adaptive 2-d wavelet transform based on the_lifting scheme with preserved vanishing moments_

8/7/2019 Adaptive 2-D Wavelet Transform Based on the_Lifting Scheme with Preserved Vanishing Moments_

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IEEE TRANSACTIONS ON IMAGE PROCESSING, MIROSLAV VRANKIC, DAMIR SERSIC, AND VICTOR SUCIC 1

Adaptive 2-D Wavelet Transform Based on the

Lifting Scheme with Preserved Vanishing MomentsMiroslav Vrankic*, Member, IEEE, Damir Sersic, Member, IEEE, and Victor Sucic

AbstractIn this paper, we propose novel adaptive waveletfilter bank structures based on the lifting scheme. The filterbanks are nonseparable, based on quincunx sampling, withtheir properties being pixel-wise adapted according to the localimage features. Despite being adaptive, the filter banks retaina desirable number of primal and dual vanishing moments.The adaptation is introduced in the predict stage of the filterbank with an adaptation region chosen independently for eachpixel, based on the intersection of confidence intervals (ICI) rule.The image denoising results are presented for both syntheticand real-world images. It is shown that the obtained waveletdecompositions perform well, especially for synthetic imagesthat contain periodic patterns, for which the proposed method

outperforms the state of the art in image denoising.

Index TermsWavelets, second generation wavelets, adaptivelifting scheme, quincunx sampling, interpolating filters, intersec-tion of confidence intervals, image denoising.

I. INTRODUCTION

In many applications, such as image denoising or compres-

sion, transforms are used to obtain a compact representation

of the analyzed image. The wavelet transform relies on a set

of functions that are translates and dilates of a single mother

function, and provides sparse representation of a large class

of real-world signals and images. Wavelet thresholding has

been shown to be a very efficient method for image denoising[1]. The choice of the threshold determines the compromise

between the noise reduction and the degradation of the original

features in the reconstructed image.

In order to obtain a wavelet transform that gives more

compact representation of an image, we propose a pixel-wise

adaptation of the analysis functions according to its local

properties. A locally adaptive wavelet transform that is tuned

to the image features results in transform coefficients which

predominantly contain the noise, i.e. the content that is not

inherent to the original image. In that case, application of a

rather high threshold reduces the noise efficiently, with lower

degradation of the original image.

The properties of the wavelet decomposition depend on thesubsampling scheme used in the corresponding filter bank

[2]. To obtain a more isotropic 2D wavelet transform than

the usual separable approach, we used the quincunx scheme,

M. Vrankic and V. Sucic are with the Faculty of Engineering, Uni-versity of Rijeka, Vukovarska 58, HR-51000 Rijeka, Croatia (e-mail:[email protected]; [email protected])

D. Sersic is with the University of Zagreb, Faculty of Engineering andComputing, Unska 3, HR-10000 Zagreb, Croatia (e-mail: [email protected])

Manuscript received xxxx, 2008; revised xxxxxxEDICS category: TEC-MRS (Multiresolution Processing of Images &

Video - Wavelets; Filter banks)

which is the simplest nonseparable sampling [3]. A compre-

hensive overview of two- and multi-dimensional FIR perfect

reconstruction filter banks (FBs) for arbitrary sampling lattices

can be found in [4] and [5], while multidimensional wavelet

FBs design methods are presented in [6], [7].

The adaptation of wavelet transforms can generally be

performed in two ways: globally or locally. An example of

a global adaptation approach is the best basis algorithm [8],

in which, depending on the criterion (e.g. signal entropy),

the optimal basis for the whole image is obtained. Local

adaptation, on the other hand, is based on the local features

of the analyzed image, and it generally outperforms the global

adaptation methods.

A number of authors have used the lifting scheme in order to

build locally adaptive wavelet FBs. Claypoole et al. proposed

the so-called space-adaptive transform (SpAT) based on the

lifting scheme, which changes the predictor order for every

signal sample to minimize detail coefficients [9]. Therefore,

near the edges present in the image, shorter prediction filters

are used to avoid ringing effects. The update step is per-

formed first, followed by the adaptive prediction. This allows

the maintaining of multiresolution properties across different

decomposition levels in spite of the nonlinearity introduced

through the adaptive predict step.A locally adaptive filter bank structure for image coding has

been proposed by Gerek and Cetin [10]. Their FB is based

on the lifting scheme with adaptation of the prediction filter.

The least mean squares (LMS) adaptation algorithm is used to

minimize the signal at the high-pass channel output. However,

the proposed adaptive structures do not retain polynomial

annihilation properties inherent to wavelets.

An adaptive prediction stage is also employed in [11] to

improve the image decorrelation efficiently for the purpose

of lossless image compression. Similarly to the previous

approach, these FBs lack some crucial wavelet properties, such

as polynomial annihilation.

The approaches presented in [12] and [13], on the otherhand, use the update first structure with adaptation of the

update step. The update filter is changed based on the local

gradient information such that sharp variations in the signal

get less smoothened than the more homogenous regions.

Moreover, the choices of update filters can be automatically re-

produced on the synthesis side, so no bookkeeping is required

for a perfect reconstruction.

In another approach presented in [14], to obtain a multidi-

mensional nonseparable wavelet transform, the prediction step

is designed to minimize the variance of the signal, while the

update step minimizes the reconstruction error.


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Recently, many novel wavelet decomposition schemes that

use directional information from images have been introduced,

either adaptive or non-adaptive: curvelets, ridgelets, bandelets,

directional lifting-based wavelets, directionlets etc. [15][17].

Some authors have developed very useful directional algo-

rithms for compression, showing advantages over traditional

separable non-directional wavelets. Nevertheless, there is a

lack of papers that deal with wavelet adaptation in the presence

of noise. In this paper, rather than using directional information

from images, we use more isotropic quincunx 2D wavelets,

and pay attention to reducing the influence of noise on the

adaptation.

A milestone denoising method was introduced by Portilla

[18]. Pointwise shape-adaptive DCT [19] is a novel adap-

tive image decomposition with applications in denoising that

outperforms in many cases the Portillas method. The SA-

DCT changes the shape of the transforms support at each

pixel of the image, while in our work we change the support

of the adaptation region, and, according to the result, the

base functions. Both approaches rely on the intersection of

confidence intervals (ICI) rule. We used SA-DCT, as one ofthe state-of-the-art image denoising methods, for comparison.

A. The Lifting Scheme

The lifting scheme provides an effective tool to build

biorthogonal wavelet FBs [20]. Moreover, due to its inherent

perfect reconstruction properties, it is a great tool to build

second-generation wavelets [21], i.e. wavelets that are not

necessarily translated and dilated copies of a single function.

An example of the lifting scheme structure based on quin-

cunx sampling is shown in Fig. 1. It is built from three

main parts: polyphase decomposition, predict and update steps.

The image at the FB input is split into M = | det(D)|phases, where D represents the subsampling matrix [3]. For

the quincunx case, the image is split in two phases as defined

by the quincunx subsampling matrix:

Dq =

1 1

1 1

. (1)

As shown in Fig. 1, the polyphase decomposition is fol-

lowed by the prediction step. The filter P predicts the pixelvalues of the second (odd) phase, based on the pixels from

the first (even) phase. The prediction error corresponds to

the wavelet detail coefficients, e.g. to the output of the high-

pass channel. The subsequent update step corresponds to the

approximation coefficients, e.g. produces the low-pass output

of the FB.

The reconstruction is readily obtainable. The synthesis FB

is constructed using the same steps but in the reverse order

and with opposite signs. This way, the operations from the

analysis side are reversed and cancelled on the synthesis side.

B. The Proposed Method

An important property of wavelets is their ability to anni-

hilate polynomials. Many real-world signals and images can

be locally viewed as polynomial segments, so the polyno-

mial annihilation is a key to sparse representation in the

transform domain. The sparseness is a desired property for

many applications: feature extraction, denoising, compression,

compressed sensing, blind separation of dependent sources

[22], and many others. Another important property of wavelets

is their multi-resolution nature: numerous families of octave

wavelets (either bases or redundant frames) can be realized

using wavelet filter banks in a numerically efficient way. By

introducing adaptation, we do not want to lose any of the

winning properties of the wavelets!

When building the adaptive wavelet FB, one would like to

retain a number of dual and primal vanishing moments. They

are necessary for convergence of the corresponding wavelet

and scale functions across different decomposition levels,

and they contribute to their smoothness and regularity. The

convergence of decomposition and/or reconstruction functions

across different levels is a key link between wavelets and FBs.

Moreover, the number of vanishing moments is closely related

to the order of the polynomials that the wavelet FB is able to

annihilate [23].

In this paper, we propose two novel adaptive non-separable

2D filter bank structures based on the lifting scheme andquincunx sampling. By using the lifting scheme, the existing

wavelet transform is made adaptive and further improved with

additional lifting steps, while at the same time a desirable

number of vanishing moments is retained. The structures and

adaptation algorithms are improved when compared to our pre-

viously reported work in [24] and [25]. In [24], we proposed

adaptive 1D wavelet FB structures that can be generalized to

2D using a well-known separable approach.

In this work, the adaptation is introduced through an ad-

ditional predict stage with a parameter which is tuned for

each pixel separately, based on the adaptively chosen pixel

neighborhood. Unlike our work presented in [26] and [27],

which is based on changing filter supports for each signalsample, in this paper we do not change filter supports, but

rather change the region on which a FB parameter is being

calculated.

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

adaptive FB structures, named PPaU and PUPa, are pre-

sented, the first of which was proposed in [25]. The proposed

structures can provide any number of vanishing moments or

adaptive parameters. In this paper, we have chosen the FB

structure PPaU with the shortest possible (i.e. canonical)

symmetric support that provides two primal and two dual

vanishing moments, followed by a single adaptive parameter.

However, the quest for independent adaptation across differ-

ent decomposition levels, with smooth and regular synthesisfunctions, brought us to the structure PUPa. The proofs of

vanishing moment preservation are given in the Appendix.

In Section III the minimization of energy of detail coeffi-

cients is introduced as the adaptation criterion. The adaptation

is performed for every pixel of the image based on the

neighborhood of a free shape. Assuming an additive noise

model, we show that the adaptation is biased. However, we

have found a way to make the adaptation almost insensitive

to noise. The proof is given in Section III-D.

In Section IV we present a method for obtaining the

appropriate adaptation neighborhood for each pixel of the


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+

D

D

-

+ -

D

D1 2

( , )X z z

1 2( , )X z z

1 2( , )P z z 1 2( , )P z z1 2( , )U z z1 2( , )U z z

1

1z

1z

eX

oX D

Fig. 1. Quincunx filter bank based on the lifting scheme. Downsampling and upsampling operators are defined with a quincunx subsampling matrix,D

=Dq.

image. The method is based on the intersection of confi-

dence intervals (ICI) rule [28]. The ICI rule provides for the

maximum size of local neighborhoods of similar statistical

properties, thus providing for the best unbiased estimation. To

find the confidence intervals limits, we used the Monte Carlo

approach.

Finally, in Section V the proposed adaptation algorithm is

used within the framework of image denoising by wavelet

thresholding. The image denoising results are presented for

both synthetic and real-world images. Our trade-off between

more vanishing moments and local adaptation using the pro-posed structures has shown to be useful for images containing

periodic components, as expected.

II. THE ADAPTIVE LIFTING SCHEME

A. Quincunx Interpolating Filter Bank

The construction of compactly supported biorthogonal

wavelets and perfect reconstruction filter banks for any lattice,

in any dimension, and with any number of primal and dual

vanishing moments is presented in [29].

The filters used to build the prediction step, called Neville

filters, are based on the polynomial interpolation. Filtering the

polynomial of degree lower than N with the Neville filter P ofdegree N or higher gives the samples of the same polynomialbut sampled on the lattice shifted by . Following the notation

in [29], we define to be a multivariate polynomial, and (Z2)the sequence formed by evaluating the polynomial on the

lattice Z2. Then, applying the Neville filter P to the sampledpolynomial yields

P (Z2) = (Z2 + ), (2)

where R2.The construction of such filters relies on solving the inter-

polation problem. The solution of the interpolation problem in

multiple dimensions is given by de Boor and Ron in [30]. In

[29] the authors give Neville filter coefficients for the quincunx

case obtained by using the de Boor-Ron algorithm. The predict

filter coefficients are given in Table I. Coefficients with equal

values are grouped into rings. The ring numbering scheme is

shown in Fig. 2.

B. Vanishing Moments

An analysis FB has N vanishing moments if its high-passchannel annihilates polynomial sequences of degree lower than

N. Also, a synthesis FB has N vanishing moments if its syn-thesis high-pass channel annihilates polynomial sequences of

degree lower than N. The vanishing moments of the analysis

TABLE IQUINCUNX NEVILLE FILTER COEFFICIENTS [29]. THE COEFFICIENTS ARE

GROUPED INTO RINGS, AS SHOWN IN FIG . 2.

Ring 1 2 3 4 5 6 7

Order

2 1 22

4 10 -1 25

6 174 -27 2 3 29

8 23300 -4470 625 850 -75 9 -80 216

filter bank are called dual, while the vanishing moments ofthe synthesis FB are called primal.

It was shown in [29] that in order to have N dual vanishingmoments, the predict filter P of the lifting scheme has to bea Neville filter of order N and shift

= D1t, (3)

where D represents the subsampling matrix and t defines

the shift between the two phases. For the quincunx sampling

matrix from (1), where t = [1 0]T, the shift is = [0.5 0.5]T.As it was also shown in [29], the primal vanishing moments

are defined with the update filter U of the lifting scheme. Inorder to obtain N primal vanishing moments, with P beinga Neville filter of order N and shift = D1t, 2U hasto be a Neville filter of order N and shift . Consequently,2U has to be a Neville filter of order N and shift . Thesimplest choice is to make the update filter the adjoint of the

same-order predict filter divided by two:

U = UN =1

2PN, (4)

where PN is a Neville filter of order N and shift [29].It is important to note that the prediction filter by itself

defines the number of dual vanishing moments, and the update

filter influences only the number of primal vanishing moments

as long as N N [29].For example, the FB with 2 dual vanishing moments and

2 primal vanishing moments will be constructed using theprediction filter P2 and the update filter U2 = 1/2P2 (seeTable I):

P2(z1, z2) =1

4(1 + z11 + z

12 + z

11 z

12 ), (5a)

U2(z1, z2) =1

8(1 + z1 + z2 + z1z2). (5b)

C. The PPaU structure

The adaptive lifting scheme structure, of which a simpler

version has been proposed in [25], is shown in Fig. 3. It

consists of a fixed prediction step PN that guarantees N


4/17


5/17




X

D

D

D

-

z1-1

Xe

Xo

UN

A

+

PN~

-

z1

D

D

+

PN~UN

Pa

-

p

+

Pa

p

X

Fig. 4. The PUPa analysis and synthesis adaptive filter bank structure.

take, if the Pa filter is chosen as in (6), where

R N , and S N. (10)

Also, in Appendix B we show that N primal vanishingmoments are preserved as long as

N N. (11)

E. The p parameter

In this paper, we use FBs that have 2 primal and 2 dualvanishing moments fixed, obtained by the following filter

selection:

PN = P2, (12a)

Pa = P4 P2, (12b)

UN = U2. (12c)

This is a canonical (i.e. simplest possible) structure with

symmetric analysis and synthesis 2D functions. As the value

of parameter p is changed, the FB properties will change ac-cordingly. The magnitude frequency responses of the analysis

high-pass filters of the PPaU structure for different values of

parameter p are shown in Fig. 5. As shown in Fig. 5, with

the increase of parameter p the zero ditch appears and growswider in the resulting magnitude response of the analysis high-

pass filters. Therefore, by changing the value of parameter p,it is possible to cancel some frequency components present

in the input image. Similar results are obtained for the PUPa

structure as well.

The limit analysis and synthesis wavelet functions for both

PPaU and PUPa structures for various values of parameter pare shown in Figs. 6-9. The limit functions are obtained by

setting a single value of parameter p at all decomposition levelsand calculating the impulse response of the 7-level iterated FB.

As shown in Figs. 7 and 9, the limit synthesis wavelet

functions are regular for smaller values of parameter p. For

absolute values ofp greater than 5, the limit functions becomeincreasingly irregular. However, for a limited range ofp values,well-behaved analysis and synthesis functions are obtained.

The analysis wavelet functions behave differently for the

two proposed structures. For the PPaU structure, the analysis

wavelet functions change significantly with increasing para-

meter p. Yet, for the PUPa structure, the analysis waveletfunctions are similar for the whole range of p values. Thereason for such behavior lies in the fact that the analysis low-

pass filter of the iterated PUPa filter bank does not depend on

the value ofp, so the iterated analysis high-pass filter changesonly due to the value ofp in the last level of the FB cascade.

III. ADAPTING THE FILTER BANK PARAMETERS

A. Noise Model

An input image corrupted with additive zero-mean white

Gaussian noise w can be modeled as

x(n) = x0(n) + w(n), (13)

where the original image x0 will be treated as deterministic,and the samples of the noise are independent and identically

distributed (IID) random variables, w(n) N(0, 2). Forsimplicity of notation, we will use linear indexing of the image

pixels such that the two-dimensional indexes (j, k) will bereplaced by a single index n.

In deriving the method for adapting the values of parameter

p we will use the undecimated FBs, since it has been shownthat for denoising purposes they outperform decimated filter

banks [31]. The results obtained for the undecimated FB will

be straightforwardly applicable to the decimated FB structure

as well. The adaptation of parameter p will first be derivedfor the PPaU structure and then generalized for the PUPa

structure.

B. The Undecimated Filter Bank and Pixel Numbering

The undecimated analysis PPaU filter bank is shown in Fig.

10(a) and the simplified equivalent of its high-pass channel is

shown in Fig. 10(b). In the figures, the upsampling of the

filters impulse response is denoted by P2(zD), where z =[z1 z2]T. For the case of quincunx subsampling matrix Dgiven in (1) we therefore have P2(z1z

12 , z1z2).

The xp represents the image at the output of the filter P42 =P4 P2 which is quincunx-upsampled, having the impulseresponse:

p42 =1

32

0 1 0 1 0

1 0 2 0 10 2 0 2 0

1 0 2 0 1

0 1 0 1 0

. (14)

The image denoted as xh, resulting after the first predictionstep, is obtained by convolving the input image with the

impulse response of the high-pass filter

h =

0 14

0

14

1 14

0 14

0

. (15)


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0

3 0

30

0.5

1

1.5

(a)

0

3 0

30

1

2

(b)

0

3 0

30

1

2

(c)

-3 0 3

-3

0

3

(d)-3 0 3

-3

0

3

(e)-3 0 3

-3

0

3

(f)

Fig. 5. Magnitude frequency responses of the analysis high-pass filters of the PPaU structure with 2 vanishing moments fixed, obtained for different valuesofp (first row), and the appropriate zero locations of the responses (second row). p = 1 for the first column, p = 2 for the second, and p = 5 for the thirdcolumn.

X

P2(zD)

D-

z1-1

P42(zD)

-

+

Xh

A

U2(zD

)

Xh^p

Xp

X

D

P42(zD)

-

p

H(zD)Xh

Xh^

Xp

(a)

(b)

Fig. 10. (a) The first decomposition level of the adaptive PPaU filter bankwhere P42 = P4P2, and (b) the simplified scheme of its high-pass channel.

In order to enumerate pixels from the even quincunx phase

relative to the prediction center n located in the odd phase, we

introduce an alternative notation x(n, i). The sample x(n, i) islocated at position i of the constellation in Fig. 11(a), whichis centered on pixel x(n). Therefore, a pixel xh(n), as shown

5

2

4

1

6

3

9

7

8

10

11

12

13

5

2

4

1 3

(a) (b)

Fig. 11. Filter supports for (a) p42 filter, and (b) h filter are marked with graycircles. The relative pixel numbering starts from the support center (markedin bold).

in Fig. 11(b), is a weighted sum of 5 pixels from the input

image x

xh(n) = x(n, 1) 14

(x(n, 2) + x(n, 3) + x(n, 4) + x(n, 5)).

(16)

The xp(n) and xh(n) from Fig. 10(b) can be expressed interms of the original input image x0(n) and the random noisew(n) as:

xp(n) = (x0(n) + w(n)) p42(n) = x0p(n) + wp(n),

xh(n) = (x0(n) + w(n)) h = x0h(n) + wh(n).(17)

The signals x0p and x0h, which result from filtering theoriginal image, are considered to be deterministic components

ofxp and xh respectively. However, the components wp andwh are stochastic, coming from filtering the input noise.


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(a) (b) (c)

(d) (e) (f)

Fig. 6. The analysis limit wavelet functions of the PPaU structure obtained for a range ofp values. (a) p = 1, (b) p = 0 , (c) p = 1 , (d) p = 2 , (e) p = 3 ,(f) p = 5 .

(a) (b) (c)

(d) (e) (f)

Fig. 7. The synthesis limit wavelet functions of the PPaU structure obtained for a range ofp values. (a) p = 1, (b) p = 0, (c) p = 1, (d) p = 2, (e)p = 3 , (f) p = 5 .

C. Minimizing the Energy of Detail Coefficients

The image xh represents detail coefficients resulting froma dual wavelet with two vanishing moments. In general, the

filters used in the prediction branch are Neville filters of

degree N. Hence, if the input to the FB is a polynomial ofdegree lower than N, the prediction is perfect, leading to zerodetail coefficients. Our goal is to improve the prediction by

additionally annihilating non-polynomial components, such as

periodic ones. This would lead to smaller detail coefficients

and finally to a more compact image representation. In order

to obtain a sparse image representation we minimize the

detail coefficients, by changing the value of parameter p. Weminimize the energy of detail coefficients, which leads to the

well-known least squares solution.

The prediction error for the n-th pixel is defined as

e(n) = xh(n) xh(n), (18)

where

xh(n) = p(n) xp(n). (19)

The adaptation of parameter p is performed for every pixelof the analyzed image based on an appropriately chosen

neighborhood. A given pixel and its neighborhood form a


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(a) (b) (c)

(d) (e) (f)

Fig. 8. The analysis limit wavelet functions of the PUPa structure obtained for a range ofp values. (a) p = 1, (b) p = 0 , (c) p = 1 , (d) p = 2 , (e) p = 3 ,(f) p = 5 .

(a) (b) (c)

(d) (e) (f)

Fig. 9. The synthesis limit wavelet functions of the PUPa structure obtained for a range ofp values. (a) p = 1, (b) p = 0, (c) p = 1, (d) p = 2, (e)p = 3 , (f) p = 5 .

region on which the value of the parameter p is calculated.For a pixel indexed with n, the appropriate region is denotedas Rn. (Determination of regions Rn is dicussed in SectionIV). For every region Rn the adaptation algorithm sets thevalue of the parameter p(n) in order to minimize the sum ofsquared errors. So, the function

F(p(n)) =iRn

e2(i) =iRn

(xh(i) p(n) xp(i))2 (20)

needs to be minimized with respect to p. The least squares (LS)solution minimizes the energy of detail coefficients. Optimal

parameter p(n) is the one for which

F(p(n))

p(n)=iRn

2(xh(i)p(n) xp(i))(xp(i)) = 0, (21)

which gives

p(n) =

iRn

xh(i) xp(i)

iRn

x2p(i). (22)

The numerator and denominator in (22) are functions of

pixels xp(i) and xh(i) belonging to the region Rn. If the


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image does not contain the noise, the local adaptation results

in a well-adapted sparse representation. Parameter p is usuallyset to cancel periodic components that have remained after the

fixed polynomial predictor. In the presence of white noise, one

would expect that the estimator (22) is statistically unbiased

and that a larger region should decrease the variance of the

estimation.

The random variable wp(i), which is a part of xp(i), is afunction of a number of independent random variables of the

additive noise present in the input image from the constellation

given in Fig. 11(a). In a similar manner, wh(i), which is apart of xh(i), is a weighted sum of 5 independent randomvariables (see Fig. 11(b)). Therefore, the summations in the

numerator and denominator ofp in (22) are dependent randomvariables. Hence, the statistical properties of the estimated

p are influenced not only by the input noise properties, butalso by the shape of the adaptation region and the value of

the original image, which is generally unknown in image

denoising applications. Unfortunately, the estimator (22) is

biased, and we need some extra steps to cope with the problem.

D. Reducing the Bias of the Estimator for p

The bias ofp is defined as

bias(p(n)) = E[p(n) p(n)]. (23)

If one knew the estimation bias, then an unbiased estimator

ofp could be calculated as

pu(n) = p(n) + bias(p(n)). (24)

Unfortunately, the unbiased estimator is not available, since

there is no closed form expression for the expectation of p[32].

The true value of parameter p is obtained as a function ofthe original image pixels only, i.e.

p(n) =

iRn

xh0(i) xp0(i)iRn

x2p0(i)=

q(n)

r(n). (25)

The estimate ofp is obtained as a function of the image pixelscorrupted with noise

p(n) =

iRn

xh(i) xp(i)

iRn

x2p(i)=

q(n)

r(n). (26)

In Appendix C the closed form expressions for the biases

of the numerator and denominator of p are obtained and it

is shown that the values of bias(q) and bias(r) are neitherdependent on the value of the original signal x0 nor on theshape of the region. The biases depend only on the size of the

region on which the estimation is based and the variance of the

additive zero-mean noise present in the image. It is reasonable

to assume that the variance of the additive noise is known (or

can be easily estimated).

Therefore, we introduce an improved estimator pc as

pc(n) =q(n) + bias(q(n))

r(n) + bias(r(n))=

qc(n)

rc(n), (27)

which is not exactly compensating (24), but it gives acceptable

estimates of parameter p.

Using expressions (75) and (79) for the bias compensation,

the improved estimator of parameter p is obtained as

pc(n) =

iRn

xh(i) xp(i) +N

162w

iRn

x2p(i) 6N

2562w

, (28)

where N represents the number of samples included in theadaptation region Rn. Acceptable results are obtained as longas the denominator is not close to 0. Fortunately, in that case

the adaptation is not necessary at all, since the signal is already

well suppressed by the fixed polynomial part.

The histograms of p(n) and pc(n), calculated for a singlepixel of a test image, are shown in Figs. 12(a) and 12(b)

respectively. The test image used is composed of a single sine

wave pattern with period T = 10 and amplitude of 100. Ascan be seen from the figures, the improved estimator has a

significantly reduced bias.

Similar results are obtained for the PUPa structure as well.

The adaptive PUPa analysis FB is shown in Fig. 13(a), and

the equivalent simplified structure is shown in Fig. 13(b).

X

P2(zD)

D-

z1-1

P42(zD)

-Xh

A

Xh^p

Xp

+

U2(zD)

X

D

P242(zD)

-

p

H(zD)Xh

Xh^

Xp

(a)

(b)

Fig. 13. (a) The first decomposition level of the adaptive PUPa filter bank,and (b) the simplified scheme of its high-pass channel.

Similarly to the derivation in Section C, the improved

estimation of parameter p can be obtained as

pc(n) =

iRn

xh(i) xp(i) +13N

2562w

iRn

x2p(i) 2533N

1310722w

. (29)


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0 0.5 10

100

200

300

(a)

0.5 1 1.50

100

200

300

(b)

Fig. 12. Histograms of (a) p(n) and (b) improved pc(n) for a single pixel of an image with T = 10 sine-wave calculated for a square region of size 5 5.The example is given for the PPaU structure. Input noise is white Gaussian zero-mean with w = 10. The true value of parameter p = 1.106 is shown bythe vertical dash-dot line. The bias of p(n) is 0.2273, while for pc(n) the bias equals to 0.0031.

E. Adaptation in the Iterated Filter Bank

For the iterated PPaU filter bank, the adaptation of parame-

ter pj at the j-th level is based on the corresponding Pj42and Hj filters, which are based on the j-times quincunx-upsampled predict and update filters. Therefore, pj42 is theimpulse response of the j-level FB structure up to the input tothe multiplier pj . Also, hj is the impulse response of the j-level FB structure up to the summation operator at the output

of the pj multiplier, much as in Fig. 10(b).It is important to note that the adaptation at lower levels

affects the Pj42 and Hj filters at all the successive levels. Since

the adaptation is being performed pixel-wise, the impulse

responses pj42(n) and hj(n) are different for every pixel

position n. Therefore, the bias correction at level j will havedifferent values for every pixel of the image, depending on

the adaptation at the previous j 1 levels. The estimation ofparameter p at level j and for the position n is obtained as

pjc(n) =qj(n) + bias(qj(n))

rj

(n) + bias(rj

(n))

. (30)

The numerators bias is obtained as

bias(qj(n)) = iRn

E

wjh(i)wjp(i)

= N 2wk=l

hj(n, k)pj42(n, l),(31)

and the denominators bias is obtained as

bias(rj(n)) = iRn

E

(wjp(i))2

= N 2wk

(pj42(n, k))2,

(32)

where N is the number of pixels included in the region.In contrast, in the iterated PUPa structure the Pj242 and

Hj filters at level j do not depend on the adaptation at thelower decomposition levels. The p parameters at level j arecalculated by using (30), but the numerator and denominator

biases are independent of the pixel position:

bias(qj(n)) = N 2wk=l

hj(k)pj242(l), (33)

bias(rj(n)) = N 2wk

(pj242(k))2. (34)

IV. THE ICI RULE

So far, we have not discussed a way of determining the

neighborhoods of similar statistical properties for every pixel

of the image.

Let us suppose that a compact representation of an analyzed

image can be obtained by setting the appropriate values of

the FB parameter p in distinct and appropriate regions of theimage. In the simplest setting, the adaptation of parameter pcan be performed on a fixed-size region. It is important to

note that wider adaptation regions lead to smaller varianceof p. So, in order to obtain more reliable estimates we wantto have parameter p calculated on a region that is as big aspossible, but still not incorporating parts of the image with

different statistical properties. To obtain such regions, we use

a statistical method based on the intersection of confidence

intervals (ICI) rule [28], [33], which is applied for every pixel

in order to obtain the most appropriate adaptation region for

it.

The ICI rule is based on a number of estimates of some

signal-related parameter. The estimates are based on different

window sizes. We define a set of growing window sizes:

H

= {hk|hk > hk1, k = 1, 2, . . . , K }. (35)Window hk1 is a subset of the next window hk. The Kestimates are calculated for K successive windows. Next, wedefine confidence intervals of the estimate xk as

Ck = [pc,k(n) p, pc,k(n) + p], (36)

where p = std(pc,k(n)), and is an empirically-set constantthat defines the width of the confidence interval.

The ICI algorithm starts with the smallest window size

and calculates the appropriate confidence interval C1. Thenthe second estimate is calculated, and its confidence interval

C2 is obtained. The intersection of confidence intervals for ksuccessive estimates is

Ik = k

i=1Ci. (37)One wants to obtain an estimate that is based on the biggest

possible window size but still statistically related to the previ-

ous estimates. In terms of confidence intervals this means that

the closest-to-optimal scale, which will be denoted as k+, willbe the largest of indices for which Ik = . Therefore, the ICIalgorithm consists of the following steps:

First estimate. Obtain the estimate pc for the smallest window.Calculate L1 and U1 as the lower and the upper bounds of thefirst confidence interval obtained from (36).

Successive estimates. Calculate successive estimates based

on the growing windows. Track the value of the highest


11/17


12/17




(a)

pc

=1.099

(b)

pc

=1.12

(c)

Fig. 16. (a) The test SineCircle image. The test image was additionally corrupted with Gaussian white noise with w = 10. (b) Adaptive wedge-shapedregion and (c) the overall region obtained for the PUPa structure with 95% confidence level are shown by black dots. The origin pixel is shown by the whitecircle. The regions are plotted over the noisy test image to outline the true region borders. True values ofp (obtained for the original image without noise)are 1.111 and 1.742 for the inner and outer sine waves respectively.

(a) (b) (c)

Fig. 17. (a) Part of the Barbara image and (b) the corresponding detail coefficients obtained for the first decomposition level of the fixed P2U2 structure,and (c) for the adaptive PUPa structure with 5 5 window adaptation.

As mentioned earlier, for denoising purposes we have used

the undecimated FB. The denoising results are given for 4-

level adaptive wavelet decomposition based on the quincunx

PUPa filter bank with 2 dual and 2 primal vanishing moments.

Therefore, in each decomposition level, fixed P2 and U2 stagesare applied, followed by the adaptive P42 branch in which thevalue of parameter p is calculated based on the 5x5 pixelswide neighborhood of each pixel. Since undecimated FB is

employed, we have used j-times quincunx-upsampled predictand update filters, where j stands for the decomposition level.The synthetic test image used was the 512x512 SineCircle

image. The sine wave inside the circular region has period

T1 = 9 and angular orientation 1 = /8, while for the outersine wave T2 = 5 and 2 = /4.

A part of the noisy SineCircle image, and the images

obtained with the three denoising methods are shown in

Fig. 19. The resulting detail coefficients obtained with the

adapted p parameters are shown in Fig. 19(c). When comparedto the detail coefficients obtained with nonadaptive wavelet

decomposition (Fig. 19(b)), it is obvious that the adaptation

introduced significant improvements, setting the detail coef-

ficients to zero in the whole image except at the circular

boundary of the two sine-wave patterns.

Results from Figs. 19(d) and 19(e) show that the adaptation

introduced in the FB significantly improves denoising results.

The adaptation transferred the information on the harmonic

components to the filter parameters, and the noise present

in the image was more efficiently suppressed by the wavelet

thresholding operation. Fig. 19(f) shows the result obtained

by using the state-of-the-art pointwise shape-adaptive DCT

method (SA-DCT) [19]. It is clearly visible that our method

more accurately restores the periodic pattern in the image.

Note that the SA-DCT method was used for comparison since

it also uses the ICI rule to obtain appropriate adaptation

regions.

The obtained denoising results are summarized in Table

II. For comparison purposes we have added denosing results

obtained with BLS-GSM denoising method [18] and wavelet-

domain Wiener filtering (WDWF) [35], which uses separable

db2 and sym6 wavelets. For the synthetic SineCircle and

Ripples image, the improvement in terms of peak signal-to-

noise ratio (PSNR) when using adaptive wavelets is signifi-

cant, hence indicating that the proposed method would be of

particular interest in the analysis of images with dominant

periodic components. All the test images are available at

http://www.riteh.hr/mvrankic/images.


13/17




(a) (b) (c)

(d) (e) (f)

Fig. 19. (a) Part of the noisy SineCircle image with w = 10 . (b) Corresponding detail coefficients for the first decomposition level of the unadapted P2U2structure, and (c) for the PUPa structure with adaptation using a fixed 5 5 window. Denoising results for (d) 5 5 window-based adaptation, (e) ICI-basedadaptation, and (f) the result of the point-wise SA-DCT method from [19].

VI. CONCLUSIONS

In this paper, we have presented a novel design technique forlocally adaptive wavelet decompositions based on an adaptive

lifting scheme structure. The main purpose of the adaptation

was to obtain a more compact representation of an analyzed

image when compared to the nonadaptive wavelet transform.

The adaptation involves a modification of the predict stage

in the lifting scheme, and two new schemes, named PPaU and

PUPa have been proposed, both of which retain a desirable

number of dual and primal vanishing moments. The main

advantage of the PUPa structure over the PPaU structure

is that the adaptation at one level does not influence the

following levels, leading to its computationally less demanding

implementation.

The adaptation of the filter bank parameters ensured theminimization of the squared prediction error on the neighbor-

hood of a given pixel, which then led to the minimization of

the energy of the wavelet detail coefficients. The influence of

the additive zero-mean Gaussian noise on the estimation of the

adaptive filter parameter p has been also studied. To reducethe bias of p we have proposed the improved estimator pc,whose bias is significantly reduced.

In order to obtain an appropriate adaptation region for every

pixel, we employed a statistical method called the intersection

of confidence intervals (ICI) rule. The ICI method gives re-

gions of similar statistical properties for estimating parameter

p. While it is computationally demanding, the ICI methodsignificantly improves the estimation of parameter p. Wehave shown that, for synthetic images composed of localized

periodic components, the proposed adaptive wavelet FBs give

high-quality image denoising results, even outperforming the

current state-of-the-art pointwise SA-DCT method.

APPENDIX A

PRESERVATION O F VANISHING MOMENTS IN THE PPAU

STRUCTURE

The analysis polyphase matrix for the general lifting scheme

structure from Fig. 1 can be written as:

Hp =

1 U

0 1

1 0

P 1

. (38)

Therefore: H0e H0o

H1e H1o

=

1 U P U

P 1

, (39)

where H0e, H1e, H0o, H1o respectively denote even and oddpolyphase components of the low-pass and high-pass H0 andH1 filters.

The dual vanishing moments (DM) condition requires that

[29]

( D)H1(Z2) = 0, for N, (40)

where N denotes the space of all polynomial sequences oftotal degree strictly less than N. By using the filter polyphase


14/17




TABLE IIDENOISING RESULTS IN TERMS OF PSNR. P2U2 STANDS FOR NONADAPTED QUINCUNX WAVELETS WITH 2 DUAL AND 2 PRIMAL VANISHING MOMENTS.

Image name P2U2 PUPa BLS-GSM Pointwise SA-DCT WDWF

SineCircle 5 34.35 41.97 40.40 36.03 36.66

10 28.43 36.46 35.30 31.37 32.15

20 22.74 30.40 30.05 26.75 27.53

40 17.10 21.98 23.78 23.54 20.99

Ripples 5 35.20 43.29 41.27 35.32 38.03

10 29.02 37.36 36.49 30.22 33.12

20 24.12 31.74 31.27 26.04 28.98

40 17.23 25.93 25.74 22.42 25.27

Barbara 5 35.48 35.73 37.60 37.42 37.28

10 30.26 30.98 33.51 33.58 32.87

20 25.87 26.68 29.22 29.88 28.24

40 20.34 23.45 25.41 26.25 24.33

Textile 5 33.01 34.12 33.95 25.34 34.09

10 27.93 28.21 28.54 24.02 28.12

20 22.14 22.57 23.35 21.49 22.56

40 16.05 18.58 18.88 18.42 18.15

Brick wall 5 33.05 34.14 33.98 27.11 34.18

10 27.89 28.25 28.74 25.19 28.30

20 22.08 22.96 23.67 22.24 22.91

40 16.35 19.21 19.60 19.38 18.87

0 5 10 15 20

-2

-1

0

1

2

3

(a)

0 5 10 15 20

0

0.5

1

1.5

2

(b)

Fig. 15. (a) The ICI graph and (b) its magnified view for pc calculatedfor w = 10 and 95% confidence level, resulting in the final wedge-shapedregion in Fig. 16(b). The dashed horizonatal line in (b) represents the truevalue ofp (which is unknown to the ICI algorithm).

representation, the output of the analysis high pass filter can

be obtained with each polyphase component affecting one

(a)

1.8 2 2.2 2.4 2.6 2.80

5

10

15

20

25

(b)

1.8 2 2.2 2.4 2.6 2.80

5

10

15

20

25

(c)

Fig. 18. The histograms of pc parameters for the part of Barbaras scarf,noisy with w = 10, which is shown in (a), for the first decompositionlevel of the undecimated PUPa structure. The histograms are shown for (b)adaptation using a fixed 5 5 window and (c) the ICI-based adaptation with95% confidence level.

sublattice ofZ2, which transforms the DM condition to

H1e(DZ2

) + H1o(DZ2

+ t) = 0. (41)By using (39), the DM condition becomes

P (DZ2) + (DZ2 + t) = 0, (42)

which yields

P (DZ2) = (DZ2 + t). (43)

As given in [29], (43) can be expressed in the downsampled

domain as

P (Z2) = (Z2 +D1t), for N. (44)

Now, introducing the overall prediction filter from Fig. 3,

which consists of the fixed part PN and the adaptive part pPa,


15/17




we obtain(PN + pPa)(Z

2) = (Z2 +D1t),

PN(Z2) + pPa(Z

2) = (Z2 +D1t).(45)

Since PN is a Neville filter of order N and shift = D1t,

we have

PN(Z2) = (Z2 +D1t), for N, (46)

which combined with (45) gives

(Z2 +D1t) + pPa(Z2) = (Z2 +D1t), (47)

which further simplifies the DM condition to

pPa(Z2) = 0, for N. (48)

So, in order to keep N dual vanishing moments introducedwith PN, the filter Pa must cancel polynomials of degree lowerthan N. As stated in Section II-C, we propose filter Pa suchthat

Pa = PR PS , (49)

where PR and PS are Neville filters of shift = D1t and

orders

R N, S N, (50)

respectively. Applying the filter to a polynomial sequence N gives

p(PR PS)(Z2) = 0, (51)

since

PR(Z2) = PS(Z

2) = (Z2 +D1t). (52)

Therefore, with the choice of the Pa filter from (49), the Ndual vanishing moments are guaranteed no matter what values

the p parameter may take.Now, we will show that for the PPaU filter bank the number

of primal vanishing moments N that is fixed with the updatefilter is not influenced by the adaptive prediction branch as

long as N N

. The synthesis polyphase matrix follows

directly from the analysis polyphase matrix as Gp = H1p

Therefore,G0e G0o

G1e G1o

=

1 P

N+

UN 1 UNP

N+

, (53)

where G0 and G1 denote synthesis low-pass and high-passfilters respectively, and PN+ is the overall prediction filter asgiven in (8). The primal vanishing moments (PM) condition

states that in order to have N vanishing moments of thesynthesis FB, the following relation must be satisfied

G1( D) = 0 for N, (54)

where N denotes the space of all polynomial sequences of

total degree strictly less than N. Stated in terms of polyphasecomponents, the PM condition becomes

G1e(DZ2) + G1o(DZ

2 + t) = 0, (55)

which combined with (53) gives

UN(DZ2) + (1 UNP

N+)(DZ2 + t) = 0, (56)

UN(DZ2) + UNP

N+(DZ2 + t) = (DZ2 + t). (57)

IfN N, R N, and S N, thenPN+

(DZ2 + t) = (PN

+ p(PR

PS

))(DZ2 + t)

= PN

(DZ2 + t)

= (DZ2),

(58)

since

p(PR

PS

)(DZ2 + t) = 0. (59)

Therefore, the PM condition from (57) becomes

2UN(DZ2) = (DZ2 + t), for N, (60)

which is always true, since 2UN = PN, as stated in (4).

APPENDIX B

PRESERVATION O F VANISHING MOMENTS IN THE PUPA

STRUCTURE

The polyphase matrix of the analysis PUPa filter bank from

Fig. 4 is

Hp =

1 0

pPa 1

1 UN

0 1

1 0

PN 1

= 1 PNUN UNpPa(1 PNUN) PN pPaUN + 1

,(61)

where filter Pa is given as in (49). Therefore, the polyphasecomponents of the analysis high-pass filter are

H1e = pPa(1 PNUN) PN,

H1o = pPaUN + 1.(62)

The DM condition can be stated in the polyphase form as

H1e(DZ2) + H1o(DZ

2 + t) = 0, for N. (63)

Since, as stated in (48), it holds that

Pa(Z2) = 0, for N, (64)

using (62) the DM condition from (63) becomes

PN(DZ2) = (DZ2 + t), for N, (65)

which is always true, and therefore the N dual vanishingmoments are guaranteed for any value of parameter p.

The synthesis polyphase matrix of the PUPa filter bank from

Fig. 4 can be obtained from the analysis polyphase matrix (61)

as

Gp =

pPaUN + 1 pPa(1 P

NUN) + P

N

UN 1 P

NUN

. (66)

Therefore, the polyphase components of the synthesis high-

pass filter G1 are

G1e = UN,

G1o = 1 P

NUN,

(67)

which are the same as those for the PPaU structure; so,based on the proof in Section A, we conclude that N primalvanishing moments are preserved as long as N N.

APPENDIX C

NUMERATOR AND DENOMINATOR BIAS FOR THE

ESTIMATOR OF p

The bias of the numerator q from (26) is

bias(q(n)) = E[q(n) q(n)] = q(n) E[q(n)] (68)


16/17




where

E[q(n)] = E

iRn

xh(i)xp(i)

= E

iRn

(x0h(i) + wh(i))(x0p(i) + wp(i))

= iRn

(x0h(i)x0p(i) + E[wh(i)wp(i)])

= q(n) +iRn

E[wh(i)wp(i)] ,

(69)

since E[wp(i)] = 0 and E[wh(i)] = 0. Combining (68) and(69) the bias can be expressed as

bias(q(n)) = iRn

E[wh(i)wp(i)] . (70)

Knowing that (see Fig. 11)

wp(n) = (w p42)n =2

32

5i=2

w(n, i) +1

32

13i=6

w(n, i), (71)

wh(n) = (w h)n = w(n, 1) 1

4

5n=2

w(n, i), (72)

and by using the fact that

E[w(n, k)w(n, l)] =

0 for k = l

2w for k = l, (73)

we can write

E[wh(i)wp(i)] = E

1

4

2

32

5k=2

w2(i, k)

= 1

4

2

32 42w =

1

162w.

(74)

By combining (70) and (74), the bias of q can be expressed

as

bias(q(n)) = iRn

E[wh(i)wp(i)] = N2w16

, (75)

where N is the number of pixels included in the region Rn.The bias of the denominator r from (26) is

bias(r(n)) = E[r(n) r(n)] = r(n) E[r(n)] (76)

where the expectation of r(n) is:

E[r(n)] = E

iRn

x2p(i)

= E

iRn

(x0p(i) + wp(i))2

= EiRn

x20p(i) + 2x0p(i)wp(i) + w2p(i)

=iRn

(x20p(i) + E[w2p(i)])

= r(n) +iRn

E[w2p(i)],

(77)

since from E[w(i)] = 0 it follows that E[wp(i)] = 0. Next,since wp = w p42,

E[w2p(i)] = 2w

n

p242(n) =6

2562w. (78)

By using (76), (77), and (78) the denominators bias can now

be expressed as

bias(r(n)) = iRn

E[w2p(i)] = N6

2562w, (79)

where again N is the number of pixels included in the regionRn.

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adaptive 2-d wavelet transform based on the_lifting scheme with preserved vanishing moments_

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