14-Bayesian Wavelet-Based Image Denoising Using the Gauss–Hermite Expansion.pdf

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IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 17, NO. 10, OCTOBER 2008 1755

Bayesian Wavelet-Based Image Denoising Using theGaussHermite Expansion

S. M. Mahbubur Rahman, M. Omair Ahmad, Fellow, IEEE, and M. N. S. Swamy, Fellow, IEEE

AbstractThe probability density functions (PDFs) of thewavelet coefficients play a key role in many wavelet-based imageprocessing algorithms, such as denoising. The conventional PDFsusually have a limited number of parameters that are calculatedfrom the first few moments only. Consequently, such PDFs cannotbe made to fit very well with the empirical PDF of the waveletcoefficients of an image. As a result, the shrinkage functionutilizing any of these density functions provides a substandarddenoising performance. In order for the probabilistic model of theimage wavelet coefficients to be able to incorporate an appropriatenumber of parameters that are dependent on the higher ordermoments, a PDF using a series expansion in terms of the Hermite

polynomials that are orthogonal with respect to the standardGaussian weight function, is introduced. A modification in theseries function is introduced so that only a finite number of termscan be used to model the image wavelet coefficients, ensuring atthe same time the resulting PDF to be non-negative. It is shownthat the proposed PDF matches the empirical one better thansome of the standard ones, such as the generalized Gaussian orBessel K-form PDF. A Bayesian image denoising technique is thenproposed, wherein the new PDF is exploited to statistically modelthe subband as well as the local neighboring image wavelet coef-ficients. Experimental results on several test images demonstratethat the proposed denoising method, both in the subband-adaptiveand locally adaptive conditions, provides a performance betterthan that of most of the methods that use PDFs with limitednumber of parameters.

Index TermsBayesian image denoising, GaussHermite,higher order moments, orthogonal series expansion, probabilitydensity function, wavelet coefficient.

I. INTRODUCTION

DENOISING of images corrupted by additive white

Gaussian noise (AWGN) is a classical problem in

signal-processing. The distortion of images by noise is

common during its acquisition, processing, compression,

storage, transmission, and reproduction. The aim of denoising

is to remove the noise while keeping the signal features as muchas possible. Traditional algorithms perform image denoising

Manuscript receivedNovember 9, 2007; revisedMay 5, 2008. Current versionpublished September 10, 2008. This work was supported in part by the NaturalSciences and Engineering Research Council (NSERC) of Canada and in partby the Regroupement Stratgique en Microlectronique du Qubec (ReSMiQ).The associate editor coordinating the review of this manuscript and approvingit for publication was Dr. Michael Elad.

The authors are with the Center for Signal Processing and Communications,Department of Electrical and Computer Engineering, Concordia University,Montral, QC, H3G 1M8 Canada (e-mail: [email protected];[email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TIP.2008.2002163

in the pixel domain. However, the application of the wavelet

transform in image denoising, compression, etc. has shown

remarkable success over the last decade. Wavelet shrinkage

method proposed by Donoho [1][3] is the pioneering work

for signal denoising using the wavelet transform. The method

described in [3] provides a mini-max optimal solution. Another

criterion commonly used is the mean squared error (MSE),

even though it does not match well with the characteristics of

the human visual system [4]. However, since it is a measure of

the noise power [5], and is mathematically convenient in the

context of optimization, it is employed as a denoising criterionin many image denoising applications. The discrete wavelet

transform (DWT) used in image denoising can be of many

types, such as orthogonal/biorthogonal, real/complex-valued,

separable/nonseparable, or decimated/nondecimated. Due

to the shift-invariance property, the nondecimated separable

DWT [6] improves the image denoising performance in MSE

approximately by 1 dB as compared to that of the decimated

representation [7][10]. To reduce the computational load of

the nondecimated wavelet transform as well as the Gibbs arti-

facts, cycle-spinning has been proposed [11]. Image denoising

by using transforms with improved directional selectivity,

such as the dual-tree complex wavelet transform (DT-CWT)[12], ridgelet [13], curvelet [14], contourlet [15], and steerable

pyramid [16], [17] have also been introduced.

There are two major approaches for statistical wavelet-based

denoising [18], [19]. The first approach is to design a sta-

tistically optimal threshold parameter for some nonlinear

thresholding or shrinkage functions. Image denoising methods

using such an approach include the VisuShrink [2], SureShrink

[3], BayesShrink [20], amplitude-scale-invariant Bayes esti-

mator [21], NeighCoeff [22], Spatial-correlation thresholding

[23], and empirical Bayes thresholding (EBayesThresh) [24].

Standard choices for nonlinear thresholding or shrinkage

functions are soft- and hard-thresholding [1], firm-shrinkage

[25], and non-negative garrote shrinkage [26]. Methods usingsuch shrinkage functions are computationally simple, but have

certain drawbacks in view of the fact that the functions repre-

senting the nonlinearity are arbitrarily chosen. For example,

the soft-thresholding technique yields a biased estimate with

a moderate variance, whereas the hard-thresholding technique

yields a less biased estimate but with a higher variance [26]. In

the second approach, instead of using an arbitrary function to

represent the nonlinearity, the shrinkage function is designed

by minimizing a Bayesian risk, typically under the minimum

mean squared error (MMSE) criterion [27], minimum mean

absolute error criterion [28] or maximum a posteriori (MAP)

criterion [29], [30]. The development of an optimal threshold

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1756 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 17, NO. 10, OCTOBER 2008

or shrinkage function is usually based on the probability den-

sity function (PDF) or simply the prior function of the image

wavelet coefficients. By exploiting the nature of the histogram

of the wavelet coefficients, different noise reduction algorithms

try to incorporate the prior model that best matches the true

image coefficients. The marginal PDF of a true image and its

self-similar subband wavelet coefficients are unimodal andsharply peaked in nature [31], [32]. Therefore, higher order

statistics are necessary to describe the sharp central cusp of

the density function of the image coefficients. The parameters

of a PDF and their estimation play a significant role in the

Bayesian denoising technique, since the computational com-

plexity of a subband-adaptive Bayesian denoising is ,

being the number of parameters of the PDF and

the size of the image. Subsequent to the work of Mallat [33],

the generalized Gaussian (GG) distribution with the shape

parameter has been commonly used to model the

image wavelet coefficients [8][10], [20], [27], [34][36]. For

example, the threshold factor in the BayesShrink method [20]

and the parametersignal of interestin the ProbShrink method[10] assume that the wavelet coefficients in each subband are

independent and identically distributed (i.i.d.) random variables

having a GG distribution. Under the simplest prior GG with

, and the i.i.d. assumption, both the MMSE and MAP

estimators provide the same closed-form solution, the well

known Wiener filter. A general class of prior functions used

for the purpose of denoising is the Gaussian scale mixture

(GSM), wherein the wavelet coefficients are modelled as a

zero-mean Gaussian function multiplied by random positive

state variables [7], [16], [37][42]. For a simple two-state GSM

prior, an expectation maximization (EM) algorithm can be used

to estimate the multipliers [7], [41]. In [43], the EM algorithmhas been extended for a finite number of state variables and

it has been shown that four to eight state variables of the

GSM density function are sufficient for a good approximation

of the prior function of the subband wavelet coefficients of

images. In general, the estimation of the GSM prior requires

another PDF, such as the log-normal density function [40]

or Jeffreys noninformative prior [16], to estimate the set of

multipliers. In order to obtain a Bayes estimator for image

denoising, Figueiredo and Nowak [21] have used Jeffreys

prior itself as a statistical model for the wavelet coefficients,

since this prior is free of any parameter. In [44] and [45], a

class of heavy-tailed densities, namely, the -stable family with

characteristic exponent has been proposed as a

prior of the image coefficients. This PDF has no closed-form

expression even though its characteristic function has one, and

its parameter estimation is poor, specially in the presence of

noise [43]. Another family of sharply peaked PDFs, called the

Bessel K-form (BK), introduced to model the filtered image in

[32], has been applied in the wavelet-domain MMSE estimator

by Fadili and Boubchir [46]. The wavelet coefficients of a

log-transformed image (for example, synthetic aperture radar

or medical ultrasound image) are modelled by the normal

inverse Gaussian PDF for removing the multiplicative noise

using a MAP estimator in the homomorphic approach [47],

[48]. The parameters of a PDF can be estimated using severalstrategies, such as the method of moments [49], maximum

likelihood method [50], and entropy-matching method [51].

Since the image wavelet coefficients of a subband can be treated

as a data set having a large sample size, the moment-based

parameter estimation provides a consistent result [52]. In order

to capture the higher order statistics, most of the PDFs used in

modeling the image wavelet coefficients use afinite number of

parameters that are usually derived from the second and fourthorder moments of the image coefficients. As a result, in most

of the cases these density functions become much more lep-

tokurtic than the empirical one. Since the wavelet coefficients

are locally stationary [53], the parameters of some of the PDFs

are refined with respect to the local spatial context within a sub-

band [8], [16], [20], [22], [40], [54][57]. In such a case, when

the denoising technique is locally adaptive within a subband,

the computational complexity increases to ,

being the size of a squared-shape local window. A few of the

PDFs have also been extended to the bivariate [29], [58] [60]

and multivariate cases [36], [42] or refined in the hidden

Markov tree framework [7], [41] to take into consideration the

intersubband dependency. In such a case, the computationalload of a denoising technique increases exponentially with the

number of subbands, and this is substantially higher than the

corresponding locally adaptive method.

In this paper, a novel marginal statistical model is proposed

for the wavelet coefficients of images by using the GaussHer-

mite (GH) expansion. This expansion is used in view of the fact

that it allows higher order moments to be incorporated in the

probabilistic modeling of the wavelet coefficients. A method of

choosing a finite number of terms in the GH series such that

the resulting PDF matches well with the empirical PDF of the

wavelet coefficients of the images and at the same time remains

non-negative, is proposed. A closed-from expression is derivedfor the parameters of the modified PDF in terms of the higher

order sample-moments. The proposed PDF is then applied as

a prior function in the subband-adaptive and locally adaptive

Bayesian MMSE-based noise reduction algorithms, in view of

the fact that they offer lower computational burden as com-

pared to the method using intersubband dependency. It is shown

that the introduction of the more accurate prior in the proposed

noise reduction technique results in a performance better than

that of the subband-adaptive and locally adaptive wavelet-based

image denoising algorithms that use standard PDFs having lim-

ited number of parameters, in terms of both the visual perception

and standard indices.

The paper is organized as follows. In Section II, the modifica-

tion of the GH expansion in the context of statistical modeling

of the wavelet coefficients of images is presented. A comparison

with some of the standard PDFs on a number of test images from

a database is also given. A Bayesian image denoising method

that exploits the proposed PDF is described in Section III. In

Section IV, the performance of the method is evaluated and

compared with some of the existing denoising methods. Finally,

Section V gives some conclusions.

II. GAUSSHERMITE EXPANSION FOR PDF OF IMAGE

WAVELET COEFFICIENTS

This section introduces a probabilistic model for the crit-ically sampled real DWT coefficients as well as for the real


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RAHMANet al.: BAYESIAN WAVELET-BASED IMAGE DENOISING USING THE GAUSSHERMITE EXPANSION 1757

TABLE I

WEIGHTFUNCTIONS ANDREGIONS OFSUPPORT FOR THE SERIESEXPANSIONSOFSOME ORTHOGONAL POLYNOMIALS

and imaginary components of the DT-CWT coefficients of im-

ages. To derive the PDF of the image wavelet coefficients, we

use a series expansion so that the derived PDF enables one to

use an arbitrary number of parameters that are dependent on

higher order moments. In practice, the weight functions for the

various expansions are represented in terms of standard PDFs.

Table I lists a few orthogonal polynomials along with their cor-

responding weight functions and regions of support. Among

the various possible series expansions, we choose the one that

uses the Hermite polynomials, which are orthogonal with re-

spect to the standard Gaussian function. There are two major

reasons for such a choice. First, this expansion is valid for the

image wavelet coefficients that do not have any compact sup-

port. Second, the Gaussian weight function resembles the em-

pirical PDF, and hence there is a high likelihood that the ex-

panded PDF would have a better match with the empirical one

[61]. There are some other advantages in using the Hermite se-

ries expansion: it has a uniform mean square convergence [62],

the parameters can be expressed in terms of higher order mo-

ments in closed-form (as seen later in this section), and the poly-

nomials can be estimated recursively.

Let represent a wavelet coefficient at spatial locationof a subband in the decomposition level with a transfor-

mation matrix applied on the true image. Let represent the

vector containing all the coefficients of a subband, wherein each

coefficient is a sample of that subband. Let represent

the PDF of the random variable . This PDF may be expressed

in terms of the Hermite polynomials, which are orthogonal with

respect to a Gaussian weight function, in the form [61]

(1)

where is the standard deviation of and is the Her-mite polynomial of order that satisfies the recurrance relation

[63]

(2)

The resulting PDF will be called as the GH PDF. A closed-form

expression of the series coefficients can be obtained in terms

of the th order moment as (see Appendix A)

(3)

where denotes the largest integer contained in . The se-

ries coefficients are determined only by the even-order mo-

ments for even , and the odd-order moments for odd . Hence,

the parameter estimation of the GH PDF becomes relatively

simple and straight forward. Although the PDF of the image

coefficients described in (1) is an infinite series expression, in

practice, only thefirst few terms are used for a reasonably goodapproximation. For example, if the highest value of order in

(1) is chosen as 4, the resulting density function reduces to the

GramCharlier (GC) PDF [57], [64], [65]

(4)

where and , respectively, are the

skewness and kurtosis of . However, the approximation of the

right side of (1) by only three terms may introduce some er-rors in , and the error can be reduced by including more

number of terms in the series. In order to include a larger number

of terms in the series, higher order moments are necessary for

estimating the series coefficients . But, the deviation between

the estimated sample-moments and the true-moments is not in-

significant when the order becomes high [64]. Therefore, the

error introduced by the estimation of is not negligible when

a large number of terms of the series is included. In addition,

the decay rate of the density function is proportional

to or simply for suf-

ficiently large . The larger the order , the slower the decay of, thus making it a challenge to fit the tail part of

with that of the empirical PDF of the image coefficients. As-

suming the largest value of in (1) to be , the right side of (1)

may introduce real zeros in the expansion of , the max-

imum number of such zeros being . In essence, the PDF defined

by the truncated GH series may not be always non-negative. For

example, the symmetric (i.e., the skewness is zero) GC PDF is

non-negative only if [66].

Table II shows the average values of the magnitude of the

skewness and the kurtosis of the DWT coefficients in thefirst

three levels for some test images. It can be observed that the

skewness of the image wavelet coefficients is insignificant. It

has also been observed that other odd-order moments of interest

are also insignificant. Hence, the PDF of the image coefficients

can be treated as symmetric. As a result, all in the infinite

series of (1) for odd disappear and the truncated GH PDF for

the image wavelet coefficients can be written as

(5)

The kurtosis of the subband wavelet coefficients of images isusually more than 7 (see Table II); therefore, cannot


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model these coefficients sufficiently well. In addition, each sub-

band of the image wavelet coefficients usually has a sufficiently

large number of data samples, and hence, the higher order mo-

ments would be useful for modeling. Recall that the truncated

GH PDF with a sufficiently large value of is not ensured to be

non-negative. However, the model of the subband coefficients

defined by the truncated GH PDF in (5) is an even function,and therefore, the real zeros are symmetrically situated around

the mode. Fig. 1 shows the statistical modeling of the HL1band

DWT coefficients by the empirical PDF and by the truncated GH

PDF with varying from 4 to 14 for the imagesLenaand Bar-

bara. The empirical PDF is obtained from the histogram using

a nonparametric density estimation technique, wherein the bin

width of the histogram plays a significant role. If the bin width is

too small, then the estimated PDF suffers from undersmoothing,

i.e., the density function has a small bias but a large variability.

On the other hand, if it is too large, then the estimated PDF suf-

fers from oversmoothing, i.e., the density function has a small

variability but a large bias. In order to obtain a tradeoff be-

tween the bias and variance, an appropriate choice of the binwidth is necessary. The bin width can be fixed or variable in

size. Variable width may be better, when the density function

is asymmetric and multimodal in nature. Since the PDF of the

image wavelet coefficients is unimodal and symmetric in na-

ture, afixed bin width is chosen for the histogram. The value of

this bin width is chosen to be , being the

interquartile range1 and the length of the data, since such a

choice asymptotically minimizes the mean integrated squared

error of the empirical PDF that is non-Gaussian [67]. It can be

observed from Fig. 1 that, as increases, the kurtotic behavior

of the truncated GH PDF approaches that of the empirical one,

even though it results in zeros in the tail part of the PDF. Letbe the two zeros closest to the mode for a particular value

of . The value of may be found by using any rootfinding

routine, but we calculate it as

(6)

where is a unit step function and is logicalequal

operation. It is evident from Fig. 1 that when the coefficients lie

in the interval , then t hese c oef ficients can be modelled

well by the proposed GH PDF with an appropriate choice of .

Outside this range, i.e., in the tail part, such a truncated GH PDF

introduces negative probability, which is not possible. Hence, inthe tail part of the truncated GH PDF, a modification is necessary

1The IQR is the difference between the third and first quartiles of the data.

so that the probability becomes non-negative and at the same

time the modified GH PDF fits well with the empirical PDF.

For such a modification, the use of a GH PDF is preferred to

that of using a different PDF, since the estimated parameters

and Hermite polynomials can also be used for the tail part of the

PDF. Examples of the GH PDFs with non-negative probability

are the Gaussian PDF and the symmetric GC PDF having. Since the latter has a longer tail than the former and

the PDFs of the image wavelet coefficients are known to have

long-tails, we choose the GC PDF as the modified tail. In order

to ensure that the total area under the curve of the modified GH

PDF is unity, we multiply the GC PDF by a weight parameter

. Henceforth, this density function will be referred to as the

modified GH (MGH) PDF. It can be mathematically formulated

as (7), shown at the bottom of the page, where is the

weight parameter, and . Since the

total area under the MGH PDF is unity, the parameter satisfies

(8)

After some manipulations, the value of can be obtained as (see

Appendix B)

(9)

where

(10)

and represents the error function. In summary, the MGH

PDF for a particular subband is estimated using the following

steps.1) Select as the highest-order even moment. Estimate the

moments .

if

if

(7)


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Fig. 1. Probability density function of theHL1 band DWT coefficients in terms of the GH series expansion with increasing value of . Test images are (a)Lenaand (b) Barbara.

TABLE IIAVERAGEVALUES OF THE MAGNITUDE OF THESKEWNESS

AND THE

KURTOSIS

OF THEDWT COEFFICIENTS FORSOMETESTIMAGES

2) Estimate the series parameters by

using (3).

3) Select the bin width . Set the dis-

crete points , where and

, respectively, are the minimum and maximum values

of .

4) Estimate the Hermite polynomials

by using (2) for each .

5) Estimate using (5) and (6).

6) Estimate using (9) and (10).7) Obtain the MGH PDF using (7).

The proposed MGH PDF requires the estimation of ,

and , and the computational complexities of these are

, and , respectively, where is

total number of discrete points resulting from the selected bin.

Since and , the computational complexity

of the proposed MGH PDF is . Fig. 2 shows the

modeling of theHL1band DWT coefficients by the MGH PDF

in the linear-scale using different values of for theLena and

Barbara images. It can be observed from this figure that the

MGH PDF is ensured to be non-negative and for a sufficiently

large value of , the tail part of the MGH PDF matches very

well with that of the empirical PDF. Therefore, in the proposedMGH PDF the choice of is of importance, since it determines

the highest-order of the Hermite polynomials and of the mo-

ments used in the parameter estimation. In order to make a good

choice of , the MSE between the empirical and proposed MGH

PDFs is plotted as a function of the highest-order moments used

in (7) for theHL1andHL2 band DWT coefficients for some test

images, and shown in Fig. 3. It can be observed from thisfigure

that the inclusion of an increased number of higher order mo-

ments in the parameter estimation of the proposed MGH PDF

lowers the MSE, and a value of 12, 14, or 16 is a good choice

for . In the log-scale, Fig. 4 shows the modeling performanceof three PDFs, namely, the GG, the BK and the MGH that

uses for theHL1 band DWT coefficients of the Lena

and Barbara images. The closeness of the fit of the proposed

PDF with the empirical one, both in its peak and tail parts, is

obvious as compared to the others. Moreover, to quantify the

performance of the PDFs, we use, the Kolmogorov-Smirnov

distance (KSD), , and the Kull-

back-Leibler divergence (KLD), ,

as the two metrics. Table III shows the results concerning

the metrics KSD and KLD for the GG, BK, and MGH PDFs

of the first and second level image DWT coefficients. The

metrics are calculated by averaging the results obtained using96 images, each of size 512 512, from the database given in

[68]. Standard deviations of the KLD metric are also included

in the table. For the results shown here, the MGH PDF uses

the even-order moments from 2 up to 14, whereas the GG and

BK PDFs use only the second and fourth order moments in

view of their limited number of parameters. From the table, it

is evident that in most of the cases the MGH PDF, compared to

the GG and BK PDFs, has the lowest distance and divergence

from the empirical one. It should be mentioned that similar

kind of results have been obtained from the prior modeling of

the DT-CWT coefficients and are not included here for lack

of space. This high degree of matching between the empirical

and MGH PDFs is mainly due to the fact that the proposedPDF can incorporate more number of higher order moments


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Fig. 2. Modified GH probability density function of the HL1 band DWT coefficients with increasing value of . Test images are (a)Lenaand (b)Barbara.

Fig. 3. Mean squared error between the empirical and the modified GH probability density functions for the DWT coefficients as a function of the highest-orderof the moments that has been used for the parameter estimation. The subbands are (a)HL1 and (b)HL2.

in its parameter estimation process as compared to the others.This feature, along with the modification proposed through the

introduction of the parameter , results in the proposed MGH

PDF to match the empirical PDF very well both in the peak and

tail segments. In order to demonstrate the effectiveness of the

proposed PDF, we next consider the case of Bayesian denoising

with the MGH prior function.

III. DENOISINGALGORITHM

In this section, we assume that the image pixels are corrupted

by AWGN with a known variance . If is unknown, it may

be estimated by applying the median-absolute-deviation method

[3] in the highest frequency subband of the noisy wavelet coeffi-cients. Since the noise is uncorrelated with the true image signal,

the wavelet coefficient of the noisy image at the spatial locationof a subband is given by

(11)

where is the wavelet coefficient of the additive noise at

the reference location. We propose an MMSE-based denoising,

wherein the Bayes-risk estimator uses the proposed MGH PDF.

In a Bayesian framework, , , and in (11) are consid-

ered as samples of the independent random variables , , and

, respectively. The signal component is modelled as the i.i.d.

MGH random variable with subband-adaptive parameters. The

random variable of the noise component is modelled as a zero-

mean Gaussian random variable with a subband-adaptive vari-ance , which depends on , , and the arithmeticoperations


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Fig. 4. Modified GH probability density function in log-scale for the HL1band DWT coefficients. Hermite polynomials are up to the fourteenth order. Test imagesare (a)Lena, where

and (b)Barbara, where

.

TABLE IIIVALUES OF THE METRICSKSD AND KLD FOR THE PRIOR FUNCTION

MODELING OF THE IMAGEDWT COEFFICIENTS. T HE AVERAGEVALUES OFKLD (STANDARDDEVIATIONS INPARENTHESES) AND KSD ARE OBTAINED

FROM A96 IMAGE DATABASE

of the corresponding wavelet-domain. In this case, we assume

that is known; however, if is unknown, it may be esti-

mated using the method given in [16]. The Bayes-risk estimator

for the denoised wavelet coefficient under a quadratic cost

function that minimizes the MSE is the conditional mean given

by

(12)

where is the posterior density function conditioned

on the subband coefficients under consideration. According to

the Bayesrule, the prior function of the image wavelet

coefficients can be used to compute the posterior density func-

tion as

(13)

where is the density function of the wavelet coeffi-

cients corresponding to the noise component. It can be seen

from (12) and (13) that the MMSE-based shrinkage functionis independent of the weight parameter that has been

estimated in Section II. An analytical expression of the pro-

posed MMSE-based shrinkage function can be obtained as

in (14), shown at the bottom of the page (see Appendix C

for proof), where ,

, and is the parabolic

cylinder function [69]. It is to be noted that for ,

and . The shrinkage function

given by (14) may be directly used for estimating each of

the noisy wavelet coefficients. The computational complexity

of this function is, where is the

computational complexity of . Since the computational

load increases exponentially with , an implementation of such

(14)


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a function may not be practical in view of the excessive com-

putational effort involved. Hence, we represent the shrinkage

function in terms of a linear convolution as

(15)

where is the convolution operator. Taking the computa-

tional advantage of the discrete Fourier transform (DFT), the

shrinkage function can be expressed as

(16)

where is the forward DFT and is the inverse DFT. Using

a fast implementation of the DFT, the complexity of computing

the shrinkage function can be reduced to .The function given in (16) provides the estimates of the

noisy wavelet coefficients at the discrete points , where the

probabilities of the MGH PDF are estimated. Therefore, an

interpolation function is required to estimate each of the noisy

wavelet coefficients. In our case, we choose the cubic Hermite

spline [70] as an interpolation function so that the estimated

Hermite polynomials for the MGH PDF can also be used

for interpolation. This spline function requires the estimation

of four parameters and has a computational complexity of

. Since and , the compu-

tational complexity of the shrinkage function given by (16) for

subband-adaptive denoising is . From simulationresults, we have observed that the denoising performances

using the two shrinkage functions, given by (14) and (16), are

essentially the same. Hence, we prefer to use (16) for the pro-

posed Bayesian denoising in view of its lower computational

complexity.

The proposed MMSE estimation requires that the parameters

of the proposed prior function be estimated as accurately as pos-

sible from the noisy observation. Since the parameters of the

MGH prior function are expressed in terms of , the esti-

mation of the th order moment of the signal coefficients from

the noisy observation is sufficient for such a parameter esti-

mation. The noise is independent of the signal and additive in

nature, since it is AWGN. Therefore, the th order moment of

the signal coefficients is estimated as by utilizing,

and , which are, respectively, the th order moments of the

noisy signal and noise coefficients. Since the noise is zero-mean

Gaussian distributed, the odd-order moments of the noise coef-

ficients are zero and the even-order moments can be expressed

in terms of the noise standard deviation . In general, the th

order moment of the noise coefficients is [71]

for odd

for even. (17)

Therefore, the th order moment of the signal coefficients is

evaluated as (18), shown at the bottom of the page. The param-

eter estimation of the MGH prior function is relatively simple,

since these can be substituted in (3) to obtain . On the

other hand, the parameter estimation of traditional prior func-

tions that have fewer number of parameters are often not so

simple. For example, the shape and width parameters of the GG

prior function are estimated numerically that involves iteration.

In practice, the convergence of the iterative procedure depends

on the initial choice of the parameter. Fig. 5 shows an example

of the nonlinear shrinkage function obtained from the pro-

posed denoising technique that uses three different values of

in the MGH prior function. It can be observed from thisfigure

that the amount of shrinkage can be controlled by changing .This feature of being able to vary the amount of shrinkage may

be useful, in particular, in producing a denoised image that is

visually acceptable from the users perspective.

Since the wavelet coefficients of images are locally stationary,

an improved performance of the Bayesian denoising method can

be obtained by considering the proposed prior function to be lo-

cally adaptive during the MMSE estimation. The higher order

moments of the MGH PDF, however, will not be effective for

modeling the local neighboring coefficients as compared to that

of the subband coefficients, since the error introduced by the

estimated higher order sample-moments in the former is not in-

significant as in the case of the latter. This is due to the factthat the number of data samples for the local neighboring co-

efficients is much less than that of the subband coefficients. In

our earlier work [57], it has been shown that the symmetric GC

PDF, which is a particular case of the MGH PDF having ,

performs better than the GG PDF for modeling the local neigh-

boring wavelet coefficients for images. A better quality MMSE

estimator for image denoising can, therefore, be obtained by

using the symmetric GC prior function as the prior function in

the proposed shrinkage function. It was observed from exten-

sive simulations that the difference in denoising performance in

terms of the MSEs of the MMSE estimators designed using the

MGH PDF with and is n o more than 0.25%. H ence,

in order to savethe computational effort, MGH PDF with ,

i.e., the GC PDF, is chosen as the prior function for the case of

locally adaptive denoising. In such a case, the number of pa-

rameters of the MGH prior function is the same as that of the

conventional prior functions, namely, GG and BK. The compu-

tational complexity of the locally adaptive version of the pro-

for odd

for even. (18)


9/17


Fig. 5. Shrinkage function obtained from the modified GH probability density function for the LH2 subband DWT coefficients as a function of with .The test images are (a) Lena and (b)Boat.

Fig. 6. Output PSNR as a function of the input PSNR for different subband-adaptive DWT-based Bayesian denoising algorithms using the test image Lena

( for the proposed method).

posed denoising method is , where is the

average number of discrete points in a window obtained from

the selected bin.

IV. SIMULATION RESULTS

Extensive experimentations have been carried out in order

to compare the performance of the proposed denoising method

with that of the other methods. Due to space limitation, how-

ever, we give in this section results concerning five 512 512

grayscale images, namely,Lena,Barbara,Boat,Bridge, andBa-

boon and two types of wavelet representations, namely, the deci-

mated DWT and redundant DT-CWT. Thefirst three test imagesare obtained from the same sources as mentioned in [16] and

Fig. 7. Output PSNR as a function of the input PSNR for different subband-adaptive DWT-based Bayesian denoising algorithms using the test image Bar-bara(

for the proposed method).

the last two from [68]. The DWT-based experiments use the or-

thogonal wavelet filter Symmlet-8, whereas the DT-CWT-based

ones use the -tap biorthogonal filters at level ,

and 6-tap Q-shift orthogonalfilters at , as have been pro-

posed in [72]. The proposed algorithm is applied to the 4-level

decomposition of the DWT and the 3-level decomposition of the

DT-CWT, since any further decomposition beyond these levels

does not produce a significant increase in the denoising perfor-

mance. The performance of the Bayesian denoising method is

tested using two sets of experiments. In the first set of experi-

ments, the MGH prior function is used for the subband-adaptive

denoising. In the second set of experiments, the symmetric GC

prior function, i.e., MGH prior function with , is used forlocally adaptive denoising.


10/17


Fig. 8. Output PSNR as a function of the input PSNR for different subband-adaptive DWT-based Bayesian denoising algorithms using the test image Boat

(


Fig. 9. Output PSNR as a function of the input PSNR for different subband-adaptive DWT-basedBayesian denoising algorithms usingthe testimageBridge(


A. Subband-adaptive Bayesian Denoising

Seven different wavelet-based Bayesian denoising methods,

which are subband-adaptive in nature, have been considered in

this set of experiments. These are the classical Wiener filter,

SureShrink [3], BayesShrink [20], EBayesThresh [24], Prob-

Shrink [10], BKShrink [46] and proposed method. As pointed

out in Section I, these methods enjoy a computational com-

plexity that is lower than that of the corresponding locally adap-

tive one by a factor of ; and that the computational com-

plexity of those methods that employ intersubband dependency

is even higher than those of the corresponding locally adap-

tive ones. To the best of authorsknowledge the BKShrink and

ProbShrink methods provide the best results amongst the sub-

band-adaptive denoising methods. The denoising performanceof the various methods are evaluated using two indices, namely,

Fig. 10. Output PSNR as a function of the input PSNR for different subband-

adaptive DWT-based Bayesian denoising algorithms using the test image Ba-

boon ( for the proposed method).

Fig. 11. Output PSNR of subband-adaptive Bayesian denoising algorithmsusing redundant DWT, wherein test images are corrupted with ( for the proposed method).

the peak signal-to-noise ratio (PSNR) and the structural simi-larity (SSIM) [4]. In this context, it is to be noted that the authors

in [4] have claimed that SSIM can assess the similarity between

the original and processed images better than the PSNR. De-

noising performance of the proposed method is tested by using

the two shrinkage functions, given by (14) and (16) in Sec-

tion III. The evaluation of the parabolic cylinder function for

the analytic expression in (14) is performed by the same source

code as mentioned in [46]. It is to be noted that the processing

time of (16) is very fast as compared to that of (14). For example,

in a 3 GHz and 512 MB machine, the processing time of (16)

using the MATLAB code is only 6.6 seconds for and the

time is virtually the same for values of . On the other hand, the

processing time of (14) is 1 hour and 24 minutes for andthe time increases exponentially with . It is also observed that


11/17


Fig. 12. Comparison of the denoised images obtained from the different subband-adaptive Bayesian DWT-based denoising algorithms on Lenawith :(a) original image, (b) noisy image (

dB,

), and denoised images using (c) BayesShrink method [20] (

dB,

), (d) ProbShrink method [10] ( dB, ), (e) BKShrink method [46] ( dB, ),and (f) proposed method with ( dB, ).

for a given value of , the denoising performances of both (14)

and (16) are essentially the same. Hence, we provide the results

of the proposed method that uses the shrinkage function given

by (16). From our experiments, we have observed that

provides the best performance in terms of the PSNR, whereas

provides the best performance in terms of the SSIM, for

the proposed method. These values of are consistent with thevalues of mentioned in Section II as providing the best statis-

tical modeling of the image wavelet coefficients by the MGH

PDF.

The output PSNR as a function of the input PSNR obtained

from various DWT-based denoising methods are shown in

Figs. 610, respectively, for the test images Lena, Barbara,

Boat, Bridge, and Baboon. From these figures, it can be

observed that the proposed denoising method with

that employs the DWT provides the highest output PSNR

consistently as compared to the other DWT-based methods.

According to the output PSNR values, the improvement of the

denoising performance provided by the proposed method is

substantial, when the input PSNR is low, i.e., the noise strengthis significant. Fig. 11 shows a comparison between the output

PSNR values obtained from the SureShrink [3], BayesShrink

[20], EBayesThresh [24], ProbShrink [10], and proposed

methods employing the redundant form of the DWT when the

images are corrupted with . From thisfigure, it can also

be seen that the output PSNR values of the proposed method

are consistently higher than that of the other methods. Table IV

shows the SSIM values obtained from the various DWT-baseddenoising methods for three different noise standard deviations.

It can be observed from the table that, except for two values of

in the case of the Barbaraimage, the proposed method with

provides the best SSIM compared to that provided by

the other denoising methods. It should be noted that the PSNR

values for are about the same as those for , the

difference being not more than 0.1 dB, and that these values are

at least as good as those provided by the other methods. Fig. 12

shows the original cropped image Lena, its noisy version with

, and the DWT-based denoised versions provided

by the BayesShrink method [20], ProbShrink method [10],

BKShrink method [46], and proposed method with .

From this figure, it can be observed that the proposed methodcan reduce the noise significantly with the least amount of


12/17


TABLE IV

VALUES OF THE INDEXSSIM FOR VARIOUSSUBBAND-ADAPTIVEBAYESIANDENOISINGMETHODSUSING THEDWT

artifacts (e.g., in the eye and the edge of the hat) as compared

to the other competing methods.

B. Locally Adaptive Bayesian Denoising

Six different wavelet-based denoising methods, which are lo-

cally adaptive in nature, have been considered in this set of ex-

periments. These are the BivariateShrink [58], locally adaptive

window-based MAP (LAWMAP) [54], NeighShrink [22], Prob-

Shrink [10], GSM method [16] and proposed method. In the

first method, both the inter and intrasubband dependencies are

considered. The second, third, and fourth methods inherently

use only the intrasubband dependency. In the case of the fifth,

we provide the results obtained from the intrasubband depen-

dency only in order to make a fair comparison with our pro-

posed method. The parameters of the symmetric GC prior func-

tion for the proposed denoising method is refined locally using

different sizes of square-shaped neighborhood regions. Fig. 13

shows the improvement in the PSNR (IPSNR) values2 asa func-

tion of the window size with a reference of 3 3 window, i.e.,

as a function of , for de-

noising different test imagesbeing corrupted with a noise having

. From this figure, it can be observed that as the window

size is increased from 3 3, the IPSNR value increasesfirst and

then drops slightly when the window exceeds a certain size. Thisisdue tothe fact thata sufficient number of samples are required

for improved parameter estimation of the prior function, but, if

the neighborhood region becomes relatively large, the estima-

tion process loses its local contextual information. The experi-

ments reveal that a 9 9 window provides the highest IPSNR

values in most cases. Considering this fact, we report the re-

sults provided by our proposed denoising method using a 9 9

window.

Table V shows the output PSNR values obtained from the

various locally adaptive denoising methods for the test images

considered here. From the tabular results, it can be observed that

among all the DWT-based denoising methods, the proposed one2IPSNR is the difference between output PSNR and input PSNR.

Fig. 13. IPSNR values in dB as a function of the window size relative to theIPSNR values corresponding to a 3

2

3 window.

provides the highest output PSNR values. The PSNR perfor-

mance of the proposed method improves further, if the DT-CWT

is employed. Fig. 14 shows the original cropped image Bar-

bara, its noisy version with , and the DWT-based de-

noised versions provided by four different denoising methods,

namely, the BivariateShrink method [58], ProbShrink method

[10], GSM method [16] and proposed method. From this figure

also, it can be observed that the proposed method can reduce the

noise significantly with the least amount of artifacts (e.g., in the

eyes and scarf ofBarbara) as compared to the other competing

methods.

V. CONCLUSION

The traditional probability density functions (PDFs) of the

wavelet transform coefficients of images have a limited number

of parameters, which are calculated by using only thefirst few

moments. As a result, these PDFs cannot provide a sufficiently

good statistical match for the empirical PDF of the waveletcoefficients of a true image. In this paper, a new PDF, wherein


13/17


Fig. 14. Comparison of the denoised images obtained from the different locally adaptive DWT-based denoising algorithms onBarbarawith

: (a) originalimage, (b) noisy image dB , and denoised images using the (c) BivariateShrink method [58] dB , (d) ProbShrink method [10] dB , (e) GSM method [16] dB , and (f) proposed method dB .

a more appropriate number of parameters can be incorporated,

has been proposed in order to overcome this shortfall. This PDF

uses a series expansion in terms of the Hermite polynomials,

which are orthogonal with respect to the standard Gaussian

weight function. Such an orthogonal series expansion has been

utilized, since the Gaussian density function bears a resem-

blance with the empirical PDF of the wavelet coefficients of a

true image. A closed-form expression for the parameters of thePDF (which has been referred to in this paper as the GH PDF)

has been derived in terms of the higher order sample-moments.

A modification has been proposed in the GH PDF so that an

appropriate number of parameters can be used to model the

true image wavelet coefficients while keeping the resulting

PDF to be non-negative. It has been shown that this modified

GH (MGH) PDF matches the empirical ones of the test images

used, more closely than some of the standard PDFs such as the

generalized Gaussian and Bessel K-form.

In order to demonstrate the efficacy of the proposed MGH

PDF, we have designed and tested the minimum mean squared

error-based shrinkage function for its use in an image noise re-

duction technique as a prior function of the subband as well aslocal neighboring wavelet coefficients. The parameters required

for the prior function have been estimated from the noisy obser-

vation. The simulation results using a number of standard test

images have shown that the performance of the proposed de-

noising method, both in subband-adaptive and locally adaptive

conditions, is better than that of most of the methods that use

PDFs with limited number of parameters for statistical modeling

of the wavelet coefficients.

APPENDIXA

PARAMETERS OF PDF IN TERMS OF THE

GAUSSHERMITEEXPANSION

Let be the normalized version of with the relation

and its PDF be . Since the Hermite polynomials are

orthogonal with respect to the zero-mean unit variance Gaussian

weight function, can be written as [64]

(19)


14/17


TABLE IV

OUTPUTPSNR, IN DB, WHERE IS THEERROR STANDARD

DEVIATION, FOR DIFFERENTLOCALLYADAPTIVEDENOISINGMETHODS

where are the series coefficients. Multiplying both sides of

(19) by and integrating from to , can be eval-

uated as

(20)

Using the fact that has a zero mean and unit variance, one can

obtain and . Finally, changing the variable

of to in (19) yields [61]

(21)

The Hermite polynomials can be expressed as

(22)

Using the identity and the expression of , the series coefficients in (20) can be obtained in a

generalized form

(23)

where is the th order moment of .

APPENDIXB

PARAMETER FOR THEMODIFIEDGAUSS-HERMITEPDF

Using the normalized variable , we can solve (8) as

, where

(24)

Using the identity and the

recurrance relation in (2), we obtain

(25)

Hence, for

(26)

(27)

Using the above, and , and hence, the parameter can be

expressed as

(28)


15/17


16/17


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S. M. Mahbubur Rahman received B.Sc. andM.Sc.degrees in electrical and electronic engineering fromthe Bangladesh University of Engineering andTechnology (BUET), Dhaka, Bangladesh, in 1999

and 2002, respectively. He is currently pursuingthe Ph.D. degree in the Department of Electricaland Computer Engineering, Concordia University,

Montreal, QC, Canada.He was a Lecturer from 1999 to 2002 and an As-

sistant Professor from 2002 to 2003 at BUET. Hisresearch interests are in the areas of statistical signal

processing, digital image/video processing, and waveletfilters.Mr. Rahman hasreceivedseveral awardsduringhis Ph.D. program, including

the Concordia University Graduate Fellowship, the Faculty of Engineering andComputer Science Graduate Scholarship, the Carolyn and Richard Renaud

Teaching Assistantship, and the Doctoral Teaching Assistantship.

M. Omair Ahmad (S69M78SM83F01) re-ceived the B.Eng. degree from Sir George Williams

University, Montreal, QC, Canada, and the Ph.D.degree from Concordia University, Montreal, both

in electrical engineering.During 19781979, he was a member of the Fac-

ulty of the New York University College, Buffalo. InSeptember 1979, he joined the Faculty of ConcordiaUniversity, where he was an Assistant Professor ofcomputer science. Subsequently, he joined the De-partment of Electrical and Computer Engineering of

thesame university, where he wasthe Chair of thedepartment from June 2002 toMay 2005 and where he is presently a Professor and holds the Tier I Concordia

University Research Chair in Multimedia Signal Processing. He has publishedextensively in the area of signal processing and holds four patents. His current

research interests include the areas of multidimensionalfilter design, speech,image and video processing, nonlinear signal processing, communication DSP,artificial neural networks, and VLSI circuits for signal processing. He was afounding member of Micronet from its inception in 1990 as a Canadian Net-work of Centers of Excellence until its expiration in 2004.

Dr. Ahmad was anAssociate Editor of the IEEET RANSACTIONS ON CIRCUITSANDSYSTEMSPARTI: FUNDAMENTALTHEORY ANDAPPLICATIONS from June1999 to December 2001. He was the Local Arrangements Chairman of the 1984IEEE International Symposium on Circuits and Systems. During 1988, he wasa member of the Admission and Advancement Committee of the IEEE. He hasalso served as the Program Co-Chair for the 1995 IEEE International Con-ference on Neural Networks and Signal Processing and the 2004 IEEE Inter-

national Midwest Symposium on Circuits and Systems. Presently, he is theChairman of the IEEE Circuits and Systems Chapter (Montreal Section). He

is a recipient of the Wighton Fellowship from the Sandford Fleming Founda-tion.

M. N. S. Swamy (S59M62SM74F80) re-ceived the B.Sc. (Hons.) degree in mathematics fromMysore University, India, in 1954, the Diploma inElectrical Communication Engineering from theIndian Institute of Science, Bangalore, in 1957, theM.Sc. and Ph.D. degrees in electrical engineeringfrom the University of Saskatchewan, Saskatoon,SK, Canada, in 1960 and 1963, respectively, and the

Doctor of Science degree in engineering (HonorisCausa) from Ansted University In recognitionof his exemplary contributions to the research in

Electrical and Computer Engineering and to Engineering Education, as wellas his dedication to the promotion of Signal Processing and CommunicationsApplications in August 2001. As a recognition of his research contributions,as well as his contributions to Concordia University, the University has recentlyestablished the M. N. S. Swamy Chair in Electrical Engineering.

He is presently a Research Professor and the Director of the Center for SignalProcessing and Communications, Department of Electrical and Computer En-gineering, Concordia University, Montreal, QC, Canada, where he served asthe Chair of the Department of Electrical Engineering from 1970 to 1977 andDean of Engineering and Computer Science from 1977 to 1993. Since July2001, he has held the Concordia Tier 1 Research Chair in Signal Processing. Hehas also taught in the Electrical Engineering Department, Technical Universityof Nova Scotia, Halifax, NS, Canada, and the University of Calgary, Calgary,AB, Canada, as well as in the Department of Mathematics at the University

of Saskatchewan. He has published extensively in the areas of number theory,circuits, systems and signal processing, and holds five patents. He is the co-au-thor of two book chapters and four books: Graphs, Networks and Algorithms

(Wiley, 1981),Graphs: Theory and Algorithms(Wiley, 1992),Switched Capac-itor Filters: Theory, Analysis and Design (Prentice Hall International U.K. Ltd.,1995), andNeural Networks in a Softcomputing Framework(Springer, 2006). ARussian translation of the first book was published by Mir Publishers, Moscow,Russia, in 1984, while a Chinese version was published by the Education Press,Beijing, China, in 1987. He was a founding member of Micronet from its incep-tion in 1990 as a Canadian Network of Centers of Excellence until its expirationin 2004, as well as its Coordinator for Concordia University.

Dr. Swamy is a Fellow of the Institute of Electrical Engineers (U.K.), the En-gineering Institute of Canada, the Institution of Engineers (India), and the Insti-tution of Electronic and Telecommunication Engineers (India). He has servedthe IEEE in various capacities such as the President-Elect in 2003, President in2004, Past-President in 2005, Vice President (Publications) during 20012002,VicePresident in 1976, Editor-in-Chief of theT RANSACTIONS ON CIRCUITS AND

SYSTEMSPART I: FUNDAMENTAL THEORY AND APPLICATIONS fromJune 1999to December 2001, Associate Editor of the TRANSACTIONS ON CIRCUITS AND

SYSTEMSfrom June 1985May 1987, Program Chair for the 1973 IEEE CASSymposium, General Chair for the 1984 IEEE CAS Symposium, Vice-Chair forthe 1999 IEEE CAS Symposium, and a member of the Board of Governors ofthe CAS Society. He is the recipient of many IEEE-CAS Society awards in-cluding the Education Award in 2000, Golden Jubilee Medal in 2000, and the1986 Guillemin-Cauer Best Paper Award.

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