher energies as the variation in current copies increases. Moreover, the energy resolution becomes poorer since the width of the photopeak increases with increasing current copy variation.

Table 5.1 Noise contributions of the Medipix3 detector in Single Pixel Mode and Charge Summing Mode (design parameters).

Noise Contribution SPM CSM − − σelectronics (Gaussian) 70 e 190 e σgain (Gaussian) 2.58 % 4.56 % σcurrentcopy (Gaussian) - 1.47 % − − σTHL (Min to Max, ﬂat distribution after equalisation) 190 e 375 e

5.5 Conclusion

This chapter gave an overview of the Monte Carlo Tool ROSI. It also presented how the energy response of photon counting detectors can be implemented taking into account all physical steps to energy deposition and all known noise contributions. The implementation in case of Medipix2 (Single Pixel Mode of Medipix3) demonstrated a good agreement between simulation and measurement. The response of Medipix3 in Charge Summing Mode has to be conﬁrmed experimentally. Nevertheless, the simulation results show an improvement in correction for energy distortion and suppression of charge sharing.

6 Characterization of the Medipix3 Detector

Contents 6.1 Test Pulses Measurements to determine Optimal DAC Settings 33 6.2 Equalisation Procedure of the Medipix3 Detector ...... 37 6.3 Energy Calibration of the Medipix3 Detector ...... 40 6.4 Energy Resolutionof the Medipix3 Detector ...... 42 6.5 Parameters Extraction for Monte Carlo Simulations . . . . . 43 6.6 Energy Response of the Medipix3 Detector in Charge Sum- mingMode ...... 43 6.7 Count Rate Linearity of the Medipix3 Detector ...... 45 6.8 Conclusion ...... 47

This chapter focuses on the characterization of the Medipix3 detector. Section 6.1 con- cerns the measurements performed using test pulses in order to understand parameters used to control the detector. The second section describes the threshold adjustment procedure and presents results of measurements performed to calibrate the detector. Finally, some parameters like the gain variation from pixel-to-pixel or the threshold dispersion are extracted from measurements and used as input to validate the Monte Carlo Simulations of Charge Summing Mode of Medipix3.

6.1 Test Pulses Measurements to determine Optimal DAC Settings

It is possible to send test pulses to the entire pixel matrix. Test pulses are used for example to test the functionality of each pixel without an X-ray source. They can also be utilized for energy calibration [27]. An interface panel written in C++ routine was developed [5] to control the Medipix3 detector. Figure 6.1 shows a picture of the panel. This routine was recently implemented in Pixelman [28], the software developed at the CTU of Prague, that is used to control the Medipix detectors. Several Digital to Analog Converters (DAC) are used to set globally currents or voltage levels in the detector via Pixelman. The parameters defined for measurements were the bias voltage, the threshold scan range, the height and number of test pulses to be injected to the test capitance of each pixel. In addition the detector was programmed in High Gain Mode. The varying DAC- parameters were the Shaper (that controls the shaper current), the IKRUM (responsible for the rise time and return to zero time of the signal above a defined threshold) and the Preamp (sets the preamplifier current Ipreamp). Figure 6.2 shows a 3D plot with the noise (at the ouput of the shaper) and gain as function of the Shaper value. Note that the definition of the gain used here (e−/DAC step) is the reverse of the

33 6 Characterization of the Medipix3 Detector

Figure 6.1 Panel for Test pulse measurements conventional definition of the gain (DAC step/e−). Decreasing the value of the Shaper leads to a higher gain but to an increasing noise. A closer look at the gain variation as function of the Shaper (figure 6.3) denotes an increasing gain variation up to 10% at lower Shaper values. As a result, it is necessary to use Shaper values above 150 in order to achieve a relative gain variation below 5%. The second step in the investigation concerned the impact of the IKRUM on the gain and noise as depicted in figure 6.4. Gain and noise increase monotonically with increasing IKRUM values. It is therefore better to use smaller IKRUM values in order to have a higher gain (conventional gain) and less noise. The last parameter of interest was the Preamp, that was adjusted while the other DAC-parameters remained unchanged. Figure 6.5 shows the impact of the Preamp on noise and gain. In fact the gain and noise remain almost constant for Preamp values above 100. Below this range noise and gain increase abruptly.

34 6.1 Test Pulses Measurements to determine Optimal DAC Settings

150

125

100

75 Noise (e− rms)

0 300 200 250 180 160 200 140 120 150 100 80 100 60 Shaper (DAC step) 40 50 20 Gain (e− / DAC step) 0 0

Figure 6.2 Shaper versus noise and gain variation. Small Shaper values lead to higher noise and to an increasing gain.

Gain variation in % 7

4 0 50 100 150 200 250 300 Shaper in DAC step

Figure 6.3 Shaper versus gain variation. Lower Shaper values lead to higher pixel-to-pixel variation.

35 6 Characterization of the Medipix3 Detector

250

200

150 Noise (e− rms)

100

0 300 270 100 240 210 80 180 60 150 120 40 90 60 IKRUM (DAC steps) Gain (e− / DAC step) 20 30 0 0

Figure 6.4 IKRUM versus gain and noise. Lower IKRUM values lead to higher gain and lower noise level at the output of the Shaper.

140

120

100

80 Noise (e− rms)

20 300 250 0 200 50 40 150 30 100Preamp (DAC steps) 20 50 Gain (e− / DAC step) 10 0 0

Figure 6.5 Preamp versus gain and noise. Noise and gain remain almost unchanged for Preamp values above 100. Below this value noise and gain increase abruptly.

36 6.2 Equalisation Procedure of the Medipix3 Detector

6.2 Equalisation Procedure of the Medipix3 Detector

In order to perform measurements with the detector, it is necessary to equalize the thresholds in the pixel matrix. This procedure is necessary since the DC level of the shaper output current IN (see fig. 6.7) is different from pixel to pixel due to mismatch of the front end transistors. Mismatch is a process that causes random variations in the physical quantities of identically designed devices [5]. The difference caused by mismatch leads for example to threshold dispersion that should be corrected. Figure 6.7 illustrates the 3 current sources necessary to have a better understanding of the equalisation method. The three currents ITHP, ITHADJ and ITHN, useful for the equalisation procedure, are controlled with DACs which are implemented in the chip periphery.

ITHP sets the current common to all pixel. It sets the global threshold for the pixel matrix.

ITHADJ and ITHN: The ﬁrst one is implemented as a 4 bit DAC and is used to equalise the threshold mismatch. The latter also sets a global current source and is implemented as a 1 bit DAC. It is used to adjust the DC level of the shaper output current.

In order to understand the following, a simultaneous attention should be given to ﬁgures 6.7 and 6.8. The procedure is performed using the routine mentioned above or via a plug in implemented in Pixelman (see Figure 6.6). A simpliﬁed description of the algorithm follows:

Step 1: The DACs for adjustment are inhibited (see ﬁg 6.7). Following this, threshold (ITHP) scans are performed for diﬀerent values of ITHN (activating B TH<4>) and ITHADJ (activating B TH<0..3>) and the threshold value where the noise is registered.

Step 2: Based on the resulting distributions, the optimal value for the ITHN (current source) is calculated and the Least Signiﬁcant Bit current of the threshold adjustment DAC ITHADJ is computed and the optimal value for B TH<4> as well.

Step 3: At this stage, the optimal THN current value and B TH<4> are set, for those pixels with positive current flowing into the zero crossing block (fig 6.8). For pixel with negative current (see fig 6.8), the 4-bit Threshold Adjustment DAC suffices to adjust the threshold.

Step 4: Finally, the value of the on pixel DAC is adjusted to move the pixel pedestal as close as possible to zero. Fig 6.9 shows a distribution of adjusted thresholds at the end of the equalisation process. This distribution exhibits a spread of 500 e− rms for a Shaper-DAC of 150. For the same detector settings, the value of the spread can strongly vary from chip to chip.

Detailed information on the equalisation procedure can be found in [5].

37 6 Characterization of the Medipix3 Detector

Figure 6.6 Pixelman Medipix3 equalisation panel. In this version of Pixelman it is possible to equalise at the same time the two thresholds available per pixel.

Figure 6.7 IN is the shaper output current. IZX is the current entering the Zero Crossing Block (ZX). The Zero Crossing Block changes its digital output voltage with the sign of its input current. ITHP sets the global threshold. ITHADJ and ITHN adjust the threshold in the matrix [5].

38 6.2 Equalisation Procedure of the Medipix3 Detector

Figure 6.8 Scheme of the procedure for threshold adjustment. IN is the output current from the shaper. Its DC level is a consequence of the mismatch of the front-end transistors. When the DC level is negative the adjustment DAC suﬃces for adjusting the threshold. When the DC level is positive, the combination of ITHN and ITHADJ is necessary to adjust the threshold. IZX is the input to the zero crossing circuit. The DC level of this signal is the equalized threshold [5].

Figure 6.9 Histogram of adjusted thresholds at the end of the equalisation procedure.

39 6 Characterization of the Medipix3 Detector

6.3 Energy Calibration of the Medipix3 Detector

For the calibration of the Medipix chip we usually use fluorescence lines of material like copper, lead, molybdenum, silver, tin, iodine, gadolinium or radioactive sources like Cd-109 or Am-241. Figure 6.10 shows a diagramm of the setup used to detect the fluorescence photons generated in the target material when irradiated by the X-ray Source. Depending on the energy of the fluorescence photon, the tube voltage was varied from 40 kV to 100 kV and the tube current from 19 mA to 40 mA. The acquisition time was set to 10 s. In the first step a threshold scan is performed in the threshold range of interest. After that the derivative of the threshold scan is calculated and a Gaussian function is fitted to the data in order to determine the position (THL-value) associated to the peak of interest. Figure 6.11 shows full energy peaks of different fluorescence materials and the Gaussian functions fitted to the data. The responses reveal a nice symmetrical shape as the charge sharing part is effectively suppressed. Figure 6.12 depicts the energy calibrations obtained for different shaper settings and gain modes of the chip: in the figure, calibration curves for 2 different Shaper values are presented. Higher Shaper values denote a steeper slope, enabling a reduced energy range, e.g. from 4 keV to 78 keV for a Shaper-DAC value of 150. Although the slope is coarser for Shaper-DAC of 100, we have a broader energy range from 5 keV to 100 keV. A broad energy range can also be achieved by operating the chip in Low Gain Mode (this means that we have a lower signal amplification at the output of each pixel shaper). The drawback is the more gentle slope which implies a bigger binning step and of course a worse energy resolution.

Figure 6.10 Setup to produce and detect the ﬂuorescence photons generated in the target material when irradiated by an X-ray Source [29].

40 6.3 Energy Calibration of the Medipix3 Detector

8 x 10

Ag K 2.5 α

Mo Kα

Sn Kα 2 Cu Kα

Pb L 1.5 α

Derivative count 1

I Kα Gd K 0.5 α

0 0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 THL (DAC value)

Figure 6.11 Full energy peaks of diﬀerent ﬂuorescent targets (Cu, Pb, Mo, Ag, Sn, I, Gd) measured with a Medipix3 detector operating in CSM Low Gain Mode. The peaks have a nice symmetrical shape. The tail due to charge sharing is suppressed.

500 CSHG, Shaper 150: THL=6.411*E+19.878, 1 THL ~ 0.156 keV 450 CSHG, Shaper 100: THL=5.125*E+9.925, 1 THL ~ 0.195keV CSLG, Shaper 150: THL=3.571*E+13.074, 1 THL ~ 0.280 keV 400

350

300

250

200 THL (DAC value) 150

100

0 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 Energy (keV)

Figure 6.12 Energy calibration curves of Medipix3 in Charge Summing Mode for Shaper- DAC of 150 and 100. A calibration curve in Low Gain Mode is also presented. Higher Shaper values lead to a steeper slope of the calibration curve and to a better energy resolution.

41 6 Characterization of the Medipix3 Detector

6.4 Energy Resolution of the Medipix3 Detector

In order to evaluate the relative energy resolution of Medipix3, Gaussian functions were fitted to full energy peaks (obtained from measured data) of well known fluorescent targets. The relative resolution is the ratio of the spread σE obtained at a given fluorescence energy E to that energy. The result is expressed in percent. The uncertainty on the resolution is deduced from the fitting procedure. Table 6.1 shows the comparison between the resolution of the detector in Single Pixel Mode and Charge Summing Mode for a Shaper-DAC of 100. We can denote an improvement of the resolution when using Charge Summing Mode. In SPM, the resolution is almost the same for 22.10 keV and 28.51 keV. In CSM, we notice a small improvement of the resolution at the mentioned energies. The resolution in Charge Summing Mode can be improved by using higher Shaper-DAC values as confirmed by table 6.2, where the resolution for Shaper-DAC 100 and 150 is depicted. The resolution becomes better with increasing energy. At 42.76 keV, we have a relative resolution of 5 % for Shaper-DAC 150 and 6 % for Shaper-DAC 100. In Charge Summing Mode, the detector can be operated in Low Gain Mode (LGM) or in High Gain Mode (HGM). High/Low Gain Mode means that we have a higher/lower signal amplification at the output of each pixel shaper. Table 6.3 depicts the resolution in the two mentioned modes. The energy resolution in HGM is almost 2 times better than in LGM as expected since we have half signal in this mode. This leads to broadened full energy peaks. Therefore, having a good resolution with the Medipix3 detector is achieved by operating it in Charge Summing High Gain Mode and at Shaper-DAC values of at least 150.

Table 6.1 Energy resolution of Medipix3 in Single Pixel Mode and Charge Summing Mode for a Shaper-DAC = 100. An improvement of the resolution is noticeable in CSM.

Energy Resolution in SPM Resolution in CSM 8.04 keV - (18.7 0.6) % ± 17.42 keV (11.2 0.5) % (10.3 0.3) % ± ± 22.10 keV (9.2 0.4) % (9.0 0.2) % ± ± 28.51 keV (9.2 0.6) % (8.3 0.3) % ± ± 42.76 keV (7.0 1.0) % (6.0 0.6) % ± ±

42 6.5 Parameters Extraction for Monte Carlo Simulations

Table 6.2 Relative energy resolution of Medipix3 in Charge Summing Mode for diﬀerent Shaper-DAC values. The energy resolution is better at Shaper-DAC 150. A good resolution is achieved by using higher Shaper values.

Energy Resolution for Shaper-DAC 150 Resolution for Shaper-DAC 100 8.04 keV (16.0 0.3)% (18.7 0.6) % ± ± 17.42 keV (9.1 0.2) % (10.3 0.3) % ± ± 22.10 keV (8.0 0.2) % (9.0 0.2) % ± ± 28.51 keV (6.8 0.2) % (8.3 0.3) % ± ± 42.76 keV (5.2 0.1) % (6.0 0.6) % ± ±

Table 6.3 Relative energy resolution of Medipix3 in Charge Summing High Gain and Low Gain Mode for a Shaper-DAC of 150. The resolution in low Gain Mode is poorer as expected since we have half signal in this mode. This leads to broadened full energy peaks.

Energy Resolution in High Gain Mode Resolution in Low Gain Mode 8.04 keV (16.0 0.3) % - ± 17.42 keV (9.1 0.2)% (19.5 0.5) % ± ± 22.10 keV (8.0 0.2) % (16.0 0.4) % ± ± 28.51 keV (6.8 0.2) % (16.0 0.3)% ± ±

6.5 Parameters Extraction for Monte Carlo Simulations

For each of the ﬂuorescence materials, we performed a pixelwise Gaussian ﬁt to determine the mean value and the spread of the derivative threshold scan in DAC units. The spread of the distribution of the obtained means provide the threshold dispersion. It has been found that this spread is energy dependent. So an energy dependent linear function is used in the simulations to describe the threshold dispersion. From the distribution of the spreads of the single pixel, the Equivalent Noise Charge (ENC) can be evaluated as the mean of the resulting distribution. By performing a pixelwise energy calibration, the gain factor in each pixel can also be estimated. From the distribution of these gains a gain variation from pixel-to-pixel can be determined. It was determined to 3.7% in CSM and 3.5% in SPM.

6.6 Energy Response of the Medipix3 Detector in Charge Summing Mode

Based on the parameters determined in the previous section, Monte-Carlo simulations were performed in order to compare simulation and measurement. Figure 6.13 and 6.14 compare simulated and measured spectra for 22 keV and 60 keV incoming photons. The plateau between 8 keV and 15 keV in the measurement curve (fig. 6.13) represents the L-fluorescence lines of lead used as wall material in our laboratory which is not included in the simulation. Moreover, the source spectrum used is only the spectrum of the fluo-

43 6 Characterization of the Medipix3 Detector

rescent target. The peak at 25 keV (ﬁg. 6.14) is the Kα-ﬂuorescence line of tin, present in solder bumps. The Compton edge is clearly visible at about 11 keV. Simulations and measurements are in good agreement.

3e+08

2e+08 Simulation Measurement Intensity [a. u.] 1e+08

0 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 Energy in keV

Figure 6.13 Energy responses of Medipix3 to 22 keV incoming silver ﬂuorescence photons in simulation and measurement. The two curves are in a good agreement. The plateau between 8 keV and 15 keV in the measurement curve represents the L-ﬂuorescence lines of lead used as wall material in our laboratory.

44 6.7 Count Rate Linearity of the Medipix3 Detector

1e+06

Simulation Measurement

Intensity[a.u.] 5e+05

0 10 15 20 25 30 35 40 45 50 55 60 65 70 75 Energy in keV

Figure 6.14 Energy responses of Medipix3 to 59.5 keV incoming gamma photons emitted by an Am-241 source in simulation and measurement. The two curves exhibit a good agreement. The energy peak at 25keV denotes the Kα-ﬂuorescence line of tin, present in solder bumps. The Compton edge is clearly visible at about 11 keV.

6.7 Count Rate Linearity of the Medipix3 Detector

In order to investigate the count rate linearity of Medipix3, the detector was placed 22 cm in front of our Megalix X-ray source. The applied voltage was 40 kV and the current was varied from 2 mA up to 45 mA. The IKRUM DAC was set at 20. About 65000 pixels were used in the count rate calculation as the masked pixels were not taken into account. Figure 6.15 shows a comparison between the count rate of the detector in SPM and CSM as function of the tube current. One can notice an overall linearity of the detector in SPM for the investigated count range, whereas an upper limit for the linear behaviour of about 8 106counts/sec/mm2 is reached in CSM. This difference in the count rate behaviour at × high flux rates is ascribed to the pixel intercommunication used in the Medipix3 charge summing algorithm as when a photon is recorded by a pixel, its adjacent 8 neighbours are inhibited from getting the hit [30]. An estimate of the relative count rate loss in CSM is depicted in figure 6.16. The relative count rate loss, for a value of the tube current, is given by RL RM − RL

The result is expressed in percent. RL denotes the count rate obtained assuming a linear behaviour of the detector and RM is the measured count rate. A loss of about 10 % is observable for an applied tube current of 20 mA corresponding to a rate of 8 106counts/sec/mm2. ×

45 6 Characterization of the Medipix3 Detector

4e+07

3e+07 Single Pixel Mode Charge Summing Mode 2

2e+07 counts /sec / mm

1e+07

0 0 5 10 15 20 25 30 35 40 45 50 Tube current [mA]

Figure 6.15 Comparison between the count rate of the Medipix3 detector in CSM and SPM. The linearity range of the count rate is larger in SPM. In CSM, for the applied settings, an upper limit for the linear behaviour is observed at a rate of 8 106counts/sec/mm2). ×

30 Charge Summing Mode 25

Relative count rate loss in % 10

0 10 15 20 25 30 35 40 45 50 Tube current [mA]

Figure 6.16 Relative count rate loss in CSM. A deviation of about 10 % is observable at 20 mA corresponding to a rate of 8 106counts/sec/mm2) ×

46 6.8 Conclusion

6.8 Conclusion

In this chapter the Medipix3 detector was characterized: The test pulses measurements gave us a better understanding of the new DAC called Shaper. Small Shaper values enable a wide energy range (see section 6.3). The measurements confirm that Charge Summing Mode works. Moreover, parameters like gain variation or threshold dispersion were extracted from measured data and used as input for the simulations. A good agreement between simulation and measurement was achieved. This confirms our precise modeling of the responses of the Medipix detectors. These are necessary for deconvolution procedures to determine the true incident spectrum from the measured spectrum. The count rate linearity of the Medipix3 detector was also investigated. An overall linearity was observed in SPM for the given count rate range whereas a relative count rate loss of 10% at a rate of 8 106counts/sec/mm2 was observed in CSM. This difference in the behaviour at × high fluxes is due the pixel intercommunication architecture used in the Medipix3 charge summing algorithm.

7 Spectrum Reconstruction with hybrid Photon counting Detectors

Contents 7.1 Motivation ...... 49 7.2 Theory...... 50 7.3 Methods...... 51 7.3.1 MatrixInversionMethod ...... 51 7.3.2 SpectrumStrippingMethod...... 51 7.3.3 IterativeSpectrumStripping ...... 54 7.3.4 PointJacobiMethod...... 56 7.3.5 Richardson Lucy algorithm (Bayesian deconvolution) ...... 56 7.4 Reconstruction with the Medipix2 Detector ...... 57 7.4.1 Simulation ...... 58 7.4.2 Measurement ...... 61 7.5 Reconstruction with the Medipix3 Detector in Charge Sum- mingMode ...... 67 7.5.1 Simulation ...... 67 7.5.2 Measurement ...... 67 7.6 Conclusion ...... 71

7.1 Motivation

Since their discovery, X-rays have been used in a wide range of applications e.g. in the field of Non Destructive Testing or in medical radiography. As the attenuation is energy dependent the determination of the spectrum of an X-ray tube is an important task. This is possible with commercial detectors only with a very low flux of photons. In order to determine the spectrum for a high flux of photons, the used detector must be able to process a high event rate and to supply energy information. The detectors of the Medipix family fulfill these requirements. They are able to count individual quanta. Moreover, their adjustable energy thresholds enable to gain information about the energy of the incident photons. However, the Medipix detectors have a drawback: They suffer from charge sharing because of their small pixel size, i.e, the energy deposited by an interacting X-ray quantum can be detected by more than one pixel. It becomes obvious that the measured energy deposition spectrum differs substantially from the incoming X-ray spectrum (see figure

49 7 Spectrum Reconstruction with hybrid Photon counting Detectors

7.1). The energy responses of the detectors to monochromatic radiation can be calculated with the Monte-Carlo simulation tool ROSI. These spectral response functions can be used in deconvolution methods to determine incident X-ray spectra from measured energy deposition spectra.

0.05 Incident Spectrum Measured Spectrum

0.04

0.03

0.02 Intensity in a.u.

0.01

0 10 20 30 40 50 60 70 80 90 100 Energy / Energy deposition in keV

Figure 7.1 Comparison of the measured spectrum (red curve) using a detector of the Medipix family and the theoretical incoming spectrum (black curve, [31]). The two spectra diﬀer considerably from each other due to charge sharing.

7.2 Theory

In general, the deposited energy spectrum M(E′) in a detector results from a convolution of the incoming spectrum S(E) and the detector response function R(E′, E) and can be expressed as:

Emax M(E′)= R(E′, E) S(E) dE (7.1) · · ZEmin Emin and Emax are the minimal and maximal photon energy, E denotes the primary photon energy and E′ denotes the energy deposition in the detector. The response functions R(E′, E) are normalized to the number of incoming photons and can be interpreted as the probability for an incident photon with a primary energy E to be detected with an energy E′ in deposition spectrum. Because of the lack of monoenergetic sources, only few monoenergetic responses were measured, while a large number of mono energetic responses have been simulated using the Monte-Carlo tool ROSI. The simulated and measured response functions were in good agreement (see chapter 5 and chapter 6). We rely on these response functions to determine the incoming spectrum. For known response functions R the measured spectrum M can be deconvoluted to obtain the incoming spectrum S. By decomposing the energy deposition E′ into I energy channels

50 7.3 Methods with subscript i and the primary photon energy E into J energy channels with index j, the integral equation can be converted into a set of linear equations which can be written in a discrete matrix notation as follows:

J Mi = RijSj (7.2) j=1 X Several unfolding methods and techniques are proposed in the literature [32–36] to solve equation 7.1. This list is not exhaustive. The following methods were applied to solve the equation: methods like the matrix inversion employing the pseudo inverse, the Spectrum Stripping method, and iterative methods like the iterative Spectrum Stripping, the point Jacobi method and the Richardson-Lucy (Bayesian deconvolution) algorithm.

7.3 Methods

7.3.1 Matrix Inversion Method

The Rij from the equation above can be considered as elements of a I J matrix R, that × can be inverted for equally sized bins of response functions and energy deposition. In this case it is necessary that the matrix should have a full rank, which implies that the number of energy bins have to be small enough to ensure the linear independence of the response functions [23]. This is the case if the determinant of R is non-zero. In the general case where I

S~ = R+ M~ · An estimate of the statistical error for the results of this reconstruction method is determined by:

+2 2 Si = R ( Mi) △ ij · △ q where Mi denotes the statistical error on the measured spectrum for bin i. The statistical △ error on the responses have been neglected since they are simulated with high statistics.

7.3.2 Spectrum Stripping Method It is a prerequisite for the Spectrum Stripping method that the matrix R must be square. To achieve this, the measured spectrum and the response functions are rebinned to have the same energy bin size, which is also the bin size of the unfolded spectrum. The stripping procedure is often used for Ge detectors [33]. The method is based on the successive substraction of the scaled monoenergic response functions from the measured spectrum. The following example explains step by step how the method works: Figure 7.2 shows the spectrum of an X-ray tube with tungsten anode at 60 kV. After the interaction of photons with sensor material, the detector delivers M(E′) as response function (ﬁgure 7.3). This response should be unfolded to recover the incoming spectrum S(E).

51 7 Spectrum Reconstruction with hybrid Photon counting Detectors

0.04 S(E)

0.03 S 1 S2

0.02 1 2 3 Intensity [a.u.]

0.01

0 10 20 30 40 50 60 70 Energy [keV]

Figure 7.2 Decomposition of a 60kV tungsten anode spectrum in 3 energy bins. Bin 1 comprises energies from 10keV to 30keV, bin 2 from 30keV to 50 keV, bin 3 from 50 keV to 70 keV. The coeﬃcients S1, S2, S3 shall be determined from the deposited spectrum.

Let us decompose this deposited spectrum into 3 energy channels centered at 20 keV, 40 keV and 60 keV (blue arrows in ﬁgure 7.2). The respective bins are numbered by 1, 2, 3. In order to perform the reconstruction we need the spectral response functions R(E′, 20 keV), R(E′, 40 keV), R(E′, 60 keV) to 20 keV, 40 keV, 60 keV incoming photons. Fig 7.4 illustrates the energy response to 60 keV photons. The factors R13, R23 and R23 in ﬁgure 7.4 represent R(20, 60 keV), R(40, 60 keV), R(60, 60 keV), respectively. The equation can therefore be written as:

M1 = R11S1 + R12S2 + R13S3 (7.3)

M2 = R21S1 + R22S2 + R23S3 (7.4)

M3 = R31S1 + R32S2 + R33S3 (7.5)

The Spectrum Stripping method makes use of the approximation that the counts M3 in the highest energy bin of the deposited spectrum M are produced by photons with the highest energy E3 of the incoming spectrum. This results in the spectral contribution

S3 = M3/R33

We know the relative contributions Ri3 of this energy to all energy bins i of the deposition spectrum and subtract it accordingly:

′ M3 = M3 R33S3 = 0 (7.6) ′ − M2 = M2 R23S3 ′ − M = M1 R13S3 1 −

52 7.3 Methods

1.5e+05

M1 ’ M(E )

1e+05 Counts 1 2 3 50000 M2 M3

0 10 20 30 40 50 60 70 ' E [keV]

Figure 7.3 Decomposition of the detector response to a 60kV tungsten anode spectrum in 3 energy bins. Bin 1 comprises energies from 10keV to 30keV, bin 2 from 30 keV to 50 keV, bin 3 from 50 keV to 70 keV. M1, M2, M3 represent the Counts in the respective bin.

The remaining spectrum M ′ can be considered as the energy deposition spectrum without the contribution of bin number 3. We continue the procedure with the next lower bin (bin number 2). S2 is determined as:

′ M2 M2 R23S3 S2 = = − (7.7) R22 R22

After the determination of S2, the new energy deposition spectrum can be written as:

′′ ′ M2 = M2 R22S2 = 0 (7.8) ′′ ′ − M = M R12S2 = R1 A13S3 R12S2 1 1 − − −

M ′′ is now the new energy deposition spectrum. The highest and last bin is bin number 1. The associated coeﬃcient S1 is therefore given by :

′′ M1 M1 R13S3 R12S2 S1 = = − − (7.9) R11 R11

The intensity Si in bin i can be generalized as: m Mi j=i+1 RijSj Si = − (7.10) PRii In this example the matrix R deduced from the response functions is a triangular matrix:

R11 R12 R13 R = 0 R22 R23 (7.11)   0 0 R33  

53 7 Spectrum Reconstruction with hybrid Photon counting Detectors

Response function to 60 keV incoming photons 0.06

’ R( E , 60) 0.04

R13

Intensity [a.u.] R R 33 0.02 23

0 10 20 30 40 50 60 70 ' E [keV]

Figure 7.4 Spectral response function of the detector to 60 keV photons. It is decomposed in 3 energy bins. Bin 1 comprises energies from 10keV to 30keV, bin 2 from 30 keV to 50 keV, bin 3 from 50 keV to 70 keV. The coeﬃcients R13, R23, R33 are normalized over the number of incoming photons.

Spectrum Stripping always delivers a solution. The basic idea of the stripping method is to set all Rij with i > j to zero in order to eliminate noise contributions and threshold dispersion eﬀects that lead to registration of events in energy channels that are higher than the primary energy channel of interest.This reconstruction method works well for energy bin size above 1 keV for polychromatic spectra and down to 0.5 keV for monoenergetic spectra. From the equation 7.10, we see that the Si are functions of Sj, M and R. In order to determine the error for this reconstruction method, it is necessary to express each Si just as a function of the measurement M and response functions R. Neglecting the statistical error of the response functions, we can determine the statistical error for the coeﬃcient Si using the following formula based on Gaussian error propagation:

2 m 2 Mi det(R(i : k 1, i + 1 : k)) 2 Si = △ + α − Mk (7.12) △ v R 2 · diag(R(i : k, i : k)) ·△ u ii k=i+1 u X t with α = 1 for i = m and α = 0 for i = mQ. M denotes the error on deposition spectrum 6 △ M. R(i : k, i : k) are the elements of the submatrix formed by rows from i to k and columns from i to k. diag of a matrix gives its diagonal elements.

7.3.3 Iterative Spectrum Stripping The Stripping method as described in the previous section does not work well for the Charge Summing Mode since the basic assumption of the Stripping is no more valid: The full energy peak to monoenergetic photons is very broad due to several noise contributions

54 7.3 Methods

(current copy variations, threshold dispersion, electronic noise). This leads to contributions in bins above the maximal photon energy. An iterative Spectrum Stripping Method was developed in order to overcome this problem. This iterative method works as follows: From the measured spectrum M an estimate of the incoming spectrum S0(E) is determined using the Spectrum Stripping procedure as described in the previous section :

S0(E)= Stripping(M(E)) (7.13)

After that a forward“calculation is performed to determine the measured spectrum asso- ” ciated to the reconstructed spectrum S0(E):

M0(E)= R(E, Ej)S0(Ej) (7.14) j X The next step is to compare this calculated energy deposition spectrum with the original measured deposition spectrum by subtraction:

M0(E)= M0(E) M(E) △ − This difference is in general not equal to zero, but the aim is to minimize this difference or in best case to reduce it to zero. In the third step we perform a spectrum reconstruction of the difference M0(E): △

S0(E)= Stripping(K M0(E)) (7.15) △ ·△ with 0

S1(E)= S0 + S0(E) △ and the associated measured spectrum of deposited energies is given by:

M1(E)= j R(E, Ej)S1(Ej)

Then we compare once again this calculatedP spectrum and the original measurement by calculating:

M1(E)= M1(E) M(E) △ − This procedure is repeated until suﬃcient accuracy is reached. If this is the case after N iterations, the reconstructed spectrum is given by:

N−1 SN (E)= S0(E)+ Sk(E) k=0 △ Suﬃcient accuracy is usually reached after 50-200P iteration steps depending on the value of the weighting factor K.

55 7 Spectrum Reconstruction with hybrid Photon counting Detectors

7.3.4 Point Jacobi Method

The process of computing Si from the following equation is known as the point Jacobi method as described in [36]:

N+1 N k N Si = Si + Mi Pi (7.16) Rii − n N N Pi = RijSj j=1 X k represents a relaxation parameter, N is the number of iterations and n denotes the number of energy bins. In order to achieve an optimal reconstruction the relaxation parameter should be smaller than 1. Furthermore this parameter should not be constant for all channels as suggested in equation 7.16. The following modiﬁed procedure was then computed taking into account migration in neighbouring bins [36]:

N N+1 N Si N Si = Si + N Mi Pi (7.17) Pi − N+1 This equation indicates that no negative value of the Si appears in the iterative procedure if the spectrum used as initial guess for the iteration procedure has only positive values. After each iteration m, the sum of squared deviations

m 2 (Mi Mi ) − i X is calculated. When the rate of change has fallen below a speciﬁed value, the iteration process is stopped. However, in order to determine the quality of the result obtained, the reduced m 2 (Mi Mi ) χ2 = i − n Mi P · is also calculated. In general, the method converges after approx. 30 iterations.

7.3.5 Richardson Lucy algorithm (Bayesian deconvolution) The Richardson Lucy algorithm ( [38], [39]) has been widely used for restoring hubble space telescope images for optical data. The algorithm works better when data approxi- mates Poisson statistics. This method is based on the Bayes theorem. For a better understanding of the method it is better to recall the Bayes’ theorem: the probability of an event A given an event B can be written as:

P (B A) P (A) P (A B)= | · (7.18) | P (B)

P (B) is named the a priori and P (A B) the a posteriori probability. We will derive | the iteration formula using a maximum likelihood approach as described in [40]: the

56 7.4 Reconstruction with the Medipix2 Detector probability to measure M given an expected value R S (convolution of the response ∗ Matrix R with the real spectrum S) can be written using the likelihood function L :

Mi (R S)i L = P (M R S)= exp ( (R S)i) ∗ (7.19) | ∗ − ∗ M ! i i Y Applying the natural logarithm to equation 7.19 and setting all partial derivatives of the obtained Log-likelihood function with respect to Sj to zero, yields the maximum likelihood solution:

F = log L = (R S)i + Mi log((R S)i) log(Mi!) (7.20) − ∗ ∗ − i X ∂F ∂ =0= RikSk + Mi log( RikSk) log(Mi!) = 0 (7.21) ∂S ⇒ ∂S − − j j i " #! X Xk Xk Simplifying equation 7.21 gives the following relationship:

Mi Rij 1 Rij Mi · Rij = 0 1= · (7.22) R S − ⇒ R R S i k ik k k kj i l il l ! X · X From the equationP above the following RichardsonP -Lucy iterationP formula can be derived to determine the incoming spectrum S .

N Sj Rij Mi S N+1 = · j R N k kj i l RilSl ! X N PSj RijPMi = · (7.23) ǫ N j i l RilSl ! X N P Sj is the estimate of Sj after N iterations. ǫj = k Rkj is the eﬃciency of the detector at the energy Ej. The Rij represent the probability for a photon to be registered with P energy Ei in deposition spectrum knowing that its initial energy was Ej. At the beginning 1 of the iteration procedure, Sj = Mj. The iteration formula above is also called Bayesian deconvolution by some authors [41]. So we will use this expression to refer to Richardson Lucy algorithm in the one dimensional case. The main advantage of the method is that the solutions are positive. For each iteration step, the reduced χ2 (see section 7.3.4) is calculated. As soon as its value is almost constant, the iteration process is stopped. The major drawback of the method remains the large number of iterations. The method leads to reasonable solution after 500 - 2000 iterations.

7.4 Reconstruction with the Medipix2 Detector

The following plots are reconstruction results with the matrix inversion and the Spectrum

Stripping methods for a Medipix2 detector bump bonded to a 300 ñm thick silicon sensor.

57 7 Spectrum Reconstruction with hybrid Photon counting Detectors

The incident spectra used here were measured by Dierker [31]. These spectra have contributions above the maximal acceleration voltage. But this does not have an inﬂuence on the quality of the reconstruction.

7.4.1 Simulation In order to ensure the reliability of the deconvolution techniques, the methods were tested using simulated data. The response functions to monoenergetic exposure were simulated with 10 millions of photons. In order to test the methods with simulated spectra, an X-ray source and the Medipix2 detector (implemented with all its components) were used as setup. The incident spectra had 100 millions of photons. These spectra have been measured and published by Dierker et al. [31] and used as input to the simulations. Figures 7.5 and 7.6 show in comparison the reconstructed and incident spectra of a tungsten anode tube at 40 kV and 80 kV. As we can see the incident and reconstructed spectra are in good agreement qualitatively but also quantitatively. For energy bin widths smaller than 1.5 keV the reconstructed spectrum using the Stripping method is closer to the incident spectrum (fig. 7.5). This is expected because the response functions are nearly dependent for small binning, so the matrix inversion becomes inaccurate. For higher energies the reconstruction is overall successful (see fig. 7.7). The reconstructed and incident spectra are in agreement within the statistical error bars. Some fluctuations can be observed in the reconstructed sprectra for energies above 80 keV. This is ascribed to the poor statistics in the deposition spectra as the efficiency of the silicon detector is small at those energies. Both reconstruction algorithms deliver quite similar results and deviate only for fine energy binning (see fig. 7.13).

58 7.4 Reconstruction with the Medipix2 Detector

1e+07 Spectrum Stripping Matrix Inversion Incident Spectrum

8e+06

6e+06

4e+06 Number of photons / 1.5 keV

2e+06

0 10 15 20 25 30 35 40 45 50 Energy in keV

Figure 7.5 Reconstruction of a 40 kV incident tungsten anode spectrum from simulated data. The two reconstruction methods give almost identical results.

3.5e+06 Incident spectrum Spectrum Stripping Matrix Inversion 3e+06

2.5e+06

2e+06

1.5e+06

Number of photons / 2 keV 1e+06

5e+05

0 10 20 30 40 50 60 70 80 90 Energy in keV

Figure 7.6 Reconstruction of a 80 kV incident tungsten anode spectrum from simulated data. The two reconstruction methods give almost identical results. The Kα and Kβ ﬂuores- cence lines of tungsten at 58.85 keV and 67.23 keV are visible.

59 7 Spectrum Reconstruction with hybrid Photon counting Detectors

1e+07 Spectrum Stripping Matrix Inversion Incident Spectrum 8e+06

6e+06

4e+06 Number of photons / 3 keV

2e+06

0 0 10 20 30 40 50 60 70 80 90 100 110 120 130 Energy in keV

Figure 7.7 Reconstruction of a 125 kV incident tungsten anode spectrum from simulated data. The two reconstruction methods give almost identical results. The Kα and Kβ ﬂuores- cence lines of tungsten at 58.85 keV and 67.23 keV are visible.

60 7.4 Reconstruction with the Medipix2 Detector

7.4.2 Measurement Reconstruction of X-ray tube Spectra The reconstruction methods were applied to measured data with X-ray tube spectra. Our Siemens Megalix tube was operated at an intensity of 10-40 mA and the Medipix2 detector was positioned in the primary photon field at a distance of 1m resulting in a count rate of approximately few kHz on the Medipix for the lowest energy threshold. With a total number of 2 107 counts on the Medipix for the lowest energy threshold, a threshold scan × was performed with energy steps of 0.75 keV. We applied our reconstruction methods and obtained the spectra shown in fig. 7.8 and fig. 7.9. The incident spectra here were scaled with the integral of the reconstructed spectra as the real number of photons coming out of our Megalix tube is unkown. Obviously, both methods give very similar results and both are able to reconstruct the spectra. It is important to point out that the precision is especially due to the detailed modeling of the response function of the detector in the simulation.

4e+08 Spectrum Stripping Matrix Inversion Incident Spectrum (Literature)

3e+08

2e+08 Number of photons / 3 keV 1e+08

0 10 20 30 40 50 60 Energy in keV

Figure 7.8 Reconstructed spectra for a 50 kV incident tungsten anode spectrum. The energy bin size used in the reconstruction is 3 keV. Spectrum Stripping and matrix inversion deliver almost the same result.

Reconstruction of radioactive Source Spectra and Energy Resolution of the Medipix2 Detector The algorithms described above can also be applied to determine the spectrum emitted by a radioactive source i.e. they were applied on data of Am-241 and Cd-109. The Ameri- cium source was positioned 4 mm in front of the detector. Because of the weakness of the source, the measurement took 8 days to achieve an acceptable statistics. Figure 7.10 is composed of 3 subﬁgures: ﬁgure 7.10 (a) depicts a measured threshold scan

61 7 Spectrum Reconstruction with hybrid Photon counting Detectors

4e+08 Spectrum Stripping Matrix Inversion Incident Spectrum (Literature)

3e+08

2e+08 Number of photons / 3keV 1e+08

0 10 20 30 40 50 60 70 80 90 100 110 120 Energy in keV

Figure 7.9 Reconstructed spectra for a 120 kV incident tungsten anode spectrum. The two reconstruction methods give almost identical results. The Kα and Kβ fluorescence lines of tungsten at 58.85 keV and 67.223 keV are visible. of the spectrum emitted. Figure 7.10 (b) represents the energy deposition obtained by derivation of the measured threshold scan. It is important to point out that the full energy peak is almost invisible in the deposition spectrum because of the low efficiency of silicon at that energy. After deconvolution using the Spectrum Stripping method with a bin width of 0.5 keV, we obtained the incident emitted spectrum (figure 7.10 (c)) . The main lines are clearly visible at 59.4 keV, 26.3 keV. The lines at 13.9 keV, 17.5 keV and 21.0 keV represent the Lα, Lβ and Lγ-fluorescence lines of Neptenium-237 (daughter product of Americium). It is noticeable that the Stripping method (fig. 7.10 (c)) also delivers negative number of photons. This is of course physically senseless. To correct for this, iterative methods were applied. Figure 7.11 and figure 7.12 show the reconstructed spectra emitted by an Am-241 source and the associated calculated spectra for each reconstruction method, respectively. The iterative methods provide almost comparable results. However the point Jacobi method and the Bayesian deconvolution deliver the best results since their associated calculated measurements are closer to the original measurement. Addi- tionaly, the full energy peaks in the reconstructed spectra are well positioned whereas a small shift to higher energies in peak positions is noticeable in case of the iterative Spec- trum Stripping due to smoothing performed before each iteration step. Moreover, those peaks are also lower in case of iterative Stripping due to smoothing. Fig 7.13 depicts reconstructed spectra of Cd-109 using the Spectrum Stripping method (s1-s4) and matrix inversion method (m1-m4) for different reconstruction bin widths δE (indicated on the right-hand side) in comparison. One can see that for reconstruction bin width δE larger than 1 keV the matrix inversion and Spectrum Stripping method almost give the same results. For reconstruction energy bin widths smaller than 1keV the matrix inversion method fails due to the numerical instability of the solution of the set of equations. This

62 7.4 Reconstruction with the Medipix2 Detector instability increases for smaller reconstruction bin widths. Spectrum Stripping leads to good reconstruction results even for a small reconstruction bin width of δE = 0.5 keV.

Table 7.1 gives the energy resolution obtained by the Spectrum Stripping method. The recon energy resolution σE was obtained as the standard deviation of a Gaussian ﬁtted to the reconstructed spectra. The energy resolution obtained with the Spectrum Stripping method seems to improve for smaller bin widths δE. Due to the sampling of the energy deposition spectrum in steps of 0.45 keV during the threshold scan no further improvement can be obtained here. The limiting factor here is the sampling density of the energy deposition spectrum in the measurement.

63 7 Spectrum Reconstruction with hybrid Photon counting Detectors

a intensity [a.u.]

0 10 15 20 25 30 35 40 45 50 55 60 65 b intensity [a.u.]

0 10 15 20 25 30 35 40 45 50 55 60 65 c

0 intensity [a.u.]

10 15 20 25 30 35 40 45 50 55 60 65 70 energy [keV]

Figure 7.10 Reconstruction of the radiation emitted by an Am-241 source. a: measured threshold scan i.e. counts as function of threshold; b: energy deposition spectrum obtained by derivation of the function in ﬁgure a, i.e. diﬀerence between consecutive intensity values; c: reconstructed energy spectrum by application of the Spectrum Stripping method.

64 7.4 Reconstruction with the Medipix2 Detector

5e+07 Point Jacobi method Bayesian deconvolution Spectrum Stripping 4e+07 Iterative Spectrum Stripping

3e+07

2e+07

1e+07 Number of photons / 0.5 keV

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 Energy in keV

Figure 7.11 Reconstruction of the radiation emitted by an Am-241 source for Medipix2 using 4 reconstruction methods. The iterative methods deliver better results as negative number of photons are not permitted. Moreover, the reconstructed spectra with Jacobi and Bayesian methods show well positioned lines whereas a shift of 1 keV is observed for Stripping methods due to smoothing performed at each iteration step.

recon δE (keV) σE (keV) 1.0 1.05 0.05 ± 0.6 0.97 0.05 ± 0.5 0.97 0.05 ± Table 7.1 Energy resolution of the Medipix2 detector. A resolution of approx. 1.0 keV can be achieved in the reconstruction.

65 7 Spectrum Reconstruction with hybrid Photon counting Detectors

Iterative Spectrum Stripping Spectrum Stripping Bayesian Inversion 3e+06 Point Jacobi method Original measurement

2e+06 Counts

1e+06

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 Energy in keV

Figure 7.12 Comparison between the measured energy spectrum of an Am-241 source and the calculated spectra from the reconstructed spectra using 4 deconvolution methods. The reconstruction bin width is 0.5 keV. The computed spectra associated to the Bayesian deconvolution is closer to the original measurement.

Spectrum stripping Matrix inversion

s1 m1 1.5keV

0 0 intensity [a.u.]

s2 m2 1.0keV 0 0 intensity [a.u.]

s3 m3 0.6keV

0 0 intensity [a.u.]

s4 m4 0.5keV 0

intensity [a.u.] 0 10 12.5 15 17.5 20 22.5 25 27.5 30 10 12.5 15 17.5 20 22.5 25 27.5 30 energy [keV] energy [keV]

Figure 7.13 Reconstructed spectra emitted by a Cd-109 source obtained with the Spectrum Stripping (left-hand side) and matrix inversion (right-hand side) method for reconstruction bin widths of 1.5 keV, 1.0 keV, 0.6 keV and 0.5 keV.

66 7.5 Reconstruction with the Medipix3 Detector in Charge Summing Mode

7.5 Reconstruction with the Medipix3 Detector in Charge Summing Mode

7.5.1 Simulation The response functions R to monoenergetic exposure were simulated with 10 millions of photons. An X-ray source and the Medipix3 detector were used as setup. The incident spectra S had 10 millions of photons. These spectra have been measured and published by Dierker et al. [31] and used as input to the simulations. Figures 7.14 and 7.15 show a non iterative Spectrum Stripping (green curve) can not work for the Medipix3 (green). The reason is the width of the full energy peak in the response of the Medipix3 to monoenergetic exposure: the Spectrum Stripping method assumes that the events with a certain energy deposition stem from photons with that primary energy. This assumption is acceptable for a narrow photopeak. Due to the large width of the photo peak in case of Medipix3 wrong photon energies are attributed to the energy deposition spectra. The reconstructed spectra are then shifted to higher energies and the number of photons are overestimated due to lower detection eﬃciency at higher energies. The application of iterative methods yields better reconstruction results. These results were obtained using reconstruction bin widths of at least 3 keV. The iterative methods provide almost the same results.

2e+07 Iterative Spectrum Stripping Spectrum Stripping Incident Spectrum Bayesian deconvolution Point Jacobi Method 1.5e+07

1e+07 Number of photons / 3 keV 5e+06

0 10 20 30 40 50 60 70 80 Energy in keV

Figure 7.14 Reconstructed spectra for a 70 kV incident tungsten anode spectrum. The energy bin width in the reconstruction is 3 keV. The iterative methods deliver almost comparable results.

7.5.2 Measurement The measurement was performed using a Megalix Siemens tube and a Medipix3 detector operating in Charge Summing Mode and placed 1 m away in front of the X-ray tube. Depending on the applied tube voltage, the tube current was adjusted so that the power did not not exceed the maximal allowed power of 2 kW. In order to achieve high statistical

67 7 Spectrum Reconstruction with hybrid Photon counting Detectors

2e+07 Bayesian deconvolution Point Jacobi method Iterative Spectrum Stripping Incident Spectrum Spectrum Stripping 1.5e+07

1e+07 Number of photons / 4keV 5e+06

0 0 10 20 30 40 50 60 70 80 90 100 110 120 130 Energy [keV]

Figure 7.15 Reconstructed spectra for a 110 kV incident tungsten anode spectrum. The energy bin width in the reconstruction is 4 keV. The iterative methods deliver almost comparable results. precision the Medipix3 chip was programmed in Charge Summing Mode using 24 bit counter depth. This results in a count rate of 5.5 kHz per pixel on the detector matrix at a threshold of 4 keV. After performing a threshold scan, the derivative of the recorded spectrum is calculated and used as measured spectrum in the deconvolution algorithms. Figures 7.16 and 7.17 depict the results of the reconstruction using the Spectrum Stripping and the iterative methods. The iterative methods deliver the best results as expected. The reconstructed spectra deviate from the incoming spectrum at lower energies: this might be due to the modeling of the response functions at these energies. Also the real spectrum emitted by the tube could be slightly different from the literature spectrum used here for comparison. Moreover, the X-ray tube used in this experiment was a brand- new Megalix Siemens tube and thus we expect slightly different X-ray emission spectra compared to the emission spectra used in the experiments in section 7.4.2 because of less surface roughness of the brand-new anode. In order to determine the energy resolution of the detector in Charge Summing Mode and to compare the three iterative methods, the fluorescence lines of silver were reconstructed with a binning of 0.5 keV. It is important to recall that the measurements were taken with an energy bin step of 0.5 keV. For such a binning, oscillations appeared in the spectrum provided by the point Jacobi method. This issue was addressed by performing a simple smoothing before each iteration step but not in the last step. Fig 7.18 shows the results of the reconstruction. One can see that the two lines at 22.1 keV and 25 keV are clearly visible in the spectrum reconstructed using Bayesian deconvolution. The Iterative Spectrum Stripping delivers comparable result, but the lines are shifted to the right and are a bit broader. The Bayesian method seems to be the best method since it converges to a stable state. The energy resolution achieved here is of the same order of that obtained with the Medipix2 detector.

68 7.5 Reconstruction with the Medipix3 Detector in Charge Summing Mode

6e+09 Iterative Spectrum Stripping Spectrum Stripping Incident Spectrum (Literatur) 5e+09 Bayesian deconvolution Point Jacobi Method

4e+09

3e+09

2e+09 Number of photons / 2keV

1e+09

0 0 10 20 30 40 50 60 Energy in keV

Figure 7.16 Reconstructed spectra for a 50 kV incident tungsten anode spectrum with the Medipix3 detector in Charge Summing Mode. The reconstruction bin width is 2 keV.

8e+09 Incident Spectrum (Literatur) Point Jacobi Method Iterative Spectrum Stripping Spectrum Stripping Bayesian deconvolution 6e+09

4e+09

Number of photons / 2keV 2e+09

0 10 20 30 40 50 60 70 Energy in keV

Figure 7.17 Reconstructed spectra for a 70 kV incident tungsten anode spectrum with the Medipix3 detector in Charge Summing Mode. The reconstruction bin width is 2 keV.

69 7 Spectrum Reconstruction with hybrid Photon counting Detectors

Iterative Spectrum Strippping Bayesian deconvolution Point Jacobi Method 1e+10

5e+09 Number of photons / 0.5 keV

0 10 12.5 15 17.5 20 22.5 25 27.5 30 32.5 35 Energy in keV

Figure 7.18 Reconstruction of ﬂuorescence lines of silver with the Medipix3 detector in Charge Summing Mode using three iterative methods. The reconstruction bin width is 0.5 keV. Two ﬂuorescence lines of silver are clearly visible and well positioned for Bayesian deconvolution. The two other methods show a shift to higher energies due to the smoothing performed at each iteration step. Hence, they provide a slightly poorer energy resolution

70 7.6 Conclusion

7.6 Conclusion

In this chapter, it was shown that the Medipix detectors can be used as spectrometers to determine incoming X-ray spectra or emitted gamma ray spectra. Since the detectors suﬀer from charge sharing the measured and the impinging spectra diﬀer from each other substantially. In order to recover the true incoming spectrum, a precise knowledge of the response functions of the detectors to monoenergetic photons is necessary. It is important to stress that a high quality of the responses is the key to a sucessfull unfolding. The mentioned responses were simulated using Monte Carlo methods and all the components of the detector were taken into account. Direct methods (matrix inversion, Spectrum Stripping) and iterative methods (iterative Spectrum Stripping, Point Jacobi method and Bayesian deconvolution) were applied to recover the incident spectrum. The matrix inversion works for energy bin width above 1 keV-2 keV depending on the incoming spectra. The disadvantage of direct methods is that also negative solutions are sometimes unavoidable. This is of course unphysical. Thus the necessity arises to compute the solution iteratively. The applied methods provide almost the same results for high energy bin width (above 1.5 keV for the Medipix2 detector) but the Bayesian deconvolution provides reasonable solution down to 0.5 keV. Before applying the methods to experimental data, it is important to have a precise energy calibration of the detectors, since a small shift in the energy can lead to a shift in the reconstructed spectra. These reconstruction methods can be used for qualiy assurance in medical imaging or can be applied in spectral imaging.

8 Determination of the kVp with the Medipix2 Detector

Contents 8.1 Determination of the kVp with multiple Filter Combinations 74

8.1.1 Simulation ...... 74

8.1.2 Measurement ...... 77

8.2 Determination of the kVp with only one Filter Combination 80

8.3 Conclusion ...... 81

Peak kilovoltage (kVp) is the maximum tube voltage occuring during an X-ray shot and governs the maximum energy of radiation produced. In the diagnostic X-ray imaging the peak kilovoltage of an X-ray tube is one of the most important parameters since it has a big impact on dose and image quality: As the beam intensity is approximately proportional to the square of the tube voltage, small changes in the high voltage supply can cause significant modification in the absorbed dose and image quality. For these rea- sons the peak voltages should be routinely monitored [42]. Several methods have been applied in order to determine the kVp. These methods can be grouped into direct invasive and indirect non invasive methods: the direct method makes use of high voltage divider that is connected between the generator and the X-ray tube and delivers isolated analog voltage signals (proportional to the voltage applied across the X-ray tube) that can be displayed and analyzed on an external oscilloscope. These devices are typically intended for calibration of X-ray generators. Non invasive methods employ X-ray spectrometers, techniques that utilize X-ray fluorescence, and kVp meters that are based on the change in X-ray transmission through varying thicknesses of filtration [43]. Because of their ease of use they are widely employed for routine quality control. The Medipix2 detector has already been used as spectrometer [44]: the authors have fitted a power law function to the falling edge of the measured energy deposition spectrum in order to determine the end point of the energy spectrum. An accuracy better than 2% could be achieved. In this chapter another approach is shown: results of simulations and measurements to determine the kVp based on combination of filters are presented. Additionally, it is shown that the peak kilo voltage can be determined with good precision using only one filter combination.

73 8 Determination of the kVp with the Medipix2 Detector

8.1 Determination of the kVp with multiple Filter Combinations

8.1.1 Simulation In principle, the kVp meter consists of a pair of closely spaced diodes that are filtered by different thicknesses of material, e.g., copper. The ratio of the signals of the filtered diodes depends on the energies of the X-ray beam. Figure 8.1 shows the setup that was implemented in the simulations and used in the measurements: it comprises an X-ray source, a 0.4 mm thick aluminium prefilter, 2 copper filters with thicknesses varying from 0.2 mm up to 6 mm (depending on the kVp range of interest), and the Medipix2 detector which replaces the diodes (present in conventional kVp meters). Table 8.1 shows filter thicknesses that are usually used and the associated tube voltage ranges. For data processing, two regions of interest(ROI) are defined, one on each side of the detector behind the copper filters. The sum of counts in each region - N1 for region 1 and N2 for region 2 - is determined. The ratio N R= 1 N2 is calculated and plotted as function of the applied tube voltage. The statistical error R △ on the ratio, assuming the counts are Poisson distributed, is calculated as:

2 N1 N1 R= 2 + 3 △ sN2 N2

Since the Medipix2 detector has an adjustable low discriminator threshold, the first step in the simulations was to investigate the influence of the threshold in determining the peak voltage. Figure 8.2 shows the ratio of the counts recorded behind the copper filters as function of the applied tube voltage in the range of 40kV up to 80 kV. The spectra used are literature spectra that were measured by Dierker [31]. As we can see, the ratio depends on the threshold. The higher the threshold, the higher is the ratio at a given tube voltage. It is noticeable that the statistical error on the ratio increases at higher threshold values. This is expected as the number of detected photons decreases with increasing threshold. Therefore, it is better to use lower threshold values to achieve high statistics in order to minimize statistical errors in determining the ratio.

Table 8.1 Tube voltage range and the corresponding copper thickness combinations.

Tube voltage range [kV] Copper ﬁlter1 [mm] Copper ﬁlter2 [mm] 35 - 55 0.4 0.2 45 -75 0.8 0.4 60 -100 2.4 1.4 80 -120 4 2 100 -160 6 3

74 8.1 Determination of the kVp with multiple Filter Combinations

Figure 8.1 Setup for kVp determination. It consists of an X-ray tube, an aluminium prefilter, two copper filters of different thicknesses and the Medipix detector. N1 and N2 denote the sum of counts recorded behind copper filter1 and copper filter2, respectively. Their ratio R is a function of the tube voltage applied across the X-ray tube.

0.5 THL: 10 keV THL: 15 keV THL: 30 keV THL: 35 keV THL: 40 keV 0.4 Ratio = 0.0076 * Voltage - 0.11 Ratio = 0.0072 * Voltage - 0.15 Ratio = 0.0074 * Voltage - 0.19 Ratio = 0.0075 * Voltage - 0.21 Ratio = 0.0078 * Voltage - 0.23 0.3 Ratio

0.2

0.1 40 45 50 55 60 65 70 75 Voltage [kV]

Figure 8.2 Ratio of the counts recorded behind copper ﬁlters with the Medipix2 detector. For tube voltages ranging from 45 kV to 75 kV. The ratio is a linear function of the kVp.

75 8 Determination of the kVp with the Medipix2 Detector

In a second step, values of the ratio R have been determined for other tube voltage ranges. Figures 8.3 and 8.4 depict the ratio obtained for voltages between 60 kV and dR 100 kV, and between 80 kV and 120 kV, respectively. One can notice that the slope dV decreases as the tube voltage range increases. This fact is ascribed to the weak energy dependence of Compton scattering cross section. The low statistics at higher energies implies that the statistical uncertainty in determining the ratio increases. As a result the accuracy in determining the kVp decreases.

0.4 Simulation Ratio = 0.0060 * Voltage - 0.25

0.35

0.3

0.25 Ratio

0.2

0.15

0.1 60 65 70 75 80 85 90 95 100 105 Voltage [kV]

Figure 8.3 Ratio R versus tube voltage ranging from 60 kV to 100 kV.

Simulation Ratio = 0.0036 * Voltage - 0.19

0.25

0.2 Ratio

0.15

0.1

80 85 90 95 100 105 110 115 120 Voltage [kV]

Figure 8.4 Ratio R versus tube voltage ranging from 80 kV to 120 kV.

76 8.1 Determination of the kVp with multiple Filter Combinations

8.1.2 Measurement In the measurements, the Medipix detector was placed at a distance of 1 m from the X-ray tube. A tube current of 15 mA was applied through the tube and an acquisition time of 1 s was chosen. In a first step, copper filters of 0.2 mm and 0.4 mm thickness were used. Figure 8.5 depicts the ratio from measurement and simulation as functions of the tube voltage ranging from 40 kV to 120 kV. The simulation and measurement show a similar functional behaviour. Meanwhile the ratio values obtained are slightly different. This is probably due to the fact that the intrinsic filtration of the tube is not exactly known. Moreover the uncertainty on the value of the tube voltage applied across the tube is not known. In a second step, copper filters with thicknesses varying from 0.2 mm to 4 mm were used depending on the tube voltage range (see table 8.1). Figures 8.6, 8.7, 8.8 show the ratio obtained R as function of the tube voltage ranging from 40 kV to 55 kV, 60 kV to 100 kV and 80 kV to 120 kV, respectively. The curves show the same functional behaviour as the associated simulated data. The slopes and intercepts resulting from linear regressions to the simulated and measured data are almost the same as illustrated in table 8.2 for 2 tube voltage ranges. This validates the good modelling of the detector response in the simulation.

Table 8.2 Comparison between the linear regressions to simulated and measured data for 2 tube voltage ranges.

Tube voltage range [kV] Slope Intercept Simulation 60 - 100 0.0060 -0.25 Measurement 60 - 100 0.0056 -0.18 Simulation 80 - 120 0.0036 -0.18 Measurement 80 - 120 0.0038 -0.17

77 8 Determination of the kVp with the Medipix2 Detector

Measurement Simulation

0.5

0.4

0.3 Ratio

0.2

0.1

0 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 Voltage [KV]

Figure 8.5 Ratio of the number of counts recorded behind copper ﬁlters with the Medipix2 detector from simulation and measurement for tube voltage values ranging from 40kV to 120 kV. The two curves denote the same functional behaviour.

0.4 Ratio = 0.0093 * Voltage - 0.18

0.35

0.3

0.25 Ratio

0.2

0.15

0.1 35 40 45 50 55 60 Voltage [kV]

Figure 8.6 Ratio R versus tube voltage ranging from 40 kV to 55 kV.

78 8.1 Determination of the kVp with multiple Filter Combinations

0.4 Ratio = 0.0056 * Voltage - 0.18

0.35

0.3

0.25 Ratio

0.2

0.15

0.1 55 60 65 70 75 80 85 90 95 100 105 Voltage [kV]

Figure 8.7 Ratio R versus tube voltage ranging from 60 kV to 100 kV.

0.4 Ratio = 0.0038 * Voltage - 0.17

0.35

0.3

0.25

0.2 Ratio

0.15

0.1

0.05

0 75 80 85 90 95 100 105 110 115 120 125 130 Voltage [kV]

Figure 8.8 Ratio R versus tube voltage ranging from 80 kV to 120 kV.

79 8 Determination of the kVp with the Medipix2 Detector

8.2 Determination of the kVp with only one Filter Combination

Using so many filter combinations in order to determine the peak kilo voltage is cum- bersome. From the figures of the previous section, it can be noted that higher statistical accuracy (lower possoiny fluctuations) could be achieved by using the 0.4 mm / 0.2 mm copper combination (see figure 8.5). From the measured data the relative statistical error of the kVp measurement can be given as

kVp R △ = △ kVp ( dR ) V dV ∗ R denotes the uncertainty in determining the ratio, V is the tube voltage. dR repre- △ dV sents the voltage dependent slope that was determined as the derivative of a 3th degree polynomial fitted to the data. An acurracy below 2.1 % can be reached as shown in figure 8.9. This is satisfying since some protocols used in hospital mentioned a relative precision of 10 % [44]. However, it should be kept in mind that systematic errors due to threshold fluctuations and pile up of events also influence the precision that can be achieved. It was found that threshold fluctuations above 5 keV can have a small impact on the value of the ratio obtained. Since the threshold is almost stable in case of Medipix such strong changes are unlikely. Hence the contribution of this systematic error to the global error can be neglected.

2.75

2.5

2.25

1.75

1.5

1.25

0.75 Relative statistical error of kVp in % 0.5

0.25

0 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 Voltage [KV]

Figure 8.9 Relative statistical error of the kVp measurement versus tube voltage. A precision below 2.1 % is achievable.

Some disadvantages of using a kVp-meter based on the ratio of the signals behind copper filters arise: their readings depend on the voltage waveform, the filtration utilized and total filtration of the X-ray tube. Therefore, they need a constant verification of their calibration [43]. The method described in the introduction of [44] could be used as complementary method since it is less sensitive to the filtration utilized.

80 8.3 Conclusion

8.3 Conclusion

In this chapter, it was shown that photon counting detectors such as Medipix2 can be used as kVp meter. Moreover determining the peak voltage accross an X-ray tube can be done with a precision below 2.1 % with only one filter combination. This is an improvement since different filter combinations are usually necessary depending upon the tube voltage of interest.

9 Introduction to X-ray Imaging

Contents 9.1 ImageQualityMetrics ...... 83 9.1.1 ImageContrast...... 83 9.1.2 Signal-to-noiseRatio...... 83 9.1.3 DetectiveQuantumEﬃenciencyDQE ...... 84 9.1.4 Point Spread Function PSF, Line Spread Function LSF, Modu- lationTransferFunctionMTF ...... 84 9.2 ImagingwiththeMedipix3Detector...... 85 9.3 Conclusion ...... 89

The ﬁrst part of this thesis focuses on the spectroscopic properties of the Medipix detectors. In fact, the Medipix detectors are designed for imaging: several publications show that the Medipix2 detector can be sucessfully used for X-ray radiography or bio- medical imaging, K-edge imaging or computed tomography ( [45], [46], [47], [48], [49]). The ﬁrst part of this chapter reviews the quantities used to characterize an imaging system. The second part will present some images that were taken using the Medipix3 detector in Single Pixel Mode and Charge Summing Mode.

9.1 Image Quality Metrics

9.1.1 Image Contrast The constrast of an object relative to its background is deﬁned as

Ib I C = − (9.1) Ib where I is the image intensity in the region of the object of interest, and Ib is the image intensity in the background surrounding the object [50].

9.1.2 Signal-to-noise Ratio This parameter is deﬁned as the ratio of the signal to noise. It is a measure of how good signal and noise can be separated from each other. For a given number of detected photons N, and a stochastical Poisson distributed signal with ﬂuctuation σ = √N , the SNR is given by:

N N SNR = = = √N (9.2) σ √N

83 9 Introduction to X-ray Imaging

Generally, the SNR is less than the ideal value √N corresponding to Poisson statistics. Often of more interest is the differential signal-to-noise ratio, SNRdiff , which uses the difference in signal behind an object of interest relative to its background [50]:

∆N CN SNRdiﬀ = = = C √N (9.3) σ √N Higher signal-to-noise ratio gives the imaging system stronger ability to discriminate a signal from background.

9.1.3 Detective Quantum Eﬃenciency DQE This quantity is one of the most important parameters used to characterise an imaging system. It is a measure for information transfer through the imaging chain, including the detector and is deﬁned for a given spatial frequency f, as follows:

2 (SNRout(f)) DQE = 2 (9.4) (SNRin(f))

SNRin and SNRout denote the input and output SNR, respectively. It describes the degradation of information of the signal in the imaging system. The DQE ranges between 0 and 1. The DQE at zero spatial frequency (DQE(0)) describes the fraction of photons that are detected and contribute to the image [5].

9.1.4 Point Spread Function PSF, Line Spread Function LSF, Modulation Transfer Function MTF The spatial resolution of a system is its ability to distinguish small objects that are situated closely. Several test objects have been developed to measure resolution. Fig.9.1 shows a so-called Huttner¨ grid composed of blocks of black and white line pairs of different frequencies. The number of line pairs that can be resolved by an imaging system constitutes its resolution. The Point Spread Function PSF describes the response of an

Figure 9.1 Photograph a Huttner¨ grid [11]. imaging system to a point source or point object. The knowledge of the PSF gives information on how sharp details can be processed by the system. The PSF is usually derived from the Line Spread Function LSF. The LSF, a one dimensional intensity proﬁl, represents the response of an imaging system to an irradiated inﬁnitely long, narrow slit [5]. The line can be realised in simulation using a fan source (as source of photons) slightly

84 9.2 Imaging with the Medipix3 Detector tilted with respect to the pixel symmetry axes at a steep angle so that aliasing errors can be avoided. Projecting the detector response on an axis perpendicular to the slit yields the LSF. In practice, since narrow slits are diﬃcult to realise, the so called edge method [51] is used to determine the LSF, as derivative of the Edge Spread Function. The ESF is calculated by projecting the detector response on an axis perpendicular to the edge axis. For pixelated detectors, the detector response to a pencil beam depends on the relative position of the signal with respect to the pixel. This means that translation invariance holds only if the translation distance is a multiple of the pixel size. The line spread function should then be modiﬁed as follows [21]:

1 p LSF (x)= T (δ(x a)) da (9.5) p · − · Z0 p is the distance between the center of the pixel and the point of impact of the pencil beam. T denotes the transfer function of the detector. In oder to determine the resolution of a system, it is better to analyse the LSF in the frequency domain. This is achieved by performing a Fourier transformation of the LSF. The Modulation Transfer Function MTF is proportional to the Fourier transform of the LSF and is deﬁned as:

F T [LSF (x)] MT F (f)= | | (9.6) F T [LSF (x)] (0) | | In fact this function measures how faithfully an imaging system reproduces or transfers structures from the object to the image. A detailed investigation of the MTF of Medipix3 will be presented in the next chapter.

9.2 Imaging with the Medipix3 Detector

Figures 9.3 and 9.5, 9.4 and 9.6 are raw and flatfield corrected images of a cash card taken using the Medipix3 operating in Charge Summing Mode and in Single Pixel Mode, respectively. The threshold in both cases was set to 10 keV. The sample (see figure 9.2) was placed about 1m away from the Megalix X-ray tube and a voltage of 40 kV was applied accross the tube. The tube current was 20 mA and the acquisition time was 25 s. On the raw image taken in CSM, lines present on the surface of the cash card can be recognized. However they seem to be highlighted due to high node to node efficiency variations. A simple flatfield correction technique [49] helps to gain some homogeneity. Meanwhile more details cannot be resolved. It is obvious that far more internal structures (wires, electronics) can be recognized in the raw SPM image as depicted in figure 9.5. The CSM image presents lots of granularity that hinder a better resolution. This granularity is due to the fact that the charge collected after the summing procedure is allocated to the wrong node: let us assume that the reconstructed charge after the summing process is the same for 4 neighbouring nodes. In this case, the charge should be allocated randomly to the node with the highest electronic noise level. The granularity points out that noise is not the process, which decides which node counts but the high threshold dispersion even after equalisation does. Hence some nodes seem to be preferred, thus they count more

85 9 Introduction to X-ray Imaging

Figure 9.2 Photograph of a cash card.

5 x 10

3 50

2.5

100 2

150 1.5

1 200 0.5

250 0 50 100 150 200 250

Figure 9.3 Raw image of an electronic cash card taken with the Medipix3 detector in CSM. The threshold was set to 10 keV. than their neighbours. A redesign of Charge Summing Mode of Medipix3 is proposed in order to solve this issue. This new architecture will be presented in chapter 12. Figures 9.7 und 9.8 show two images of a human bone taken with Medipix3 in SPM. The sample was positioned so that a high magnification could be achieved. The X-ray tube used was a micro focus tube manufactured by the YXLON company. The tube voltage was set at 50 kV. The first image depicts the image of the bone at a threshold of 10 keV. One can notice regions of low and high absorption. This fact is confirmed in figure 9.8 where regions of high absorption can be selected. This was achieved by defining an energy deposition window from 5 keV to 25 keV.

86 9.2 Imaging with the Medipix3 Detector

4 x 10

50 14

12 100 10

8 150

200 4

250

50 100 150 200 250

Figure 9.4 Flatﬁeld corrected image of a cash card recorded with the Medipix3 detector in CSM at a threshold of 10 keV.

4 x 10

50 8

100 7

6 150

200 4

250 3 50 100 150 200 250

Figure 9.5 Raw image of a cash card taken with the Medixpix3 detector in SPM at a threshold of 10keV. Some wires connected to a chip can be recognized.

87 9 Introduction to X-ray Imaging

4 x 10

8.5

50 7.5

100 6.5

150 5.5

200 4.5

250 3.5 50 100 150 200 250

Figure 9.6 Flatﬁeld corrected image of cash card recorded with the Medipix3 detector in SPM at a threshold of 10 keV. Some wires connected to a chip can be recognized.

5 x 10 2

1.8

50 1.6

1.4

100 1.2

150 0.8

0.6

200 0.4

0.2 250 0 50 100 150 200 250

Figure 9.7 Image of a human bone taken with the Medipix3 detector in SPM. The threshold used was 10 keV.

88 9.3 Conclusion

9.3 Conclusion

4 x 10 10

50 8

100 6

150 4

200 2

1 250 0 50 100 150 200 250

Figure 9.8 Image of a human bone taken with the Medipix3 detector in SPM. An energy window from 15 keV to 25 keV was used.

This chapter focused on 2 parts: the first part gave an overview of the quantities used to characterize imaging systems. The second part was on images taken with the Medipix3 chip. The SPM provides best quality images. Imaging with the Medipix3 chip programmed in Charge Summing Mode has some problems due to granularity. A flatfield correction does not really help to get rid of the problem completely. This matter is now addressed in redesigning the Charge Summing Mode. More details can be found in the last chapter.

10 Spatial Resolution of the Medipix3 Detector

Contents 10.1Methods ...... 91 10.2 Spatial Resolution of the Medipix3 Detector in Single Pixel Mode...... 93 10.2.1 Simulation ...... 93 10.2.2 Measurement ...... 95 10.3 Spatial Resolution of the Medipix3 Detector in Charge Sum- mingMode ...... 96 10.3.1 Simulation ...... 96 10.3.2 Measurement ...... 100 10.4 Conclusion ...... 101

Single Pixel Mode of Medipix3 is equivalent to the counting mode of the Medipix2 detector as each pixel acts independently from its neighbours. Simulations and measurements have been performed few years ago to determine the LSF and the MTF of Medipix2. The spatial resolution was found to be 9.1 lp/mm at Nyquist frequency [11]. Moreover, it was shown that parameters like the discriminator threshold, the tube voltage applied across the X-ray tube, the bias voltage (applied for charge carriers transport) and fluorescence photons originating from some materials contained in the detector had an influence on the shape of the MTF and somewhat on the effective pixel size ([21], [52], [53]). In addition to that, preliminary simulations were performed in CSM of Medipix3 [21] to estimate the spatial resolution when Medipix3 was in design phase. Since the detector is now available some new simulations have been carried out using the Photoncounting Class [16] taking into account parameters such as the gain variation from pixel-to-pixel or the threshold dispersion extracted from measurements. Presenting results of simulations and measurements of the spatial resolution of Medipix3 constitutes the goal of this chapter.

10.1 Methods

The slit and edge methods have been applied in order to determine the spatial resolution of Medipix detectors. The slit method was implemented in the simulation routine. This method measures the response of the system to an impulse function, but rather than a delta function it uses a slit. In order to realise such a narrow slit in the simulation, a fan source, slightly tilted with respect to the pixel matrix, was used as source of photons and was placed 1 m away in front of the detector implemented with all its electronics. Moreover, relevant physical

91 10 Spatial Resolution of the Medipix3 Detector processes like diffusion or repulsion of charge carriers and noise contributions (see chapter 5) were taken into account. The simulation was performed with 107 incident photons. Figure 10.1 depicts an image of the slit placed over the pixel plane and inclined at an angle of 3 degrees with respect to the pixel axes. Since the orientation of the slit influences the spatial sampling of the LSF and MTF, it is recommended for Medipix detectors to use angles between 1-3 degrees to achieve a sampling interval much finer than that provided by the pixel-to-pixel distance. The LSF is obtained by projection of the simulated image on an axis perpendicular to the direction of the line input signal [51]. Since it is difficult to prepare a sufficiently collimated fan source in real world measurements, the so called edge method was used in the measurements for determining the spatial resolution of Medipix3. The edge function is in fact an integrated delta function. The setup consists of a tungsten plate of 3 mm thickness that covers about half of the detector so that a sharp edge in the count rate along the detector rows is produced. Futhermore the tube voltage of the tube was set to 100 kV. In order to minimize blurring due to finite focus size of the tube, the distances (detector to plate, plate to X-ray source) were chosen so that the magnification of the focal spot was as small as possible. Figure 10.2 shows an image of an edge taken with a Medipix3 detector programmed in Single Pixel Mode. After the determination of the edge inclination angle, the pixel center distances from the edge can be calculated. Projecting the edge image along the edge axis on a perpendicular axis yields the so called Edge Spread Function (ESF). Taking the derivative of the ESF delivers the LSF. The MTF is then obtained by Fourier transformation of the LSF and normalization.

1800

1600 50

1400

100 1200

1000

150 800 Pixel Number 600

200 400

200

250 0 50 100 150 200 250 Pixel Number

Figure 10.1 Image of a slit acquired with the Medipix3 detector in SPM (simulation).

92 10.2 Spatial Resolution of the Medipix3 Detector in Single Pixel Mode

6 x 10 3.5

3 50

2.5

100 2

1.5 Pixel Number 150

1 200

0.5

250 0 50 100 150 200 250 Pixel Number

Figure 10.2 Image of a tungsten edge acquired with a Medipix3 detector in SPM (measurement).

10.2 Spatial Resolution of the Medipix3 Detector in Single Pixel Mode

10.2.1 Simulation

Figure 10.3 shows the simulated LSF of the Medipix3 detector in SPM at different threshold values. It can be seen in the figure that the effective pixel size depends on the threshold.

A pixel size of 55 ñm is expected for the ideal case. From the ﬁgure one can notice that the pixel size is bigger than the ideal value at lower threshold as a blurring of the LSF is observed. This is ascribed to multiple counting of a photon as result of charge sharing between adjacent pixels. Increasing the threshold leads to a reduction of multiple counts and thus an improvement of the LSF as observed for a threshold of 40 keV where the pixel

size almost matches the 55 ñm. The intensity of the LSF is quite small at this threshold due to the poor eﬃciency of silicon at higher energies and the dominance of Compton scattering. The peak present in the middle of the pixel proﬁle and observed in a period of

55 ñm is caused by the L-fluorescence photons (energy: 10.45 keV ;14.76 keV) of lead and K-fluorescence photons (energy: 25 keV) of tin contained in the solder bumps. The peak is also influenced by the applied threshold. For threshold values above the energy of the fluorescence photons of the mentioned materials, no peak is observed. A Fourier transform of the LSF data provides the MTF as depicted in figure 10.4 for different threshold values. A drop of the MTF at low spatial frequencies is observed as in case of Medipix2 [21]. This is due to fluorescence photons originating from tin contained in the solder bumps. Another effect that can be observed is the improvement of the spatial frequency with increasing threshold values. This is expected as the influence of charge sharing decreases at higher thresholds. The presampled MTF, calculated using a MTF value of 0.3, reaches a value of 12.45 lp/mm at a threshold of 5 keV. It can be noticed that the first zero point of the MTF moves to higher frequencies as the threshold increases. This can be understood as the zero point is inversely proprotional to the effective pixel size. Hence

93 10 Spatial Resolution of the Medipix3 Detector a small eﬀective pixel size leads to higher zero point of the MTF. As a result, a pseudo

resolution is obtained although the pixel size of 55 ñm remains the same.

3000 5 keV 10 keV 20 keV 2500 40 keV

2000

1500 LSF

1000

500

0 -0.11 -0.0825 -0.055 -0.0275 0 0.0275 0.055 0.0825 0.11 x in mm

Figure 10.3 Simulated LSF of Medipix3 in SPM for diﬀerent threshold values. Lower

threshold values lead to an eﬀective pixel size above 55 ñm due to charge sharing. Increasing

the threshold yields pixel size close to the expected value of 55 ñm because single counts are promoted.

1 5 keV 10 keV 20 keV 40 keV 0.8

0.6 MTF

0.4

0.2

0 0 5 10 15 20 25 30 35 40 Frequency in lp / mm

Figure 10.4 Simulated MTF of Medipix3 in SPM for diﬀerent THL values. A drop at low spatial frequencies is observed at lower thresholds. Responsible for the drop are the ﬂuorescence photons originating from lead and tin. Increasing the threshold leads to a shift of the zero point of the MTF to higher frequencies.

94 10.2 Spatial Resolution of the Medipix3 Detector in Single Pixel Mode

10.2.2 Measurement

For the measurements, the Medipix3 detector was programmed in SPM, High Gain Mode and the acquisition time was set to 600 s and the threshold was 10 keV. The setup has already been described in section 10.1. A flatfield correction of the images recorded was performed before the MTF calculation. This is necessary because of the gain variation from pixel-to-pixel. Moreover, for the edge method applied, pixel are assumed to behave all in the same way. Figure 10.5 depicts the measured MTFs of the Medipix3 in SPM for Shaper DAC 100, 150 and 255. The simulated MTF is also presented. The drop at low frequencies due to the K-fluorescence photons of lead and to the L-fluorescence photons of tin contained in the solder bumps can be clearly seen in the measured data. This drop is well reproduced by the simulation. The presampled MTF, calculated using a MTF value of 0.3, reaches a value of 12.45 lp/mm as estimated from simulated data. This value is independent from Shaper DAC. Nevertheless, higher fluctuations are observed at Shaper DAC value 100 since noise is increased and the gain is decreased for lower Shaper DAC values. The influence of the gain mode used, has also been investigated. Figure 10.6 shows the measured MTF in SPM, High Gain Mode and Low Gain Mode (see chapter 4, section 4.3). The result remains almost unchanged. However, more fluctuations are observed in LGM since we have half signal which implies lower statistics compared to High Gain Mode. The MTF curve in LGM lies slightly above the curve in HGM. This is ascribed to the reduction of multiple counting in LGM as photo-electric effect events are promoted in this mode. An optimal configuration of the Medipix3 detector for MTF measurements is realised in using Shaper DAC values of at least 150 and in operating the detector in High Gain Mode.

0.9 Shaper: 100 (Measurement) Shaper: 150 (Measurement) 0.8 Shaper: 255 (Measurement) Simulation 0.7

0.6

0.5 MTF

0.4

0.3

0.2

0.1

0 0 2.5 5 7.5 10 12.5 15 17.5 20 Frequency in lp/mm

Figure 10.5 Measured MTF of Medipix3 in SPM for Shaper DAC 100, 150, 255 and a threshold of 10 keV. The simulated MTF is also presented. A good agreement between simulation and measurement is observed for frequencies below 12.45 lp/mm. The drop in lower frequencies is well reproduced in the simulation. The MTF is independent from the Shaper DAC value.

95 10 Spatial Resolution of the Medipix3 Detector

1 Low Gain Mode 0.9 High Gain Mode

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Figure 10.6 Measured MTF of Medipix3 in SPM, HGM and LGM. The threshold was 10 keV.

10.3 Spatial Resolution of the Medipix3 Detector in Charge Summing Mode

10.3.1 Simulation In the simulations the impact of parameters such as the bias voltage, the energy threshold, the variations in gain and current copies on the spatial resolution of Medipix3 in CSM has been investigated. Figure 10.7 is an illustration of simulated LSFs of the Medipix3 detector in CSM. Bias voltages of 30 V and 90 V were applied. The shape of the LSF for 90 V bias differs considerably from the expected square shape. This is due to the charge allocation procedure: In CSM some ambiguities arise in the hit allocation scheme if the charge is collected by only one pixel or is collected by two adjacent pixels as shown in figure 10.8. At the end of the summing procedure, 4 or 2 nodes have the same values. In this case electronic noise should decide which pixel should get the hit assigned. In case 2 nodes have the same value the hit can be assigned with the same probability to the pixel on the left or on the right hand side in figure 10.8(a) or up and down in figure 10.8(b). This leads to a broadening of the LSF of 2 times the pixel size. The peak in the middle of the LSF is due to hits that were clearly assigned to one node (see figure10.8(c)). The LSF can be improved by decreasing the bias voltage as the worst case is suppressed and the case where the charge is deposited among many pixels is promoted. The LSFs are not centered around zero because the summing nodes are situated at the edge of each pixel. Figure 10.9 depicts the MTF of the Medipix3 detector in CSM for different threshold values. One can notice that the MTF is almost threshold independent. This is due to the charge summing scheme that permits an effective reduction of charge sharing. The gain variation from pixel to pixel does not really lead to a notable degradation of the MTF (see figure 10.10). The two parameters that lead to an improvement or a degradation of

96 10.3 Spatial Resolution of the Medipix3 Detector in Charge Summing Mode the MTF are the variations in current copy procedure and the bias voltage. Increasing the variations in copy procedure leads to a shift to lower frequencies of the zero point of the MTF which implies a degradation of the MTF (see ﬁgure 10.11). Figure 10.12 depicts the MTF for diﬀerent values of the bias voltage. Lower bias voltages lead to an improvement of the MTF. This is once again ascribed to the summing scheme that corrects for charge sharing occuring between adjacent pixels, thus improving the MTF.

1 Bias voltage: 30 V Bias voltage: 90 V

0.8

0.6 LSF

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0 -0.0825 -0.055 -0.0275 0 0.0275 0.055 0.0825 0.11 x in mm

Figure 10.7 Simulated LSF of Medipix3 in CSM for diﬀerent bias voltage values

Figure 10.8 Cases of ambiguity in the hit allocation in CSM. Red sum symbols indicate that the largest partial charge was registered by the pixel. The letters in red display the pixel that can probably get the hit assignment [52].

97 10 Spatial Resolution of the Medipix3 Detector

1 8 keV 20 keV 30 keV 60 keV 0.8

0.6 MTF

0.4

0.2

0 0 5 10 15 20 Frequency in LP / mm

Figure 10.9 Simulated MTF of the Medipix3 detector in CSM for diﬀerent threshold values. The MTF is almost threshold independent as charge sharing is eﬃciently suppressed.

1 3.7 % 4.56 % 7 % 15 % 0.8 50 %

0.6 MTF

0.4

0.2

0 0 2.5 5 7.5 10 12.5 15 17.5 20 Frequency in LP / mm

Figure 10.10 Simulated MTF of the Medipix3 detector in CSM for diﬀerent values of the gain variation. The threshold does not depend on the gain variation.

98 10.3 Spatial Resolution of the Medipix3 Detector in Charge Summing Mode

1 1.47 % 6 % 15 % 30 % 0.8

0.6 MTF

0.4

0.2

0 0 2.5 5 7.5 10 12.5 15 17.5 20 Frequency in LP / mm

Figure 10.11 Simulated MTF of the Medipix3 detector in CSM for diﬀerent values of current copy variation. The zero point of the MTF is shifted to lower frequencies as the variations increase. This leads to a degradation of the MTF.

1 30 V 60 V 90 150 V 0.8

0.6 MTF

0.4

0.2

0 0 2.5 5 7.5 10 12.5 15 17.5 20 Frequency in LP / mm

Figure 10.12 Simulated MTF of the Medipix3 detector in CSM for diﬀerent bias voltages. Lower bias voltages lead to an improvement of the MTF.

99 10 Spatial Resolution of the Medipix3 Detector

10.3.2 Measurement The measurements to determine the MTF in CSM have been performed using the same setup, acquisition time and the same tube voltage as for SPM. The Medipix3 used was

also an assembly with 300 ñm thick silicon sensor. Figure 10.13 depicts the result obtained for a threshold value of 10 keV. As can be seen the result is far way from the design value as the presampled MTF has a value of 2.5 lp/mm instead of 12.45 lp/mm expected. Since some nodes count much more than the others, this eﬀect degrades the edge image used as input to determine the MTF. Figure 10.14 shows the ESF for SPM and CSM. The edge can hardly be recognized in CSM once again due to the bad hit allocation.

0.9

0.8

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0.6

0.5 MTF

0.4

0.3

0.2

0.1

0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Frequency in LP / mm

Figure 10.13 MTF of Medipix3 in CSM for a threshold value of 10 keV. The presampled MTF is only about 2.5 LP/mm.

100 10.4 Conclusion

2.5

2 Charge Summing Mode Single Pixel Mode

1.5 ESF

0.5

0 -5 -4 -3 -2 -1 0 x in mm

Figure 10.14 ESF in CSM and SPM for a threshold value of 10 keV

10.4 Conclusion

In this chapter, results of simulations and measurements of the MTF of the Medipix3 detector in SPM and CSM have been presented. A good agreement between simulations and measurements was found in case of SPM. Moreover, it was found that the measured MTF are smoother when the chip is programmed in High Gain Mode and Shaper DAC values of at least 150 are used. In case of CSM, a value of 2.5 lp/mm was determined from the measured data. This is far away from 12.45 lp/mm expected from simulation. The main reason is the inhomogeneous hit allocation that degrades the spatial resolution. A new allocation scheme has been proposed in order to improve imaging with Medipix3 operating in CSM. This will be described in chapter 12.

101

11 Material Reconstruction with Photon counting Detectors

Contents 11.1 Theory of Material Reconstruction ...... 103 11.2 CombinationMethod ...... 105 11.3 MinimizationAlgorithms...... 106 11.3.1 DownhillSimplexMethod ...... 106 11.3.2 Simulated Annealing Method ...... 108 11.4 Reconstruction Results with Medipix2 and Medipix3 . . . . . 108 11.4.1 Simulation ...... 108 11.4.2 Measurements ...... 112 11.5 Material Reconstruction of the Solder Bumps of Medipix . . 115 11.6 Conclusion ...... 118

In conventional X-ray imaging, the image of a compound sample is registered as cu- mulative attenuation of all the materials which are present in the object. As a result the speciﬁc contribution of materials contained in the sample cannot be resolved. Several techniques like dual energy CT [54] or Principle Components Analysis (PCA) [55] have been applied to exploit the energy information accessible with the broad spectrum of an X-ray tube. The PCA method has the disadvantage that it does not deliver quantita- tive information. The energy resolved material reconstruction makes use of the energy sensivity and good spatial resolution of Medipix detectors. The main focus of this chapter is to present results of simulations and measurements achieved with the Medipix detectors in order to distinguish diﬀerent materials in an X-ray image of an object by reconstructing their individual areal densities.

11.1 Theory of Material Reconstruction

In general the transmittance of a compound object comprising M materials, derived from the Lambert Beer’s law, that can be measured with a photon counting detector, is expressed as:

m=M N(E) ′ T (E)= = N0(E) exp µ (E) am (11.1) N (E) · − m · 0 m=1 ! X N0(E) is the number of incident photons of energy E. N(E) is the energy dependent number of photons transmitted through the compound object. m is the subscript of

103 11 Material Reconstruction with Photon counting Detectors

the material component (basis material). am denotes the areal density of the material component m and has the unit mass per area. It is deﬁned as the projection of the density ρm of material m along the X-ray path S through the object :

am = ρm ds ZS

′ µm(E) represents the energy dependent mass attenuation coeﬃcient of the basis material m. Taking equation 11.1 into account and introducing discrete energy channels i yields the following relation [23] :

m=M Ni ′ ti = ln = µ (E)am (11.2) − N im 0i m=1 X or in matrix equation: t = B a · The elements of the matrix B are the energy dependent mass attenuation coefficients of the basis materials. Figure 11.1 shows the mass attenuation coefficients of nickel, silicon, tin and lead. If a compound object contains these four materials, they should be chosen as basis materials for the reconstruction. The set of linear equations obtained delivers a solution when the columns/rows of the matrix B are linearly independent: If the compound material contains two materials, the linear independence holds as for the energy range of interest Compton and photo-electric effect contribute most to the total attenuation coefficient and have different dependence on photon energy [23]. If the compound material consists of more than two materials, the K- and L-absorption edges of the material can additionally be used to ensure the linear independence since the edges are material specific and cannot be obtained through linear combination of other basis materials. Solving the set of linear equations 11.2 requires that the energy dependent transmittance ti should be measured for several channels in the primary photon energy. Using a highly segmented detector like Medipix adds the difficulty that the energy deposition differs significantly from primary photon energy. Therefore a spectrum reconstruction which requires high statistics would be necessary. This is of course feasible but time consuming. The combination method presented in the next section is therefore a good alternative since it requires less channels and spectrum reconstruction is performed only once without an object in the X-ray beam.

104 11.2 Combination Method

Nickel Silicon Lead / g

2 Tin 100

10 Mass attenuation coefficient in cm

8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 Energy in keV

Figure 11.1 Mass attenuation coeﬃcient of nickel, silicon, tin and lead as function of photon energy [15].

11.2 Combination Method

This method developed in our group [23] combines the spectrum reconstruction and the actual material reconstruction in one step. The spectrum reconstruction (see chapter 7) is only needed to determine the incident spectrum. The combination method is realised by including the detector response to monochromatic radiation in the likehood function. The expected number of events recorded by a pixel ˜ ˜ in an energy deposition channel Ni can be expressed as: Ni = j RijNj where R is the energy response function of the detector to monochrome radiation. The Lambert Beer’s P law can then be rewritten as follows:

′ ′ − P µ am − P µ am Nj = N0,je m jm = N˜i = RijN0,je m jm . (11.3) ⇒ j X To account for the relatively broad energy channels used in measurements, ﬁner channels are internally used for calculations. The L adjacent large channels meet at the energies El = E1, E2,...,EL. The ﬁner channels i can be added up to match the broader channels, ˜ ˜ so that counts in the broad channels become Nl = i Ni . The negative log-likelihood El≤Ei

˜ − Pm µjmam Nl i j RijN0,je − El≤Ei

105 11 Material Reconstruction with Photon counting Detectors

Table 11.1 Overview of the subscripts used in this section [23].

index maximum description i I energy deposition channel j J primary photon energy channel m M basis materials l L energy deposition channel at measurement pixels of the detector. Table 11.1 summarizes the subscripts used in this section for the sake of clarity.

11.3 Minimization Algorithms

The downhill simplex method and simulated annealing method have been used to ﬁnd the minimum of the likehood function deﬁned in equation 11.4.

11.3.1 Downhill Simplex Method This minimization method requires only function evaluations, not the derivatives. It is based on the Nelder and Mead scheme. In N dimensions, a simplex is a polyhedron with N+1 points. The method iteratively updates the worst point (where the value of the function is the highest) by four operations: reﬂection, expansion, one-dimensional contraction and multiple contraction [56].

Reﬂection: this operation involves moving the worst point of the simplex to a point reﬂected through the remaining N points.

Expansion: If the result of the previous operation is better than the best point, then the method attempts to expand the simplex along this line.

Contraction: If the new point is not much better than the previous point, then the simplex is contracted along one dimension from the highest point.

Multiple contraction: In case the new point is worse than the previous points, the simplex is contracted along all dimensions toward the best point and steps down the valley.

Figure 11.2 illustrates the 4 operations described above. These series of operations are repeated until the optimal solution is found. This routine is implemented in the function fminsearch in MATLAB [57].

106 11.3 Minimization Algorithms

Figure 11.2 Basic operations in the downhill simplex method [56].

107 11 Material Reconstruction with Photon counting Detectors

11.3.2 Simulated Annealing Method The method of simulated annealing is suitable for optimization problems in which a desired global extremum is hidden among many poorer local extrema. At the heart of the method of simulated annealing is an analogy with thermodynamics, specifically with the way that liquids freeze and crystallize. At high temperatures, the molecules of a liquid move freely with respect to one another. If the liquid is cooled slowly, thermal mobility is lost. The atoms are often able to line themselves up and form a pure crystal. So the essence of the process is slow cooling to ensure that a low energy state will be achieved. The probability to find a system in an energy state E can be expressed using the Boltzmann probability distribution as follows: E P (E) exp (11.5) ∼ −k T T is the temperature and k is the Boltzmann’s constant. With this distribution, there is a corresponding chance for the system to get out of a local energy minimum in favor of finding a global one. To implement the method following elements should be provided:

A description of possible system conﬁgurations.

A generator of random changes in the conﬁguration; these changes are the options presented to the system.

An objective function E (here the likelihood function in equation 11.4) whose minimization is the goal of the procedure.

A control parameter T (analog of temperature) and an annealing schedule which describes how it is lowered from high to low values.

The annealing algorithm is very time consuming compared to the simplex method. For material reconstruction at least 2 days of computation time were necessary. Detailed information on the annealing method can be found in [56].

11.4 Reconstruction Results with Medipix2 and Medipix3

11.4.1 Simulation Figure 11.3 (a) presents the phantom and the detector implemented in the simulation. The phantom is made of water and has a cubic structure with as side length of 10 mm. It contains two plates made of iodine and gadolinium. Both are used as contrast agents in medical imaging. Each of the plates comprises three regions of different areal density (table 11.2). The energy of the incident photons was distributed according to an X-ray tube with tungsten anode operated at 100 kV with 2.5 mm aluminum filtering. The X-ray tube (not illustrated) is placed on the left hand side of the phantom. The simulation was performed with 109 incident photons. Several counting images are recorded for threshold values ranging from 11.7 keV to 76.7 keV in steps of 7 keV. Images are calculated for different defined energy deposition windows. In this process it is important to have energy bins containing the K-absorption edges of iodine (33.2 keV) and gadolinium (50 keV) to ensure the linear independence of the basis materials and to perform a good separation

108 11.4 Reconstruction Results with Medipix2 and Medipix3

1200 50

1000

100 800

600 150

400 200

200

250 0 50 100 150 200 250 (a) Setup (b) photon counting image.

Figure 11.3 Sketch of the phantom and the resulting counting image at a threshold of 18.7 keV. The counting image is tilted with respect to the setup image at an angle of 90 degrees: gadolinium is on the left hand side and iodine on the right hand side of the materials contained in the compound object. The result can therefore be used as input for the combination method mentioned in section 11.2. The reconstruction was performed using the minimization algorithms described in the previous section. Figure 11.3 (b) shows the photon counting image. This image is tilted with respect to the setup-lay-out picture at an angle of 90 degrees so that gadolinium is on the left hand side and iodine on the right hand side. Figure 11.4 shows the reconstructed areal density images for the Medipix2 detector (SPM of Medipix3) and for CSM of Medipix3, reconstructed with simulated annealing. A notable diﬀerence in reconstruction results is observed for both modes. The reconstructed images in SPM contain some artifacts: In the iodine image, gadolinium can also be seen. This is due to charge sharing and Compton scattering as no anti-scatter-grid were implemented in the simulation. The separation works better in CSM as only the expected material is observed in the associated images: only gadolinium is visible in the gadolinium image and iodine in the iodine image, respectively. This is ascribed to the summing scheme as charge sharing between adjacent pixels is almost suppressed. However the water image for CSM presents some artifacts. This is due to Compton scattering in the phantom. Table 11.2 shows a comparison between the input areal densities and the reconstructed areal densities for SPM and CSM of Medipix3. The reconstructed areal densities are in the same order of magnitude as the known densities. Meanwhile, small densities couldn’t be reconstructed. This is expected as they are almost not visible in the counting image (see image 11.3(b)). Figure 11.5 shows a comparison between reconstructed areal density images of the phantom using the simplex and annealing method for SPM of Medipix3. The images obtained using the simplex method seem to be better as more areal densities can be recognized. But at the same time, the level of artifacts is also higher than in the images delivered by the annealing method.

109 11 Material Reconstruction with Photon counting Detectors

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(e) Water image (SPM) (f) Water image (CSM)

Figure 11.4 Results of the material reconstruction: gadolinium, iodine and water images are g shown for SPM and CSM. The color bar represents the reconstructed areal densities in cm2 . The separation is better for CSM.

110 11.4 Reconstruction Results with Medipix2 and Medipix3

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Figure 11.5 Results of the material reconstruction: gadolinium, iodine, and water images are shown for SPM. The reconstruction was performed using the simplex and simulated g annealing methods. The color bar represents the reconstructed areal densities in cm2 .

111 11 Material Reconstruction with Photon counting Detectors

g Table 11.2 Comparison between the reconstruced areal densities given in cm2 for SPM and CSM and the original input densities. The reconstruction was performed using the simulated annealing algorithm. The statistical errors of the reconstructed areal denstities are not shown as they are negligible.

g real areal density [ cm2 ] SPM CSM Galolinium 0.008 - - Galolinium 0.040 0.026 0.032 Galolinium 0.080 0.041 0.050 Iodine 0.050 0.030 0.050 Iodine 0.025 - 0.030 Iodine 0.005 - - Water 1.000 0.053 0.05

11.4.2 Measurements Measurements have been performed to test the reconstruction method. The setup in the measurements was a standard radiography setup comprising an X-ray tube, the phantom to be irradiated and the Medipix2 detector. The phantom had a cubic shape and was made of PMMA and placed one meter away from the X-ray tube. It had four cylindrical cutouts. Each of them has a diameter of 5 mm and a depth of 7 mm. Two of the cutouts were filled with gadolinium and iodine solutions. The other two cutouts were empty. The X-ray tube voltage was set at 80 kV. A tube current of 20 mA was applied and the acquisition time was set at 1s. Figure 11.6 shows two counting images recorded with the detector at a threshold of 18.7 keV and for an energy window from 32.7 keV to 39.7 keV. The absorption edge of iodine is at 33.2 keV. In the images the cylinder with the stronger absorption is filled with gadolinium (blue). The other one is iodine. The areal density of iodine seems to be small since it cannot be separated from the background (figure 11.6(b)). The simulated annealing and the simplex method were applied to reconstruct the areal densities. The result is shown in figure 11.7. The areal densities could not be properly reconstructed using the simplex method as the reconstructed materials cannot be seen in their associated images. The simulated annealing delivers the best result: gadolinium appears in the gadolinium image, PMMA in the PMMA image. The reconstructed areal density of gadolinium is estimated to 0.029 g /cm2. Iodine could be not reconstructed with these methods. This is due to the small areal density of iodine contained in the compound object (see figure 11.6(b)).

112 11.4 Reconstruction Results with Medipix2 and Medipix3

4 x 10

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(a) 18.7 keV

4 x 10 2.5

50 2

100 1.5

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200 0.5

250 0 50 100 150 200 250

(b) 32.7 keV-39.7 keV

Figure 11.6 Photon counting image at a threshold of 18.7 keV and for an energy window from 32.7 keV to 39.7 keV. In each ﬁgure, the upper cutout is ﬁlled with iodine and the lower cutout contains gadolinium.

113 11 Material Reconstruction with Photon counting Detectors

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Figure 11.7 Reconstructed gadolinium, iodine, PMMA images with the simplex method on the left hand side and the simulated annealing algorithm on the right hand side. The color g bar represents the reconstructed areal densities in cm2 .

114 11.5 Material Reconstruction of the Solder Bumps of Medipix

11.5 Material Reconstruction of the Solder Bumps of Medipix

In this section, the reconstruction method is applied to the solder bumps connecting ñ sensor (300 ñm silicon) and ASIC (780 m silicon) of a Medipix assembly. Figure 11.8 shows the setup of the measurement. The tube was a micro focus tube manufactured by YXLON company. The tube voltage was 70 kV and the power was fixed to 10 W. In order to achieve a high magnification the sample was placed 1 cm in front of the tube and the detector - tube distance was set at 68 cm. Figure 11.9 shows two counting images recorded with the Medipix3 detector operating in SPM for energy windows ranging from 8 keV to 17 keV and from 24 keV to 35 keV. The L-absorption edge of lead is at 12.5 keV and the K-edge of tin is at 29.1 keV. In the two images the solder bumps are clearly visible. The reconstruction was performed using the simplex method. The material reconstruction is quite successful as depicted in figure 11.10 where images of lead, tin and silicon are shown. Unfortunately, the obtained silicon areal density image is not homogenous because some artifacts from the bumps are observed. This is once again ascribed to charge sharing and to Compton scattering in the region of g the bump balls. The mean value of the reconstructed areal density is about 0.015 cm2 for g lead and 0.025 cm2 for tin. The height of the bumps in direction of the X-rays is obtained

by multiplication of the areal densities of the materials contained in the bumps with their ñ respective density. The height calculated this way ranges from 14 ñm to 22 m.

Figure 11.8 Setup for bump bond material reconstruction

115 11 Material Reconstruction with Photon counting Detectors

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116 11.5 Material Reconstruction of the Solder Bumps of Medipix

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117 11 Material Reconstruction with Photon counting Detectors

11.6 Conclusion

In this chapter, theory, methods and results of material reconstruction using photon counting detectors were presented. It was shown it is possible to exploit the energy sensitivity of the detectors to separate materials in a compound object. From simulation we deduce that reconstructed images in Charge Summing Mode were better than the ones obtained in the single pixel counting mode. This is ascribed to the summing scheme as charge sharing is corrected. The simulated annealing algorithm and the simplex method were applied for reconstruction. They delivered diﬀerent results in the investigated cases. The method of material reconstruction might be of value in medical imaging where iodine and gadolinium are used as contrast agents.

118 12 Redesign of Charge Summing Mode of Medipix3

Contents 12.1 New Architecture of Charge Summing Mode of Medipix3 . . 119 12.2 Simulation of the proposed architecture ...... 120 12.3 Impact of the new Charge Summing Mode architecture on Imaging ...... 120

The currently implemented Charge Summing Mode (CSM) works well for spectroscopy since the deposited charge is correctly reconstructed and the tail in energy deposition spectra to low energy deposition due to charge sharing is suppressed. However the unex- pectedly high pixel-to-pixel variation in threshold in CSM even after equalization has a severe influence in the arbitration decision: pixels with a lower threshold with respect to their neighbours have a far higher probability to get the hit assigned. The hit is in most cases allocated to the wrong summing circuitry. This is why some granularity is present in images taken in CSM. The image quality is destroyed and the MTF is degraded. Even after correction of the pixel efficiency, the image quality remains poor compared to the images recorded in the Single Pixel Mode (SPM). Since advanced correction techniques might perhaps not help to get rid completely of this problem, it is necessary to find a solution in redesigning the CSM. This issue is solved with a new hit allocation scheme which hopefully should be more robust to pixel-to-pixel threshold variation.

12.1 New Architecture of Charge Summing Mode of Medipix3

The basic idea is to combine the SPM and CSM scheme in the same pixel as follows:

a channel in Single Pixel Mode, on which the arbitration algorithm is applied

a channel in Charge Summing Mode

The channel working in Single Pixel Mode contains a discriminator which compares the signal in the local preampliﬁer output with an energy level (THSPM). For a given pixel, the counter associated with this channel is incremented if the locally induced signal is above the threshold and if the local signal is the largest in the neighbourhood. The channel working in Charge Summing Mode contains a discriminator which compares the sum of a cluster of four pixels (reconstructed charge) with an energy level (THCSM). The counter associated with this channel is incremented if the reconstructed signal is above the threshold and if-and-only-if the ﬁrst channel has been allocated the same event [58]. In this architecture it is important to note that the hit is therefore assigned to the

119 12 Redesign of Charge Summing Mode of Medipix3 pixel with the biggest charge induced but not to the summing circuit as in the currently implemented architecture. It should also be kept in mind that the SPM channel is used only for arbitration purposes. In the currently implemented CSM architecture, 2 thresholds are available per pixel. In the new CSM architecture, we will have one threshold in SPM for hit allocation and one threshold for summing.

12.2 Simulation of the proposed architecture

The simulation was performed using the Monte carlo Tool ROSI. This was realised in two steps: ﬁrst, the position of the pixel in SPM that has the largest energy deposition is recorded. In the second step, the result of the summing procedure is attributed the pixel with the largest energy deposition.

12.3 Impact of the new Charge Summing Mode architecture on Imaging

Figure 12.1 depicts the results from simulation of the currently implemented and the proposed Charge Summing Mode, respectively. The simulation was performed with a statistics of 100 millions of photons and for a threshold dispersion in CSM of 375 e− rms and 1000 e− rms. The simulation results agree with our expectation. The evident granularity caused by high pixel-to-pixel variation in the threshold in ﬁgure 12.1(left) is smoothed in ﬁgure 12.1 (right), especially in those parts with low absorption and the background. The edges between the high and the low absorption parts are more distinct. In summary a great improvement on image quality in Charge Summing Mode could be achieved, should the new hit allocation architecture be implemented in the future Medipix3 assemblies.

120 12.3 Impact of the new Charge Summing Mode architecture on Imaging

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121

13 Summary and Outlook

The Medipix2 and Medipix3 detectors are hybrid pixelated photon counting detectors

with a pixel pitch of 55 ñm. The sensor material used in this thesis was silicon. Because of their small pixel size they suffer from charge sharing i.e. an incoming photon can be registered by more than one pixel. In order to correct for charge sharing due to lateral diffusion of charge carriers, the Medipix3 detector was developed: with its Charge Summing Mode, the charge collected in a cluster of 2 2 pixel is added up and attributed × to only one pixel whose counter is incremented. The adjustable threshold of the detectors allows to count the photons and to gain information on their energy. The main purposes of the thesis are to investigate spectral and imaging properties of pixelated photon counting detectors from the Medipix family such as Medipix2 and Medipix3. The investigations are based on simulations and measurements. In order to investigate the spectral properties of the detectors measurements were performed using fluorescence lines of materials such as molybdenum, silver but also some radioactive sources such as Am-241 or Cd-109. From the measured data, parameters like the threshold dispersion and the gain variation from pixel-to-pixel were extracted and used as input in the Monte Carlo code ROSI to model the responses of the detector to monoenergetic photons. The measured data are well described by the simulations for Medipix2 and for Medipix3 operating in Charge Summing Mode. Due to charge sharing and due to the energy dependence of attenuation processes in silicon and to Compton scattering the incoming and the measured spectrum differ substantially from each other. Since the responses to monoenergetic photons are known, a deconvolution was performed to determine the true incoming spectrum. Several direct and iterative methods were successfully applied on measured and simulated data of an X-ray tube and radioactive sources. The knowlegde of the X-ray spectrum is important for quality assurance and constancy checks in hospitals. The second part of the thesis is about the imaging properties of the Medipix detectors. Images of samples (cash card, human bone) were taken with the Medipix3 chip in Single Pixel Mode (equivalent to the counting mode of the Medipix2 detector) and in Charge Summing Mode. The images in Single Pixel Mode were sharper than the ones taken in Charge Summing Mode. The latter show high granularity. This is due to high pixel- to-pixel variation in threshold in Charge Summing Mode. A redesign of the Medipix3 detector is proposed in order to correct for this problem. The determination of the spatial resolution confirms that Single Pixel Mode is better for imaging. Energy resolved material reconstruction was also performed with Medipix3 programmed in Single Pixel Mode and Charge Summing Mode. The combination method was applied to determine the concentration of elements in a compound object. The Downhill Simplex and Simulated Annealing methods were used to minimize the likelihood function delivered by the combination method. In a first step, the reconstruction method was tested using simulated

123 13 Summary and Outlook data. The results of the reconstruction show that the reconstruction is better in Charge Summing Mode than in Single Pixel Mode. The method of material reconstruction was also applied with success to data taken with the Medipix3 detector programmed in Single Pixel Mode. In summary, the Medipix detectors were successfully used in spectroscopy and imaging. An improvement of Charge Summing Mode of Medipix3 is necessary in order to reach at least the same image quality as in Single Pixel Mode.

124 14 Zusammenfassung und Ausblick

Die Medipix2 und Medipix3 Detektoren sind hybride photonenzählende Halbleiterdetek- ñ toren mit 256 256 Pixel mit einer Pixelgröße von 55 ñm 55 m. Fur¨ die Anfertigung × × dieser Arbeit wurden Medipix-Detektoren mit 300 ñm dickem Sensor aus Silizium verwendet. Aufgrund der geringen Pixelgröße, leiden die Detektoren unter Charge Sharing“, das ” heißt ein einfallendes Photon kann von mehr als einem Pixel detektiert werden. Um den Einfluss des Charge Sharings (hervorgerufen durch laterale Diffusion von Ladungsträger) zu verringern, wurde der Medipix3 entwickelt: mit dem ladungsummierenden Modus (Charge Summing Mode) sollten Ladungsträgermengen, die in einem Cluster von 2 2 × Pixel registriert wurden, wieder zusammengefasst und nur einem Pixel zugewiesen werden. Mit einstellbaren Diskriminatorschwellen, können diese Detektoren Photonen zählen und gleichzeitig Information uber¨ die Energie der Photonen liefern. Die Hauptziele dieser Dissertation sind die Untersuchungen von spektralen und bildgebenden Eigenschaften von pixelierten photonenzählenden Detektoren der Medipix Familie am Beispiel von Medipix2 und Medipix3. Diese Untersuchungen wurden mit Simulationen und Messungen durchgefuhrt.¨ Um die spektralen Eigenschaften der Detektoren zu untersuchen, wurden Messungen von Röntgenfluoreszenzlinien von Silber und Molybdän durchgefuhrt.¨ Spektren von radioak- tiven Stoffen wie Am-241 oder Cd-109 wurden ebenfalls verwendet. Aus den gemessenen Daten wurden Parametern wie die Schwellendispersion oder die Variation in den Verstärkungsfaktoren der Pixel bestimmt, um damit in Monte Carlo Simulationen die Antwortspektren der Detektoren auf Einstrahlung monoenergetischer Photonen zu model- lieren. Gute Ubereinstimmung¨ zwischen simulierten und gemessenen Antwortspektren wurde erzielt sowohl fur¨ den zählenden Modus des Medipix2 (entspricht dem Single Pixel Mode des Medipix3) als auch fur¨ den Charge Summing Modus des Medipix3. Aufgrund des Charge Sharings, der Energieabhängigkeit von Absorptionsprozessen in Silizium sowie der Verteilung von Elektronenenergien beim Compton-Effekt, unterscheidet sich das einfallende Spektrum vom gemessenen Spektrum. Mittels Entfaltungsalgorithmen kombiniert mit simulierten Antwortspektren kann das tatsächlich einfallende Spektrum rekonstruiert werden. Mehrere Entfaltungsmethoden wurden mit Erfolg auf simulierte und gemessene Spektren einer Röntgenröhre und auf gemessene Spektren radioaktiver Ma- terialien angewendet. Das Wissen uber¨ das Spektrum einer Röntgenröhre ist wichtig fur¨ Qualitätssicherung und Konstanzprufung¨ von Röntgenanlagen, z.B. in Krankenhäusern. Der zweite Teil dieser Arbeit befasst sich mit den bildgebenden Eigenschaften der Medipix- Detektoren. Bilder von verschiedenen Proben (menschlicher Knochen, EC Karte) wurde mit dem Medipix3 im einfachen Zählmodus (Single Pixel Mode) und im Charge Summing Modus aufgenommen. Die Bilder in Single Pixel Mode zeigten eine höhere Ortsauflösung als die im ladungssummierenden Modus. Letztere zeigen eine hohe Granularität hervorgerufen durch die große Schwellendispersion in Charge Summing Modus. Ein Vergleich der Modulationtransferfunktionen (MTF) in beiden Modi bestätigt, dass der Single Pixel

125 14 Zusammenfassung und Ausblick

Mode wesentlich besser fur¨ die Bildgebung geeignet ist. Ein Redesign der Charge Sum- ming Pixelelektronik ist vorgeschlagen, um die Bildqualität in diesem Modus zu erhöhen. Die Energiesensitivität der Detektoren wurde ausgenutzt, um Flächendichte von Mate- rialien in einem zusammengesetzen Objekt zu bestimmen. Eine sogenannte kombinierte Methode wurde angewendet um die Flächenbeläge zu bestimmen. Downhill Simplex und Simulated Annealing Methoden wurden benuzt um die verwendete Likelihood-Funktion zu minimieren. Die Methode der Materialrekonstruktion wurde zunächst an simulierten Daten, gewonnen mit dem Medipix2 Detektor und mit dem Medipix3 im Charge Sum- ming Modus, untersucht. Die Rekonstruktion zeigte im Charge Summing Modus bessere Ergebnisse. Die Methode wurde auch erfolgreich an mit Medipix3 im Single Pixel Mode gewonnenen Daten angewendet. Die Medipix-Detektoren konnten mit Erfolg in der Röntgen-Spektroskopie und Bildgebung eingesetzt werden. Ein Redesign der Pixelelektronik des ladungssummierenden Modus ist notwendig um mindestens eine vergleichbar gute Bildqualität wie im Single Pixel Modus zu erzielen.

126 Acknowledgements

At the end of this thesis I would like to take the opportunity to express my gratitude to everyone, who in the past years kindly and generously oﬀered me their help in every aspect of my work in the institute. My sincere thanks to:

Prof. Dr. Gisela Anton and Dr. Thilo Michel for giving me the chance to work in this wonderful group and for the good, patient and pleasant supervision of my thesis.

Friedrich Ebert Stifung and DAAD for the ﬁnancial support.

Dr. Jurgen¨ Durst and Wilhelm Haas for constructive discussion and help in pro- gramming tasks.

every member of the work team for all the profession and non-profession related talks and activities.

Medipix3 collaboration, especially the designer of Medipix3, Dr. Rafael Ballabriga Su˜n´e, for his patient ear to all my questions.

all the present and former colleagues in the institute: Michael B¨ohnel, Ulrike Gebert, Zhang Qiang, Thomas Gleixner, Ewald Guni, Ina Munster,¨ Peter Sievers, Bj¨orn Kreisler, Thomas Weber, Florian Bayer, Peter Bartl, Andre Ritter, Markus Firsching, Anja Loehr.

my wife for making my life so colourful and enjoyable.

my family in cameroon, my family in Europe (Rachel Djeukeng and the children, Leonel Zebaze, Clemence Emilie Feugang Talla, Yannick Talla Mba). Thank you for the continuous support and everything!

all my friends especially Steve Muriel Djouaka, Kadijda Mbango for their support.

127

List of Figures

2.1 Illustration of photoelectric effect [4]...... 4 2.2 Illustration of Compton scattering [4]...... 4 2.3 Illustration of Rayleigh scattering [4]...... 5 2.4 Illustration of pair production [4]...... 6 2.5 Regions of predominance of photoelectric effect, Compton effect, and pair production as function of photon energy and for different atomic numbers Z. The region of predominance of photoelectric effect increases with the atomic number Z. As a result, high Z materials are indicated as sensor material for radiation detection [5]...... 6

3.1 Schematic view of a Medipix assembly [11]. The sensor is connected to the ASICviabumpbonds...... 10 3.2 Electronic circuitry of a Medipix2 pixel cell. The analog part consists of a preamplifier and two discriminators. The digital part receives the output of the discriminators and decides in the DDL (Double Disc Logic), if the counter will be incremented. This circuitry is also responsible for the readoutofthedata[12]...... 11 3.3 Illustration of the path of an electron in silicon. The extension of that path over several pixel leads to charge sharing...... 12 3.4 CSDA range for electrons in silicon versus electron energy [15]...... 13 3.5 Illustration of one effect leading to charge sharing [16]. An incoming photon (0) interacts with the sensor and creates at (1) through Compton and at (2) through photo effect a free electron. The bold dashed lines represent the path of the electron during the energy loss process. The dashed circles represent the end diameter of a charge cloud at the end of the drift process. The diameters due to diffusion are bigger than the initial diameters (solid circles) at the beginning of the drift process...... 14 3.6 Illustration of diffusion process for a punctual charge cloud while drifting towards the pixel electrodes adapted from M. Boehnel [19]...... 15

4.1 Simulated energy response of Medipix2 to 20 keV incoming photons. The observed low energy tail is caused by the charge sharing eﬀect. The edge visible at 5 keV represents the discriminator threshold in the simulation. . 18 4.2 Medipix3 chip connected to an USB-Readout...... 19 4.3 Medipix3blockdiagram[5]...... 19 4.4 Charge Summing Mode adapted from P. Bartl [21]...... 21

5.1 Flow chart of the signal generation algorithm for the simulation of a photon countingdetector[16]...... 26

129 List of Figures

5.2 Energy response of the Medipix2 detector to 59.3 keV incoming photons. Simulation and measurement are in good agreement [4]...... 27 5.3 Simulated energy responses of the Medipix3 detector to 20 keV photons in CSM and SPM. The simulation was performed with 107 incident photons. The full energy peak in CSM is higher and symmetrical since charge sharing is almost completely suppressed (red curve). The plateau between 8 keV and 15 keV represents the L-fluorescence lines of lead contained in the solder bumps...... 28 5.4 Simulated energy responses of the Medipix3 detector to 80 keV incoming photons in CSM and SPM. The simulation was performed with 107 incident photons. Noise observed on the curves is quantum noise. The full energy peak in CSM in higher and symmetrical since charge sharing is almost completely suppressed (red curve). The peak at 25 keV is due to the Kα- fluorescence line of tin contained in the solder bumps. This new architecture providesabetterenergyresolution...... 29 5.5 Energy responses of the Medipix3 detector to 60 keV photons in Charge Summing Mode for different values of gain variation. The simulation was performed with 107 incident photons. Although the position of the full energy peak remains unchanged, the energy resolution becomes poorer since the broadness of the peak increases with increasing gain variation. . . . . 30 5.6 Energy responses of the Medipix3 detector to 60 keV photons in Charge Summing Mode for different values of current copy variation. The simulation was performed with 107 incident photons. The full energy peak shifts to higher energies as the variation in current copies increases. Moreover, the energy resolution becomes poorer since the width of the photopeak increases with increasing current copy variation...... 31

6.1 PanelforTestpulsemeasurements ...... 34 6.2 Shaper versus noise and gain variation. Small Shaper values lead to higher noiseandtoanincreasinggain...... 35 6.3 Shaper versus gain variation. Lower Shaper values lead to higher pixel-to- pixelvariation...... 35 6.4 IKRUM versus gain and noise. Lower IKRUM values lead to higher gain and lower noise level at the output of the Shaper...... 36 6.5 Preamp versus gain and noise. Noise and gain remain almost unchanged for Preamp values above 100. Below this value noise and gain increase abruptly. 36 6.6 Pixelman Medipix3 equalisation panel. In this version of Pixelman it is possible to equalise at the same time the two thresholds available per pixel. 38

6.7 IN is the shaper output current. IZX is the current entering the Zero Cross- ing Block (ZX). The Zero Crossing Block changes its digital output voltage with the sign of its input current. ITHP sets the global threshold. ITHADJ and ITHN adjustthethresholdinthematrix[5]...... 38

130 List of Figures

6.8 Scheme of the procedure for threshold adjustment. IN is the output current from the shaper. Its DC level is a consequence of the mismatch of the front- end transistors. When the DC level is negative the adjustment DAC suffices for adjusting the threshold. When the DC level is positive, the combination of ITHN and ITHADJ is necessary to adjust the threshold. IZX is the input to the zero crossing circuit. The DC level of this signal is the equalized threshold[5]...... 39 6.9 Histogram of adjusted thresholds at the end of the equalisation procedure. 39 6.10 Setup to produce and detect the fluorescence photons generated in the target material when irradiated by an X-ray Source [29]...... 40 6.11 Full energy peaks of different fluorescent targets (Cu, Pb, Mo, Ag, Sn, I, Gd) measured with a Medipix3 detector operating in CSM Low Gain Mode. The peaks have a nice symmetrical shape. The tail due to charge sharing issuppressed...... 41 6.12 Energy calibration curves of Medipix3 in Charge Summing Mode for Shaper- DAC of 150 and 100. A calibration curve in Low Gain Mode is also presented. Higher Shaper values lead to a steeper slope of the calibration curve andtoabetterenergyresolution...... 41 6.13 Energy responses of Medipix3 to 22 keV incoming silver fluorescence photons in simulation and measurement. The two curves are in a good agreement. The plateau between 8 keV and 15 keV in the measurement curve represents the L-fluorescence lines of lead used as wall material in our laboratory...... 44 6.14 Energy responses of Medipix3 to 59.5 keV incoming gamma photons emitted by an Am-241 source in simulation and measurement. The two curves exhibit a good agreement. The energy peak at 25 keV denotes the Kα- fluorescence line of tin, present in solder bumps. The Compton edge is clearlyvisibleatabout11keV...... 45 6.15 Comparison between the count rate of the Medipix3 detector in CSM and SPM. The linearity range of the count rate is larger in SPM. In CSM, for the applied settings, an upper limit for the linear behaviour is observed at a rate of 8 106counts/sec/mm2)...... 46 × 6.16 Relative count rate loss in CSM. A deviation of about 10 % is observable at 20 mA corresponding to a rate of 8 106counts/sec/mm2) ...... 46 × 7.1 Comparison of the measured spectrum (red curve) using a detector of the Medipix family and the theoretical incoming spectrum (black curve, [31]). The two spectra differ considerably from each other due to charge sharing. 50 7.2 Decomposition of a 60 kV tungsten anode spectrum in 3 energy bins. Bin 1 comprises energies from 10 keV to 30 keV, bin 2 from 30 keV to 50 keV, bin 3 from 50 keV to 70 keV. The coefficients S1, S2, S3 shall be determined fromthedepositedspectrum...... 52 7.3 Decomposition of the detector response to a 60 kV tungsten anode spectrum in 3 energy bins. Bin 1 comprises energies from 10 keV to 30 keV, bin 2 from 30 keV to 50 keV, bin 3 from 50 keV to 70 keV. M1, M2, M3 represent theCountsintherespectivebin...... 53

131 List of Figures

7.4 Spectral response function of the detector to 60 keV photons. It is decomposed in 3 energy bins. Bin 1 comprises energies from 10 keV to 30 keV, bin 2 from 30 keV to 50 keV, bin 3 from 50 keV to 70 keV. The coefficients R13, R23, R33 are normalized over the number of incoming photons. . . . 54 7.5 Reconstruction of a 40 kV incident tungsten anode spectrum from simulated data. The two reconstruction methods give almost identical results. . . . 59 7.6 Reconstruction of a 80 kV incident tungsten anode spectrum from simulated data. The two reconstruction methods give almost identical results. The Kα and Kβ fluorescence lines of tungsten at 58.85 keV and 67.23 keV are visible...... 59 7.7 Reconstruction of a 125 kV incident tungsten anode spectrum from simulated data. The two reconstruction methods give almost identical results. The Kα and Kβ fluorescence lines of tungsten at 58.85 keV and 67.23 keV arevisible...... 60 7.8 Reconstructed spectra for a 50 kV incident tungsten anode spectrum. The energy bin size used in the reconstruction is 3 keV. Spectrum Stripping and matrix inversion deliver almost the same result...... 61 7.9 Reconstructed spectra for a 120 kV incident tungsten anode spectrum. The two reconstruction methods give almost identical results. The Kα and Kβ fluorescence lines of tungsten at 58.85 keV and 67.223 keV are visible. . . 62 7.10 Reconstruction of the radiation emitted by an Am-241 source. a: measured threshold scan i.e. counts as function of threshold; b: energy deposition spectrum obtained by derivation of the function in figure a, i.e. difference between consecutive intensity values; c: reconstructed energy spectrum by application of the Spectrum Stripping method...... 64 7.11 Reconstruction of the radiation emitted by an Am-241 source for Medipix2 using 4 reconstruction methods. The iterative methods deliver better results as negative number of photons are not permitted. Moreover, the reconstructed spectra with Jacobi and Bayesian methods show well positioned lines whereas a shift of 1 keV is observed for Stripping methods due to smoothing performed at each iteration step...... 65 7.12 Comparison between the measured energy spectrum of an Am-241 source and the calculated spectra from the reconstructed spectra using 4 deconvolution methods. The reconstruction bin width is 0.5 keV. The computed spectra associated to the Bayesian deconvolution is closer to the original measurement...... 66 7.13 Reconstructed spectra emitted by a Cd-109 source obtained with the Spec- trum Stripping (left-hand side) and matrix inversion (right-hand side) method for reconstruction bin widths of 1.5 keV, 1.0 keV, 0.6 keV and 0.5 keV. . . 66 7.14 Reconstructed spectra for a 70 kV incident tungsten anode spectrum. The energy bin width in the reconstruction is 3 keV. The iterative methods deliveralmostcomparableresults...... 67 7.15 Reconstructed spectra for a 110 kV incident tungsten anode spectrum. The energy bin width in the reconstruction is 4 keV. The iterative methods deliveralmostcomparableresults...... 68

132 List of Figures

7.16 Reconstructed spectra for a 50 kV incident tungsten anode spectrum with the Medipix3 detector in Charge Summing Mode. The reconstruction bin widthis2keV...... 69 7.17 Reconstructed spectra for a 70 kV incident tungsten anode spectrum with the Medipix3 detector in Charge Summing Mode. The reconstruction bin widthis2keV...... 69 7.18 Reconstruction of ﬂuorescence lines of silver with the Medipix3 detector in Charge Summing Mode using three iterative methods. The reconstruction bin width is 0.5 keV. Two ﬂuorescence lines of silver are clearly visible and well positioned for Bayesian deconvolution. The two other methods show a shift to higher energies due to the smoothing performed at each iteration step. Hence, they provide a slightly poorer energy resolution ...... 70

8.1 Setup for kVp determination. It consists of an X-ray tube, an aluminium prefilter, two copper filters of different thicknesses and the Medipix detector. N1 and N2 denote the sum of counts recorded behind copper filter1 and copper filter2, respectively. Their ratio R is a function of the tube voltage applied across the X-ray tube...... 75 8.2 Ratio of the counts recorded behind copper filters with the Medipix2 detector. For tube voltages ranging from 45 kV to 75 kV. The ratio is a linear functionofthekVp...... 75 8.3 Ratio R versus tube voltage ranging from 60 kV to 100 kV...... 76 8.4 Ratio R versus tube voltage ranging from 80 kV to 120 kV...... 76 8.5 Ratio of the number of counts recorded behind copper filters with the Medipix2 detector from simulation and measurement for tube voltage values ranging from 40 kV to 120 kV. The two curves denote the same func- tionalbehaviour...... 78 8.6 Ratio R versus tube voltage ranging from 40 kV to 55 kV...... 78 8.7 Ratio R versus tube voltage ranging from 60 kV to 100 kV...... 79 8.8 Ratio R versus tube voltage ranging from 80 kV to 120 kV...... 79 8.9 Relative statistical error of the kVp measurement versus tube voltage. A precision below 2.1 % is achievable...... 80

9.1 Photograph a Huttnergrid[11]...... ¨ 84 9.2 Photographofacashcard...... 86 9.3 Raw image of an electronic cash card taken with the Medipix3 detector in CSM.Thethresholdwassetto10keV...... 86 9.4 Flatﬁeld corrected image of a cash card recorded with the Medipix3 detector inCSMatathresholdof10keV...... 87 9.5 Raw image of a cash card taken with the Medixpix3 detector in SPM at a threshold of 10 keV. Some wires connected to a chip can be recognized. . . 87 9.6 Flatﬁeld corrected image of cash card recorded with the Medipix3 detector in SPM at a threshold of 10 keV. Some wires connected to a chip can be recognized...... 88 9.7 Image of a human bone taken with the Medipix3 detector in SPM. The thresholdusedwas10keV...... 88

133 List of Figures

9.8 Image of a human bone taken with the Medipix3 detector in SPM. An energywindowfrom15keVto25 keVwasused...... 89

10.1 Image of a slit acquired with the Medipix3 detector in SPM (simulation). 92 10.2 Image of a tungsten edge acquired with a Medipix3 detector in SPM (measurement)...... 93 10.3 Simulated LSF of Medipix3 in SPM for diﬀerent threshold values. Lower

threshold values lead to an eﬀective pixel size above 55 ñm due to charge sharing. Increasing the threshold yields pixel size close to the expected

value of 55 ñm because single counts are promoted...... 94 10.4 Simulated MTF of Medipix3 in SPM for different THL values. A drop at low spatial frequencies is observed at lower thresholds. Responsible for the drop are the fluorescence photons originating from lead and tin. Increasing the threshold leads to a shift of the zero point of the MTF to higherfrequencies...... 94 10.5 Measured MTF of Medipix3 in SPM for Shaper DAC 100, 150, 255 and a threshold of 10 keV. The simulated MTF is also presented. A good agreement between simulation and measurement is observed for frequencies below 12.45 lp/mm. The drop in lower frequencies is well reproduced in the simulation. The MTF is independent from the Shaper DAC value. .... 95 10.6 Measured MTF of Medipix3 in SPM, HGM and LGM. The threshold was 10keV...... 96 10.7 Simulated LSF of Medipix3 in CSM for different bias voltage values . . . 97 10.8 Cases of ambiguity in the hit allocation in CSM. Red sum symbols indicate that the largest partial charge was registered by the pixel. The letters in red display the pixel that can probably get the hit assignment [52]. . . . . 97 10.9 Simulated MTF of the Medipix3 detector in CSM for different threshold values. The MTF is almost threshold independent as charge sharing is efficientlysuppressed...... 98 10.10Simulated MTF of the Medipix3 detector in CSM for different values of the gain variation. The threshold does not depend on the gain variation. . . . 98 10.11Simulated MTF of the Medipix3 detector in CSM for different values of current copy variation. The zero point of the MTF is shifted to lower frequencies as the variations increase. This leads to a degradation of the MTF...... 99 10.12Simulated MTF of the Medipix3 detector in CSM for different bias voltages. Lower bias voltages lead to an improvement of the MTF...... 99 10.13MTF of Medipix3 in CSM for a threshold value of 10 keV. The presampled MTFisonlyabout2.5LP/mm...... 100 10.14ESF in CSM and SPM for a threshold value of 10 keV ...... 101

11.1 Mass attenuation coeﬃcient of nickel, silicon, tin and lead as function of photonenergy[15]...... 105 11.2 Basic operations in the downhill simplex method [56]...... 107

134 List of Figures

11.3 Sketch of the phantom and the resulting counting image at a threshold of 18.7 keV. The counting image is tilted with respect to the setup image at an angle of 90 degrees: gadolinium is on the left hand side and iodine on therighthandside ...... 109 11.4 Results of the material reconstruction: gadolinium, iodine and water images are shown for SPM and CSM. The color bar represents the reconstructed g areal densities in cm2 . The separation is better for CSM...... 110 11.5 Results of the material reconstruction: gadolinium, iodine, and water images are shown for SPM. The reconstruction was performed using the simplex and simulated annealing methods. The color bar represents the recon- g structed areal densities in cm2 ...... 111 11.6 Photon counting image at a threshold of 18.7 keV and for an energy window from 32.7 keV to 39.7 keV. In each ﬁgure, the upper cutout is ﬁlled with iodine and the lower cutout contains gadolinium...... 113 11.7 Reconstructed gadolinium, iodine, PMMA images with the simplex method on the left hand side and the simulated annealing algorithm on the right g hand side. The color bar represents the reconstructed areal densities in cm2 .114 11.8 Setup for bump bond material reconstruction ...... 115 11.9 Photon counting images for an energy window from 8 keV to 17 keV and energywindowfrom24keVto35keV ...... 116 11.10Reconstructed areal densities images of lead, tin and silicon. Images were reconstructed using the simplex algorithm...... 117

12.1 Simulated images of a Siemens star for Medipix3 in the currently implemented (left) and the proposed Charge Summing Mode (right). The Threshold dispersion in CSM was 375 e− rms (top) and 1000 e− rms (bottom)...... 121

135

List of Tables

3.1 Characteristics of the Medipix2 detector [5]...... 10

4.1 Modes of operation of the Medipix3 detector [5]...... 20

5.1 Noise contributions of the Medipix3 detector in Single Pixel Mode and ChargeSummingMode(designparameters)...... 31

6.1 Energy resolution of Medipix3 in Single Pixel Mode and Charge Summing Mode for a Shaper-DAC = 100. An improvement of the resolution is no- ticeableinCSM...... 42 6.2 Relative energy resolution of Medipix3 in Charge Summing Mode for different Shaper-DAC values. The energy resolution is better at Shaper-DAC 150. A good resolution is achieved by using higher Shaper values...... 43 6.3 Relative energy resolution of Medipix3 in Charge Summing High Gain and Low Gain Mode for a Shaper-DAC of 150. The resolution in low Gain Mode is poorer as expected since we have half signal in this mode. This leadstobroadenedfullenergypeaks...... 43

7.1 Energy resolution of the Medipix2 detector. A resolution of approx. 1.0 keV can be achieved in the reconstruction...... 65

8.1 Tube voltage range and the corresponding copper thickness combinations. 74 8.2 Comparison between the linear regressions to simulated and measured data for2tubevoltageranges...... 77

11.1 Overview of the subscripts used in this section [23]...... 106 g 11.2 Comparison between the reconstruced areal densities given in cm2 for SPM and CSM and the original input densities. The reconstruction was performed using the simulated annealing algorithm. The statistical errors of the reconstructed areal denstities are not shown as they are negligible. . . 112

137

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