Fundamentals of Imaging Light and Radiometry
Prof. Dr. Charles A. Wüthrich, Fakultät Medien, Medieninformatik Bauhaus-Universität Weimar caw AT medien.uni-weimar.de Today’s lesson
• Today’s themes: – Light, its definition – Characteristics of light: coherence and polarization • How to measure light: radiometry – Spectrometry – Perfect diffusors – The radiant theorem – How light reflection is measured through a blackbody
April 17 Charles A. Wüthrich 2 Light
• What is light? – A full explaination of light can be found in Quantum ElectroDynamics – Very abstract and difficult to understand • We don’t want to get to that depth in this course • However, we can calculate and predict its behaviour from phenomena that we can observe: – Light can be seen as electromagnetic waves between the frequencies of 4x1014 (740nm) and 7.8x1014 Hz (380nm). – Light carries energy in discrete amounts – Light can be seen as photons, has linear momentum, and therefore exerces some force on the surfaces it reaches – Light of the same frequency has different polarizations, which can be filtered by some materials (polarizers)
April 17 Charles A. Wüthrich 3 Light
• There are therefore 2 models • Schrödinger’s wave equation is for light: NOT the electromagnetic wave – Geometrical optics: described by Maxwell: light behaves as rays – For classical phenomena, like – Wave model: interference, the time-averaged based on Maxwell’s equations for electromagnetic waves Poynting vector calculated from Maxwell’s equations • Quantum theory uses two
objects to describe a physical
April 17 Charles A. Wüthrich 4 Light emission
• When an electron of molecules makes the transition from one level of energy to a lower one a photon is emitted Atomic nucleus • The time it takes for this transition is very short… electron • …and short is the length of the wave train of emitted light (10-8s)
• Remember the wave propagation equations: – For a sinus wave, λ=v/f – Thus, for light λ=c/f where c is the speed of light
photon
April 17 Charles A. Wüthrich 5 Light coherence
• The electromagnetic field at • There are two special cases two points in space-time can of coherence which have fluctuate completely been studied through simple independently experiments – In this case, they are said to – Temporal coherence: be incoherent. here field fluctuation is • If the fluctuations are not measured at the same completely independent they location in space are said to be partially or – Spatial coherence: completely coherent. here field fluctuation is measured at the same • The degree of coherence instant can be measured through statistical correlation
April 17 Charles A. Wüthrich 6 Temporal coherence
• Michelson’s interferometer: a half reflecting mirror splits the light – Some of the light (the refracted one through the mirror) will take a shorter path than the reflected light – The light paths are then
joined again, but having Wikipedia © and Courtesy travelled different distances their phase differs – Interference fringes at the detector
April 17 Charles A. Wüthrich 7 Spatial coherence
• Young’s double slit interference R experiment – A coherent light source S sends Δs light through a pinhole d – In front of the pinhole there are two slits symmetrically shifted from the perpendicular to the Light source pinhole Pinhole Slit(s) Screen – Because the source is the same, the light travels different distances, and will form interference on the observation screen – If one uses only one slit, no interference can be seen – For interference to be observable, it must be that d≤(Rλ)/Δs • Normal light sources, such as bulbs or the sun, can be treated as incoherent Wikipedia © and Courtesy
April 17 Charles A. Wüthrich 8 Maxwell’s equations
• Because light can be seen as – Gauss laws for the electric electromagnetic waves, one can and magnetic field: use for it Maxwell’s equations • Maxwell equations describe the electric and magnetic fields arising from various distributions • Here, of charges and current, and how – E and H are electric and these fields change in time magnetic field intensities • They are four equations: – D and B are electric and – Faraday’s law of induction: magnetic flux densities
– ρ and J are volume charge density and electric current density of any external – Ampere’s law amended by charges (the sources) Maxwell: – and ∇ is the nabla operator
Displacement current
April 17 Charles A. Wüthrich 9 Polarization
• Far from their source, the electric and magnetic fields are – perpendicular to each other – perpendicular to the direction of Direction propagation. of propagation • This fact leaves one degree of freedom for the pair, the rotation angle δ around the direction of propagation • When the light is such that this
degree of freedom is not there δ any more, then the light is said to be polarized • In other words, once δ is fixed, the light is polarized
April 17 Charles A. Wüthrich 10 Polarization and interference
• Fresnel and Arago performed – Two beams of linearly polarized again Young’s slit experiment, light, derived from a linearly this time with a polarized light polarized light beam in source perpendicular directions, can • Results obtained were: produce interference patterns if – Two beams of light linearly they are brought back to the polarized in the same plane can same polarization plane by produce interference patterns polarizers. – Two beams of light linearly polarized perpendicularly to each other cannot produce R interference patterns. – Two beams of linearly polarized light, derived from an Δs d unpolarized light beam in perpendicular directions, cannot produce interference patterns Light source even if they are brought back to the same polarization plane by Pinhole Slit(s) Screen polarizers.
April 17 Charles A. Wüthrich 11 Non-Linear Polarisation
• When we talked about polarization, we said that the angle of the plane of the waves has to be fixed • In fact, this is not completely true: the degree of freedom has to be eliminated. • If it is with the fixed angle, we speak of linear polarization • In the case illustrated at the right, here the polarization is done on a circular pattern • Note that the drawing shows • We know now the nature of light: only the electrical field, and not Next we will introduce how to the magnetic one (which is measure light perpendicular to it)
April 17 Charles A. Wüthrich 12 Radiometry
• It is important to know how • The energy flow per unit time much light energy reaches a through a point (x, y, z) in space sensor in a direction (θ, φ) is called the • Sensors have limited operating radiant flux, range: controlling how much (x, y, z, θ, φ). light reaches them is essential. • Radiant flux is a quantity • The study and measurement of dependent on position and the optical energy flow is the direction, which means, it is subject of radiometry. associated with the light beam • Radiometry units for light • At a far away distance from a measurement are standardized light source, the source can be by the CIE (Commission modeled as a point source Internationale de l’Eclarage) • In practice, if the light source • The unit of radiant light energy size is less then 1/10 of the Q is the Joule distance between light source and object, the error in considering a point light source for radiometric calculations is less than 10%
April 17 Charles A. Wüthrich 13 Projected area - Solid angle
• Projected area: area of an object seen from a direction different from the surface normal • Solid angle of a cone: area cut by the cone onto a unit sphere with center at the apex of the cone • In other words, the solid angle of a cone is the area of its projection onto a unit sphere • A sphere has a solid angle of 4π. • If the sphere has a radius of r, then the solid angle is the surface of the projection divided by r2. • The unit of a solid angle is the steradian. • The steradian is the solid angle that, having its vertex at the center of a sphere with radius r, cuts off an area of the surface of the sphere equal to r2
April 17 Charles A. Wüthrich 14 Computing solid angles
• To calculate the solid angle • Exercise for home: of an area of arbitrary shape, – Calculate the solid angle one can use the differential subtended by a circular disk definition of solid angle of radius R to the point source P at a distance z to 2 the center of the disk. dω=dAproj/r
• And integrate it on the whole area
where Aproj is the projected area onto the ray connecting A and the point light source
April 17 Charles A. Wüthrich 15 Intensity
• Let dΨ be the radiant flux • Intensity is an idealization leaving a point source not taking into account through a cone of solid angle diffraction and other dω. parameters • Then the quantity that • For example: stars are point describes the light output sources, and exibit from the point source at this diffraction on telescope cone is called the radiant images intensity – I: radiant flux per unit solid angle
April 17 Charles A. Wüthrich 16 Radiance
• If the light source is close, it is • Radiance is used to measure not possible any more to treat it how much light is reflected by as a point light source walls, or how much light comes • Dividing the area source in from the sun small elements dA one can treat each one as a point source, but the intensity is proportional to its area dA. • And if the surface is not perpendicular to the rays, then its projected area dAcosθ has to be used • The quantity describing the amount of light coming from a surface is called radiance L: it is the light flux per solid angle per projected surface area
April 17 Charles A. Wüthrich 17 Radiant flux, Irradiance, Radiant Exitance
• To describe how much light • W, E and M are dependent on passes through a surface area, the direction and on the position the term radiant flux (surface) • Many times, one is interested in density is used the total power received by a W(x,y,z,θ,φ) sensor surface measured in [Wm−2] • In this case, the flux is • The radiant flux irradiating onto integrated over all angles and is a surface instead is called no longer dependent on the irradiance direction E(x, y, z, θ, φ)
measured in [W m−2] • For the radiant flux leaving a surface one uses radiant exitance
M(x, y, z, θ, φ)
measured in [W m−2]
April 17 Charles A. Wüthrich 18 Lambertian surfaces
• Many reflecting surfaces appear • A surface whose radiance is to be approximately equally independent from the viewing angle bright, independently of the is said to be a Lambertian surface viewing angle. (it can reflect or transmit) • A source whose radiance is • There are two reasons for this: independent from the viewing angle one Physical and one is called a Lambertian source phsychophysical (it emits light) – except for highlights, most • A Lambertian surface that reflects surfaces reflect light with (transmits) 100% of the incident light approximately the same is called a perfect reflecting radiance in all directions, and (transmitting) diffuser. the retinal image irradiance is • This idealized reflector is used in the proportional to the radiance of definition of the radiance factor, the scene viewed which describes how bright a surface is in an imaging system
• On the book: several example of how to compute these quantities.
April 17 Charles A. Wüthrich 19 Radiant exposure
• Some sensors allow reading • Here, one has to use the of the light signal integral of the product of image continously, for example the irradiance and time, which gives eye, others no. energy per unit area • This is called radiant exposure • For these a shutter has to be H, which for an exposure time used to control the time Δt is defined as interval during which the sensors are exposed to the image signals • Such sensors respond to the -2 energy per unit area, and not and is measured in [Jm ] to the power per unit area • All photographic films, and most CCDs work this way
April 17 Charles A. Wüthrich 20 Reflectance
• Light reaching an object surface • To take this into account, one is defines the reflectance factor: – Reflected – ratio of radiant flux reflected in – Absorbed the direction delimited by the – Transmitted measurement cone to that • The ratio between the flux of reflected in the same direction reflected light and the flux of by a perfect reflecting diffuser incident light is called the identically irradiated reflectance ρ. • When the measure cone is very • In a passive material, it is small, and one is measuring always ≤1. radiances, it is called the • A lambertian surface having a radiance factor. reflectance of 1 is called a perfect diffuser. • A similar definition is given for the transmittence factor. • For calculating the reflectance, the whole light reflected is used: • The radiance factor correlates surfaces reflect differently in better with what we see than the different directions reflectance
April 17 Charles A. Wüthrich 21 Responsitivity
• Suppose you have a detector • In most cases, the terms (sensor): responsitivity and sensitivity are the ratio between the input used interchangeably. power (light flux) and its output • In vision science, instead, sensitivity signal is called responsitivity s. is the inverse of the input power • In case of an electrical sensor, required to produce a threshold this ratio depends on the sensor response. itself and its characteristics. • If the system response is a • There are detectors, such as nonlinear function of the input film, that respond to the total power, measured sensitivity is a light energy, I.e. the exposure: function of the threshold response here, responsivity is defined as that is chosen for the measurement. the ratio of the output signal to • Radiometric measurements are the input light energy. difficult to do: they depend from • Sometimes, responsitivity is wavelength, variation of the measured with respect to a radiation in time, direction of single wavelength λ, and is incident radiation, position of the called s . detector, temperature, coherence λ and polarization of the radiation…..
April 17 Charles A. Wüthrich 22 Spectral radiometry
• Until now, we have not mentioned the spectral composition of light being measured. • If we want to measure light energy flow with respect to certain light frequency intervals we have to measure in small intervals. • In this case, we speak of spectral quantities, and their units become per unit wavelength interval (e.g., nanometer) or per unit frequency interval.
April 17 Charles A. Wüthrich 23 The radiance theorem
• In geometrical optics, light is • The radiometric quantity treated as rays associated with each ray is • Tracing rays from one point the so-called basic radiance: in space to another is an – it is L/n2, the radiance operation carried out in divided by the index of diverse applications, such as refraction squared. lens design, thermal • It can be proved that the radiative transfer, scanner basic radiance is conserved engineering, and computer when light is propagated graphics. through non-absorbing and • An implicit assumption in ray non-scattering media. tracing is that each ray • This conservation property of carries with it some measure the basic radiance is called of energy flux through space the radiance theorem. and time.
April 17 Charles A. Wüthrich 24 Integrating Cavities
• An integrating cavity is a physical cavity enclosed by reflecting surfaces • They allow to integrate non- uniform radiant fluxes to produce a source of uniform radiant flux. • Most common shape: spherical • The study of the radiant flux distribution in an integrating cavity involves radiometric calculation of the mutual • A sample is placed, and reflection of surfaces. illuminated by a light beam. • This is done • The light is reflected to and – Either with a finite element integrated by the sphere method • This allows to measure irradiance, – or through Monte-Carlo which is proportional to the raytracing amount of reflected light • An integrating sphere can be used to measure reflectances
April 17 Charles A. Wüthrich 25 Blackbody radiation
• A blackbody is an idealized object • h Planck constant that absorbs all incident radiant k Boltzmann constant energy K: absolute temperature – No reflection, no transmission • A blackbody is a Lambertian • If it absorbs all incoming energy, its radiator temperature must rise • Its color variation is similar to that of • This until it reaches thermal daylight equilibrium, which is ruled through the heat transfer equations • Planck derived the energy spectrum of blackbody radiation
2 −16 2 C1=hc =0.59552x10 Wm −2 C2=hc/k=21.438769x10 mK c: speed of light λ: wavelength in the medium
April 17 Charles A. Wüthrich 26 Recap
• In this lesson we have touched the following themes: – Light, its definition – Characteristics of light: coherence and polarization • How to measure light: radiometry – Spectrometry – Perfect diffusors – The radiant theorem – How light reflection is measured through a blackbody • Granted, this sounds a little bit dry, but it is the basis of both imaging and colour, ultimately of Computer Graphics
April 17 Charles A. Wüthrich 27 Thank you!
• Thank you for your attention! • Web pages http://www.uni-weimar.de/medien/cg
• First exercitation: Thursday in B11, SR13
April 17 Charles A. Wüthrich 28