Design of signal-adapted multidimensional lifting ... - Semantic Scholar

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Design of signal-adapted multidimensional lifting scheme for lossy coding A. Gouze, M. Antonini, M. Barlaud, Fellow Member, IEEE and B. Macq, Senior Member, IEEE

Abstract This paper proposes a new method for the design of lifting filters to compute a multidimensional nonseparable wavelet transform. Our approach is stated in the general case, and is illustrated for the 2-D separable and for the quincunx images. Results are shown for the JPEG2000 database and for satellite image acquired on a quincunx sampling grid. The design of efficient quincunx filters is a difficult challenge which has already been addressed for specific cases. Our approach enables the design of less expensive filters adapted to the signal statistics to enhance the compression efficiency in a more general case. It is based on a two-step lifting scheme, and joins lifting theory with Wiener’s optimization. The prediction step is designed in order to minimize the variance of the signal, and the update step is designed in order to minimize a reconstruction error. Application for lossy compression shows the performances of the method. Index Terms Lifting scheme, nonseparable filtering, multidimensional, quincunx, biorthogonal wavelet transform, filter design, signal statistics.

I. I NTRODUCTION Wavelet transform provides a multiresolution representation of signals and enables decorrelation in space and frequency. Through its regularity and decorrelation properties, wavelets are powerful and adapted tool to data compression. Mallat [1] develops a pyramidal wavelet transform using of a numerical filter bank. Since last decades, many works proposed some filter constructions. The most popular of them is the Daubechies (9,7) filter bank. Its success is due to its nice regularity and energy compaction properties [2]. To generalize the concept of filter bank, work had started on the definition of multidimensional filter banks, and particularly nonseparable two-dimensional filter banks. The construction of Siohan is based on the definition of 2D half-band filters [3]. That of Moreau de Saint Martin [4] uses the Gröbner bases. Constructions of Ansari, Guillemot,

Kim [5], [6], Antonini, Barlaud [2], [7], [8], Kova evi and Vetterli [9], [10] use 1D-2D extension transforms, such as the rotational transform or the McClellan transform. All these methods are providing direct filter bank design algorithms. Sweldens introduced the lifting scheme as an efficient and powerful tool to compute wavelet transform [11], [12]. Daubechies showed in [13] how to factorize 1D biorthogonal wavelet filter bank in lifting scheme. In [14], we showed how to obtain quincunx lifting filters from 1D lifting filters. The idea of building A. Gouze, M. Antonini and M. Barlaud are with the Université de Nice Sophia Antipolis and the CNRS. A. Gouze and B. Macq is with the Université Catholique de Louvain la Neuve.

March 16, 2004

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2

wavelets without using of Fourier transform appeared before lifting. Indeed, the lifting scheme is both inspired by the work of Donoho [15] and by the one of Lounsbery [16] concerning the transform on meshes. In their work [17], Sweldens and Schröder designed 1D lifting operators by integrating primal and dual lifting steps into a basic lifting scheme (for more details, see paragraph II-A or [17]). For example, a lifting step can be designed in order to increase the number of vanishing moments of the wavelet, or to preserve the signal

approximation mean on each level [17]. Kova evi [18] defines a multidimensional lifting scheme based on the same interpolation principle. In this paper we will follow the lifting representation of the filter banks but we choose to rely the design on the statistics of the images. This idea was presented in [19], [20], for the design of 1D para-unit filters and leads to optimal compaction of the wavelet transform. Hampson and Pesquet built M-band non-linear filters banks with perfect reconstruction and maximal decimation [21]. Gerek and Çetin introduced adaptive filter banks with perfect reconstruction [22]. They proposed a method to update filter coefficients by a reduction around edges. Majority of this work proposed method for the design of 1D filter bank. Claypoole introduced a technique of adaptive filtering which enables to choose the prediction operator according to the local properties of the image [23]. In the same way, Taubman develops a technique of adaptive lifting scheme in [24]. His method consider a 1d filtering on lines and a 1D filtering on row and takes into account the direction of 1D vertical filtering to determine the predictor step and objects to minimize the quantization errors. The main difference between our work and the Claypoole or Taubman one is that our method provides linear transforms. Claypoole uses the principle of lifting to associate it with methods of nonlinear and adaptive filtering. Boulgouris presented an adaptive non-linear filtering for lossless coding [25]. The majority of these approaches base itself primarily on the use of a known filter, and define thereafter a non-linear filtering by the means of various methods. Boulgouris proposes to define the lifting operators according to condition on the sum of the coefficients and while seeking to reduce the variance of the signal high frequency like in [19], [20]. Their design is proposed for first order Markov processes with a correlation factor of 0.95. Their filters are therefore adapted for a specific signal model. Our goal is to build lifting filters adapted to the original data without any a priori model of the image autocovariance [26]. The prediction stage of the lifting scheme is designed to achieve best compaction of the signal, i.e. minimization of the high frequency energy like [19], [20] and [25]. While [25] optimized their filter bank for lossless compression, we propose a second criterion to optimize the update step. This second criterion is useful to rise performances in case of lossy compression. We propose to optimize update lifting step in order to minimize the distortion between the original image and the reconstructed image, when the high frequency is removed. The method is defined in the general case of multidimensional nonseparable filtering, and aims to design a M-dimensional two-step lifting scheme. We proposed applications related to the particular case of quincunx filtering. The development of new quincunx sampling image acquisition devices and quincunx displays justifies research in order to improve the performances of quincunx filtering. This new way of image acquisition and display offers an increase of

in the resolution of the images compared to an orthogonal sampling. This path is

followed by the earth observation satellite SPOT 5 and by some recent video cards which have a quincunx antialiasing mode. March 16, 2004

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3

Our contribution is described in 4 parts. In section II, we introduce the problem and state the criteria definition. Section III develops the constrained criteria and presents methodology employed to obtain lifting operators. Section V presents the particular case of bidimensional signal sampled on a quincunx grid. Section VI presents some application to lossy compression. Comparison with method of Boulgouris and all. [25] and (9,7) used for JPEG2000 [2], [27] are also presented. Numerical experimentations show the efficiency of the proposed method by a gain of quality for compression. II. P ROBLEM

STATEMENT: DEFINITION OF THE CRITERIA

In this section, we first give a brief overview on the lifting scheme construction. Then, we explain the selection of the two criteria. Finally, we give constraints on the obtained filter bank. A. Background on the lifting scheme x e = s(0) x

s

P

SPLIT

U d

xo = d (0) Fig. 1.

Canonical two steps lifting scheme.

The lifting scheme implementation is simple, fast, efficient and easily invertible [28]. Indeed, the wavelet transform is performed by a simple undoing of operations. The canonical scheme consists in three main steps shown in Fig. 1 [29]. In a first step, the original signal, , is split into two disjoint subsets. The split is a 2-band polyphase transform, which classifies the signal samples in two components, and , defined by the even and odd index signal samples. Those two components are highly correlated, then one can apply an operator

on to predict . The details resulting from the prediction correspond to the set of the wavelet

coefficients. In a third stage, the signal can be seen as a signal representation on a lower resolution. A simple subsampling provides a signal occupying the totality of the frequential band of the original signal. The condition of Shannon is not observed what cause a spectral folding. To get a more adequate low frequency representation, one can apply an operator

on wavelet coefficients to update .

This yields the following equations:

(1) (2)

where and denote the two subsets resulting from the split, and and respectively denote the low and high frequency signals at resolution

!#"$

. The general scheme can include several lifting steps. The lifting

is built by alternating different prediction and update steps. The inverse lifting scheme consists to undo the operation and a merge (inversion of the polyphase transform).

March 16, 2004

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4

B. Problem statement The optimization algorithm we propose is based on a simple two-step lifting scheme (one prediction and one update step). Such a scheme is sufficient enough to have a broad panel of filters. The first stage consists in optimizing the predictor, and the second, the update step. The filters are systematically transmitted to the decoder in order to proceed to the inverse transform. The bit-rate of the coded signal is thus increased to a negligible amount. C. Definition of the optimal predictor The prediction step provides the details as illustrated by figure 1. A good predictor

has to generate a

signal of coefficients with suitable properties for compression. According to the coding theory, a signal with minimum variance will be coded with the highest efficiency [30], [31]. Indeed, quantization error is a function of the signal energy. The optimal bit allocation defines quantization steps for a given bit-rate, called target bitrate. The model-based bit allocation characterizes the high frequency signal by a Gaussian or other model of probability density [32]–[34]. They define the quantization steps, which minimize the distortion. The distortion is therefore dependent on the signal variance. Then, a reliable and adequate criterion for the application of a coder is to minimize the variance of the wavelet coefficients signal:

%'& The optimization of the criterion

% &

( )

$

(3)

specifies a predictor which minimizes the variance of the wavelet coeffi-

cients. All the discussion leads us to take the same criterion than [19], [25]. Although the criteria are identical, our approach differentiates in the method used to find the optimal operator

.

D. Definition of the optimal update operator s (0)

s

U

P

MERGE

d =0

Fig. 2.

Backward lifting scheme for

* +-,.+

and

x

d (0)

/*

,.0

.

Following the design of the prediction operator, one can compute the wavelet coefficients. An update operator

is then applied to these coefficients, in order to approximate the signal on a lower resolution. An efficient

operator returns low frequency signal offering an accurate representation of the original signal at this lower resolution. The image of wavelet coefficients is supposed to undergo more or less big degradations, when the wavelet transform is associated to a coder which generates losses of information. It is thus of primary importance to be able to ensure the most reliable signal reconstruction, by starting from the single low frequency image. We propose an original design of the filter

that determines the minimal reconstruction error, by vanishing

the high frequency signal 1 . The criterion is defined as the quadratic error of reconstruction:

%32

March 16, 2004

546798 ;:

$3
= ?A@ = ?#BCD ? B; @

P E @ =CE F@ = E HG IJ'KMLON

(5)

I = I P E B E QB E HG IJ'KML N

(6)

I B I >P E = E = E HG IJ'KRTS

(7)

I @ = I P E @B E @B E HG IJ'K R S & 2 (resp V ) the support of the operator where U is the scaling factor defining the subsampling, and V

(8) (resp.

) of dimension W . These theorems are applicable in M dimensions [12], [35]. They state that the analysis and synthesis filter banks corresponding to the lifting scheme are biorthogonal in two conditions: the lifting scheme is composed of a basic biorthogonal transform and the operators

and have a compact support. The proposed

lifting operators have finite impulse response. They are biorthogonal and thus, with perfect reconstruction. 2) Filtering “normalization” constraints : It is meaningful to require two further conditions for the twostep biorthogonal lifting scheme. The first constraint is to vanish the wavelet coefficients mean in the detail band. The second condition imposes the mean conservation on each low frequency subband, and enables to avoid capacity overflow during the implementation. These two constraints are associated with their counterpart constraints on the corresponding filter bank. They impose that corresponding wavelet filters @ = , B@ , = and B check, for the analysis:

Y

XY J'KMZ =

and for the synthesis:

\[ Y

XQY J'KMdZ @ =

Y

X5Y

J'KMa B

7^]_[`:

7c]

5b

7AV

Y

X5Y 7AVe[`:

5b

J'KMda B@

:

:

Let us define by V-f , V>g , VCh f , V;hg the “M-dimensional” index sets being used as sampling spaces for the different filters. These domains are included in i

for the case of one-dimensional signals, in i

$

for the case of two-

dimensional signals, and in ij for the case of three-dimensional signals. One can note that the mean conservation

March 16, 2004

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6

condition on low frequency band involves the filters normalization as in (A1) and in (S1). For a two step lifting scheme with the first step given by (1), and from the Sweldens theorems, we can deduce:

Y G

J'KMdZ

Y

@ =lk mn o[

G

k mn

J'K L N

(9)

With the second lifting step 2, and from Sweldens theorem, we deduce the second sum on coefficients :

Y G

J'Kda

Y

B@ k mn 5b_\[

G

k mnrq [

J'K RpS

G k stnvu E 'J K L N

One can associate the preceding sums respectively with constraints 7cVe[w: and 7cV conditions on operators

(10)

: . These two conditions imply

and , deduced from previous calculations (9) and (10). we retrieve the same

conditions stated for the quincunx case by Boulgouris in [25]:

Y

and

Y G

G

J'Kwx N

J'K'R S

k mn y[

(11)

[

k mn

(12)

The conditions (11) and (12) will be added to constrain the previous criteria (3) and (4). 3) Symmetrical linear phase constraints: Among the constraints added to our criteria, we impose the zerophase condition on the associated low pass filters, = and @ = . By extension, the condition implies the linearity of the phase of the high-pass filters B and B@ [36]. This condition is very significant in image compression, it allows to avoid noticeable visual distortions [2]. A nonlinear phase filter will cause significant degradation near contours. The operators

and

must be symmetric linear phase FIR filters, such as the corresponding

wavelet filters @ = , B@ , = and B are also symmetric linear phase FIR filters. In the case of supports z{i , zero phase implies by definition (see [36])

=-k mn =|k mn

and @ =|k mn @ =|k mn9}

(13)

From equations (9) and (10), we deduce:

In the case of support z~i

$

N

km [ n N

k mn

and

S

k mn S

k m [ nA}

(14)

, the linear phase imply in general case a two-fold symmetry (centrosymmetry) [9],

[36]. But we restricted the set of linear phase filter to octagonal filers to have less discontinuity [3], [37]. From this assumption, equations (9) and (10), we deduce:

N

k m! ? m!`n

k m! ? ! m [n N

k m ['? m [ n

N

k m [? m n

March 16, 2004

k ! m ['? m!n N

N

N

N

N

k m ? m n

k m ? m [ n

k m! [? m! [ n

(15)

DRAFT

7

and

S

k m! ? m! [ n

k m! ? m!n

D k m! ['? m!n S

k m [? m [ n

S

S

S

S

k m! [? m!wn S

D k m ? m n S D k m! ? ! m [n

k m! [? m! [ n

(16)

These constraints result in a significant reduction of our criterion degree of freedom. Thus, for a quincunx operator of size . , the number of coefficients to be defined is not any more about III. D EFINITION

$

but about 7^

:.

AND DESCRIPTION OF THE OPTIMIZATION METHOD UNDER CONSTRAINTS

Considering the constraints modifies criteria stated in 3 and 4. In this section, we first define the new criteria to design lifting operator of a two-step lifting scheme. Then, one can optimize the criteria according to the coefficients of each operator. Previously, we pointed out some concepts of statistics occurring in our operations. A. Background on statistics - Notations The image statistic computation is based on the assumption that each pixel of the image is considered as a random variable

[38]. The image can be seen as the ordered set of all these random variables and thus as a

random process with mathematical expectation: 47 k mn : , covariance 7 k mn ?# k mn : , and variance: (

$

7 k mn : .

In the case of a stationary process, these values are independent from m :

47 k mn :rQ

(17)

7 k mn ?# k m n :r4 k 7^ k mn :>7 k m n : n 7 m m :

(18)

and

(

$

7 k mn :r497 k mn :

$

$ $ Q 7^bt:r5(

(19)

Moreover, we assume ergodicity. B. Design of the predictor

1) Constrained criterion: The minimization of the variance of the high frequency signal, under the constraint (11) is performed in order to define the prediction operator. Let be a vector gathering in an ordered manner all the filter coefficients

, let

&

be the index set of

coefficients and let be the Lagrange operator.

The new criterion can be expressed as follows:

%t& $

$ 7 ? C:5( ) q [ G E u E JwLON

The variance ( ) depends on the signal , determined on theY3first lifting step. The

(20) and

filters are supposed

to be symmetrical. Consequently, we define the variable V E gathering all pixels of image ' to which the same coefficient E is applied (because of the symmetry) when

N

March 16, 2004

is centered out of m (m “indexing” a point

DRAFT

8

of the support of the signal ' ). This formulation allows to both simplify notations and generalize method to other types of regular sampling grid. Let us define by

&-

the set of index of coefficients of

Thus, the first lifting step (1) is generalizedY by:

restricted to non-identical coefficient by symmetry.

Y

Y

Q G E V E E Jw L3 N

Let be the vector gathering all

(21)

coefficients, which are non-identical by symmetry. We have y

The criterion (20) is then expressed as follows:

% &

$ 7^? C:5( ) q [

G E§¦ tK ¨ 7 s : u E JwLtwN

N

E¢¡Ms¤£

&|`¥

(22)

¦ tK ¨ 7 s : is the number of coefficients equal to the E value by symmetry ( E included). If ¦ Lt is size of N N &- , the system has 7 ¦ L' [`: equations with 7 ¦ L3 [`: unknowns. The minimal value vanishes the partial &- derivatives (23) and (24), and verifies for s in : © %'& &- 7T?;: © Qb«ª s£ (23) E © %'& N 7T?;: © Qb (24) 2) Criterion expansion: Before differentiating the criterion (22) with respect to the 7¬

[w: variables (¬

represents the size of the vector ), it is necessary to expand the latter:

%'&

Y

7^? C:4

) :

7^

$

q [ ®G ¦ K ¨ 7 s : Ewu N E Jw L3

(25)

The use of equation (19) enables to rewrite the criterion in the following way:

% &

Y

7^? C:497^

:

$

$ ) q [

The criterion expansion gives:

%'&

Yy²

7T?;:4°¯±

$³

Y

(26)

Y3

4µ´ G G E E ¶ V E V E ¶¸· E Jw tL E ¶ Jw tL N N Y 3Y Y ² \ 4°´ G E V E · E Jw L' N

G ¦ Kt¨ 7 s : Ewu N E JwL3

´94¹±` D

Y

$

G E 4yºAV E§» · E Jw Lt N

q [ G (27) ¦ K ¨ 7 s : Ewu N E Jw L3 Y3 sense stationary, i.e. ¼¹y )½ 9¨ ¾ y ½ ¨9¾ . This assumption allows us One can assume that the signal is wide K ¨ ¦ Kt¨ 7 s : ½ Ä¾ ¦ Kt¨ 7 s :!¼ , where ¦ Kt¨ 7 s : represents the to define the signal mean by 4¿tV EÀ Á '

March 16, 2004

DRAFT

.

9

Y number of coefficients contained in Vr E . Following this assumption,

% &

Y3 ²

$ 7T?;:( ) ½ ¨9¾

Y

G

E JwL'wN

G

can be defined by:

$ ¼ ¦ 'K ¨ 7 s : Y 3Y

± V E

E Â4

%

$ º V E V E ¶ » ¦ Kt¨ 7 s : ¦ Kt¨ 7 s : ¼Ä

G

E E ¶Ã 4 E JwMLt E ¶ JwL'wN N

q [ G ¦ Kt¨ 7 s : E u N E JwL3

(28)

3) Minimization of the criterion: 1) Criterion differentiation (28) with respect to the variable leads us to the equation (29):

© %t&

©

7T?;:

q [ ÅG ¦ K ¨ 7 s : Ewu N E Jw L3

(29)

The minimum value vanishes the partial derivative about , and thus it checks equation (24). We deduce

from it the condition on the sum of the coefficients of the operator

G ¦ K ¨ 7 s : E \[ N E Jw Lt &-

2) The equation (23) is solved, for all Æ in

© % & ©

7T?;: N

Â4

$ ¼ ¦ K'¨ 7^Æc: ¦ K'¨ 7^Æc: G E Ã4 E JwMLtwN

± V

The coefficients of the optimum operator

bÁ7¸ª>Æ £

&-

(30)

, as follows:

Y3 ²

Y

:

Y3

Y

ºV V E » $ ¦ K'¨ 7^Æc: ¦ Kt¨ 7 s : ¼ Ä

: implies: Y Y3

$ E Ã 4yºAV V ME » ¦ K ¨ ^7 Æc: ¦ K ¨ 7 s : ¼ Ä E wJ L' N G

By setting,

Y

Y ¦ K ¨ c7 Æc:r4

Y3 ²

± V

È

(33) (34)

[ ¦ Kt¨ 7^Æc:?

(35)

the equations (30) and (32) can be rewritten under the form of a linear system with 7 ¬

ÉÊÊ

Ë

(32)

Y3

March 16, 2004

$ ¼ ¦ K ¨ ^7 Æc:

$ ] E 5 4 º V V Y ME » 3Y ² ¦ K ¨ 7^Æc: ¦ K ¨ 7 s : ¼ ? $ Ç

V ¼ ¦ K'¨ 7^Æc:? 4

±

and

(31)

© '% & © %'& & annul the partial derivatives © , ª>Æ £ . Thus, © N N

[w: variables:

Ê

G ¦ K ¨ 7 s : E o[ N È ÊÊ E Jw L3 Ç ÊÌ G ] E E Í N E JwML3

ªÆ £

&-

(36)

DRAFT

10

The linear system (36) is rewritten under the form of a matrix system:

ÎÍÏ Ò Ò Ò Ò ÒÓ

Ï Ü

(37)

ÑÒ

with

Î

Ð

Ú Ú

] E ¨ ¨ .. .

ÔÔÔ

] E ¨ Õ ¦ Kt¨ 7 s :

ÔÔÔ

N

..

Ç

] EÕ ¨ .. .

.

¨ÛÔÔÔ

N

E Ç

ÔÔÔ N

ÔÔÔ

Ø Ø Ø Ø

. È ..

(38)

Ù ¥

[

Õ

Ø

Õ b

EÕ Ç

× Ø

¨

] EÕ Õ ¦ Kt¨ 7 s'Ö :

ÔÔÔ

E ¨ÛÔÔÔ

È

(39)

¥

(40)

The solution of system (37), is obtained from the inversion of a square matrix and is given by:

Ï The matrix inversion

Î

Î > Ð

(41)

is realized by classical algorithm. The vector

the system, is identified with:

Ï

y OÞßà?

Ï

¥ y E ¨ ? EÝ ? }}} ? EÕ ? , solution of N N N

¥

(42)

áÞ9ßà corresponds to the optimal vector and gather the coefficients of the optimal filter

.

C. Design of the update operator The update operator acts on the low frequency component of the image. A good low resolution approximation must allow the reconstruction of a higher resolution image as reliable as possible, and this in spite of the loss of the high frequency signal. The definition of the criterion suggest to annul the high frequency coefficients, by

8 Qb . Let us define by , , D , and the different signals occurring for the forward transform, and setting 8 , 8 , 8 , 8 and 8 the signals occurring for the backward transform (see Fig. 2). For the reconstruction stage, the backward transform takes as inputs the low frequency image on a lower level of resolution and a vanishing image of wavelet coefficients (see Fig. 2). The criterion defining the update operator quadratic error between the original image and the reconstructed image.

must minimize the

has to verify symmetry constraints

and (12). The criterion is defined as the reconstruction mean quadratic error under constraints:

$ [ G (43) ¦_ã 7 s : Ewu Ã 7c8 : Ä q S E Jw'Rt where â represents the vector gathering all the coefficients of the operator non-identical by symmetry and 2> the set of index of the coefficients of , restricted to the coefficients non-identical by symmetry. ¦ ã 7 s : % 2

7Aâ? C:r4

is the number of coefficients equal to

S

E by symmetry ( E included). To minimize , we have to take into S

account the imposed symmetries on . This consideration allows a reduction of the degree of freedom and allows to respect the assumption of stationarity. The optimal values of the coefficients of

© %2

©

7^â?;:

5bäª>Æ £

© % 2 S 7^â?;: © b 5 March 16, 2004

2>

verifies: (44) (45)

DRAFT

11

If ¦ 'Rt is size of

2>

, the system has 7 ¦ Rt

[`: equations with 7 ¦ Rt [`: unknowns. Since:

¤

å 8 {8 8

and

(46)

The criterion (43) is then identified to:

%32

7^â?;:

[

Y ² $ ³ Y Y ² $³t³ 4°¯± 8 ¯4°¯±8 Y

[ q ®G ¦ ã 7s : E u S E Jw'R'

(47)

The sets 8 and 8 are defined by the inverse lifting scheme and are given by the following equations:

8 8 D Q8

Q8

8 8 1) Computation of

% 2

: By definition,

% 2

(48) (49)

Y

is given by the equation (47). The signal can be recovered

from filter . The equation of the inverse update step is given by: Y Y Y

G (50) E E E wJ 'R S 2 Y3 operator . To minimize % 2 , we have to take where represents the set of indexes of the coefficients of the Y into account all the symmetries imposed on . Let us set 1 of the signal , to E the sum of the coefficients 2! which the factor E ( sÍ£ ) is applied, for the computation of Y3the coefficient . We thus have: Y Y S G (51) E 1 E E Jw'R' S

Considering the equations (48), (49), and (51) gives:

ætçéèê|ëíìî!ïð ð ñ ñ ¶cõ÷ö øùú ñ ø ùMú ñ ¸¶ ûü÷õþý è¸ÿ ù ñòó Rt ñ ¶ òó Rttô ô ñ ð ð ñòó Rt ô òó ð ü ñòó Rt ñ

î ÿ õ ö ù ø ù úñ û L ð ð ñ ñ¶ ð ò¶ ó Rt ô ô ò ó L ¶ ò ó L

¶ õ ö ø ù

ú ñ ø ù

¶ úñ ¶ û

(52)

From (47) and (52), the criterion can be expressed as follows:

% 2

7^â?;:

[

Y3

G G E E ¶ k4 k1 E JwRt E ¶ wJ 'R' S S

Y3

E 1

E ¶n

I I ¶ 4 k1 G G IJwL I ¶ JwLON N

Y

I E 1 Y Y

I 4 k 1 G E G E Jw R'S IJw L N

Y

I ¶ E ¶n·

I En

[ q G ¦ ã 7 s : Eù S E Jw Rt

March 16, 2004

$ $ ( ) )

(53)

DRAFT

12

%2

2) Minimization of the criterion

: The filter , which minimizes the reconstruction mean squared error,

is defined by resolving the system defined by the conditions (44) and (45). The derivative of the criterion with respect to and the equation (45) return:

© %32

©

7^â?;:

[

G ¦ ã ^7 ÆA: 5b S J Rt

The resolution of this equation leads to:

[

G ¦_ã c7 Æc: S wJ Rt

(54)

(55)

The partial derivatives of the criterion with respect to the different coefficients of the vector â are defined, for all Æ in

2>

, by:

© %32 ©

Y3

7^â?;:

G E k 4Q7^1 E J Rt S

S

Y3 E 1

: Y

I

:

Y

I E 1

I I ¶ 4Q7^1 G G IJw L I ¶ Jw L N Y N Y

I 4Q7c 1 HG I JwL N

I ¶ :·

¦ ã 7^Æc:í

(56)

The cancellation of the derivatives give us the following equation:

Y

G

E J Rt3S

Y

I E 1

I I ¶ 4Q7^1 G E ´ G I JwL I ¶ JwL N N

I ¶ : Y3 3Y

4{7^1

: n ¦_ã 7^ÆA:#

E 1

Y

Y

I Q G 4 7c 1 I wJ L N

I : (57) Y3 3Y

Y can Y be written Y Y system. The equations (55) and (57) in the form of a linear ByÈ setting ] E 4 k1 E 1 n Ç I I ¶ 4 k1 I E 1 I ¶ n, I 4 k 1 I n and G G G ¦_ã 7ÂÆ : , we obtain the IJwML I ¶ JwML N

system:

N

ÉÊÊ Ë

Ê ÊÊ ÊÌ

IJ L N

[ G ¦_ã ^7 ÆA: S È wJ 'Rt G ] E E S E wJ Rt

Ç

Í

ªÆ £

(58)

&|

As for the predictor case, the linear system (58) can be rewritten in the form of a matrix system:

ÎÍÏ If the squared matrix

Î

(59)

is invertible (see section IV), the matrix system (59) admits for unique solution:

Ï Ï

Ð

Î > Ð

(60)

Ï

¥ ¥ Ú E ¨ ? EÝ ? }}} ? EÕ ? \ âOÞßà ? . The process of resolution of this system S S S is the same as for the operator (37). The solution ârÞ9ßà corresponds to the optimal vector â and contains the The vector

is identified by

different coefficients (not-identical by symmetry) of the filter .

March 16, 2004

DRAFT

13

D. Transmission cost of the filter coefficients The application of adaptive lifting scheme to the data implies the transmission of the coefficients of the lifting operators. We have to add coefficients to the header of the bit stream to allow the decoding with the appropriate filters. For the MD general case, one sent the coefficients only one time of two operators (P and U). Because of the zero-phase condition (symmetries), the transmission of all coefficients is not necessary. One can transmit only the coefficients which are non-identical by symmetry. So, the header arises from a value of:

È

p

È

&- ¥

2! ¥

(61)

float coefficients (coded on 24 bits). This value are insignificant in comparison to the global data size. For example, in one dimension, we have:

for all filter

of size

m

(62)

and all filter

of size m . In general, we take

, m o[ or 2. Thus

For the bidimensional non-separable case, the growth of the header is equal to:

for a filter

! , and a filter

of size

w[ : m 7 m [`:

7

"

of size m m . thus,

IV. A LGORITHM

¦

is in general inferior to # .

ROBUSTNESS

Î

The proposed optimization method assumes the square matrix of size ¦

(63)

is invertible. In other words, the matrix

Î

must be of rank ¦ . The matrix size depends on the number ¦%$ of different coefficients of the

[.

operator to be optimized. This number is equal to ¦ In order to assert that the matrix matrix rank. If the rank m of of the criterion. Else, until the new matrix

Î

Î

Î

Î

is invertible, the algorithm includes a functionality which computes the

is equal to ¦ , then the matrix is invertible and we can proceed to the optimization

is not invertible. The imposed size e. is thus reduced until m is invertible. In the worst case, the optimized filters have size

¦

¦ $

[ , i.e.

and the constraint

on the sum of coefficients reduces the degree of freedom until zero. The obtained scheme corresponds to the lifting (2,2) defined in [14]. The matrix terms depend of the measure of the covariance between the data samples. The reductions of degree of freedom and of the operator size allow to be more adapted to the data. V. S AMPLING A. Definition of the predictor

EXAMPLE :

O PTIMIZATION

OF QUINCUNX FILTERS

1) Definition of parameters for the quincunx sampling grid: The goal of this section is to adapt the general criterion allowing to optimize the lifting filters for images sampled on a quincunx grid. In order to find the operator

which minimizes the criterion, let us consider the first quincunx lifting step:

Y Y

Ý

March 16, 2004

&

Y Y

Ý &

'

>

G EÝ)(

>

'

'

E

G

& (

t' N

Y

EÝ E & Ý EÝ

Y

&

E &

(64)

DRAFT

14

where the value

is defined so as the size of the operator

is equal to

. As the filters

and

are

assumed to be symmetric, we can rewrite the first step as follows:

Y Y

Ý

Y Y

&

Ý

' &

>

EÝ

Y

Y

G G EÝ E & 6 Ý E Ý & E & E Ý ( E & ( YN Y Y Y Y Y Ý *|EÝ+* & E & DY Ý E Ý Y & *-E & * Y Ý+ *-EÝ)* Y & *-E 7í[ -, EÝ E & :±w Ý E & & E Ý Y Ý *|E & * Y & E Ý

Ý E

& *

Y Ý+ *|E Y3

& *-EÝ)* &

s o7 s ? s $ : (with m and s in i Y3 Y Y Y V E

Ý E Ý &

E &

Ý *-E YÝ

), by:

Y

Y

Ý *-E Ý

& *-E & *

Y

? m $ : and

7m

Y

& : ±w Ý í7 [ Y , EÝ E Y YE

& *|E & * Y *

(65)

Y

E & Ý E Ý Y &

*

?

ü

è>?

: X

ü

6rè

: X

ï

ü

5

ð

è

: X

ü

òó L è >1? ü

ü

C !E :BXMè >

and

Ý

ü &

ü ü

8 : X

è>?

? ë+>

G@

ö

@

&

ü @

ü

Y

îè

î ö

: X

ü

? ë+> ? G@

ü

C @

Q

Î

: X

ü

ë> ? +

è>

$ :

(87)

7^Æ ?íÆ $ : and H

è>?

Y ?

ü

ë)> C

ü

Y

î+D C

C !E

@

5

The coefficients of the matrix

î C

?) $

î

!Y

ñ Ý ñ & î 83: X è > !Y ë+> ? !Y ? î Y C ë)>1? !Y ?#î ü : X è > !Y ë+> ? ü Y ? C î : X è > ë > ? Y ? î+D ) ü ü Y C ü C î8;: X è > ? !Y +ë > AY ? î : X è > ? Y ë)> AY ? î ü ü C ë)> ? î : è 1 > ? ) ë > ? îD + X ü Y C ü ü Y C ü Y C 6rè !E î 8;: X è > AY ? +ë > ? AY î C 5 Ý H5 & ë > ? !Y î :PXè > Y ? ë)> ? Y î Y ? C ) ü ü C : X è > ë ) > î + D ? ? ü ü Y C ü Y C ð ð 8 ¶cõ ü ø ù

ò Z ò Z L ¶ L

Y

:PXMè >

ü

J

è>?

ë)> C

C AE 5 Ý H

5&

ñÝ ñ

C AE

ü

è>

: X

ü ü

ë> Y ? +

6rè

ü

Y ?

ü

7s ?s $ :, Æ

Then, the three factors, occurring in the equation (81), are given for s $ 7 ? $ : £ i by: 4

[?)

î

and the vector Ð

ü

ë>? ? + C

ï

ü

î ü

ü

@

î

: X

è >1? @

:PXMè >

: X

ü î

!@

ü

: X

A@

? C

è> ?

!@

6rè RAE 5 Ý H

5&

è>?

î

C

?

ë)> C

ü

ë)>

!@

A@

?

ë)> ?

ë)>1?

!@

C

î ü

û î

¶ ú5

D

(88)

î @

C

ú ñ ø ù

î @

C

û

D

(89)

(90)

only depend on the covariance functions between samples

of the signal .

March 16, 2004

DRAFT

18

VI. A PPLICATION This section introduces image compression application. In a first part, we present performance of quincunx filters adapted to satellite image acquired on a quincunx grid (section VI-A), and in a second part we present the filtering performance for images from the database JPEG2000 (section VI-B). A. Quincunx sampled images 1) Study framework: There is an increasing demand of high quality satellite images. In this context, the earth observation satellite SPOT51 [39], provides a quincunx sampling images by using a pair of CCD linear arrays shifted each other by b } pixel in the direction of linear arrays and by m

b } pixels (mQ£\[ ) in the

satellite motion direction. SPOT5 acquisition system combines CCD linear arrays in a quincunx arrangement to improve image resolution. The German company, LH Systems, makes an airborne use of the double linear arrays SPOT5 [40]. The interest for such sampling techniques is due to their Modulation Transfer Function (MTF) of optical systems equipped with CCD instruments. This MTF corresponds roughly to a low pass filter and has a frequency support close to the quincunx one [39]. The double linear arrays make a denser sampling grid with an optimal frequency support for this kind of acquisition scheme [39]. Each CCD linear array generates an image sampled on a square grid. The combination of the two square grid images provides a quincunx sampling image (see Fig. 3 4 5). The optimal way of reducing the redundancies between the two CCD images is to process the quincunx sampled image with a suited transform as quincunx transform, in order to avoid processing of the two images separately as in SPOT5. Although, work is carried out on the development of a quincunx image compression well-adapted to quincunx sampling for SPOT5 successors [41]. In simulations, quality evaluation is given for satellite images, called N ICE S TATION, N ICE N ORTH, and N IME coded on 10 bits per pixel (see Fig. 3 4 5)2 .

Fig. 3.

N ICE S TATION : Satellite image provided by CNES.

1 Satellite 2 The

SPOT5 of the French Space Agency CNES (Centre National d’Etude Spatial) was launched on May 2002

authors wish to thank the French National Space Agency (CNES of Toulouse) to have provided the satellite images

March 16, 2004

DRAFT

19

Fig. 4.

N IMES : Satellite image provided by CNES.

Fig. 5.

N ICE N ORTH : Satellite image provided by CNES.

The evaluation of the optimization method is supported by the comparison of the filtering performance between optimized and non-optimized quincunx lifting scheme. The quincunx filtering is included in a lossy compression scheme. We use the encoder EBWIC, developed by Parisot and all. [42], [43]. The EBWIC coder is a model based bit allocation method [44]. We use the quincunx version of EBWIC with computation of optimal weighted distortion [45]. The coding follows the application of wavelet transform to the original data. The principle of bit allocation is to define the optimal quantization steps for each subband so as to optimize the trade-off rate-distortion. Following the allocation, the uniform scalar quantization is applied with the optimal steps. The coder used thereafter is a context based arithmetic coder [27] from the JPEG2000 [46], [47] standard. 2) Adaptive quincunx lifting scheme : Values of reference filters of quincunx lifting scheme and of optimized filters on image N ICE S TATION are shown in tables VI-A.2 and VI-A.2. 3 One can notice that the optimal operators are valid only for the image N ICE S TATION, because of the data dependency of the filter coefficients. 3 QLS

is for Quincunx Lifting Scheme and OQLS is for Optimized Quincunx Lifting Scheme

March 16, 2004

DRAFT

20

f e d d e f

e c b b c e

d b a a b d

d b a a b d

e c b b c e

f e d d e f

QSL (4,2) ]

^

,!_aÌbdcIe

h

s

h

,jik_Ibdc

n

,ji mbdc

^

QSL (6,2) ^

e e

,lmbdo

,facagIbdc

?? ?

,li m)fagIbdc ?1p n ,likoagIbdc ? / , m)gIbdc ?1p j q , m)gIbdc ?1p j r , _Ibdc ^ , mbdo l

TABLE I L IFTING OPERATOR COEFFICIENTS FOR A

TWO STEP QUINCUNX LIFTING SCHEME .

The different filters are the quincunx (4,2) and (6,2) lifting schemes, introduced in [14] and called QLS(4,2) and OQSL (4,2) ]

^ h

s

OQSL (6,2)

,.0t _af0dumam ,ji-0 t a0 faoaàgIv n ,. 0 t a0 cIvdga0 _

^

,wmbdo

^

,å0t _Ivdgaàca` ,li-0 t a0 oaom)g0 n ,á,å 0 t 0a0 `duIoa_ / ,.0 t a0 cm+um)f q ,. 0 t 0a0 ca0 odu r , i-0 t 0a0 fmam)_ j ^ , mbdo j h

OQSL (6,4) ^

,.0t _Ivdgaàca` ,ji-0 t a0 oaom)g0 n ,á,. 0 t 0a0 `duIoa_ / ,å0 t a0 cm+um)f q ,å 0 t 0a0 ca0 odu r , i-0 t 0a0 fmam)_ l ^ ,.0 t cdua0 à_Iv h , i-0 t a0 fm)oIvd` j n ,¤ 0 t 0I0 vdoacm h

TABLE II O PTIMIZED LIFTING OPERATOR COEFFICIENTS FOR A

TWO STEP QUINCUNX LIFTING SCHEME APPLIED ON IMAGE

N ICE S TATION .

QLS(6,2), and the new optimized quincunx lifting filters (4,2), (6,2) and (6,4), noted OQLS(4,2), OQLS(6,2), OQLS(6,4). The QLS(4,2) and QLS(6,2) are obtained by the application of the McClellan transform to each lifting operator of the 1D lifting scheme [14], [48]. Figures~6, 8, 9 and 7 show frequential response of the quincunx (9,7) filter bank [2], called QDWT(9,7), and filter banks corresponding to QSL(6,2), OQSL(6,2) and OQSL(6,4). The QDWT(9,7) filter bank is obtained by the application of the McClellan transform to the wellknown one-dimensional (9,7) filters [2], [7]. These frequential responses show a sharper cut-off frequency for

optimized filters. The filters @

are sharper because of the definition of the optimal predictor.

is defined to

minimize the variance of the high frequency band, thus a maximum of energy is conserved in the low frequency band. 3) Quality evaluation : The evaluation of the optimization method is supported by the comparison of the transform performance between optimized and non-optimized quincunx lifting scheme, in a lossy compression scheme, using the encoder EBWIC [43]. Since the operators are rotated by 45 degrees between two levels of resolution in the quincunx case, we choose to optimized filters on two levels of resolution. In our method, the

March 16, 2004

DRAFT

21

a. Fig. 6.

b.

Frequential response of corresponding filters x

(a.) and x y

a. Fig. 7.

b.


(a.) and x y

a. Fig. 8.

(b.) of OQLS(6,4).

b.


(a.) and x y

(b.) of QLS(6,2).

b.

a. Fig. 9.

(b.) of QDWT(9,7).


(a.) and x y

(b.) of OQLS(6,2).

optimized lifting operators have to be transmitted to the decoder. The cost of the coefficients coding is negligible in comparison to the image coding. If one assumes that the coefficients are floating points coded on 32 bits and thanks to symmetries, the transmission of filter coefficients only requires [ bits for the OQLS(6,2), and

TT

T

bits for the OQLS(4,2),

.

bits for the OQLS(6,4). The different tested filters are the QLB, QNLB, the

Q(9,7), QLS(4,2), QLS(6,2), OQLS(4,2), OQLS(6,2), OQLS(6,4). The QLB and QNLB represent the quincunx linear and non-linear lifting schemes of Boulgouris and all [25]. The first one is obtained by is optimization and the second is the same filter more their proposed non linear filtering. The method is evaluated for the images N ICE S TATION, N ICE N ORTH and N IME. A qualitative study determines the PSNR v.s. to the compression ratio. Tab. VI-A.3 presents the performances of different lifting filters, for compression ratios equal to 2.5, 5 and 7.

March 16, 2004

DRAFT

22

CR

Q(9,7)

QNLB

QLB

QLS(4,2)

QLS(6,2)

OQLS(4,2)

OQLS(6,2)

OQLS(6,4)

2,5

43.34

42.51

43.25

43.35

44.01

44.07

44.54

44.90

5

31.9

31.10

31.71

31.90

32.54

32.66

33.07

33.44

2,5

45.45

44.74

45.29

45.52

46.18

46.02

46.31

46.74

5

33.62

33.04

33.54

33.71

34.28

34.24

34.53

34.93

Nice North

2,5

44.72

43.53

44.61

44.71

45.37

45.53

45.95

46.27

5

33.36

33.22

33.19

33.34

33.96

34.11

34.51

34.85

NiceA

7

30.01

28.84

29.82

29.92

30.42

30.57

30.89

31.24

NiceStation

Nimes

TABLE III D IFFERENT RESULTS OF QUINCUNX LIFTING SCHEME FOR IMAGES OF THE CNES DATA BASE (CR IS FOR COMPRESSION RATE )

38 37 36 35 SNR in energy

34 33 32 31 30 29 28 27 26 25 24 3,5

4

4,5

5

5,5

Linear Boulgouris OQLS(4,2)

Fig. 10.

6

6,5 7 7,5 8 Compression rate QLS(6,2) OQLS(6,4)

8,5

OQLS(6,2) NLQ Boulgouris

9

9,5

10

10,5

11

QLS(4,2)

Quality improvement Nice with the proposed method for the (4,2) and (6,2) quincunx lifting filters. Results of well-known

quincunx (9,7) filter bank, the linear and non-linear quincunx lifting schemes of Boulgouris, QLB and QNLB, and the optimized (6,4) quincunx lifting scheme are given for comparison.

For the image N ICE S TATION at a compression rate of 5, the relative SNR (in energy)gain for a OQLS(4,2) compared to QSL(4,2) is of b }{z 2 dB, the gain between OQLS(6,2) and QSL(6,2) is of b } dB, the gain between

OQLS(4,2) and QNLB is of [ } 2 dB, the gain between OQLS(6,4) and QNLB is of } # dB, and the relative gain between OQLS(6,4) and QDWT(9,7) is of [ } |. dB. The optimization of the lifting operators induces a clear improvement of the filtering performances. The PSNR4 gains are significant since they are made only on the filtering step. Thus, the method is very efficient within the framework of high and average bit-rate application and for satellite images. Figure 11 represents a zoomed and sharpen part of image Nice. Figures 12, and 13 present the same image coded, for a compression ratio of 6, using different quincunx lifting schemes. One can note that compression using optimized lifting scheme ensures a better restoration of object edges. The non optimal filtering generate more visual artifacts. The comparison between QDWT(9,7) and OQLS(6,4) shows that QDWT(9,7) smooths image objects more (see edges of the road on Fig. 12 and Fig. 13).

March 16, 2004

DRAFT

23

Fig. 11.

Visualization of an extracted part

Fig. 12.

Visual quality results of quincunx Fig. 13.

Visual quality results of opti-

of original image N ICE with a zoom and a

lifting scheme (9,7) for image N ICE S TA - mized quincunx lifting scheme (6,4) for im-

sharpen (75%).

TION

for a compression ratio equal to 6.

age N ICE S TATION for a compression ratio equal to 6.

B. Squared sampled images from JPEG2000 database This section provides some results for images from the JPEG2000 database. These images can be processed with the quincunx lifting scheme and the separable lifting scheme. In general, we observe that the optimization improves the results of quincunx filtering and it is generally more performant than the other filterings as the Q(9,7) or QLB and QNLB. Improvements are always obtained except for Lena for which the autocovariance function estimation is less significant (on the contrary of satellite images, Lena is composed of several areas with quite different statistics). Since our filters are adapted to the whole image, our method is the most efficient for stationary images which are not too much corrupted by noise. The adapted quincunx filter banks are only matched for quincunx sampling: in case of orthogonal sampling, there is no gain from our approach. The optimization improves slightly the results provides by the (2,2), (4,2) and (6,2) lifting scheme 5 for all image, and is equivalent to the non linear filtering of Boulgouris and all [25], NLB, but the (9,7) remains the best one [2]. The (9,7) filter leads to a four-step lifting scheme. So, we conclude that if for quincunx filtering, and taking into account the filtering cost of nonseparable transforms, a simple two-step lifting filtering is sufficient, for separable transform it is necessary to have a scheme with at least two prediction-update steps. VII. C ONCLUSION We have defined a M-dimensional 2-channel lifting filtering according to statistical properties of the source signal. The proposed method can only be exploited for regular sampling scheme. Our approach consists in optimizing coding scheme performances and offering an optimal decorrelation. The optimization process defines, in a first step, a prediction operator. The choice of this operator plays an essential role, since encoding allows a bit rate gain, mainly in the high frequency subband. The criterion retained to define the prediction step consisted 4 For

images coded on 10 bits, PSNR is defined by

5 These

]~}P

,jm#0Pa

?1

ÕRÝ Õ Ý

?1

p+ &

Ý

& { F

filters are called the LS(2,2), LS(4,2), LS(6,2), and the separable optimized lifting schemes are the OLS(4,2) and OLS(6,2)

March 16, 2004

DRAFT

24

Gold

Mandrill

Boats

Lena

Cafe

Filters\CR

5

8

5

8

5

5

8

5

8

15

Q(9,7)

40.53

37.08

32.23

28,48

43.51

42.92

39.80

34.35

29.43

25.12

QNLB

39.82

36.43

31.15

27.38

43.01

42.19

39.18

34.91

30.25

25.59

QLB

40.12

36.61

31.72

28.02

43.17

42.47

39.26

34.41

29.71

25.15

QLS(4,2)

40.16

36.7

31.68

27.98

43.19

42.47

39.37

34.41

29.64

25.12

QLS(6,2)

40.18

36.72

31.64

27.95

43.15

42.44

39.35

34.48

29.74

25.21

OQLS(4,2)

40.44

36.98

31.87

28.12

43.30

42.29

39.25

34.64

29.99

25.53

OQLS(6,2)

40.41

36.99

31.84

28.15

43.30

42.15

39.11

34.58

30.03

25.54

OQLS(6,4)

40.73

37.27

32.07

28.34

43.65

34.98

30.33

25.88

D(9,7)

41.23

37.84

33.1

29.5

45.04

36.5

31.9

27.12

43.49

40.41

NLB

40.88

37.54

32.44

28.88

44.62

42.96

9.96

36.1

31.55

26.61

LS(2,2)

40.76

37.4

32.28

28.74

44.32

42.74

39.68

35,8

31.21

26.46

LS(4,2)

40.86

37.5

32.52

28.96

44,59

42.99

39.97

36.1

31.43

26.67

LS(6,2)

40.80

37.71

32.5

28.96

44.58

42.97

39.98

35.93

31.38

26.6

OLS(4,2)

40.89

37.54

32.42

28.87

44,59

42.99

39.96

36.1

31.43

26.68

OLS(6,2)

40.85

37.51

32.43

28.87

44.58

42.99

39.99

TABLE IV D IFFERENT RESULTS OF SEPARABLE AND NON

SEPARABLE LIFTING SCHEME FOR IMAGES OF THE

JPEG2000 DATABASE

in minimizing variance of the high frequency subband. The update step determines the low resolution signal. This signal is only slightly compressed, and it contains the essential information to reconstruct the signal. The update operator has been defined in order to minimize the reconstruction error when the high frequency image is removed. The operator was thus defined by imposing an optimal reconstruction starting from the low frequency image. The optimization of filters was exploited within the framework of the quincunx lifting scheme. The details of the application and the performances are given through quantitative and qualitative evaluation.Simulations show the efficiency of the proposed method, mainly for quincunx sampling image as the satellite quincunx sampled images, but less gain for orthogonally sampled images from JPEG2000 database. The results show the performance of the method: the gain is about 0.4 up to 2.4 dB with optimized quincunx lifting scheme. The proposed method allows good compression results. It involves improvements according to the subjective and objective standards of quality in comparison with the non-optimal filtering. Indeed, we show in experiments that the filters of the same length developed by optimization are more performant than filters designed using the McClellan transform, and the filters optimized by Boulgouris in most of the case. In the case of separable filtering, the (9,7) filtering corresponding to a 4-step lifting scheme remains the best one. The optimized filtering of [25] are equivalent for separable filtering. The best way to optimized a 1D filtering is to design a scheme with at least two prediction-update steps. Our method introduces a new and efficient way to design multidimensional lifting scheme. Compared to previous contributions [25] the gain is obtained at the cost of the estimation of the autocovariance function. The filters are globally adapted to the whole image. Further gain could be obtained by local adaption.

March 16, 2004

DRAFT

25

A PPENDIX Derivation of the symmetry conditions In the case of 1°zþi , zero phase implies by definition

=|k mn =|k mn

and @ =|k mn F@ =|k mn

ª m÷£ 1 }

(91)

From equations (5), (6) (7) and (8), and from the consideration of the initial filter bank defined by the split:

= D k mn ,

k mn

and B D k mn ,

km [ n

(92)

@ = D k mn ,

k mn

and B@ D k mn ,

km [ n

(93)

we deduce the equation of final filter bank (94) and (95),

=-k mn ,

k mn

G

E J'KR

,

km

s [n S

k stn

G

G

E Ý J'K L E & 'J K R

and

@ =-k mn ÉÊÊ

Thus

Ê

Æ3@ =|k Æ n

Ë

m ÊÊ ÊÌ

k mn G ,

,

X

km ,

Æ [@ =|k Æ [ n

km ,

s

$ s [`: n N

k s n

s [ n k stn N

S

k s $ n

(94)

(95)

(96)

k 7^Æ s : n k stn k Æ n N N We deduce from (96) and from the symmetry @ =|k mn @ =|k mn the relation that that we deduce: m

N

km [ n N

,

k mn

(97)

And in the same way we deduce from (94) the constraint on :

S

k mn S

k m [ n ª m÷£ V

2

(98)

The proof in the bidimensional case is resolved in the same way. R EFERENCES [1] S. Mallat, “A theory for multiresolution signal decomposition: The wavelet representation,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 11, July 1989. [2] M. Antonini, M. Barlaud, P. Mathieu, and I. Daubechies, “Image coding using wavelet transform,” IEEE Transactions on Image Processing, vol. 1, no. 2, pp. 205–220, Apr. 1992. [3] P. Siohan and V. Ouvrard, “Design of two-dimensional non-separable qmf banks,” in Procedings of the IEEE ICASSP conference, vol. 4, San Francisco, USA, Mar. 1992, pp. 645–648. [4] F. M. de Saint-Martin, P. Siohan, and A. Cohen, “Biorthogonal filterbanks and energy preservation property in image compression,” IEEE Transactions on Image Processing, vol. 8, no. 2, Feb. 1999. [5] R. Ansari and C. Guillemot, “Exact reconstruction filter banks using diamond fir filters,” Proc. Bilcon 1990, Elsevier Press, pp. 1412–1424, July 1990. [6] C. W. Kim and R. Ansari, “Subband decomposition procedure for quincunx sampling grids,” in SPIE Conference on Visual Communication and Image Processing, Nov. 1991, pp. 112–123.

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[7] M. Barlaud, Wavelet and Image Communication, ser. Jan Biemond.

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[34] G. J. Sullivan, “Efficient scalar quantization of exponential and laplacien random variables,” IEEE Transactions on Information Theory, vol. 42, no. 5, 1996. [35] M. Vetterli and C. Herley, “Wavelets and filter banks: theory and design,” IEEE Transactions on Acoustic Speech Signal Processing, vol. 40, no. 9, pp. 2207–2232, 1992. [36] P. P. Vaidyanathan, Multirate Systems and Filter Banks.

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[37] P. Siohan and V. Ouvrard, “Design of optimal non-separable 2-d half-band filters,” in Proceedings Symposium IEEE Circuits and Systems, vol. 1, Chicago, USA, May 1993, pp. 914–917. [38] A. K. Jain, Fundamentals of digital image processing.

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[40] H. Jahn and R. Reulke, “Staggered line arrays in pushbroom cameras: theory and application,” in International archives of photogrammetry and remote sensing, vol. XXXIII, Amsterdam, 2000. [41] A. Gouze, M. Antonini, and M. Barlaud, “Quincunx filtering lifting scheme for image coding,” in SPIE, Visual Communication and Image Processing, vol. 1, Jan. 1999, pp. 1–11. [42] C. Parisot, M. Antonini, and M. Barlaud, “Stripe-based MSE control in image coding,” in Proceedings of IEEE International Conference in Image Processing, Rochester, USA, Sept. 2002. [43] ——, “3D scan based wavelet transform and quality control for video coding,” EURASIP Journal on Applied Signal Processing (JASP), Special issue on Multimedia Signal Processing, Jan. 2003. [44] ——, “Ebwic: A low complexity and efficient rate constrained wavelet image coder,” in IEEE International Conference on Image Processing, vol. 1, Vancouver, Sept. 2000, pp. 653–656. [45] A. Gouze, C. Parisot, M. Antonini, and M. Barlaud, “Optimal weighted model-based bit allocation for quincunx sampled images,” in IEEE International Conference on Image Processing, Barcelona, Sept. 2003. [46] I. J. S. WG01, “JPEG-2000 part-1 standard,” ISO/IEC 15444-1. [47] C. Christopoulos, A. Skodras, and T. Ebrahimi, “The JPEG2000 still image coding system: An overview,” IEEE Transactions on Consumer Electronics, vol. 46, no. 4, pp. 1103–1127, Nov. 2000. [48] A. Calderbank, I. Daubechies, W. Sweldens, and B.-L. Yeo, “Wavelet transforms that map integers to integers,” Applied and Computational Harmonic Analysis, vol. 5, no. 3, pp. 332–369, 1998.

Annabelle Gouze was born in 1975. She received the M.S. degrees in 1998 in computer vision and the Ph.D. degrees from the University of Nice Sophia Antipolis, France, in 2002. She is currently research assistant in the Communication and Remote Sensing Laboratory from the Université catholique de Louvain. Her research interests focus on image compression, quincunx filter design and video segmentation by actifve contours.

Marc Antonini Marc ANTONINI received the Ph.D degree in electrical engineering from the University of Nice-Sophia Antipolis (France) in 1991 and the Habilitation à Diriger des Recherches from the University of Nice-Sophia Antipolis (France) in 2003. He was a postdoctoral fellow at the Centre National d’Etudes Spatiales (Toulouse, France), in 1991 and 1992. Since 1993, he is working with CNRS at the I3S laboratory both from CNRS and University of Nice-Sophia Antipolis. He is a regular reviewer for several journals (IEEE Transactions on Image Processing, Information Theory and Signal Processing, IEE Electronics Letters) and participated to the organization of the IEEE Workshop Multimedia and Signal Processing 2001 in Cannes (France). He also participates to several national research and development projects with French industries, and in several international academic collaborations. His research interests include multidimensional image processing, wavelet analysis, lattice vector quantization, information theory, still image and video coding, joint source/channel coding, inverse problem for decoding, multispectral image coding, multiresolution 3D mesh coding.

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Michel Barlaud Michel Barlaud received his These d’Etat from the University of Paris XII and Agrégation de Physique (ENS Cachan). He is currently a Professor of Image Processing at the University of Nice-Sophia Antipolis, and the leader of the Image Processing group CReATIVe of I3S Laboratory. His research topics are: Image and Video coding using Scan Based Wavelet Transform, Inverse problem using Half Quadratic Regularization and, Image and Video Segmentation using Region Based Active Contours and shape gradients. He is a regular reviewer for several journals, a member of the technical committees of several scientific conferences. He leads several national research and development projects with French industries, and participates in several international academic collaborations (Universities of Maryland, Stanford, Boston, Louvain La Neuve) and European Research Networks (COST, Schema, Similar). He is the author of a large number of publications in the area of image and video processing, and the Editor of the book "Wavelets and Image Communication" Elsevier, 1994.

Benoît Macq (M’89-SM’01) was born in 1961. He received the Electrical Engineering at the Ph.D. degrees from the Université catholique de Louvain (UCL), Belgium, in 1984 and 1989, respectively. He did his Ph.D. thesis on perceptual coding for digital TV under the supervision of Prof. Paul Delogne at UCL. He has been Professor at the same university in the Telecommunications Laboratory since 1996. From 1992 to 1996, he was a Senior Researcher of the Belgian NSF at UCL, and Invited Assistant Professor in the Telecommunications Laboratory. In 1990 and 1991, he worked in the network planning for Tractebel S.A. in Brussels. He has been a Visiting Scientist at Ecole Nationale Superieure des Telecommunications, ENST-Paris, and at the Université de Nice Sophia Antipolis, France, from 1999 until 2000. His main research interests are image compression, image watermarking and image analysis for medical and immersive communications. Prof. Macq served as a Guest Editor for the Proceedings of the IEEE as well as for the Signal Processing Journal and member of the program committee of several IEEE and SPIE conferences.

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L IST

OF

TABLES

I

Lifting operator coefficients for a two step quincunx lifting scheme. . . . . . . . . . . . . . . . . .

II

Optimized lifting operator coefficients for a two step quincunx lifting scheme applied on image N ICE S TATION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

III

20

Different results of quincunx lifting scheme for images of the CNES data base (CR is for compression rate) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

IV

20

22

Different results of separable and non separable lifting scheme for images of the JPEG2000 database 24 L IST

OF

F IGURES

1

Canonical two steps lifting scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

2

8 5b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Backward lifting scheme for 8 and

4

3

N ICE S TATION: Satellite image provided by CNES. . . . . . . . . . . . . . . . . . . . . . . . . .

18

4

N IMES: Satellite image provided by CNES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

5

N ICE N ORTH: Satellite image provided by CNES. . . . . . . . . . . . . . . . . . . . . . . . . . .

19

6 7 8

(a.) and @

(a.) and @

(a.) and @

Frequential response of corresponding filters

Frequential response of corresponding filters Frequential response of corresponding filters

(a.) and @

(b.) of QDWT(9,7). . . . . . . . . . .

21

(b.) of OQLS(6,4). . . . . . . . . . . .

21

(b.) of QLS(6,2). . . . . . . . . . . . .

21

(b.) of OQLS(6,2). . . . . . . . . . . .

21

9

Frequential response of corresponding filters

10

Quality improvement Nice with the proposed method for the (4,2) and (6,2) quincunx lifting filters. Results of well-known quincunx (9,7) filter bank, the linear and non-linear quincunx lifting schemes of Boulgouris, QLB and QNLB, and the optimized (6,4) quincunx lifting scheme are given for comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

11

Visualization of an extracted part of original image N ICE with a zoom and a sharpen (75%). . . .

23

12

Visual quality results of quincunx lifting scheme (9,7) for image N ICE S TATION for a compression ratio equal to 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

23

Visual quality results of optimized quincunx lifting scheme (6,4) for image N ICE S TATION for a compression ratio equal to 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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23

DRAFT