TECHNICAL NOTE

Comparative effects of optical-correlatorsignal-dependent and signal-independent noise onpattern-recognition performance with thephase-only filter

Jean-Christophe Terrillon

The comparative effects of optical-correlator signal-dependent and additive signal-independent noise oncorrelation-filter performance are analyzed by three different performance measures. For an identicalvalue of the signal-to-noise ratio imposed on each type of noise in a binary input image, computersimulations performedwith the phase-only filter show 1i2 that additive Gaussian signal-independent noiseyields a much larger correlation-performance degradation than signal-dependent noise and 1ii2 that thedifferent types of signal-dependent noise lead to similar correlation results because of similar effects onthe input image that are inherent to the nature of the noise.

1. Introduction

When the performance of correlation filters in thepresence of random noise is analyzed, additive signal-independent noise 1SIN2 is the most widely used noisesource. However, several noise sources inherent tooptical correlators are signal dependent.1 Specklenoise occurs in coherent diffuse illumination.2Poisson noise is characteristic of low light conditions.3Film-grain noise is produced by the recording of anoptical signal on a photographic emulsion.4 Morecomplex noise sources may also occur: for example,the combination of speckle signal-dependent noise1SDN2 and of additive SIN has been considered indetection problems.5 In this Note, I compare theeffects of the above-mentioned noise sources on pat-tern-recognition performance by using the phase-only filter 1POF2. I do not consider other noisesources, such as electronic-sensor noise or scene andtarget inherent noise. The comparison between the

The author is a fellow of the Science and Technology Agency ofJapan with the Kansai Advanced Research Center, Communica-tions Research Laboratory, Ministry of Posts and Telecommunica-tions, 588-2 Iwaoka, Iwaoka-cho, Nishi-ku, Kobe 651-24, Japan.Received 2 March 1995; revised manuscript received 12 June

1995.0003-6935@95@327561-04$06.00@0.

r 1995 Optical Society of America.

different noise sources is made possible by imposingon each type of noise an identical value of the signal-to-noise ratio 1SNRin2 in a given binary input image,which contains an object to be recognized; 104 indepen-dent noise realizations of the input image are com-puter generated for each type of noise, and eachrealization is correlated with the POF synthesizedfrom the noise-free object. Three different metricsfor assessing correlation performance are used in thisNote: the SNR of the correlation-peak value, thefalse-alarm probability, and the ratio of the meancorrelation peak intensity to the peak intensity in theabsence of noise. They are calculated over 104 result-ing noise realizations of the correlation obtained foreach type of noise. Also, in order to compare quanti-tatively the effects of the different noise sources onthe input image, we consider the ensemble average ofthe mean-square difference 1MSD2 between each noiserealization of the input image and the original image.

2. Noise Models

We assume that S1x2 is the original image signal 1orobject2 and that R1x2 is the image corrupted by SIN orby SDN. x is a vector that defines the space vari-ables or 1x, y2 coordinates in the image plane. Theunderline symbol is used here to differentiate be-tween a random variable and a deterministic signal.The SNR measured pointwise in a noisy input image

10 November 1995 @ Vol. 34, No. 32 @ APPLIED OPTICS 7561

may then be defined in general as

SNRin 5S1x2

sR

, 112

where sR is the standard deviation of R1x2 and wherewe assume that S1x2 is nonzero.Table 1 summarizes the generally accepted models

for the different types of noise 1additive SIN andSDN2 considered in this Note and the associatedstatistical parameters. It is assumed that the signalS and the noises N and Nsp are statistically indepen-dent. In all the models, the mean of R is equal to thesignal S and is not shown in the table. N is aGaussian noise with zero mean and variance sN

2

1except for film-grain noise, where sN2 5 1.02. Nsp is a

random variable with a Gamma probability-densityfunction 1pdf 2 of order M with unit mean and vari-ance 1@M, which describes fully developed speckleintegrated with an aperture. For M 5 1, the pdf ofNsp reduces to a negative exponential.2 In the case offilm-grain noise, k is a real constant that determinesthe amplitude of the noise fluctuations, and the valueof the real parameter p typically varies between 0.3and 0.7. For ideal film-grain noise, p 5 0.5.1Pl1lS2 is a stochastic process with a Poisson pdf with amean and a variance both equal to lS, where theparameter l is real.3 The dependency ofR, S,N andNsp on x is implied. In the model used for additiveSIN, RSIN, S and N may be expressed in complexamplitude or in intensity. In the model that de-scribes film-grain noise, Rfg and S are the measuredand the original film density, respectively. In all theother models, both R and S are measured in intensity.Any dependence of sR on S implies that the noise issignal dependent. Except for speckle SDN, SNRinvaries spatially with the signal S, but a single value ofSNRin is defined for an object of uniform amplitude orintensity. For speckle SDN, SNRin is independent ofS and reaches the smallest possible value when M 5

1, in which case it is equal to 1.

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3. Metrics for Assessing Correlation Performance in thePresence of Noise

A. Signal-to-Noise Ratio of the Correlation-Peak Value

The signal-to-noise ratio 1SNR2 of the correlation-peakvalue is generally defined as6

SNR 50 7C1028 02

var3C1024, 122

where the symbols 7 8 and var3 4 represent the mean1or ensemble average2 and the variance, respectively,and where C102 is the 1random2 correlation amplitudeat the origin of the coordinates 1j, h2 in the correlationplane. In the above SNRmeasure, it is assumed thatthe center of the correlation or the correlation peak islocated at the origin. When the POF is used, this isstrictly true if the object to be recognized S is centeredat the origin of coordinates 1x, y2 in the input image.

B. False-Alarm Probability

The presence of random noise in the input imageinduces random fluctuations of the correlation-peakvalue and generates sidelobes or false peaks in thecorrelation plane. We consider false alarms to occuronly when one or more false peaks are higher than themeasured correlation peak for S. In computer simu-lations the correlation peak is detected in all noiserealizations of the correlation that are generated, andover each noise realization, a search is made for thehighest sidelobe. In order that such a sidelobe beclearly distinguishable, the search excludes a windowwith a minimum size of 3 3 3 pixels centered on thedesired correlation peak. We then consider the quan-tity PFA as the percentage of all noise realizations inwhich at least one false peak is higher than thecorrelation peak for S. Hence the probability PFAcovers all the correlation plane in each noise realiza-tion. It may be considered as one of the most impor-tant performance criteria, as it measures the effectiveability of a correlation filter to detect only the object Sand not any false target in the presence of

Table 1. Models and Associated Statistical Parameters Describing Additive Signal-Independent Noiseand Four Different Signal-Dependent Noise Sources a

Noise Source Models sR SNRin

Additive SIN RSIN 5 S 1 N sN S@sN

Speckle Rsp 5 SNsp S@ŒM ŒMFilm grain Rfg 5 S 1 kSpN kSp S12p@k

Poisson RPoi 51

lPl1lS2 ŒS@l ŒlS

Speckle 1

SINRsp1SIN 5 SNsp 1 N ŒS2

M1 sN

2 S/ŒS2

M1 sN

2

aS is the original image signal; R is the noisy image. N is a Gaussian noise with zero mean and variance sN2 1except for film-grain SDN,

where sN2 5 1.02. Nsp is a random variable with a Gamma probability-density function of order M with unit mean and variance 1@M.

Pl1lS2 is a stochastic Poisson process with a mean and a variance both equal to lS. k is a real constant, and p and l are both realparameters. sR is the standard deviation of R, and SNRin is the signal-to-noise ratio 1SNR2measured pointwise in the input image.

random noise. The most severe indication of correla-tion-performance degradation is the occurrence offalse alarms in a significant number of noise realiza-tions of the correlation, which do not permit theproper identification of the object.

C. Ratio of the Mean Correlation-Peak Intensity to the PeakIntensity in the Absence of Noise

The ratio 7IP8@IP, where 7IP8 is the mean of the correla-tion-peak intensity in the presence of random noiseand where IP is the noise-free peak intensity, istheoretically equal to 1.0, as the mean of the crosscorrelation between the noise and a given object isgenerally equal to zero. In practice, if the noisesource is sufficiently strong, nonnegligible cross-correlation effects may yield a value of 7IP8@IP that isgreater than 1.0

4. Mean-Square Difference between the Noisy Imageand the Original Image

Ameasure that is generally used to assess the degra-dation of an image corrupted with random noise is themean-square difference 1MSD2 between the discretenoisy imageR1i, j2 and the original image S1i, j2. Thecalculation of theMSD requires an a priori knowledgeof the original image. For discrete images of dimen-sionsM 3 N pixels, the MSDmeasured over an entirenoisy image realization is defined as

MSD 51

MN oi51

M

oj51

N

3R1i, j2 2 S1i, j242, 132

where the underline symbol describes the MSD as arandom variable. To evaluate the average degrada-tion of an image for a given noise source and for agiven SNRin, one calculates the ensemble average ofthe MSD over many noise realizations of the image.

5. Simulation Results

Computer simulations are performed by use of thebinary image of the capital letter I, of dimensions64 3 64 pixels. The simulated value of amplitude orintensity of the object is 255, and we consider a blackbackground with a value equal to 0.Table 2 shows the correlation results when SNRin 5

1. The value of 7Ip8 is normalized with respect to Ip.Globally, one can distinguish three different situa-tions: the POF is very sensitive to additive GaussianSIN, an intermediate sensitivity is observed for thecombination of speckle SDN and of additive SIN, andthe POF is quite robust with respect to speckle,film-grain, and Poisson SDN. Moreover, the latterthree types of SDN have similar effects on the correla-tion performance of the POF. Amore detailed analy-sis shows that additive Gaussian SIN produces acorrelation-peak SNR that is smaller by a factor of 2.0than the SNR calculated for the three purely SDNsources and that it also yields a significant false-alarmprobability, whereas PFA is negligible or small for allother noise sources. Additive Gaussian SIN also

induces significant cross-correlation effects at thecenter of the correlation because 7Ip8 is larger than Ipby 11.5%. Ideal film-grain SDN and Poisson SDNyield almost identical results because their depen-dence on the signal is the same 1please see Table 12.Speckle SDN produces slightly different results fromthose of film-grain and Poisson SDN because thenegative exponential pdf that corresponds toM 5 1 isquite different from the pdf that describes film-grainor Poisson SDN.There is a direct relationship between the correla-

tion-performance degradation and the average of theMSD. The 7MSD8 calculated for additive GaussianSIN is larger by one order of magnitude than thevalue obtained for speckle, film-grain, and PoissonSDN. It is also almost twice as large as the 7MSD8 forthe combination of speckle SDN and of additive SIN.Such results indicate that the mean degradation ofthe input image that is due to additive Gaussian SINis much larger than when the image is corrupted bySDN, despite the fact that an identical value of SNRinis imposed on each type of noise.To illustrate the differences in image degradation,

and consequently the differences in correlation-performance degradation, Fig. 1 shows a particularnoise realization of the letter I for each of the fivedifferent types of noise and for SNRin 5 1.0. Obvi-ously, additive Gaussian SIN corrupts the entireimage because random intensity fluctuations appearin the background of the image, in which no signal isoriginally present. Thus the noisy background 1orscene2 increases the degradation of the image, whichresults in a much larger value of 7MSD8 and in a moresevere correlation-performance degradation thanwhen the image is corrupted with SDN. Speckle,film-grain, and Poisson SDN corrupt only the object tobe recognized and have no effect on the scene whenthe scene intensity is originally zero because of theirsignal-dependent characteristics. The combinationof speckle SDN and of additive Gaussian SIN hasintermediate effects on the image because the addi-tive SIN component corrupts the scene less than pure

Table 2. Correlation Results of the POF of the Capital Letter I Degradedwith Additive Gaussian SIN and with Four Different Types of SDN for anIdentical Value of the SNR Measured in the Input Image, SNR in 5 1.0a

Noise Source

Input Image Correlation, POF

7MSD8 SNR PFA 7Ip8n

Additive SIN 64936.0 38.36 0.1916 1.1150Speckle 6436.6 77.15 0.0030 1.0143Film grain 6425.8 81.33 0.0079 1.0158Poisson 6421.2 80.64 0.0070 1.0130Speckle 1 SIN 35705.4 47.85 0.0780 1.0598

aWith a signal S 5 S0 5 255, sN 5 255.0 for additive SIN,M 5 1for speckle SDN, p 5 0.5 and k 5 15.97 for film-grain SDN, l 5

3.9 3 1023 for Poisson SDN, and M 5 2 and sN 5 180.31 for thecombination of speckle SDN and of additive SIN. In each simula-tion, the statistics are calculated over 104 noise realizations of theinput image and of the correlation. 7Ip8 is normalized with respectto IP 1Ip 5 1,553,627 units of intensity2.

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Fig. 1. Particular noise realizations of the capital letter I degraded with additive Gaussian SIN andwith four different types of SDN for anidentical value of the SNR measured in the input image, SNRin 5 1.0: 1a2 additive SIN, 1b2 speckle SDN, 1c2 film-grain SDN, 1d2 PoissonSDN, and 1e2 combination of speckle SDN and additive SIN.

additive SIN for the same global value of SNRin. Anadditive SIN model with a different pdf 1such as aGamma pdf 2 still corrupts a scene of originally zerointensity and adds a nonzero mean to the intensityover the entire image.One concludes that, for an identical noise level in

an originally binary input image, the correlation-performance degradation observed with the POF isstrongly dependent on the inherent characteristics ofthe optical correlator noise source that is considered.

The author gratefully acknowledges the support ofthe Departement de Physique, Centre d’Optique,Photonique et Laser, Universite Laval, Ste-Foy, Que-bec G1K 7P4, Canada.

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References1. J. F. Walkup and R. C. Choens, ‘‘Image processing in signal-

dependent noise,’’ Opt. Eng. 13, 258–266 119742.2. J. W. Goodman, Statistical Optics 1Wiley, NewYork, 19852.3. C. M. Lo, ‘‘Estimation of image signals with Poisson noise,’’ Rep.

890 1Image Processing Institute, University of California, LosAngeles, Calif., 19792.

4. H. Siedentopf, ‘‘Concerning granularity, density fluctuationsand the enlargement of photographic negatives,’’ Phys. Z. 38,454 119372.

5. G. S. Mecherle, ‘‘Signal speckle effects on optical detection withadditive Gaussian noise,’’ J. Opt. Soc. Am.A 1, 68–72 119842.

6. B. V. K. Vijaya Kumar and L. Hassebrook, ‘‘Performancemeasures for correlation filters,’’ Appl. Opt. 29, 2997–3006119902.