Pattern Recognition Letters 34 (2013) 1175–1185

Contents lists available at SciVerse ScienceDirect

Pattern Recognition Letters

journal homepage: www.elsevier .com/locate /patrec

Restoration of Poisson noise corrupted digital images with nonlinear PDEbased filters along with the choice of regularization parameter estimation

Rajeev Srivastava a,⇑, Subodh Srivastava b

a Department of Computer Engineering, Indian Institute of Technology (BHU), Varanasi, Indiab School of Biomedical Engineering, Indian Institute of Technology (BHU), Varanasi, India

a r t i c l e i n f o a b s t r a c t

Article history:Received 7 January 2013Available online 6 April 2013

Communicated by M.A. Girolami

Keywords:Image restorationPDE based filtersVariational frameworkRegularization functionsRegularization parameter selectionGeneralized cross validations

0167-8655/$ - see front matter � 2013 Elsevier B.V. Ahttp://dx.doi.org/10.1016/j.patrec.2013.03.026

⇑ Corresponding author. Tel.: +91 5426702754.E-mail addresses: rajeev_sri@yahoo.com, rajeev.cse

subohresonance@gmail.com (S. Srivastava).

In this paper, the reconstruction of three nonlinear partial differential equation (PDE) based filtersadapted to Poisson noise statistics have been proposed in a variational framework for restoration andenhancement of digital images corrupted with Poisson noise. The proposed and examined PDE based fil-ters include total variation adapted to Poisson noise in L-1 framework; anisotropic diffusion; and com-plex diffusion based methods adapted to Poisson noise in L-2 framework. The resulting filters containtwo terms namely data fidelity and regularization or smoothing function. The data fidelity term is Poissonlikelihood term and the regularization functions are PDE based filters. Other choices for the regularizationfunctions have also been presented. The two terms in the proposed filters are coupled with a regulariza-tion parameter lambda which makes a proper balance between the two terms during the filtering pro-cess. The choice of method for estimation of regularization parameter lambda plays an important role.In this study, the various regularization parameter estimation methods for Poisson noise have also beenpresented and their suitability has been examined. The resulting optimization problems are furtherinvestigated for efficient implementation for large scale problems. For estimating the regularizationparameter, three choices are considered for Poisson noise case which are discrepancy principles, general-ized cross validations (GCV), and unbiased predictive risk estimate (UPRE). GCV and UPRE functions arefurther other optimization problems in addition to main image reconstruction problem. For minimizingthe GCV and UPRE functions, the methods of Conjugate Gradients (CG) is used. For digital implementa-tions, all schemes have been discretized using finite difference scheme. The comparative analysis ofthe proposed methods are presented in terms of relative norm error, improvement in SNR, MSE, PSNR,CP and MSSIM for an adaptive value of regularization parameter calculated by every methods in consid-eration. Finally, from the obtained results it is observed that the anisotropic diffusion based methodadapted to Poisson noise gives better results in comparison to other methods in consideration along withchoice of GCV for regularization parameter selection.

� 2013 Elsevier B.V. All rights reserved.

1. Introduction

In various image processing applications such as astronomicalimaging, medical imaging and fluorescence microscopy, the imageintensity is often measured by the counting of incident photonsemitted by the object of interest and in these cases the image datanoise is modeled by a Poisson distribution. To restore the imagescorrupted by Poisson noise, the Poisson maximum likelihood esti-mation can be used. Further, regularization may be needed whenthe underlying model equation for Poisson maximum likelihoodestimation is ill-posed.

ll rights reserved.

@iitbhu.ac.in (R. Srivastava),

If g is the observed blurred and noisy N � N image array, thenwe can obtain an estimate of the N � N true object array f byapproximately solving a linear system of the form [4,19,34]:

g ¼ Af þ b ð1Þ

The general form of image restoration process reads

g ¼ FðAf þ bÞ ð2Þ

where F(.) is the noise generation process; g is column stacked vec-tor of size N2 � 1; A is the known/unknown point spread function(PSF) which is often ill conditioned matrix of size N2 � N2; b isknown positive background intensity of the image being viewed;and Af is nonnegative for all f P 0. If A i.e. PSF is unknown then itbecomes a blind restoration problem which can be estimated usinggeneralized cross validations (GCV).

1176 R. Srivastava, S. Srivastava / Pattern Recognition Letters 34 (2013) 1175–1185

In fluorescence microscopy and other imaging applications suchas medical imaging and astronomical imaging, the elements ofnoisy image g belongs to noisy photon counts which followsPoisson distribution. The statistical model that models the errorin these systems reads

g ¼ PoissonðAf þ bÞ ¼ PoissonðzÞ: ð3Þ

where Poisson(Af + b) is an independent and identically distributed(i.i.d.) Poisson random vector with Poisson parameter vector z. Theprobability distribution function (pdf) of data uo corrupted by Pois-son noise [20,34] reads

pðf=gÞ ¼Yn

i¼1

ð½Af �i þ biÞðgÞi � e�ð½Af �iþbiÞ

g!ð4Þ

In general, if we substitute uo = g and u � Af + b, where uo is theobserved image and u is the estimated/restored image thenPoisson pdf reads

pðu=uoÞ ¼e�u � uuo

uo!ð5Þ

For the image data uo being generated from the model describedby Eq. (3), whose pdf is described by Eq. (5), the maximum likeli-hood estimate of u is obtained by maximizing p(u/uo) with respectto u, subject to the constraint u P 0. Alternatively, we can calculatethe maximum likelihood estimate [20,34–37] of u by minimizingthe negative log likelihood of Poisson pdf given by

uML ¼ arg minuP0

f� ln pðu=uoÞg ¼ arg minuP0

fu� uo ln ug ð6Þ

The maximum likelihood estimate of u assuming Poisson noisereads [34]

uML ¼ðuo � uÞ

uð7Þ

If PSF A is ill conditioned, the solutions of Eq. (6) given by Eq. (7) canbe noise corrupted and further regularization may be required. Fordoing so, the ensuing optimization problem can be formulatedwithin a Bayesian settings using maximum a posterior (MAP) ap-proach [23] or by casting the problem in a variational frameworkas a minimization problem [34] to obtain the solution or best esti-mate of u.

2. General framework for image restoration

2.1. MAP approach to image restoration

The proposed filter can be derived using a maximum a posterior(MAP) approach to the image reconstruction problem. In particu-lar, in Bayesian settings, a probability density p(u) for u is specifiedand the posterior density is maintained w.r.t. u as [23]:

pðu=uoÞ ¼pðuo=uÞpðuÞ

pðuoÞ: ð8Þ

The maximizer of the posterior density function p(u/uo) is called themaximum a posterior (MAP) estimator. Alternatively, given an ini-tial noisy image uo, then we reconstruct the filtered image u thatmaximizes the log-posterior probability

logðpðu=uoÞÞa logðpðuo=uÞÞ þ log pðuÞ ð9Þ

where p(u/uo) is the likelihood term of noise model and p(u) is theprior. The formulation of the filtering problem as maximization of aposterior is useful because it allows one to incorporate the Poissonlikelihood term as a data attachment which can be added to an im-age prior model. In this paper, the Gibb’s image prior model is used.The Gibbs prior model is based on the energy functional, which isdefined in terms of the gradient norm of the image, related to

anisotropic diffusion based PDE [24] for additive noise removal.The Gibb’s prior model reads [25,26]

pðuÞ ¼ expð�kEðuÞÞ ð10Þ

where energy functional [16] is defined as

EðuÞ ¼ arg min kZ

X/ðkrukÞdX

� �ð11Þ

If /(kruk) = kruk2 is defined as gradient norm of image [24]then the corresponding energy function that defines the prior reads

EðuÞ ¼ arg min kZ

Xkruk2dX

� �ð12Þ

2.2. Minimization framework

Maximizing p(u/uo) given by Eq. (8) is equivalent to minimizing[19]:

TðuÞ ¼ T0ðu=uoÞ � ln pðuÞ ð13Þ

The first term T0(u/uo) is the likelihood function that gives max-imum likelihood estimate of u and the second function �lnp(u) isthe regularization term from classical inverse problems, wherep(u) is the prior from which the unknown u is assumed to arise.

The regularization functions can be considered as negative logprior defined as:

� ln pðuÞ ¼ k2hCu;ui ð14Þ

where h.,.i denotes the Euclidean inner product, C is the symmetricpositive semi definite regularization function, and k is the regulari-zation parameter that controls the magnitudes of Eigen valuesof the covariance and it maintains a balance between fidelitymeasured by likelihood function of noise and data regularization.

Therefore, image restoration problem can be cast into followingminimization problem [19]:

uk ¼ arg minuP0

TkðuÞ ¼ T0ðu=uoÞ þk2hCu;ui

� �ð15Þ

The first term T0(u/uo) measures the fidelity given by Eq. (6) andfor Poisson noise it is ku � uo ln uk; second term is a regularizationfunction depending on the choice of C; k is the regularizationparameter that controls the degree of smoothing or regularization;and uk is the obtained solution i.e. the processed or de-noised im-age obtained by minimizing Eq. (15).

If noise in image is additive having Gaussian distributions only,then T0(u/uo) = ku � uok2 which is a popular least squares estima-tion of Gaussian Likelihood used in standard Tikhonov regulariza-tion. The classical minimization problem utilizing least squaresestimation was first introduced in 1923 by Whittaker [1] and ithas been extensively studied ever since Wahba [2]. This techniqueconsists in minimizing a criterion that balances the fidelity to thedata, measured by residual sum-of-squares (RSS), and a penalty/regularization function that reflects the roughness of the smootheddata. In this section of the paper, the classical techniques have beenmodified in a variational framework for Poisson noise.

2.2.1. Some possible choices of regularization function C

a. If C= Square integral of the pth derivative of u: The penalizedregression/regularization function is known as a smoothingspline [3,5,6].

b. If C=In (Identity Matrix): Tikhonov Regularization, L2-norm

R. Srivastava, S. Srivastava / Pattern Recognition Letters 34 (2013) 1175–1185 1177

The minimization problem given by Eq. (15) reduces to Tikonovregularization which is in L2 framework and penalizes reconstruc-tions with large L2-norm.

The Tikhonov regularization is obtained by minimizingfollowing:

uk ¼ arg minuP0

TkðuÞ ¼ ðu� uo ln uÞ þ k2juj2

� �ð15:1Þ

a. If Cu= kruk : Total Variation (TV), L1 normThe second term in minimization problem given by Eq. (15)

reduces to TV regularization [27] which is in L1 framework andpenalizes reconstructions with L1-norm.

The TV penalized Poisson maximum likelihood estimation isobtained by minimizing following:

uk ¼ arg minuP0

fTkðuÞ ¼ ðu� uo ln uÞ þ kjrujg ð15:2Þ

a. If Cu=r2u: Laplacian, L2 frameworkThe Laplacian penalized Poisson maximum likelihood estimation is

obtained by minimizing following:

uk ¼ arg minuP0

fTkðuÞ ¼ ðu� uo ln uÞ þ kjr2ujg ð15:3Þ

a. If Cu=kruk2: Anisotropic diffusion, L2 normThe second term in minimization problem given by Eq. (15)

reduces to anisotropic diffusion regularization [24] which is inL2 framework which after modification may also be converted tononlinear complex diffusion [20] regularization.

The anisotropic diffusion penalized Poisson maximum likelihoodestimation is obtained by minimizing following:

uk ¼ arg minuP0

TkðuÞ ¼ ðu� uo ln uÞ þ k2jruj2

� �ð15:4Þ

a. If Cu=kr2uk2: Fourth order PDEThe second term in minimization problem given by Eq. (15)

reduces to Fourth order PDE [28] regularization.The fourth order PDE penalized Poisson maximum likelihood

estimation is obtained by minimizing following:

uk ¼ arg minuP0

TkðuÞ ¼ ðu� uo ln uÞ þ k2r2u��� ���2� �

ð15:5Þ

a. C = kDuk2 = second order divided difference of u [1,7], whereD is a tri-digonal matrix as defined in paper [7]. In a recentwork presented in [8], this classical penalty function hasbeen modified and solved by discrete cosine transforms(DCT) that yields a solution in an efficient manner for largescale problems. This method has been developed forGaussian noise. For Poisson noise, the resulting model reads

uk ¼ arg minuP0

TkðuÞ ¼ ðu� uo ln uÞ þ k2kDuk2

� �ð15:6Þ

3. Variational approach to image reconstruction problem

In variational framework [19,34], the minimization problemgiven by Eq. (15) can be expressed as:

uk ¼ arg minuP0

EðuÞ ¼Z

XðT0ðu=uoÞ þ

k2hCu;uiÞdX; ð16Þ

where X denotes image space.

Alternatively, Eq. (16) can be rewritten as:

uk ¼ arg minuP0

EðuÞ ¼Z

XðT0ðu=uoÞ � ln pðuÞÞdX ð17Þ

If Gibbs prior given by Eq. (10) is used then Eq. (17) reduces to:

uk ¼ arg minuP0

EðuÞ ¼Z

XðT0ðu=uoÞ þ kEðuÞÞdX ð18Þ

In solution space, the first term measures the fidelity andsecond term is a regularization function depending on the choiceof C or E(u).

For Poisson pdf the first term in Eq. (18) reads (see Eq. (6)):

T0ðu=uoÞ ¼ u� uo ln u ð19Þ

In variational framework, regularization function E(u) is definedas some function of gradient norm i.e.

EðuÞ ¼ /ðjrujÞ ð20Þ

3.1. Total variation (TV) based method adapted to Poisson noise (L1framework)

The TV regularization approach was first proposed by Rudinet al. presented in [27] to de-noise an image corrupted with addi-tive white Gaussian noise. This model regards u as the solution to avariational problem that minimizes a functional which reads:

EðuÞ ¼Z

Xjruj þ k

2ju0 � uj2

� �dX ð21Þ

Using Euler–Lagrange minimization technique and gradientdescent approach the solution is given by following equation afterdiscretization by finite differences scheme:

@u@t¼ div ru

jruj

� �þ kjðuo � uÞj;with

@u

@ n!¼ 0 on @X: ð22Þ

The proposed method for Poisson noise is described as follows:In this subsection, TV model [27] given by Eq. (22) has been

modified for Poisson noise. In variational framework, the minimi-zation problem for image restoration is given by Eq. (18). TheTV-norm is the L1 functional norm of the gradient magnitude.

In TV-framework the regularization function E(u) = /(kruk) isdefined as:

/ðjrujÞ ¼ jruj ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2

x þ u2y

q; ðL1 normÞ: ð23Þ

In discrete case, TV is defined as:

TVðuÞ ¼ jruj ¼Xn

j;k¼1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijðruÞxj;kj

2 þ jðruÞyj;kj2

q; ð24Þ

where ðruÞxj;k ¼ ujþ1;k � uj;k; for j < n ¼ 0 for j ¼ n ð25Þ

and

ðruÞyj;k ¼ uj;kþ1 � uj;k; for k < n ¼ 0 for k ¼ n ð26Þ

The proposed TV based model reads

arg minuP0

EðuÞ ¼Z

Xððu� uo ln uÞ þ kjrujÞdX ð27Þ

The functional E(u) is defined on the set of u e BV(X) such thatlogu e L1(X) and u must be positive everywhere.

After applying Euler–Lagrange minimization technique, theEq. (27) reads

0 ¼ div rujruj

� �þ ðuo � uÞ

ku; with

@u

@ n!¼ 0 on @X: ð28Þ

1178 R. Srivastava, S. Srivastava / Pattern Recognition Letters 34 (2013) 1175–1185

Using gradient descent to solve Eq. (27), the solution is obtainedby

@u@t¼ div ru

jruj

� �þ 1

k� ðuo � uÞ

u; with

@u

@ n!¼ 0 on @X: ð29Þ

For numerical implementations, the derivatives can be discret-ized using standard centered difference approximations and thequantity |ru| is replaced with

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijruj2 þ eps

qfor some small posi-

tive value of eps such as 0.00000000001. The value of eps can be as-signed to lowest machine number to avoid divide by zeroconditions during implementations.

Therefore, the TV-based filter adapted to Poisson noise reads:

@u@t¼ div ruffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

jruj2 þ epsq

0B@

1CAþ 1

k� ðuo � uÞ

u; with

@u

@ n!

¼ 0 on @X: ð30aÞ

with the initial condition as uðt ¼ 0Þ ¼ uo ð31Þ

3.2. Anisotropic diffusion based method adapted to Poisson noise

In Anisotropic diffusion based framework for Poisson noise, theregularization function E(u) = /(kruk) is defined as [24]:

/ðkrukÞ ¼ kruk2; ðL2 normÞ; ð32Þ

Therefore, the proposed TV based model reads

arg minuP0

EðuÞ ¼Z

Xððu� uo ln uÞ þ kkruk2ÞdX ð33Þ

The functional E(u) is defined on the set of u e BV(X) such thatlogu e L1(X) and u must be positive everywhere.

After applying Euler–Lagrange minimization technique, theEq. (33) reads

0 ¼ r � ðcðkrukÞruÞ þ ðuo � uÞku

; with@u

@ n!¼ 0 on @X ð34Þ

Using gradient descent to solve Eq. (34), the solution is obtainedby

@u@t¼ r � ðcðkrukÞruÞ þ 1

k� ðuo � uÞ

u; with

@u

@ n!

¼ 0 on @X: ð35Þ

where the diffusion coefficient c(kruk) is defined as [24],

cðkrukÞ ¼ 1

1þ ðruk Þ

2 ð36Þ

Therefore, the proposed anisotropic diffusion based modeladapted to Poisson noise reads

@u@t¼ r � ðcðkrukÞruÞ þ 1

k� ðuo � uÞ

u; with

@u

@ n!

¼ 0 on @X: ð37aÞ

with the initial condition being the observed noisy image i.e.

uðt ¼ 0Þ ¼ uo ð37bÞ

The second term in Eq. (37a), which is first derivative of log like-lihood of Poisson probability distribution function (pdf) with re-spect to estimated image, acts as the data attachment term andmeasures the dissimilarities at a pixel between observed imageand its estimated value obtained during filtering process there bymaking the whole filtering process adapted to noise. The first termis responsible for regularization and smoothing of the image databy minimizing the variance of pixels.

For the estimation of diffusion coefficient given by Eq. (36)which is used in Eq. (37a), the adaptive value of k is proposed tobe determined [29] using the tools from robust statistics to auto-matically estimate the robust scale re of an image I. The value ofk is set to re which is minimum absolute deviation (MAD) of thegradient of an image. The adaptive value of k is estimated as:

k ¼ re ¼ 1:4826�medianu kru�medianuðkrukÞk½ �: ð38Þ

3.3. The complex diffusion based model

If, in first term of the anisotropic diffusion based model given byEq. (37a), real time factor t is replaced by the complex time factor itand the diffusion coefficient c(|ru|) by c(Im(u)) then it leads tofollowing nonlinear complex diffusion [30] based model adaptedto Poisson noise:

@u@t¼ divðcðImðuÞÞruÞ þ 1

k� ðuo � uÞ

uð39aÞ

with the initial condition

uðt ¼ 0Þ ¼ uo ð39bÞ

The diffusion coefficient c(Im(I)) is defined as follows [30]:

cðImðuÞÞ ¼ eih

1þ ImðuÞkh

� 2 ð40Þ

Here, k is edge threshold parameter and the value of k ranges from 1to 1.5 for digital images [30]. For nonlinear complex diffusion pro-cess defined by Eqs. (39a) and (39b) the evolution of real part of theimage is controlled by the linear forward diffusion, whereas evolu-tion of imaginary part of the image is controlled by both the realand imaginary equations. A qualitative property of edge detectioni.e. second smoothed derivative is described by the imaginary partof the image for small value of h, whereas real values depict theproperties of ordinary Gaussian scale -space. For large values of h,the imaginary part feeds back into the real part creating the wavelike ringing effect which is an undesirable property. Here, for exper-imentation purposes the value of h is chosen to be p

30 :

The adaptive value of edge threshold parameter is used inEq. (40). It is defined as negative exponential distribution [31]:

kt � k0 expð�atÞ ð41Þ

where a and k0 are constants, usually 1.

3.4. Discretization of the proposed models

For digital implementations, the Eqs. (30–31), (37a), (37b), and(39a), (39b) can be discretized using finite differences schemes[32]. For example, the discretized form of the proposed anisotropicdiffusion based model, given by Eqs. (37a), (37b), reads

unþ1 ¼ un þ Dt � 1k� ðuo � unÞ

unþr � ðcðkrunkÞrunÞ

�ð42aÞ

uðt ¼ 0Þ ¼ uo ð42bÞ

For the numerical scheme, given by Eq. (42a) and disretized ver-sions of Eqs. (30–31), (39a), (39b), and (42a), (42b) to be stable,the von Neumann analysis [32], shows that we require Dt

ðDxÞ2< 1/4.

If the grid size is set to Dx = 1, then Dt < 1=4 i.e. Dt < 0.25. Therefore,the value of Dt is set to 0.25 for stability of Eqs. (42a), (42b) andsimilarly others. In the similar fashion, the TV based model givenby Eqs. (30–31) and nonlinear complex diffusion model given byEqs. (39a), (39b) can be discretized using finite difference scheme.

R. Srivastava, S. Srivastava / Pattern Recognition Letters 34 (2013) 1175–1185 1179

4. Estimation of the regularization parameter k

The regularization parameter k, used in Eqs. (30), (37a) and(39a) are computed dynamically from the information present inthe image. Initially it was set to 0 and in successive iterations ofthe proposed PDE based model the specific value of k is calculated.The regularization parameter k is responsible for making a fair bal-ance between the two terms which are maximum likelihood ofPoisson pdf and regularization or penalty function in the proposedmodels.

The one of the simplest method for estimation of regularizationparameter is defined in terms of the inverse of average SNR com-puted from mean and standard deviation of the image in specificiterations. The average SNR is defined as follows:

Avg:SNR ¼ meanðuðtÞÞstd:dev iationðuðtÞÞ ¼

lffiffiffiffiffiffir2

n

p ð43Þ

k ¼ 1Avg:SNR

ð44Þ

The small value of SNR denotes that more noise is present in im-age whereas the high value of SNR denotes that the image is of bet-ter quality and contains less noise in comparison to its previousnoisier version. Therefore, for small SNR value, the value of k in-creases allowing more weight to the second term in the proposedmodel which is responsible for regularization or smoothing of data.For high SNR value, the value of k decreases allowing less weight tothe second term in the proposed model which is responsible forregularization or smoothing of data.

The various other efficient methods available in literature forestimation of regularization parameter include discrepancy princi-ples (DP) [9], generalized cross validations (GCV) criterion [10–13],the unbiased predictive risk estimate (UPRE) [14], L-curve criterion[15], and others such as presented in [16]. These methods are gen-erally developed for additive Gaussian noise case in minimum leastsquare sense and for small scale problems. The GCV method is verypopular method available in literature. For large scale problems,such as imaging applications, some authors have proposed meth-ods for its efficient implementations such as use of randomizedtrace estimation in GCV [11].

Methods for choosing regularization parameter can be dividedinto two groups depending upon their assumption about residueerror kek2 i.e. norm of the perturbation of the right sidekAuk � gk2 for the restoration model g = Au + g where g is the ob-served image, u is the true object, A is the PSF, and g is the additiveGaussian noise. The corresponding standard optimization problemi.e. Tikhonov regularization is as follows:

uk ¼ arg minuP0

kAuk � gk22 þ

k2

2kLuk2

2

( ): ð45Þ

The methods of first group for choice of regularization parame-ter are based on knowledge, or good estimate, of kek2 .The methodsof second group for choice of regularization parameter don’t re-quire kek2, but rather seek to extract the necessary informationfrom the given right hand side i.e. kAuk � gk2. Example of methodsin first group includes estimation of regularization parameter usingdiscrepancy principles [9] and examples of methods in secondgroup includes estimation of regularization parameter using gen-eralized cross validations (GCV) [10–13], L-curve criterion [5],unbiased predictive risk estimate (UPRE) [14], and many otherssuch as parameter-choice rule based on the normalized cumulativeperiodogram which is particularly suited for large-scale problems[16]. From literature survey, it has been found that GCV and UPREmethods from second group can provide good estimate for large

scale problems and due to this reason these two methods viz.GCV and UPRE have been examined, modified, and used in this pa-per for estimating regularization parameter.

The brief descriptions of these methods for additive Gaussiannoise case are as follows:

4.1. Discrepancy principles [9]

This method amounts to choosing the regularization parametersuch that the residual norm for the regularized solution satisfies

kAuk � gk2 ¼ kek2: ð46Þ

When a good estimate for kek2 is known, this method yields agood regularization parameter corresponding to a regularizedsolution.

4.2. Generalized cross validations (GCV) [10–13]

GCV is based on the philosophy that if an arbitrary element gi ofthe right hand side g is left out, then the corresponding regularizedsolution should predict this observation as well, and the choice ofregularization parameter should be independent of an orthogonaltransformation of g. This leads to choosing the regularizationparameter which minimizes the following GCV function [13]:

GCVðkÞ ¼ kAuk � gk22

½traceðIm � AAIÞ�2ð47Þ

where AI is a matrix which produces the regularized solution uk

when multiplied with g i.e. uk ¼ AIg. The denominator can be com-puted in O(n) time if Lanczos bi-diagonalization algorithm is used[17,18]. Alternatively, filter factors can be used to evaluate thedenominator [13]. In the case of large scale least square estimationproblems, an efficient method for approximating the GCV whenrandomized trace estimation is presented in [11].

4.3. Unbiased predictive risk estimate (UPRE) [14]

For the estimation of regularization parameter k using UPRE, weseek the value of k that minimizes the predictive risk:

EðT0ðuk=geÞ: ð48Þ

In this section of paper, the modified form of three methodsnamely discrepancy principles (DP) from first group; and general-ized cross validations (GCV) criterion, and the unbiased predictiverisk estimate (UPRE) from second group for Poisson noise case havebeen used [19] and efficient implementation methods such as ran-domized trace estimations have been used for large scaleproblems.

5. Regularization parameter estimation for Poisson noise

For developing the methods for regularization parameter k forthe minimization problem given by Eqs. (15), (16) a weighted leastsquares approximation of Poisson likelihood T0(u/uo) is used. Exist-ing methods for estimating the regularization parameter k in thestandard least square case can’t be applied directly to the problemin consideration because here data is Poisson distribute and stan-dard least square case is applicable for Gaussian distributed data.For the case of Poisson distributed data, a Taylor’s series argumentcan be used to obtain a quadratic approximation of T0(u/uo). Usingthis approximation, existing regularization parameter selectionmethods can be applied to Eqs. (15), (16). For simplicity ofrepresentation let z = uo then T0(u/uo) is rewritten as T0(u/z) .

1180 R. Srivastava, S. Srivastava / Pattern Recognition Letters 34 (2013) 1175–1185

In Sections 5.1, 5.2, And 5.3 brief descriptions for regularizationparameter estimation techniques as described and formulated bythe authors of the paper [9,19] are presented.

5.1. A weighted least square approximation of T0(u/uo) i.e. T0(u/z): [19]

In paper [19], the authors have studied and provided a weightedleast square approximation of T0(u/uo). Let ue be the exact objectand ze = Aue + b the background shifted exact data. Then, lettingk = z � ze and h = u � ue and expanding T0 in a Taylor series aboutue and ze, we have

T0ðu=zÞ ¼ T0ðueþh=zeþkÞ ) T0ðu=zÞ

¼ T0ðue=zeÞ þ kTrzT0ðue=zeÞ þ12

hTr2uuT0ðue=zeÞh

þ 12

hTr2uzT0ðue=zeÞkþ

12

kTr2zuT0ðue=zeÞhþ � � � ð49Þ

After solving for the gradient and Hessian of T0 and second or-der mixed partial derivatives of T0 w.r.t. u to be used in Taylor ser-ies Eq. (49), as given in paper [19], and substituting in Eq. (49),following approximation of T0 is obtained:

T0ðu=zÞ ¼ T0ðue=zÞ þ 12ðAu� ðz� bÞÞT diag

1ze

� �ðAu� ðz

� bÞÞ ð50Þ

Therefore, the quadratic Taylor series approximation of T0(u/z)about the points (ue/ze) is as follows:

Twls0 ðu=zÞ ¼ 1

2ðAu� ðz� bÞÞT diag

1ze

� �ðAu� ðz� bÞÞ ð51Þ

The Eq. (51) can be re-written as follows:

Twls0 ðu=zÞ ¼ 1

2kZ�1=2ðAu� ðz� bÞÞk2 ð52Þ

where Z = diag(z)Therefore,

T0ðu=zÞ ¼ T0ðue=zÞ þ Twls0 ðu=zÞ þ � � � ð53Þ

The least square form of Eq. (53) provides method for extendingthe standard methods of the regularized parameter estimationtechniques.

5.2. Determination of regularization parameter k using discrepancyprinciples

From Eq. (53), the expected value function can be defined as[9,19]:

EðT0ðu=zÞÞ � T0ðue=zeÞ þ EðTwls0 ðu=zÞÞ ð54Þ

where E is the expected value function. For discrepancy principle,the acceptable value of k will be those for which the weighted leastsquare value Twls

0 ðuk=zÞ is equal to the expected value function ofTwls

0 ðue=zÞ i.e.

Twls0 ðuk=zÞ � EðTwls

0 ðue=zÞÞ ð55Þ

A regularization parameter k is one which solves followingexpression [19]:

kDiscrp ¼ arg minuPo

ð2Twls0 ðuk; zÞ � nÞ ð56Þ

where uk is computed from:

Twls0 ðuk; zÞ ¼

12kZ�1=2ðAu� ðz� bÞÞk ð57Þ

and Zk ¼ diagðAuk þ bÞ ð58Þ

5.3. Determination of k using generalized cross validations (GCV)

The method of leave one out cross validation for regularizationparameter selection applied to Eq. (15) presented in [19,20] isdiscussed as follows:

Let

ukk ¼ arg min

uP0

Xi–k

ð½Au�i þ bÞ � zi lnð½Au�i þ bÞ þ k2hu;Cui

� �( )ð59Þ

Then parameter k is chosen that minimizes following:

CVðkÞ ¼ 1n

Xn

k¼1

fð½Aukk�k þ bÞ � zk lnð½Auk

k�k þ bÞg ð60Þ

GCV functional whose minimum score gives the optimal regu-larization parameter k is defined as [10,13,19]:

GCVðkÞ ¼ nTwlso ðuk=zÞ

½traceðIn � Z�1=2AAkÞ�2ð61Þ

where Ak is a matrix satisfying:

uk ¼ AkZ�1=2ðz� bÞ ð62Þ

However, for Eq. (15), the regularization operator is nonlinear,hence Ak should be a linear approximation satisfying Eq. (61). To de-rive such an approximation, we define Dk to be a diagonal matrixwith diagonal entries ½Dk�ii ¼ 1 if ½uk�i > 0 and ½Dk�ii ¼ 0 otherwise.Under this condition, the Tk is strictly convex function and uk willbe the minimum norm solution of [4,21]:

DkrTkðDku=zÞ ¼ 0 ð63Þ

After minimizing Eq. (63), we get following minimum normsolution for Ak:

Ak ¼ ðDkðAT Z�1Aþ kCÞDkÞ�1DkAT Z�1=2 ð64Þ

Now for further approximation of GCV functional given by Eq. (61),we approximate denominator for efficient implementation of GCVfor large scale problems. Since the size of matrices Ak; Dk, and Zare large, its direct implementation will not be efficient. Hence hereTrace Lemma [14] can be used to approximate the denominator.The Trace Lemma is as follows:

Given P 2 Rn�n; then v � Nð0; InÞ implies EðvT PvÞ¼ traceðPÞ ð65Þ

Hence for a given realization v from N(0, In),

traceðIn � Z�1=2AAkÞ � vTv � vT Z�1=2AAkv ð66Þ

Therefore, the GCV function for choosing k in Eq. (15) is definedas follows:

Choose the value of regularization parameter k that minimizes[19]

GCVðkÞ ¼ nTwlso ðuk=zÞ

½vTv � vT Z�1=2AAkv�2ð67Þ

For efficient implementation of Eq. (67) for large scale prob-lems, Akv is approximated by applying a truncated conjugate gra-dient iteration with x0 = 0 to

DkðAT Z�1Aþ kCÞDkx ¼ DkAT Z�1=2v ð68Þ

Another efficient method for GCV implementation usingrandomized trace estimation for large scale problems is presentedin [11].

R. Srivastava, S. Srivastava / Pattern Recognition Letters 34 (2013) 1175–1185 1181

5.4. Determination of k using UPRE

We choose the regularization parameter k that minimizesfollowing UPRE function [14,19]:

UPREðkÞ ¼ Twls0 ðuk; zÞ þ traceðZ�1=2AAkÞ �

n2

ð69Þ

As in the case of GCV, traceðZ�1=2AAkÞ must be approximatedusing the Trace Lemma and a truncated conjugate gradientiteration.

6. Results, performance comparison and discussions

In this section, the performance of the proposed three methodsviz. TV adapted to Poisson noise; anisotropic diffusion basedscheme adapted to Poisson noise; nonlinear complex diffusionbased method adapted to Poisson noise and other standard modi-fied method i.e. Tikhonov regularization have been evaluated bothqualitatively and quantitatively. The comparative study of the per-formances of various proposed scheme are presented in terms ofrelative norm error, improvement in signal-to-noise ratio (SNR),mean squared error (MSE), peak signal-to-noise ratio (PSNR), cor-relation parameter (CP) and mean structure similarity index map(MSSIM) [33].

The relative error norm is defined as [19]:

kue � ukkkuek

ð70Þ

For experimentation purposes the noisy images are generatedusing MATLAB poissrnd and randn functions and the values of back-ground count b was chosen to be 10 and standard deviation ofGaussian noise was chosen to be 5. The performances of allschemes in consideration have been compared in terms of MSE,PSNR, CP and MSSIM for the sample image lena.jpg of size200 � 200. It has been tested through experimentation that after100 iterations the proposed methods in consideration convergeto the desired level of solution i.e. produces acceptable quality ofde-noised images. After 100 iterations the PSNR values of the pro-posed schemes start decreasing. Therefore total numbers of itera-tions were fixed to 100 for all the three proposed methods. Thevalue of Dt was set to 0.25 for stability reasons. The initial condi-tion of the proposed PDE based filter is the noisy image corrupted

Table 1Performance comparison of image restoration methods for the various choices of regulariValidation, UPRE, and general method (Sample image Lena.jpg 200 � 200, number of iteratbackground count = 10, SNRnoisy = 2.7428).

Restoration Regularization parameter (lambda) Value of lambda SNRr

1 DiscrepancyGCVUPRE1/Avg.SNR

0.000635211.4073e-0066.6032e-0050.36442

2.802.802.802.80

2 DiscrepancyGCVUPRE1/Avg.SNR

0.1463170.03888910.03440480.034606

2.842.782.782.89

3 DiscrepancyGCVUPRE1/Avg.SNR

0.9928310.275110.2442880.35718

2.832.792.782.80

4 DiscrepancyGCVUPRE1/Avg.SNR

0.143050.0698710.0699720.34861

2.842.822.822.86

Methods: 1. Tikhonov regularization.2. TV regularization.3. Nonlinear complex diffusion based method (k = exp(�t)).4. Anisotropic diffusion based method.

with Poisson noise and the final solution is the restored image. Theinitial value of the dynamic regularization parameter kð0Þ is set tozero and the value of k was calculated during every iteration of theproposed PDE-based models by discrepancy principles, GCV, andUPRE methods as presented in this paper. All schemes have beenimplemented using MATLAB 7.0 software. The value of thresholdparameter k used in the diffusion coefficient C(kruk) given byEq. (36) was evaluated adaptively by Eq. (38). For nonlinear com-plex diffusion based scheme, the value of k used by diffusion coef-ficient, Eq. (40), was determined adaptively by Eq. (41).

The results for one test case are shown in this section thoughthe performances of all schemes in considerations were evaluatedfor several other test images and the performance trend remainedthe same. The test sample is the digital image lena.jpg of size200 � 20. Table 1 lists the performance measures for the sampleimage for the proposed schemes and Tikhonov regularization. InTable 1, the four methods for which the performances were evalu-ated and presented include the standard Tikhonov regularizationmethod (first method), the TV regularization (second method),nonlinear complex diffusion based method (third method), andanisotropic diffusion based method (fourth method) all modifiedfor the Poisson noise case for the three standard choices of regular-ization parameter estimation viz. discrepancy principle, GCV, andUPRE along with other simple method. Out of the three proposedrestoration methods and another modified Tikhonov regularizationbased method, the fourth method i.e. the anisotropic diffusionbased method is performing better in comparison to others in con-sideration when the regularization parameter estimation methodis discrepancy principles and GCV methods.

Fig. 1 shows relative solution error comparison of different res-toration methods for various choices of regularization parameterestimation viz. discrepancy principles, UPRE and GCV and it canbe easily observed that the minimum relative solution error isassociated with the proposed anisotropic diffusion based regulari-zation method for discrepancy principle closely followed by GCVmethod for regularization parameter estimation.

Fig. 2 shows PSNR comparison of different restoration methodsfor various choices of regularization parameter estimation meth-ods and the proposed anisotropic diffusion based method is associ-ated with largest PSNR value for discrepancy principle followed byGCV method, for regularization parameter estimation, indicatingbetter restoration quality of the Poisson noise corrupted image.

zation parameter (lambda) calculated using Discrepancy principle, Generalized Crossions for Methods 1,2, & 3 = 100, Sigma (std deviation) for Gaussian noise = 5, Poisson

est RMSE PSNR CP MSSIM Relative soln error

46252437

38.109538.104538.095838.0655

16.510116.511316.513316.5202

0.95320.95260.95270.9526

0.58150.57890.57840.5794

0.2883250.2882870.2882050.287985

93944201

13.621714.589214.759013.6258

25.446224.850224.749725.4435

0.96880.95910.95810.9720

0.72220.67180.66710.7360

0.1030570.1103770.1116620.103089

41146101

14.360614.740314.838414.5685

24.987324.760724.703124.8625

0.96260.95820.95720.9596

0.68520.66470.65940.6717

0.1115200.1165200.1122630.110805

00604389

13.452313.513213.534213.6173

25.554925.515625.502225.4490

0.97060.96870.96880.9715

0.74770.73380.73420.7490

0.1017760.1022370.1023960.102535

1 2 3 40

0.1

0.2

0.3

0.4Relative Error:Discrepancy

Restoration Methods

Rel

ativ

e E

rror

1 2 3 40

0.1

0.2

0.3

0.4Relative Error: GCV

Restoration Methods

Rel

ativ

e E

rror

1 2 3 40

0.1

0.2

0.3

0.4Relative Error:UPRE

Restoration Methods

Rel

ativ

e E

rror

Fig. 1. Relative error comparison of different restoration methods for various choices of regularization parameter.

1182 R. Srivastava, S. Srivastava / Pattern Recognition Letters 34 (2013) 1175–1185

Fig. 3 shows MSSIM comparison of different restorationmethods for different choices of regularization parameter and theproposed anisotropic diffusion based method is associated withlargest MSSIM value for discrepancy principle followed by GCVmethod, for regularization parameter estimation, indicating betterstructural property preservation in the reconstructed image.

Fig. 4 shows relative norm error and SNR comparison of differ-ent restoration methods using GCV based regularization parameterestimation technique and the anisotropic diffusion method isperforming better in comparison to other methods in

1 2 3 40

10

20

30 PSNR Comparison:Discrepancy

Restoration Methods

PS

NR

Dis

c

1 20

10

20

30 PSNR Compa

Restoration

PS

NR

upre

Fig. 2. PSNR comparison of different restoration metho

consideration. It may be noted, as discussed earlier, that the GCVis well suited for large scale problems.

Fig. 5 shows performance comparison of different restorationmethods when regularization parameter lambda is estimatedusing GCV and the anisotropic diffusion method is performing bet-ter in comparison to other methods in consideration for all perfor-mance metrics.

Fig. 6 shows comparisons of visual results of different restora-tion methods using GCV based regularization parameterestimation.

1 2 3 40

10

20

30 PSNR Comparison:GCV

Restoration Methods

PS

NR

gcv

3 4

rison:UPRE

Methods

ds for various choices of regularization parameter.

1 2 3 40

0.2

0.4

0.6

0.8 MSSIM Comparison:Discrepancy

Restoration Methods

MS

SIM

disc

1 2 3 40

0.2

0.4

0.6

0.8MSSIM Comparison:GCV

Restoration Methods

MS

SIM

gcv

1 2 3 40

0.2

0.4

0.6

0.8 MSSIM Comparison:UPRE

Restoration Methods

MS

SIM

upre

Fig. 3. MSSIM comparison of different restoration methods for different choices of regularization parameter.

1 2 3 40

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Restoration Methods

Rel

ativ

e E

rror

1 2 3 40

0.5

1

1.5

2

2.5

3

Restoration Methods

SN

Rgc

v

Fig. 4. Relative norm error and SNR comparison of different restoration methods using GCV.

R. Srivastava, S. Srivastava / Pattern Recognition Letters 34 (2013) 1175–1185 1183

Therefore, from the results obtained it can be concluded thatthe proposed anisotropic diffusion based model adapted to Poissonnoise statistics is a better choice for reduction of the intrinsic Pois-son noise from digital images. Further, the anisotropic diffusionbased filter adapted to Poisson noise statistics, also preserves theedges and other radiometric information such as luminance andcontrast of the image as shown by CP and MSSIM comparisons.

For the regularization parameter choice, the discrepancymethod is performing better for the small scale problems andwhen we already have the prior information about thresholderror.

Further, the GCV based method produces better result for thelarge scale problems and also when we do not have prior informa-tion about threshold error.

1 2 3 40

10

20

30

40

Restoration Methods

RM

SE

gcv

1 2 3 40

10

20

30

Restoration Methods

PS

NR

gcv

1 2 3 40

0.2

0.4

0.6

0.8

1

Restoration Methods

CP

gcv

1 2 3 40

0.2

0.4

0.6

0.8

Restoration MethodsM

SS

IMgc

v

Fig. 5. Performance comparison of different restoration methods when regularization parameter lambda is estimated using GCV.

Poisson Noised Image Tikhonov

Total Variation Complex Diffn based

Aniso.Diffn based

Comparison of Different Methods using GCV

Fig. 6. Comparison of visual results of different restoration methods using GCVbased regularization parameter estimation.

1184 R. Srivastava, S. Srivastava / Pattern Recognition Letters 34 (2013) 1175–1185

7. Conclusions

In this paper, three PDE based nonlinear filters adapted toPoisson noise statistics have been proposed in L-1 and L-2 frame-works for restoration and enhancement of Poisson noise corruptedimages. The proposed methods include total variation (TV) basedmethod adapted to Poisson likelihood in L-1 norm optimizationframework; anisotropic and nonlinear complex diffusion basedmethods adapted to Poisson noise in L-2 norm framework. Theresulting minimization problems include two terms coupled witha regularization parameter k: The first term being the maximumlikelihood of Poisson pdf which measures the data fidelity and

second being the regularization function responsible for producingsmooth version of image. The various other possible choices for theregularization functions were also presented. The regularizationparameter makes a fair balance between data fidelity and regular-ization function. The various choices for estimation of regulariza-tion parameter k using discrepancy principles, GCV, and UPREmethods for Poisson noise case were examined and used in imple-mentation, in addition to the simple method to estimate the regu-larization parameter defined as reciprocal of average SNR. Throughexperimentations, the performances of regularization parameterestimation for both cases were examined where the first case beingthe small scale problem with prior information about the thresholderror, and the second case being the large scale problems withoutany prior information about the threshold error. For the first case,discrepancy method is performing better and for the second caseGCV is giving better results.

For digital implementations, all schemes were discretized usingfinite difference schemes. All methods were implemented usingMATLAB 7.0 software. Comparative analysis of the three proposedmethods and standard Tikhonov regularization method are pre-sented for all choices of regularization parameter estimated withdifferent methods. Performances of all schemes were measuredin terms of relative norm error, SNR, RMSE, PSNR, CP, and MSSIM.From the results obtained it can be concluded that anisotropic dif-fusion based method adapted to Poisson noise is performing betterin comparison to other methods in consideration and is well capa-ble of reconstructing the good quality of image with good edgepreservation and radiometric quality. Regarding the choice for reg-ularization parameter estimation, the discrepancy principle is per-forming better for small scale problems and the GCV based methodis producing better result for large scale problems.

References

Whittaker, E.T., ]. On a new method of graduation. Proc. Edinburg Math. Soc. 41, 62–75.

Wahba, G., ]. Spline Models for Observational Data. SIAM, Philadelphia.Whaba, G., ]. Estimating the smoothing parameter. In: Smith, D. (Ed.), Spline Models

for Observational Data. SIAM, Philadelphia, pp. 45–65.Bardsley, J.M., ]. Stopping rules for non-negativity constrained iterative method for

ill-posed Poisson imaging problems. BIT Numer. Math. 48 (4), 651–664.

R. Srivastava, S. Srivastava / Pattern Recognition Letters 34 (2013) 1175–1185 1185

I.J. Schoenberg, Spline functions and the problem of graduation, in: Proceedings ofthe National Academy of Sciences of the United States of America, vol. 52, 1964,pp. 947–950.

Takezawa, K., ]. Introduction to Nonparametric Regression. John Wiley and Sons,Inc., Hoboken, NJ.

Weinert, H.L., ]. Efficient computation for Whittaker–Henderson smoothing.Comput. Stat. Data Anal. 52, 959–974.

Garcia, D., ]. Robust smoothing of gridded data in one and higher dimensions withmissing values. Comput. Stat. Data Anal. 54, 1167–1178.

Morozov, V.A., ]. On the solution of functional equations by the method ofregularization. Sov. Math. Dokl. 7, 414–417.

Golub, G., Heath, M.T., Wahba, G., 0]. Generalized cross-validation as a method forchoosing a good ridge parameter. Technometrics 2 (1), 215–223.

Golub, G., Von Matt, V., 1]. Generalized cross validations for large scale problems. J.Comput. Graphics Stat. 6 (1), 1–34.

Reeves, S.J., 2]. A cross-validation framework for solving imaging restorationproblems. J. Visual Commun. Image Represent. 3, 433–445.

Nikolas, P., Galatsanos, Katsaggelos, Aggelos K., 3]. Methods for choosing theregularization parameter and estimating the noise variance in imagerestoration and their relation. IEEE Trans. Image Process. 1 (3), 322–336.

Vogel, C.R., 4]. Computational Methods for Inverse Problems. SIAM, Philadelphia.Hansen, P.C., O’leary, D.P., 5]. The use of L-curve in regularization of discrete ill-

posed problems. SIAM J. Sci. Comput. 50 (6), 1487–1503.Hansen, P.C., Kilmer, M.E., Kjeldsen, R.H., 6]. Exploiting residual information in the

parameter choice for discrete ill-posed problems. BIT Numer. Math. 46, 41–59.Cullum, Willoughby, Lanczos algorithms for large symmetric eigenvalue

computations, vol. 1, ISBN 0-8176-3058-9, USA.http://en.wikipedia.org/wiki/Lanczos_algorithm.Bardsley, J.M., Goldes, John, 9]. Regularization parameter selection methods for ill-

posed Poisson maximum likelihood estimation. Inverse Prob. 25 (9).Bardsley, J.M., Laobeul, N., 0]. Tikhonov regularized Poisson likelihood estimation:

Theoretical justification and a computational method. Inverse Prob. Sci. Eng. 16(2), 199–215.

Nocedal, J., Fessler, J.A., 1]. Numerical Optimization. Springer-Verlag, New York.

Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed., NewYork, 1991.

Perona, P., Malik, J., 4]. Scale-space and edge detection using anisotropic diffusion.IEEE Trans. Pattern Anal. Mach. Intell. 12, 629–639.

Jain, A.K., 5]. Fundamentals of Digital Image Processing. PHI, India.Sonka, M., Hlavac, V., Boyle, R., 6]. Image Processing, Analysis and Machine Vision,

2nd ed. PWS Publishing, Washington, D.C..Rudin, L., Osher, S., Fatemi, E., 7]. Non linear total variation based noise removal

algorithms. Physica D 60, 259–268.You, Y.L., Kaveh, M., 8]. Fourth – order partial differential equations for noise

removal. IEEE Trans. Image Process. 9, 1723–1730.Voci, Francesco, Eiho, Shigeru, Sugimoto, Naozo, Sekiguchi, Hiroyuki, 9]. Estimating

the gradient threshold in Perona-Malik equation. IEEE Signal Process. Mag., 39–46.

Gilboa, G., Sochen, N., Zeevi, Y.Y., 0]. Image enhancement and denoising by complexdiffusion processes. IEEE Trans. Pattern Anal. Mach. Intell. 25 (8), 1020–1036.

Yongjian, Yu, Scott Acton, T., 1]. Speckle reducing anisotropic diffusion. IEEE Trans.Image Process. 11 (11), 1260–1270.

Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P., 2]. Numerical Recipes inC: The Art of Scientific Computing, second ed. Cambridge University Press, ISBN0-521-43108-5.

Srivastava, R., Gupta, J.R.P., Parthasarthy, H., 3]. Comparison of PDE based and othertechniques for speckle reduction from digitally reconstructed holographicimages. Opt. Lasers Eng. 48, 626–635.

Le, Triet, Chartrand, Rick, 4]. A variational approach to reconstructing imagescorrupted by Poisson noise. J. Math. Imaging Vision 27 (3), 257–263.

Lane, R.G., 5]. Methods for maximum-likelihood deconvolution. J. Opt. Soc. Am. A(Issue 13), 1992–1998.

Llacer, J., Nu~nez, J., 6]. Iterative maximum-likelihood and Bayesian algorithms forimage reconstruction in astronomy. In: White, R.L., Allen, R.J. (Eds.), Restorationof Hubble Space Telescope Images. Space Telescope Science Institute, Baltimore,MD, pp. 62–69.

Shepp, L.A., Vardi, Y., 7]. Maximum likelihood reconstruction for emissiontomography. IEEE Trans. Med. Imaging 6, 113–122.