Cheetham, Andrew

Huang, Xu

Madoc, A

English

Crossref

Wavelet-based Bayesian estimator for Poisson noise removal from images

Images are, in many cases, degraded even before they are encoded. Emission and transmission tomography images, X-ray films, and photographs taken by satellites are usually contaminated by quantum noise, which is Poisson distributed. Poisson shot noise is a natural generalization of a compound Poisson process when the summands are stochastic processes starting at the points of the underlying Poisson process. Unlike additive Gaussian noise, Poisson noise is signal-dependent and separating signal from noise is a difficult task. A wavelet-based maximum likelihood for a Bayesian estimator that recovers the signal component of the wavelet coefficients in original images by using an alpha-stable signal prior distribution is extended to the Poisson noise removal from a previous investigation. As we discussed in our earlier papers that Bayesian estimator can approximate impulsive noise more accurately than other models and that in the general case the Bayesian processor does not have a closed-form expression. The parameters relative to Bayesian estimators of the model are carefully investigated after an investigation of a-stable simulations for a maximum likelihood estimator. As an example, an improved Bayesian estimator that is a natural extension of other wavelet denoising (soft and hard threshold methods) via a colour image is presented to illustrate our discussion

University of Canberra Research Repository

WAVELET-BASED BAYESIAN ESTIMATOR FOR POISSON NOISE REMOVAL FROM IMAGES XHuangl. A.C. Madoc2 and A.D.Cheetham' 'School of Information Sciences and Engineering, University of Canberra, ACT 2601, Australia 'Center for Actuarial Studies, Department of Economics, University of Melboume, VIC 305 1, Australia ABSTRACT Images are, in many cases, degraded even before they are encoded. Emission and transmission tomography images, X-ray films, and photographs taken by satellites are usually contaminated by quantum noise, which is Poisson distributed. Poisson shot noise is a natural generalization of a compound Poisson process when the summands are stochastic processes starting at the points of the underlying Poisson process. Unlike additive Gaussian noise, Poisson noise is signal-dependent and separating signal from noise is a difficult task. A wavelet-based maximum likelihood for a Bayesian estimator that recovers the signal component of the wavelet coefficients in original images by using an alpha-stable signal prior distribution is extended to the Poisson noise removal from a previous investigation. As we discussed in our earlier papers that Bayesian estimator can approximate impulsive noise more accurately than other models and that in the general case the Bayesian processor does not have a closed-form expression. The parameters relative to Bayesian estimators of the model are carefully investigated after an investigation of estable simulations for a maximum likelihood estimator. As an example, an improved Bayesian estimator that is a natural extension of other wavelet denoising (soft and hard threshold methods) via a colour image is presented to illustrate our discussion. 1. INTRODUCTION It is well known that noise degrades the performance of any image compression algorithm. In many cases, degraded even before they are encoded. It is obvious that linear filtering techniques used in many image-processing applications, are attractive due to their mathematical simplicity, and efficiency in the presence of additive Gaussian noise. However, they also blur sharp edges, make some distortions of lines and tine image details, less effectively remove tailed noise, and poorly treat the presence of signal-dependent noise. For example, emission and transmission tomography images are usually contaminated by quantum noise, which is Poisson noise. Unlike additive Gaussian noise, Poisson noise is signaldependent, and separating signal from noise is difficult. Several groups have discussed that wavelet subhand coeficients have highly non-Gaussian statistics [MI and the general class of a-stable distributions has also been shown to accurately model heavy-tailed noise 15-71, The question is, how to deal with estable distributions with signal- dependent noise, such as signals contaminated by Poisson noise. Wavelet transform is a powerful tool for recovering signals from noise and has been of considerably interest [4, 8-1 I]. In fact, wavelet theory combines many existing concepts into a global framework and hence becomes a powerful tool for several domains of application. 0-7803-7965-9/03/$17.00 02003 IEEE As mentioned by Achim et al. [U], there are two major drawbacks for thresholding. One is that choice of the threshold is always done in an ad hac manner; another is that the specific dishibutions of the signal and noise may not be well matched at different scales. Donoho gives some minimum thresholds for several threshold schemes, titled "universal thresholds" [IO]. These explicitly depend on the standard deviation of noise, where the standard deviation is assumed to be known. In practice, the standard deviation can be readily estimated using the methods discussed in [9], [13]. For some applications the optimal threshold can he computed. An approach different from "universal thresholds" is presented by Nason [14], in which cross-validation is used. Two approaches to cross validation are used, namely ordinary cross validation (OCV) and generalised cross validation (GCV): each is used to minimize the least-squares error between the original (which is the unknown value) function and its estimate based an  the noisy observation. Modelling the statistics of raw images is a challenging task because of the high dimensionality of the signal and the complexity of statistical StNChIreS that are prevalent. Numerous papers discuss modelling the statistics of raw images, including Bayesian processing assuming proper modelling of the prior probability density function of the signal, but they dealt with the Gaussian noise, or with the symmetric stochastic distributions [5, 6, 15,16,17,19,20]. In this paper it is carefully discussed that a wavelet-based maximum likelihood for Bayesian estimator that recovers the signal component of the wavelet coefficients in original images from images contaminated by Poisson noise, by using an alpha- stable signal prior distribution. As an example, a colour image and its image contaminated by Poisson noise will he shown using the discussed method. 2. ALPHA-STABLE DISTRIBUTIONS AND LOG LIKELIHOOD It is well known that the symmetric alpha-stable distribution ( S a  distribution is defined by its characteristic function: @(a) = e x p ( j h  - Y Io IT), (1) The parameters a ,  y. and 6 describe completely a So5 distribution. The characteristic exponent acontrols the heaviness of the tails of the stable density. a c a n  take values in (0,2]; while a = I and 2 are the Cauchy and Gaussian cases respectively. There is not closed-form expression known for the general So5 probability density function (PDF). Thus, it is useful when using the principle of maximum likelihood estimation. The dispersion parameter y(y>O) refers to the spread of the PDF., The location parameter 6 i s  analogous to the mean of the PDF, which, for our following discussion, will be the same assumption as that in [5].  I - 593 ICME 2003 If a variable 8 is unbiased it follows that E ( 6 - 8 ) = 0  (2) j...j(B-e)fiw(3e)ax=o (3) which can be expressed as: - - where ?(() = [x l ( ( ) ,x2(5) , . . . .xN(( )JT and fES(Xi8) i s  the joint density of i ({) ,  which depends on a fixed but unknown parameter. Following [15-171 we have The function Infi;s(?;8) is well known as the "log likelihood" function of B (LLF). Its maximum likelihood estimate can be obtained from the equation: "" The first order of differential log likelihood function with respect to B is called the maximum likelihood (ML) estimate. If the efficient estimate does not exist, then the ML estimate will not achieve the lower bound and hence it is difficult to ascertain how closely the variance of any estimate will approach the bound. It is noted that the value of about 1.5 is strongly recommended if there is no information about a d u e  t o  the 2nd order simulations of the LLF for an alpha-stable [ 171. 3. WAVE-BASED BAYESIAN ESTIMATOR If we take the probability density of B as p(6) ;  and the posterior density function as J ( S  I xI, ..., x") , then the updated probability density function of Bis as follows: If we estimate the parameters of the prior distributions of the signals and noise y components of the wavelet coefficients c, we may use the parameters to form the prior PDFs of P,(s) and PJq), hence the inputloutput relationship can be established by the Bayesian estimator, namely, let inputloutput of the Bayesian estimator = BE, we have: P,(s) is the prior PDF of the signal component of the wavelet coefficients of the ultrasound image and P,(q) is the PDF of the wavelet coefficients corresponding to the noise. In order to be able to construct the Bayesian processor in (7), we must estimate the parameters of the prior distributions of the signal (s) and noise (4) components of the wavelet coefficients (d). Then, we use the parameters to obtain the two prior PDFs P,(q) and Pxs) and the nonlinear input-output relationship BE. Figure 1 shows the simulation results of inputloutput of BE with different @ values for given y (=25) and the mean of Poisson distribution (=35). It clearly shows that, for the given case, the curves with a= 0.1, 1.5, and 1.9 approximately correspond to the "hard", "soft", and "semisoft" functions respectively when compared with results in [8,18]. ,h,muV-nBEnh6"mo".,-~.~,Ihmand~d.tnbum.SI) I f I t v/: ' I I 4m 920 ?do w w I m ,  ,om ,m (coo ICM ,,m Figure I : The inputloutput of BE with different a values ( y =  25; the mean of Poisson distribution = 35). Unlike the case contaminated by Gaussian noise [15-18], the mean of Poisson noise plays a role in a B E  as shown in Figure 2, where the parameters are the same as that in Figure I except for the mean of Poisson distribution is equal to IO rather than 35 0- T* , ~ y o u l p " l d  e€wmd,"moral"rh-25 VI ma"d P o B n a U ~ b n -  to, O& Figure 2: The mean of Poisson distribution affects BE. Here all parameters are the same as that in Figure 1 except the mean of Poisson distribution, which is IO. It is clear that the curves with different a values are not significantly different and approximate a single function. This is as expected due to the relation, for Poisson distribution, between the variance and the mean. Taking the results from [15-17], we I - 594 investigated how the ratio of  mean affects BE as shown in Figures 3 and’4. The means of Poisson distribution in Figures 3 and 4 are 35 and I O  respectively. Again, they confirm the fact that the mean of Poisson distribution plays a role in BE. in Figure 4, as in Figure 2, we show the corresponding vertical axis and “identity line” for previous mean. “hey also show that y will affect the output of BE, since y ER is the dispersion of the dismbution. The ratio of 30 and 20 correspond to “soft” and “semi-soft” functions respectively. T h . m p l l ~ u o t B E l “ ~ “ r m n l o , a r i m u n n n m a n a - d a * u a n r l S )  I ’  I’ . called “dust storm” taken in NSW, Australia in Figure 5, together with a copy contaminated by Gaussian noise in Figure 6 and a copy contaminated by Poisson noise in Figure 7. Note the differences between the two noisy images. In particular that for the Poisson contaminated there is more noise in the brighter parts of the image. The Harr mother wavelet was used for this example. The output of the denoised image from BE is shown in Figure 8, where the Poisson distribution is I O  and the image with all the same conditions but changing the mean of the Poisson distribution to 35 is shown in Figure 9. . ,  I ,  I 3% ,cn ,La .m 5n 800 %n ,a) 7yI  Data ( D I  i I 5)  Figure 3: The inputloutput of BE with different ratios of jmean (with the mean of Poisson mean = 35, a= I S ) .  I 950 ua 4% 5m 5n m em 7m 754 . d P  Data ((I = 1 5)  Figure 4: The inputloutput of BE with different ratios of jmean (with the mean of Poisson mean = IO, a= 1 S). 4. SOME E X A M P L E S  When we make measurements, we have no information about the noise value of the image we obtained. The only information one may have is from experience in judgement of the noise level, which becomes the outline ofthe demising strategy. We take the parameters u=1.5, yimean = 20, in the two cases of BE with the mean of Poisson distributions equal to 35 (Figure 8) and IO (Figure 9). In order to compare, we show the original image Figure 5: An Image of the “dust storm” in NSW Australia Figure 7 Figure 5 contaminated by Poisson noise. Comparisons of other denoising results are in table I I - 595 Method I 1 1 2  1 3  1 4  1 5  SlMSE 1 14.21 I 14.30 I 14.01 I 14.87 I 14.68 Table 1:  Comparison of denoising results with BE in signal to mean square error (SIMSE) in dB. Here I =  soft thresholding; 2 = Hard thresholding: 3 = Homomomhic Wiener: 4 = BE (mean Figure 8: The denoised image from the designed BE (the mean of tbe Poisson distribution is IO). . . , .  . Fiaurc 9 The Jcnoiwd image from the desiuned RE ithe mean 0 1  thiPoisson distribution is 35). 1 5. CONCLUSION The technique using the wavelet-based Bayesian estimator has been extended to signaldependent noise obeying the Poisson distribution. The statistician’s Bayesian estimator theory is used not only to simplify the selection of parameters but also in some situations to provide more precise images than other methods. 6. ACKNOWLEDGEMENT One of the authors would like to thank his colleague, Charles Pfohl, for his kindly offering the image of “dust storm”. 1. 2. 7. REFERENCES D J Field, “What is the goal of sensory coding?”, Neural Computation, vo1.6, pp559-601, 1994. S. G. Mallat, “A theory for multiresolution signal decomposition: The wavelet representation”, IEEE Pat. Anal. Mach. Intell. Pp674-693, July 1989. 3. 4. 5.  6. 7. 8. 9. IO I I  12 13 14 I5 16 17 18 19 20 E.P. Simoncelli and E. H. Adelson, “Noise removal via Bayesian wavelet coring”, IEEE Sig. Proc. Society, Third Int’l Conf. On Image Proc. Vol.1, pp379-382, Lausanne, 1996. E.P. Simoncelli, “Bayesian Denoising of Visual Image in the Wavelet Domain”, pp291-308. ~01.141, Springer-Verlag, New York, 1999. A. Ben Hamza and Krim, “Image Denoising: A Nonlinear Robust Statistical Approach”, IEEE Tran. On Signal Proc. Vo1.49, no.12, pp3045-3054,2001. M. Popescu, P. Cristea, and A. Bezerianos, “Multiresolution distributed filtering: A novel technique that reduces the amount of data required in hiah resolution - electrocardiography,” Future Gen. Comp. Sys., no. 15, pp. 195-209. 1999. M. Shao and C. L. Nikias, “Signal processing with fractional lower order moments: Stable processes and their applications”, Proceeding of the IEEE, ~01.81, no. 7, pp986- 1010, July 1993. lmola K. Fodor and Chandrika Kamath, “Denoising Through Wavelet Shrinkage: An Empirical Study”, Lawrence Livermore national Laboratory technical report, UCRL-JC- 144258, July 27,2001. Y. XU, J.B. Weaver, D.M. Healy, and I. Lu, “Wavelet transform domain filtration technique”, IEEE Transactions on Image Processing, vol. 3,747-758, 1994. D.L. Donoho and I.M. Johnstone, “Wavelet shrinkage: Asymptopia?”, Journal of the Royal Statistical Society, Series B, vol. 57,301-369, 1995. D. L. Donoho, “Nonlinear wavelet methods for recovery of signals, densities, and spectra from indirect and noisy data”, In Proceeding of Symposia in Applied Mathematic, American Mathematical Society vol. 00, 173-205, 1993. A. Achim, A. Bezerianos, and P. Tsakalides, “An Aplha- stable based Bayesian algorithm for speckle noise removal in the wavelet domain,” Proc. NSIP-01, June 03-06, 2001, Baltimore, Maryland USA. D. L. Donoho and I. M. Johnstone, “Ideal spatial adaptation by wavelet shrinkage”, Biometrika, Also Tech. Report of Statistics, July, 1992, revised April 1993. G. P. Nason, “Wavelet regression by cross-validation”, Technical Report 447, Stanford University, Statistics Department, March 1994. X. Huang and G. A. Woolsey, “Image denoising using Wiener filtering and wavelet threshold”, IEEE International Conference, ICME 2000, Proc. Vol. 3 pp1759-1762,2000. X. Huang, “Denoising Image via Minimum Variance Bound Bayesian Estimator,” IEEE International Conf. ICME2001, Proc., pp533-536,2001. X. Huang and A. C. Madoc, “Maximum Likelihood for Bayesian Estimator Based on a-stable for Image”, IEEE lntemational Conf. ICME2002, Roc. Vol.1, pp709-712, 2002. D. L. Donoho, “Denoising by soft-thresholding,” IEEE Trans. Inform. Theory, vol. 41, pp613-627, May 1995. S. R. Peterson, Y. H. Lee, S. A. Kassam, “Some statistical properties of alpha-trimmed mean and standard type M filters,” lEEE on Acoust., Speech, Sig. Proc., vol. 36, no.5, 00707-713, 1988. B. W. Brorsen and S. R. Yang, “Maximum likelihood estimates of symmetric stable distribution parameters”, Commun. Statist.-Simuul., vol. 19, pp1459-1464, 1990. I - 596 

Wavelet-based Bayesian Estimator for Poisson Noise Removal from Images

https://researchsystem.canberra.edu.au/ws/files/29151054/fulltext_published_2_.pdf

Wavelet-based Bayesian Estimator for Poisson Noise Removal from Images

Abstract

Similar works

Full text

Available Versions

Crossref

University of Canberra Research Repository