DOCSLIB.ORG

A Sub-Band Time Delay Estimation Approach

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A Sub-Band Time Delay Estimation Approach IEEE/ACM TRANSACTIONS ON AUDIO, SPEECH, AND LANGUAGE PROCESSING 1 Frequency-Sliding Generalized Cross-Correlation: A Sub-band Time Delay Estimation Approach Maximo Cobos, Senior Member, IEEE, Fabio Antonacci, Senior Member, IEEE, Luca Comanducci, Student Member, IEEE, and Augusto Sarti, Senior Member, IEEE Abstract—The generalized cross-correlation (GCC) is regarded [6], is still today the most popular technique for TDE. By as the most popular approach for estimating the time difference using GCCs, the TDOA between two signals is estimated of arrival (TDOA) between the signals received at two sensors. as the time lag that maximizes the cross-correlation between Time delay estimates are obtained by maximizing the GCC output, where the direct-path delay is usually observed as a filtered versions of such signals. In this context, the GCC may prominent peak. Moreover, GCCs play also an important role consider different weighting functions (filters) characterized by in steered response power (SRP) localization algorithms, where a particular behavior [7], [8], [9]. Roth, smoothed coherence the SRP functional can be written as an accumulation of the transform, Eckart or Hannan-Thomson (maximum likelihood) GCCs computed from multiple sensor pairs. Unfortunately, the are examples of such weighting schemes [6], [10]. The GCC accuracy of TDOA estimates is affected by multiple factors, including noise, reverberation and signal bandwidth. In this with phase transform (GCC-PHAT) has been repeatedly shown paper, a sub-band approach for time delay estimation aimed to be a suitable alternative for TDE in real reverberant sce- at improving the performance of the conventional GCC is narios [11], [12]. However, it has also been demonstrated presented. The proposed method is based on the extraction of that its performance can only be considered optimal under multiple GCCs corresponding to different frequency bands of uncorrelated noise conditions and a high signal-to-noise ratio the cross-power spectrum phase in a sliding-window fashion. The major contributions of this paper include: 1) a sub-band GCC (SNR) [13]. The present work is particularly focused on PHAT representation of the cross-power spectrum phase that, despite weighting, thus, the terms GCC and GCC-PHAT will be used having a reduced temporal resolution, provides a more suitable indistinctly throughout this paper. Since the GCC was first representation for estimating the true TDOA; 2) such matrix proposed, many approaches have appeared to ameliorate the representation is shown to be rank one in the ideal noiseless robustness of TDE techniques, with improvements that are case, a property that is exploited in more adverse scenarios to obtain a more robust and accurate GCC; 3) we propose a mostly achieved by exploiting the spatial diversity provided set of low-rank approximation alternatives for processing the by more than two microphones [14], [15], [16], blind channel sub-band GCC matrix, leading to better TDOA estimates and estimation [17], [18], modified GCC weightings [19], or the source localization performance. An extensive set of experiments incorporation of some a priori information [20], [21]. Further is presented to demonstrate the validity of the proposed approach. improvements to deal with reverberation and correlated noise fields have also been proposed [22], addressing some of the Index Terms—Time delay estimation, GCC, SVD, weighted limitations of the GCC. In addition, other methods exploiting SVD, sub-band processing, SRP-PHAT. temporal diversity by means of Bayesian filtering have shown great potential [23], [24]. I. INTRODUCTION In contrast to the above methods, this paper proposes a IME delay estimation (TDE) refers to finding the time novel sub-band approach to TDE with two sensors which is T differences-of-arrival (TDOAs) between signals received not directly categorized within the above processing improve- at an array of sensors. Traditionally, TDE has played an im- ments. Concretely, the method is based on the exploration of portant role in many location-aware systems, including radar, the cross-power spectrum phase by following a sliding window arXiv:1910.08838v2 [eess.AS] 24 Mar 2020 sonar, wireless systems, sensor calibration or seismology. In approach, obtaining a set of sub-band GCCs that encode the acoustic signal processing, TDE is essential for localizing and contribution of different frequency bands to the estimated tracking acoustic sources [1], [2]. Automatic camera steering TDOA. The resulting sub-band GCC matrix is shown to be [3], speaker localization [4] or speech enhancement systems rank-one for a full-band signal in the noiseless single-path are application examples strongly relying on TDE [5]. case. This fact is exploited to obtain a robust GCC in adverse The generalized cross-correlation (GCC), originally pro- scenarios, proposing low-rank approximations of the sub-band posed by Knapp and Carter in their 1976 seminal paper GCC matrix that ultimately lead to better estimation accuracy, reduced level of spurious peaks and lower probability of M. Cobos is with the Departament d’Informatica,` Universitat de Valencia,` Burjassot, 46100 Spain e-mail: ([email protected]). anomalous estimates. F. Antonacci, A. Sarti and L. Comanducci are with the Dipartimento di The rest of the paper is structured as follows. Section II Elettronica, Informazione e Bioingegneria at the Politecnico di Milano, Italy. summarizes the background concerning GCC-based TDOA E-mail: (fi[email protected]) This work has been partially supported by FEDER and the Spanish Ministry estimation. Section III presents the frequency-sliding GCC of Science, Innovation and Universities under Grants RTI2018-097045-B-C21 (FS-GCC) representation. Section IV discusses our proposed and PRX19/00075. methods for TDE based on the FS-GCC. The experimental evaluation is in Section V. Finally, the conclusions of this c 2019 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works. IEEE/ACM TRANSACTIONS ON AUDIO, SPEECH, AND LANGUAGE PROCESSING 2 work are summarized in Section VI. the following estimator for time-delay estimation over signals acquired by a pair of sensors: II. TIME DELAY ESTIMATION τ^0 = arg max R[τ]: (8) τ This section summarizes the conventional GCC approach The GCC-PHAT inherently discards the magnitude infor- for TDE. To this end, the ideal anechoic model is first mation of the signals and provides a time-delay estimate presented, discussing the main problems arising in realistic based only on the phase of the cross-power spectrum, i.e. the acoustic conditions. phase of the frequency domain analysis of the cross-correlation between the two signals. It is important to remark that the A. Anechoic Signal Model measured time delay is an integer multiple of the sampling Let us consider a pair of sensors with spatial coordinates period. Note, however, that a finer resolution can be achieved 3 by interpolating between consecutive samples of the GCC given by column vectors m1, m2 2 R and an emitting acoustic source located at s 2 R3. The time difference-of- function if necessary. arrival (TDOA) measured in samples is defined as C. Problems in Realistic Scenarios ks − m k − ks − m k τ 1 2 f = η − η ; (1) 0 , c s 1 2 It is well-known that several problems arise when using GCC-PHAT for estimating the TDOA in realistic scenarios. where c is the wave propagation speed, b·e denotes the Indeed, TDOA measurements are very sensitive to reverbera- rounding operator and fs is the sampling frequency. The terms tion, noise, and the presence of potential interferers: η1 and η2 represent the time of flight (TOF) of the sound to • In reverberant environments, for certain locations and the sensors in samples. orientations of the source signal, the peak of the GCC Assuming an anechoic scenario, the signals received by the related to a reflective path could overcome that of the two sensors can be modeled as direct path. • In noisy scenarios, for some time instants, the noise level xm[n] = βms[n − ηm] + wm[n]; m = 1; 2; (2) could exceed that of the signal, making the estimated where βm 2 R+ is a positive amplitude decay factor, s[n] TDOA unreliable. is the source signal and wm[n] is an additive noise term. In • Peaks corresponding to the direct path, reflections or the discrete-time Fourier transform (DTFT) domain, the sensor combinations of interfering signals may also lead to errors signals can be written as when estimating the TDOA of a target source. The above issues can be even more problematic when the X (!) = β S(!)e−j!ηm + W (!); m = 1; 2; (3) m m m spectral characteristics of the target source or the additive noise lead to a reduced signal-to-noise ratio (SNR) at some fre- where S(!), Wm(!) 2 C, are the DTFTs ofp the source signal and the noise signal, respectively, and j = −1. quency bands [25]. The phase information can be completely lost at those frequency bands where either there is no signal information or where its content is especially affected by the B. Generalized Cross-Correlation noise. Consider, for example, the (normalized) GCCs shown in The GCC of a pair of sensor signals is defined as the inverse Figure 1. The top row corresponds to the ideal case of a noise- Fourier transform of the weighted cross-power spectrum, i.e. less full-band signal. The other two rows correspond to the 1 Z π same signal but affected by different spectral noise profiles. As R[τ] Ψ(!)ej!τ d! = F −1 fΨ(!)g ; (4) , 2π it can be observed, the noisy phase introduces many spurious −π peaks that can cause anomalous TDOA estimates.
Load more

Recommended publications
  • Moving Average Filters
    CHAPTER 15 Moving Average Filters The moving average is the most common filter in DSP, mainly because it is the easiest digital filter to understand and use. In spite of its simplicity, the moving average filter is optimal for a common task: reducing random noise while retaining a sharp step response. This makes it the premier filter for time domain encoded signals. However, the moving average is the worst filter for frequency domain encoded signals, with little ability to separate one band of frequencies from another. Relatives of the moving average filter include the Gaussian, Blackman, and multiple- pass moving average. These have slightly better performance in the frequency domain, at the expense of increased computation time. Implementation by Convolution As the name implies, the moving average filter operates by averaging a number of points from the input signal to produce each point in the output signal. In equation form, this is written: EQUATION 15-1 Equation of the moving average filter. In M &1 this equation, x[ ] is the input signal, y[ ] is ' 1 % y[i] j x [i j ] the output signal, and M is the number of M j'0 points used in the moving average. This equation only uses points on one side of the output sample being calculated. Where x[ ] is the input signal, y[ ] is the output signal, and M is the number of points in the average. For example, in a 5 point moving average filter, point 80 in the output signal is given by: x [80] % x [81] % x [82] % x [83] % x [84] y [80] ' 5 277 278 The Scientist and Engineer's Guide to Digital Signal Processing As an alternative, the group of points from the input signal can be chosen symmetrically around the output point: x[78] % x[79] % x[80] % x[81] % x[82] y[80] ' 5 This corresponds to changing the summation in Eq.
    [Show full text]
  • Random Signals
    Chapter 8 RANDOM SIGNALS Signals can be divided into two main categories - deterministic and random. The term random signal is used primarily to denote signals, which have a random in its nature source. As an example we can mention the thermal noise, which is created by the random movement of electrons in an electric conductor. Apart from this, the term random signal is used also for signals falling into other categories, such as periodic signals, which have one or several parameters that have appropriate random behavior. An example is a periodic sinusoidal signal with a random phase or amplitude. Signals can be treated either as deterministic or random, depending on the application. Speech, for example, can be considered as a deterministic signal, if one specific speech waveform is considered. It can also be viewed as a random process if one considers the ensemble of all possible speech waveforms in order to design a system that will optimally process speech signals, in general. The behavior of stochastic signals can be described only in the mean. The description of such signals is as a rule based on terms and concepts borrowed from probability theory. Signals are, however, a function of time and such description becomes quickly difficult to manage and impractical. Only a fraction of the signals, known as ergodic, can be handled in a relatively simple way. Among those signals that are excluded are the class of the non-stationary signals, which otherwise play an essential part in practice. Working in frequency domain is a powerful technique in signal processing. While the spectrum is directly related to the deterministic signals, the spectrum of a ran- dom signal is defined through its correlation function.
    [Show full text]
  • STATISTICAL FOURIER ANALYSIS: CLARIFICATIONS and INTERPRETATIONS by DSG Pollock
    STATISTICAL FOURIER ANALYSIS: CLARIFICATIONS AND INTERPRETATIONS by D.S.G. Pollock (University of Leicester) Email: stephen [email protected] This paper expounds some of the results of Fourier theory that are es- sential to the statistical analysis of time series. It employs the algebra of circulant matrices to expose the structure of the discrete Fourier transform and to elucidate the filtering operations that may be applied to finite data sequences. An ideal filter with a gain of unity throughout the pass band and a gain of zero throughout the stop band is commonly regarded as incapable of being realised in finite samples. It is shown here that, to the contrary, such a filter can be realised both in the time domain and in the frequency domain. The algebra of circulant matrices is also helpful in revealing the nature of statistical processes that are band limited in the frequency domain. In order to apply the conventional techniques of autoregressive moving-average modelling, the data generated by such processes must be subjected to anti- aliasing filtering and sub sampling. These techniques are also described. It is argued that band-limited processes are more prevalent in statis- tical and econometric time series than is commonly recognised. 1 D.S.G. POLLOCK: Statistical Fourier Analysis 1. Introduction Statistical Fourier analysis is an important part of modern time-series analysis, yet it frequently poses an impediment that prevents a full understanding of temporal stochastic processes and of the manipulations to which their data are amenable. This paper provides a survey of the theory that is not overburdened by inessential complications, and it addresses some enduring misapprehensions.
    [Show full text]
  • Use of the Kurtosis Statistic in the Frequency Domain As an Aid In
    lEEE JOURNALlEEE OF OCEANICENGINEERING, VOL. OE-9, NO. 2, APRIL 1984 85 Use of the Kurtosis Statistic in the FrequencyDomain as an Aid in Detecting Random Signals Absmact-Power spectral density estimation is often employed as a couldbe utilized in signal processing. The objective ofthis method for signal ,detection. For signals which occur randomly, a paper is to compare the PSD technique for signal processing frequency domain kurtosis estimate supplements the power spectral witha new methodwhich computes the frequency domain density estimate and, in some cases, can be.employed to detect their presence. This has been verified from experiments vith real data of kurtosis (FDK) [2] forthe real and imaginary parts of the randomly occurring signals. In order to better understand the detec- complex frequency components. Kurtosis is defined as a ratio tion of randomlyoccurring signals, sinusoidal and narrow-band of a fourth-order central moment to the square of a second- Gaussian signals are considered, which when modeled to represent a order central moment. fading or multipath environment, are received as nowGaussian in Using theNeyman-Pearson theory in thetime domain, terms of a frequency domain kurtosis estimate. Several fading and multipath propagation probability density distributions of practical Ferguson [3] , has shown that kurtosis is a locally optimum interestare considered, including Rayleigh and log-normal. The detectionstatistic under certain conditions. The reader is model is generalized to handle transient and frequency modulated referred to Ferguson'swork for the details; however, it can signals by taking into account the probability of the signal being in a be simply said thatit is concernedwith detecting outliers specific frequency range over the total data interval.
    [Show full text]
  • 2D Fourier, Scale, and Cross-Correlation
    2D Fourier, Scale, and Cross-correlation CS 510 Lecture #12 February 26th, 2014 Where are we? • We can detect objects, but they can only differ in translation and 2D rotation • Then we introduced Fourier analysis. • Why? – Because Fourier analysis can help us with scale – Because Fourier analysis can make correlation faster Review: Discrete Fourier Transform • Problem: an image is not an analogue signal that we can integrate. • Therefore for 0 ≤ x < N and 0 ≤ u <N/2: N −1 * # 2πux & # 2πux &- F(u) = ∑ f (x),cos % ( − isin% (/ x=0 + $ N ' $ N '. And the discrete inverse transform is: € 1 N −1 ) # 2πux & # 2πux &, f (x) = ∑F(u)+cos % ( + isin% (. N x=0 * $ N ' $ N '- CS 510, Image Computaon, ©Ross 3/2/14 3 Beveridge & Bruce Draper € 2D Fourier Transform • So far, we have looked only at 1D signals • For 2D signals, the continuous generalization is: ∞ ∞ F(u,v) ≡ ∫ ∫ f (x, y)[cos(2π(ux + vy)) − isin(2π(ux + vy))] −∞ −∞ • Note that frequencies are now two- dimensional € – u= freq in x, v = freq in y • Every frequency (u,v) has a real and an imaginary component. CS 510, Image Computaon, ©Ross 3/2/14 4 Beveridge & Bruce Draper 2D sine waves • This looks like you’d expect in 2D Ø Note that the frequencies don’t have to be equal in the two dimensions. hp://images.google.com/imgres?imgurl=hFp://developer.nvidia.com/dev_content/cg/cg_examples/images/ sine_wave_perturbaon_ogl.jpg&imgrefurl=hFp://developer.nvidia.com/object/ cg_effects_explained.html&usg=__0FimoxuhWMm59cbwhch0TLwGpQM=&h=350&w=350&sz=13&hl=en&start=8&sig2=dBEtH0hp5I1BExgkXAe_kg&tbnid=fc yrIaap0P3M:&tbnh=120&tbnw=120&ei=llCYSbLNL4miMoOwoP8L&prev=/images%3Fq%3D2D%2Bsine%2Bwave%26gbv%3D2%26hl%3Den%26sa%3DG CS 510, Image Computaon, ©Ross 3/2/14 5 Beveridge & Bruce Draper 2D Discrete Fourier Transform N /2 N /2 * # 2π & # 2π &- F(u,v) = ∑ ∑ f (x, y),cos % (ux + vy)( − isin% (ux + vy)(/ x=−N /2 y=−N /2 + $ N ' $ N '.
    [Show full text]
  • 20. the Fourier Transform in Optics, II Parseval’S Theorem
    20. The Fourier Transform in optics, II Parseval’s Theorem The Shift theorem Convolutions and the Convolution Theorem Autocorrelations and the Autocorrelation Theorem The Shah Function in optics The Fourier Transform of a train of pulses The spectrum of a light wave The spectrum of a light wave is defined as: 2 SFEt {()} where F{E(t)} denotes E(), the Fourier transform of E(t). The Fourier transform of E(t) contains the same information as the original function E(t). The Fourier transform is just a different way of representing a signal (in the frequency domain rather than in the time domain). But the spectrum contains less information, because we take the magnitude of E(), therefore losing the phase information. Parseval’s Theorem Parseval’s Theorem* says that the 221 energy in a function is the same, whether f ()tdt F ( ) d 2 you integrate over time or frequency: Proof: f ()tdt2 f ()t f *()tdt 11 F( exp(j td ) F *( exp(j td ) dt 22 11 FF() *(') exp([j '])tdtd ' d 22 11 FF( ) * ( ') [2 ')] dd ' 22 112 FF() *() d F () d * also known as 22Rayleigh’s Identity. The Fourier Transform of a sum of two f(t) F() functions t g(t) G() Faft() bgt () aF ft() bFgt () t The FT of a sum is the F() + sum of the FT’s. f(t)+g(t) G() Also, constants factor out. t This property reflects the fact that the Fourier transform is a linear operation. Shift Theorem The Fourier transform of a shifted function, f ():ta Ffta ( ) exp( jaF ) ( ) Proof : This theorem is F ft a ft( a )exp( jtdt ) important in optics, because we often encounter functions Change variables : uta that are shifting (continuously) along fu( )exp( j [ u a ]) du the time axis – they are called waves! exp(ja ) fu ( )exp( judu ) exp(jaF ) ( ) QED An example of the Shift Theorem in optics Suppose that we’re measuring the spectrum of a light wave, E(t), but a small fraction of the irradiance of this light, say , takes a different path that also leads to the spectrometer.
    [Show full text]
  • Introduction to Frequency Domain Processing
    MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 2.14 Analysis and Design of Feedback Control Systems Introduction to Frequency Domain Processing 1 Introduction - Superposition In this set of notes we examine an alternative to the time-domain convolution operations describing the input-output operations of a linear processing system. The methods developed here use Fourier techniques to transform the temporal representation f(t) to a reciprocal frequency domain space F (jω) where the difficult operation of convolution is replaced by simple multiplication. In addition, an understanding of Fourier methods gives qualitative insights to signal processing techniques such as filtering. Linear systems,by definition,obey the principle of superposition for the forced component of their responses: If linear system is at rest at time t = 0,and is subjected to an input u(t) that is the sum of a set of causal inputs,that is u(t)=u1(t)+u2(t)+...,the response y(t) will be the sum of the individual responses to each component of the input,that is y(t)=y1(t)+y2(t)+... Suppose that a system input u(t) may be expressed as a sum of complex n exponentials n s t u(t)= aie i , i=1 where the complex coefficients ai and constants si are known. Assume that each component is applied to the system alone; if at time t = 0 the system is at rest,the solution component yi(t)is of the form sit yi(t)=(yh (t))i + aiH(si)e where (yh(t))i is a homogeneous solution.
    [Show full text]
  • Time Domain and Frequency Domain Signal Representation
    ES442-Lab 1 ES440. Lab 1 Time Domain and Frequency Domain Signal Representation I. Objective 1. Get familiar with the basic lab equipment: signal generator, oscilloscope, spectrum analyzer, power supply. 2. Learn to observe the time domain signal representation with oscilloscope. 3. Learn to observe the frequency domain signal representation with oscilloscope. 4. Learn to observe signal spectrum with spectrum analyzer. 5. Understand the time domain representation of electrical signals. 6. Understand the frequency domain representation of electrical signals. II. Pre-lab 1) Let S1(t) = sin(2pf1t) and S1(t) = sin(2pf1t). Find the Fourier transform of S1(t) + S2(t) and S1(t) * S2(t). 2) Let S3(t) = rect (t/T), being a train of square waveform. What is the Fourier transform of S3(t)? 3) Assume S4(t) is a train of square waveform from +4V to -4V with period of 1 msec and 50 percent duty cycle, zero offset. Answer the following questions: a) What is the time representation of this signal? b) Express the Fourier series representation for this signal for all its harmonics (assume the signal is odd symmetric). c) Write the Fourier series expression for the first five harmonics. Does this signal have any DC signal? d) Determine the peak magnitudes and frequencies of the first five odd harmonics. e) Draw the frequency spectrum for the first five harmonics and clearly show the frequency and amplitude (in Volts) for each line spectra. 4) Given the above signal, assume it passes through a bandlimited twisted cable with maximum bandwidth of 10KHz. a) Using Fourier series, show the time-domain representation of the signal as it passes through the cable.
    [Show full text]
  • The Wavelet Tutorial Second Edition Part I by Robi Polikar
    THE WAVELET TUTORIAL SECOND EDITION PART I BY ROBI POLIKAR FUNDAMENTAL CONCEPTS & AN OVERVIEW OF THE WAVELET THEORY Welcome to this introductory tutorial on wavelet transforms. The wavelet transform is a relatively new concept (about 10 years old), but yet there are quite a few articles and books written on them. However, most of these books and articles are written by math people, for the other math people; still most of the math people don't know what the other math people are 1 talking about (a math professor of mine made this confession). In other words, majority of the literature available on wavelet transforms are of little help, if any, to those who are new to this subject (this is my personal opinion). When I first started working on wavelet transforms I have struggled for many hours and days to figure out what was going on in this mysterious world of wavelet transforms, due to the lack of introductory level text(s) in this subject. Therefore, I have decided to write this tutorial for the ones who are new to the topic. I consider myself quite new to the subject too, and I have to confess that I have not figured out all the theoretical details yet. However, as far as the engineering applications are concerned, I think all the theoretical details are not necessarily necessary (!). In this tutorial I will try to give basic principles underlying the wavelet theory. The proofs of the theorems and related equations will not be given in this tutorial due to the simple assumption that the intended readers of this tutorial do not need them at this time.
    [Show full text]
  • Image Analysis
    Image analysis CS/CME/BioE/Biophys/BMI 279 Nov. 8, 2016 Ron Dror 1 Outline • Images in molecular and cellular biology • Reducing image noise – Mean and Gaussian filters – Frequency domain interpretation – Median filter • Sharpening images • Image description and classification • Practical image processing tips 2 Images in molecular and cellular biology 3 Most of what we know about the structure of cells come from imaging • Light microscopy, including fluorescence microscopy ! https:// www.microscopyu.com/ ! articles/livecellimaging/ livecellmaintenance.html ! ! ! • Electron microscopy http:// blog.library.gsu.edu/ wp-content/uploads/ 2010/11/mtdna.jpg 4 Imaging is pervasive in structural biology • The experimental techniques used to determine macromolecular (e.g., protein) structure also depend on imaging X-ray crystallography Cryo-electron microscopy https://askabiologist.asu.edu/sites/default/ 5 files/resources/articles/crystal_clear/ http://debkellylab.org/?page_id=94 CuZn_SOD_C2_x-ray_diffraction_250.jpg Computation plays an essential role • Computation plays an important role in these imaging-based techniques • We will start with analysis of microscopy data, because it’s closest to our everyday experience with images • In fact, the basic image analysis techniques we’ll cover initially also apply to normal photographs 6 Representations of an image • Recall that we can think of a grayscale image as: – A two-dimensional array of brightness y values – A matrix (of brightness values) – A function of two variables (x and y) • A color image can be treated as: x – Three separate images, one for each color channel (red, green, blue) – A function that returns three values (red, green, blue) for each (x, y) pair Whereas Einstein proposed the concept of mass-energy equivalence, 7 Professor Mihail Nazarov, 66, proposed monkey-Einstein equivalence.
    [Show full text]
  • Signals and Sampling
    Signals and Sampling Chapter 7 of “Physically Based Rendering” by Pharr&Humphreys Chapter 7 7.1 Sampling Theory 7.2 Image Sampling Interface 7.3 Stratified Sampling 7.4 Low-Discrepancy Sampling 7.5 Best-Candidate Sampling Patterns 7.6 Image Reconstruction 2 Additional Reading Chapter 14.10 of “CG: Principles & Practice” by Foley, van Dam et al. Chapter 4, 5, 8, 9, 10 in “Principles of Digital Image Synthesis,” by A. Glassner Chapter 4, 5, 6 of “Digital Image Warping” by Wolberg Chapter 2, 4 of “Discrete-Time Signal Processing” by Oppenheim, Shafer Motivation • Real World - continuous • Digital (Computer) world - discrete • Typically we have to either: – create discrete data from continuous or (e.g. rendering/ray-tracing, illumination models, morphing) – manipulate discrete data (textures, surface description, image processing,tone mapping) Motivation • Artifacts occurring in sampling - aliasing: – Jaggies – Moire – Flickering small objects – Sparkling highlights – Temporal strobing • Preventing these artifacts - Antialiasing Motivation Engineering approach: nearest neighbor: Convolution with box filter linear filter: Convolution with tent filter Motivation- Graphics Original “manipulated” (continuous) signal “Graphics” (continuous) signal sampling Reconstruction sampled filter signal Motivation Engineering approach: • black-box “System” or Algorithm • discretization: Multiplication with “shah” function Convolution • How can we characterize our “black box”? • We assume to have a “nice” box/algorithm: – linear – time-invariant • then it can
    [Show full text]
  • Cross-Correlation
    ESS 522 2014 8. Cross-Correlation Cross-correlation The cross-correlation of two real continuous functions, φxy is defined by ∞ φ t = x τ − t y τ dτ (8-1) xy ( ) ∫ ( ) ( ) −∞ If we compare it to convolution ∞ x(t)* y(t) = ∫ x(t − τ )y(τ )dτ (8-2) −∞ we can see that the only difference is that for the cross correlation, one of the two functions is not reversed. Thus, φxy (t) = x(−t)* y(t) (8-3) In the frequency domain we can write the Fourier transform of x(-t) as ∞ FT ⎣⎡x(−t)⎦⎤ = ∫ x(−t)exp(−i2π ft)dt (8-4) −∞ Substituting t’ = –t yields −∞ ∞ * FT ⎣⎡x(−t)⎦⎤ = ∫ −x(t')exp(i2π ft')dt' = ∫ x(t')exp(i2π ft')dt' = X ( f ) (8-5) ∞ −∞ Time reversal is the same as taking the complex conjugate in the frequency domain. We can thus write * (8-6) Φxy = FT ⎣⎡φxy (t)⎦⎤ = X ( f )Y ( f ) Unlike convolution, cross-correlation is not commutative but we can write φxy (t) = φyx (−t) (8-7) You can show this by letting τ’ = τ - t In the discrete domain, the correlation of two real time series xi, i = 0, 1, …, M-1 and yj, j = 0, 1, …, N-1 is by analogy to equation (8-1) given by min(M −1+ k, N −1) (8-8) φxy,k = ∑ x j − k yj , k = −(M + 1), ..., 0, ..., (N − 1) j =max(0, k) In Matlab cross-correlations are computed with the function xcorr which works in the frequency domain. Note that to obtain the discrete version of φxy as defined by equation (8-8), one reverses the arguments (i.e., one calls phixy = xcorr(y,x)).
    [Show full text]