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Contents lists available at ScienceDirect

Signal Processing

Signal Processing 88 (2008) 2754– 2763

0165-16

doi:10.1

� Cor

E-m

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

Index assignment for 3-description lattice vector quantization basedon A2 lattice

Minglei Liu, Ce Zhu �

School of Electrical and Electronic Engineering, Nanyang Technological University, 639798 Singapore, Singapore

a r t i c l e i n f o

Article history:

Received 28 August 2007

Received in revised form

2 April 2008

Accepted 27 May 2008Available online 7 June 2008

Keywords:

Index assignment

Lattice vector quantization

Multiple description coding (MDC)

Sublattice

84/$ - see front matter & 2008 Elsevier B.V. A

016/j.sigpro.2008.05.020

responding author. Tel.: +65 6790 6041; fax: +

ail address: ECZhu@ntu.edu.sg (C. Zhu).

a b s t r a c t

Multiple description lattice vector quantization (MDLVQ) is an effective technique to

realize multiple description coding. In this paper, we investigate a 3-description lattice

vector quantization (LVQ) design using A2 lattice, which finds a 3-tuple combination of

sublattice points to represent each fine lattice point, based on two principles that each

edge of the triangle formed by the 3-tuple points needs to be as short as possible, and

the gravity center of the triangle is as close as possible to the fine lattice point. Following

a delicate sublattice partition, a well-designed construction and mapping of each fine

lattice point to a 3-tuple of sublattice points is developed to minimize the side

distortion. The proposed index assignment is shown to achieve smaller distortion (up to

0.58 dB for Gaussian source) than some other 3-description index assignments.

Compared with a 2-description LVQ, the 3-description LVQ achieves better expected

rate-distortion performance in most cases of high description loss rates, while it also

exhibits more graceful coding results with significantly smaller side distortions.

Numerical analysis on Gaussian source and the simulations on image source validate

the effectiveness of our proposed index assignment scheme.

& 2008 Elsevier B.V. All rights reserved.

1. Introduction

Multiple description coding (MDC) is a source codingscheme to combat transmission errors in non-prioritizednetworks. The most typical MDC implementation is a2-description coding, where an encoder generates twodescriptions or streams for one source. The two streamsare transmitted over two separate channels, respectively,where each channel may have its own rate constraint. Atthe decoder, depending on the number of descriptionsreceived correctly, different reconstruction quality will beobtained. To be more specific, if only one stream isreceived, the reconstruction quality corresponding to a so-called side distortion is expected to be acceptable, and an

ll rights reserved.

65 6793 3318.

incremental improvement will be achieved with a smallercentral distortion if both streams are received.

Multiple description lattice vector quantization(MDLVQ) is an effective technique to generate tworepresentations for a symbol. Symmetric entropy-constrained MDLVQ was introduced in [1,2] by Servetto,Vaishampayan and Sloane (known as the SVS technique)for two balanced (symmetric) channels, whereas asym-metric multiple description lattice vector quantization(AMDLVQ) was developed for possibly unbalanced (asym-metric) channels [3]. For a given uniform fine lattice andsublattice, the SVS technique maps each point in the finelattice to a pair of sublattice points (an ordered edge),where the key is to design such a mapping (also known aslabeling function or index assignment) a to minimize theside distortion. In practice, only N fine lattice points areconsidered in a Voronoi cell of a sublattice point, e.g.sublattice point ‘O’. In [4] the authors suggested that if N

edges for labeling are constructed by connecting ‘O’ to its

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M. Liu, C. Zhu / Signal Processing 88 (2008) 2754–2763 2755

nearest N sublattice points (including ‘O’ itself), the sidedistortion would be minimized for this index N. SVStechnique is used to map these N edges to N fine latticepoints in the Voronoi cell. Huang [5] further proved thatthe side distortion can be minimized if these N fine latticepoints are also the nearest N fine lattice points to ‘O’ (‘O’ isalso a fine lattice point). The proof is based on theS-similarity of some commonly used lattices. A greedylabeling algorithm with low time complexity developed in[6] is applied for the mapping function a. The main idea ofthis greedy algorithm is to partition the lattice with itsS-similar sublattice, and label fine lattice points in eachVoronoi cell of S-similar sublattice separately. There arealso some other multiple-description vector quantization(MDVQ) schemes [7,8], which do not impose a latticestructure on the quantizer codebook. However, thecomplexity for designing both the codebook and indexassignment is very high in these schemes. Therefore, inthis paper, we consider the MDVQ with a lattice codebook,i.e. MDLVQ, for which the encoding and design complex-ities are much lower than if no lattices are used.

Kelner et al. [9] developed a scheme (noted as KGKscheme for brevity) for multi-channel MDLVQ indexassignment, where an extension of the SVS technique tomore than two balanced channels was presented. How-ever, the index assignment mapping is not adequatelyaddressed in [9], in which only two examples are sketchedfor 3-description LVQ (denoted as 3DLVQ). In the KGKindex assignment scheme of designing a mapping a : L!LM

1 for M channels, a set of sublattices Li (i ¼ 2,y, M�1)should be firstly identified with L1CL2C?CLM�1CL,where Li is the sublattice for reconstruction when andonly when any i channels of descriptions are receivedcorrectly. This imposes a strong constraint to the selectionof index number Ni of Li, that is, Ni�1 is a multiple of Ni.Ostergaard et al. [10] presented in analytical expressionsof the central and side distortion under the high-resolution assumption for the symmetric entropy-con-strained MDLVQ (M42). Index assignment optimizationproblem is solved as a linear assignment problem basedon minimizing the expected distortion or side distortion.However, since these theoretical solutions are obtainedunder the high-resolution assumption by assumingN-N, they may not be appropriate for the design witha small index number N. In [6] the authors extended theirgreedy labeling algorithm to the 3-description case.However, it is not optimal for some finite index numbers,which will be shown in our experimental results. Theylater modified the algorithm by making a local adjustment[5]. In [11], we introduced an index assignment schemefor 3DLVQ design in a different view based on a 3-stepstrategy, which is, partitioning of sublattice, constructingof combinations and mapping of combinations. To mini-mize the side distortion, the principle in the indexassignment is to select the least number of the nearestsublattice points to represent a fine lattice point. For A2

lattice with an index number Np85, the index assignmentfor 3DLVQ was developed in [11]. In this paper, we give theproof of the principles in the index assignment andameliorate the 3-step index assignment for any indexnumber N by removing the constraint of Np85 in [11].

This change is realized by using different combinationconstructing strategy in the second step. The third step isalso improved by reducing computational complexity. Inthe experimental part, we will show that the proposedindex assignment can achieve smaller side distortions forGaussian source (e.g. 0.58 dB for N ¼ 31), compared with[6]. We would also like to highlight that the choice ofindex number N in the proposed design is very differentfrom that in KGK scheme in that our method does notimpose any sublattice relationship hence, removing theconstraint in the choice of the index number N1. Morespecifically, KGK method can only be used for the designwith such a sublattice index number that is a compositenumber, e.g., N1 ¼ 21, 49,y. In contrast, our developedmethod is applicable with any index number for a cleansimilar sublattice including those prime index numbers asN1 ¼ 7, 13, 19, 31,y. In this sense, ours is a general designmethod and can be used for the index assignment designwith a small index number, which KGK scheme fails to do.

This paper is organized as follows. In Section 2, somenecessary concepts and results on lattice, symmetricMDLVQ with SVS algorithm are briefly reviewed. InSection 3, the proposed index assignment based onsublattice partitioning is developed and numerical analy-sis is presented in Section 4. The paper is concluded inSection 5.

2. Background

Lattice vector quantization (LVQ) is a special categoryof vector quantization (VQ), where the VQ codebook is alattice or a subset of a lattice. A main advantage for LVQ isthat fast encoding algorithms can be developed byexploiting the geometric properties of regular lattices[12]. The lattice properties and some related concepts areintroduced first in this section before multiple-descriptionLVQ is presented.

2.1. Preliminaries

A real L-dimensional lattice L is a discrete set of pointsin the L-dimensional Euclidean space RL. Let L and L0 bean L-dimensional lattice and a sublattice, respectively,L0DL. G denotes the generator matrix for L and G0 for L0.L0 is geometrically similar to L if G0 ¼ cGU, where c is ascalar and U is an orthogonal matrix of determinant 1. Theindex number of the similar sublattice L0, denoted as N, isdefined as the number of elements (points) of L in eachVoronoi cell of L0, with N ¼ cL. Furthermore, L0 is a cleangeometrically similar sublattice if no lattice point lies onthe boundary of the Voronoi cells of L0. The ‘clean’characteristic is important to simplify index assignmentdesign, which further limits the choice of index N inMDLVQ. The current MDLVQ designs mainly consider two-dimensional lattices (L ¼ 2), such as A2 and Z2 lattices. Inthis paper, we focus on the hexagonal lattice A2. Theportion of A2 lattice and part of a geometrically similarsublattice with index 31 are shown in Fig. 1, where largerpoints are the sublattice points, and the large hexagon isthe Voronoi cell of the sublattice with the center point ‘O’.

ARTICLE IN PRESS

O

Fig. 1. A2 lattice and its sublattice with index 31.

M. Liu, C. Zhu / Signal Processing 88 (2008) 2754–27632756

It has been shown [3] that L0 is clean similar if and only ifN is a product of primes congruent to 1 (mod 6). Theseappropriate numbers are given as Sequence A004611 in[13]: 1, 7, 13, 19, 31, 37, 43, 49, 61, 67, 73, 79, 91,?.

2.2. SVS index assignment in symmetric 2DLVQ

The SVS technique is for two symmetric channels. For agiven input vector x, it is firstly mapped to a lattice pointdenoted as l, where l ¼ Q ðxÞ ¼ arg minz2Lkx� zk. In orderto generate 2 descriptions in the SVS algorithm, one cleansimilar sublattice L0 is predefined. An index assignmentmapping a: L-L0 �L0 is designed to minimize the sidedistortions, where a is an injection. Then an ordered pairof vectors is generated,

ðl01; l02Þ ¼ a1ðlÞ;a2ðlÞð Þ ¼ aðlÞ (1)

This ordered pair is also known as an edge with twoendpoints l01 and l02 (l01, l02AL0). Given an index N of L0,we need to determine the discrete Voronoi cell V0(O) of L0

and map each point lAL in V0(O) to two orderedsublattice points in L0. Then the assignment is extendedto the entire lattice using shift property. The guidingprinciple for the SVS assignment is to choose the shortestpossible edge (l01, l02) with midpoint as close as possibleto the fine lattice point l. The details of the SVS algorithmcan be found in [1,2].

2.3. 2DLVQ versus MDLVQ

It is known that 2DLVQ can better combat transmissionloss compared with the single-description LVQ. Likewise,MDLVQ (M42) exhibits better error robustness at a highdescription loss rate, since the probability of all (or mostof) descriptions being lost is further reduced in theM-description case. However, MDLVQ may result in moreredundancy with lower rate-distortion performance at noor low description loss rate.

Looking into the 2DLVQ, we find that its uniquestructure may result in a large gap between central andside distortions, due to the constraint of the selection ofsublattice index number N. Given a lattice, central

distortion D0 is fixed regardless of the choice of sublatticewith any index number. Side distortion Di, however,depends on the index number N of the selected sublatticeand the index assignment. Given a certain index N, a goodindex assignment aims to assign l01 and l02 for minimiz-ing Di. However, the side distortion increases monotoni-cally with N [14]. For a large index value N, more finelattice points are included in V0(O) and more sublatticepoints are needed to construct enough edges for these finepoints, which in turn increase the average length of theedges by including some more sublattice points far awayfrom V0(O). For example, for a uniformly distributedsource and a sublattice with index 13, the central andside distortions differ by 16.8 dB at high rate using the SVStechnique. Even if the smallest index number 7 is used,the side distortion is still 11.3 dB lower than the centraldistortion.

Large side distortion leads to a poor side reconstruc-tion, which defies the original intention of MD coding.Therefore, it is desirable to reduce the gap between thecentral distortion and side distortion, while the averagedistortion-rate performance is maintained or even im-proved. MDLVQ (M42) can substantially mitigate theproblem of large gap. Consider an M-description indexassignment a : L! LM

1 . Using K sublattice points over M

channels, KM ordered M-tuples can be generated, and thusup to KM fine lattice points close to these K sublatticepoints can be represented. For example, if 2DLVQ is used,only 22 fine lattice points in V0(O) can be represented bytheir 2 nearest sublattice points, whereas 23 fine latticepoints can be represented in 3DLVQ (M ¼ 3). In this sense,for four more fine lattice points, their side distortion canbe reduced. Therefore, more graceful coding performancecan be obtained by the MDLVQ, which will be illustratedin the experimental results in Section 4.

3. Proposed index assignment for 3DLVQ

Guiding principles for the proposed 3-descriptionindex assignment are first examined and then a 3-stepindex assignment scheme is presented. Based on theproposed scheme, the optimal index number N ofsublattice for minimizing expected distortion at high rateis investigated.

3.1. Guiding principles for 3-description index assignment

MDLVQ index assignment relates to the process ofmapping a point in a fine lattice to an M-tuple ofsublattice points to minimize side distortions whicheverchannel fails, given the fine lattice and sublattice. There-fore, for each fine lattice point, we need to find such acombination of M sublattice points from K neighboringsublattice points that can minimize the side distortions. Inthis paper, we will focus on the index assignment designfor the 3-description case. In the context of 3DLVQ, we canobtain guiding principles for the index assignment asfollows.

Theorem 1. To minimize the overall 1-description side-

distortion, each edge of the triangle formed by a 3-tuple of

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M. Liu, C. Zhu / Signal Processing 88 (2008) 2754–2763 2757

sublattice points needs to be as short as possible, while the

gravity center of the triangle is as close to the fine lattice

point as possible.

Proof. Assume that the fine lattice point l in the discreteVoronoi cell V0(O) is mapped to a 3-tuple sublattice points(l1, l2, l3) and point G is the gravity center of this 3-tuple,i.e., G ¼ (l1+l2+l3)/3 . The overall 1-description side-distortion Ds1,l for l is represented as

Ds1;l ¼ kl� l1k2 þ kl� l2k

2 þ kl� l3k2

¼ kðl� GÞ � ðl1 � GÞk2 þ kðl� GÞ � ðl2 � GÞk2

þ kðl� GÞ � ðl3 � GÞk2

¼ kl1 � Gk2 þ kl2 � Gk2

þ kl3 � Gk2 � 2½ðl� GÞ � ðl1 � GÞ

þ ðl� GÞ � ðl2 � GÞ

þ ðl� GÞ � ðl3 � GÞ� þ 3 l� G�� ��2

¼1

3ðl1 � l2Þ þ

1

3ðl1 � l3Þ

��������

2

þ1

3ðl2 � l1Þ þ

1

3ðl2 � l3Þ

��������

2

þ1

3ðl3 � l1Þ þ

1

3ðl3 � l2Þ

��������2

� 2ðl� GÞ � ½ðl1 � GÞ þ ðl2 � GÞ

þ ðl3 � GÞ� þ 3 l� G�� ��2

¼1

9½2kl1 � l2k

2 þ 2kl2 � l3k2 þ 2kl3 � l1k

2

þ ðl1 � l2Þðl1 � l3Þ þ ðl2 � l1Þðl2 � l3Þ

þ ðl1 � l2Þðl1 � l3Þ þ ðl3 � l1Þðl3 � l2Þ

þ ðl2 � l1Þðl2 � l3Þ þ ðl3 � l1Þ � ðl3 � l2Þ�

� 0þ 3kl� Gk2

¼1

3ðkl1 � l2k

2 þ kl2 � l3k2

þ kl3 � l1k2Þ þ 3kl� Gk2 (2)

To minimize the overall 1-description side-distortionDs1,l for a point l in the discrete Voronoi cell V0(O), all theterms above should be minimized. That is, the three edgesof the triangle l1l2l3 should be as short as possible whilstthe gravity center G of l1l2l3 is as close as possible to l.This completes the proof. &

Theorem 2. . Assume that l is reconstructed to the middle

point between two received sublattice points when only 2-descriptions are received. To minimize the overall 2-descrip-

tion side-distortion, each edge of the triangle formed by 3-

tuple sublattice points needs to be as short as possible, while

the gravity center of the triangle is as close as possible to the

fine lattice point.

The proof for Theorem 2 is very simple. We can replacethe variables l1, l2 and l3 in the final expression of Ds1,l in(2) by (l1+l2)/2, (l2+l3)/2 and (l3+l1)/2, respectively.Finally, we obtain

Ds2;l ¼ l�l1 þ l2

2

��������

2

þ l�l2 þ l3

2

��������

2

þ l�l3 þ l1

2

��������

2

¼1

12ðkl1 � l2k

2 þ kl1 � l3k2

þ kl2 � l3k2Þ þ 3kl� Gk2 (3)

Therefore, the same conclusion as Theorem 1 willfollow.

Based on the two results, guiding principles for 3DLVQindex assignment can be summarized as: for a triangleformed by a 3-tuple of sublattice points, (i) the sum ofsquares of three edges of each triangle needs to be assmall as possible and (ii) the gravity center of eachtriangle is as close as possible to the fine lattice point to belabeled. With the two principles, both the overall1-description side-distortion and overall 2-descriptionside-distortion can be minimized simultaneously. Theseprinciples are consistent with the results from Ostergaardet al. [10]. In [11] to facilitate realizing the first principle,we constructed 3-tuple combinations by alternativelyfinding the nearest sublattice points to the to-be-labeledfine lattice region, which may encounter some difficultiesof determining the K nearest sublattice points when KX5.However in this paper, we consider using the S-similarproperty of the A2 lattice [6] to facilitate the 3-tupleconstruction. The S-similar property is that the gravitycenter of any 3-tuple is always one of sublattice points ofthe S-similar sublattice L01/3, thus the locations of gravitycenters for 3-tuples can be easily determined. Comparedwith the construction of 3-tuples in [11], which needs todetermine the closeness between sublattice points andthe to-be-labeled fine lattice region, the new constructionstrategy in this paper is to consider the closeness betweensublattice points and a few gravity centers, which makes itmuch easier for the 3-tuple construction. In the following,we elaborate our improved scheme to realize the indexassignment, which also comprises three steps in se-quence: partitioning of sublattice, constructing of combi-nations and mapping of combinations. The last two stepsare modified accordingly, compared with the steps in [11].

3.2. The proposed scheme for index assignment

3.2.1. Partitioning of sublattice

Consider an A2 sublattice shown in Fig. 2. By exploitingthe symmetry in lattice A2, we find that if 1

6 part of V0(a),i.e., the equilateral triangle aG1G2 shown in Fig. 2(a), hasbeen assigned completely, the rest 5

6 part of V0(a) can beassigned according to the rotational symmetry in A2

lattice. Here, G1 and G2 are the gravity centers ofequilateral triangle abc and abd, respectively. In this way,we can simplify the assignment by confining the assign-ment only in the triangle aG1G2.

In [11], to determine which are the first K closestsublattice points to the triangle, the triangle aG1G2 needsto be further partitioned into two right-angled trianglesaG1P and aG2P as shown in Fig. 2(b). It is observed that thefine lattice points in triangle aG2P and triangle bG1P aresymmetrical with reference to the point P. Then, considerthe triangle abG1 instead of aG1G2 for the index assign-ment. For ease of description in the following, we denotetriangle abG1 by T1, triangle abG2 by T2, triangle acG1 by T3

and triangle bcG1 by T4. Each triangle is also referred to as

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a

b

c

d

G1

G2e

f

g

a

b

c

d

T1

T2

T3 T4

G1

G2e

f

g

P

Fig. 2. A2 sublattice and the proposed partition, e.g., fine region T1. G1 is the barycenter of the regular triangle abc: (a) aG1G2: 16 part of the Voronoi cell

region V0(a) and (b) abG1 (i.e. T1): the equivalent fine region to aG1G2, where five points of the S-similar sublattice L01/3 lying on T1 are marked by ‘+’.

M. Liu, C. Zhu / Signal Processing 88 (2008) 2754–27632758

a fine region which contains (N�1)/6 fine lattice points. Itcan be seen that the two nearest sublattice points to T1 arethe sublattice points a and b. The three nearest sublatticepoints are a, b and c, while the four nearest sublatticepoints are a, b, c and d. The closeness between sublatticepoints and T1 is applicable only when Kp4 (i.e., thenumber of nearest sublattice points selected is not largerthan 4). For K44, aG1P and aG2P have to be consideredindividually since the 5th closest sublattice point to T1

cannot be determined easily. Instead, in this paper we usethe S-similarity of the A2 lattice introduced in [6] tocircumvent this limitation. In particular, the gravity centerof any 3-tuple is always lying on the S-similar sublatticeL01/3. In our scheme, only those 3-tuples with gravitycenters lying in/on T1 are considered and these gravitycenters (totally five) are marked with ‘+’ in Fig. 2(b).

3.2.2. Constructing of combinations

From the above analysis, it can be seen that we onlyneed to consider the index assignment for the finelattice points in T1. Now we can construct 3-tuplecombinations for each of the five gravity centers on T1

by considering the nearest sublattice points around thisgravity center. All the 3-tuples for the five gravity centersare sorted according to the first term in (2). For those3-tuples with gravity centers being G1, only 1

3 of them areassigned to T1. Similarly, only 1

12 of 3-tuples are assigned toT1 with their gravity centers being a or b. This ratio is 1

2 forthe other two gravity centers. The number of 3-tuplecombinations assigned to T1 should be no smaller than(N�1)/6.

3.2.3. Mapping of combinations

In the above step of combination construction, 3-tuplecombinations have been distributed to the region T1.These combinations are not necessarily fully used in theassignment. In the following, the index assignment isdesigned to map the combinations to each fine latticepoint in L for minimizing the 1-description side-distor-

tion as follows:

minimizel;li

X3

i¼1

kl� lik2

!

3minimizel;li

1

3

X2

i¼1

X3

j¼iþ1

kli � ljk2

0@

1Aðþ3kl� Gk2Þ (4)

where liAL. Here, we consider reconstructing l to themiddle point between the two received sublattice points.Note that minimizing 2-description side-distortion issimilar to minimizing 1-description side-distortion, whichhave been shown in the above two theorems. After the finelattice points in V0(a) are assigned completely, thegenerated assignment is extended to the whole fine latticeusing shift property. The first two steps are to find the bestgroup of 3-tuple combinations for minimizing the firstterm of

P2i¼1

P3j¼iþ1kli � ljk

2 in order, while the third stepis to minimize the second termJl�GJ2 in the 1-descriptionside-distortion in (2) for the final index mapping given theordered best group of combinations. Fig. 3 shows someexamples of index assignment with different N. Given acombination in the ordered list, it should be mapped to anunallocated fine lattice point closest to its gravity centerand the process continues until all the fine lattice points inthe considered region V0(a) have been assigned.

3.3. Optimal index number at high rate

The above section addresses optimal index assignmentfor given M (number of channels) and N (index number ofsublattice). In practice, however, the requirements arenormally given as target entropy rate per-description andnumber of channels M for index assignment design.Therefore, we need to find a sublattice with an appro-priate index number N to optimize rate-distortion perfor-mance with the given requirements. The entropy rate per-description R is given by [2]

R � hðXÞ �1

Llog2ðN � vÞ (5)

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b

c

d

e

f

g

ddd

b

c

dddd

e

f

g

bg

f

e

c

dddd

Fig. 3. Three-description index assignments with different index numbers. Points on and within polygon of solid line are the fine lattice points in V0(a).

M. Liu, C. Zhu / Signal Processing 88 (2008) 2754–2763 2759

where h(X) is differential entropy of a source vector and v

is the volume of a Voronoi region of the fine lattice L. Fora given entropy rate R, achieving the optimal rate-distortion performance is boiled down to minimizing theexpected average distortion comprising side and centraldistortions. Assuming description (channel) loss rate isindependent and the same for all descriptions, denotedby pl. Then the expected distortion for 3DLVQ is given by

D3D ¼ ð1� plÞ3D0 þ 3 � ð1� plÞp

2l Ds1

þ 3 � ð1� plÞ2plDs2 þ p3

l E½kXk2� (6)

where D0 is the central distortion, Ds1 is the average1-description side-distortion, Ds2 is the average 2-descrip-tion side-distortion and E[JXJ2] is the variance of thesource. From Vaishampayan et al. [2], we knowD0EG(L)v2/L, where G(L) is the normalized secondmoment of a Voronoi cell of L and defined byGðLÞ ¼

RVðOÞ kxk

2dx=v1þ2=L. The average i-description side-distortion is given by

Dsi ¼ D0 þ1

3

Xl2L0ðaÞ

Dsi;l (7)

In contrast, the expected distortion for 2DLVQ is givenby

D2D ¼ ð1� plÞ2D̄0 þ 2ð1� plÞplD̄s1 þ p2

l E½kXk2� (8)

where D̄0 is the central distortion, D̄s1 is the average1-description side-distortion.

Optimal N is obtained by minimizing the expecteddistortions of 3DLVQ and 2DLVQ systems shown in (6) and(8), respectively. In the case of high resolution, optimal N

is obtained to be independent of the total entropy.Furthermore, optimal N can be considered as a monotonicfunction of the description loss rate pl, given the numberof channels M. It has been shown in [10] that the larger pl

the smaller N. Optimal index value N is shown in Fig. 4 asa function of pl.

4. Simulation results and analysis

We first compare the 3DLVQ using our proposed indexassignment against some other existing 3DLVQ and2DLVQ index assignments for coding Gaussian sourceand uniformly distributed source. Next, we apply theproposed 3DLVQ and 2DLVQ using SVS index assignmentfor image coding.

4.1. Gaussian source

We compare the proposed 3DLVQ index assignmentwith those in [6,10]. A two-dimensional zero-mean andunit variance Gaussian source is considered. For a faircomparison, the setting is the same as in [10], whichincludes 2�106 vectors tested with the entropy of 5 bits/dimension for each description. Index numbers of 31, 43and 61 are used for testing. The results of 1-descriptionside-distortions for different schemes are listed in Table 1,where ‘‘Theo’’ and ‘‘Exp’’ indicate the theoretical andexperimental results, respectively. For the results ofOstergaard et al. [10], the theoretical results in the columnof ‘‘Theo’’ are obtained based on the approximate1-description side-distortion formula in [10]. All the othertheoretical results are obtained based on Eqs. (5)–(7)given the designed index assignments. From Table 1,we can see our scheme achieves better results withsmaller distortions than that in [6], and similar to theresults in [10] which obtains the index assignment byusing the time-consuming linear assignment. The en-hanced method of Huang and Wu [6] in [5] is expected toachieve the same results as the proposed. We can also seethat the theoretical results of Huang and Wu [6] and theproposed method is consistent with their experimentalresults, respectively, while a difference of up to 0.8 dB(N ¼ 31) is observed for the method of Ostergaard et al.[10]. As expected, this difference decreases with increas-ing N.

Now, we compare the proposed 3DLVQ with 2DLVQusing SVS technique. The same total entropy rate Rtotal is

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Table 1Comparison of average 1-description side-distortions (dB) for different

schemes

N Ostergaard [10] Huang [6] Proposed

Theo. Exp. Theo. Exp. Theo. Exp.

31 �24.8280 �25.6729 �25.0943 �25.0901 �25.6725 �25.6767

43 �24.1396 �24.5647 �24.4120 �24.4098 �24.5879 �24.5817

61 �23.3946 �23.8757 �23.7976 �23.7941 �23.8737 �23.8704

48

1216

25

1015

20-10

-5

0

5

10

15

Total entropy rate (bit per sample)Description loss rate (%)

Diff

eren

ce D

2D-D

3D (d

B)

Fig. 5. Three-dimensional representation of the expected overall distor-

tion difference between 2DLVQ and 3DLVQ as a function of total bit-rate

and description loss rate: zero-mean and unit variance Gaussian source

is tested. The red plane in the figure is the zero plane.

Fig. 4. Optimal index number N vs. description loss rate.

M. Liu, C. Zhu / Signal Processing 88 (2008) 2754–27632760

used in the 3DLVQ and 2DLVQ. Fig. 5 shows the differenceof expected overall distortions between 3DLVQ and2DLVQ, i.e., D2D�D3D, with different total entropy ratesand different description loss rates. With a high bitrate assumption, the index numbers are selected accord-ing to the results in Fig. 4 for both systems. In particular,we tested 3-description loss rates pl of 2%, 10% and20%, respectively. Fig. 6 demonstrates the differenceof the expected distortions between 2DLVQ and 3DLVQ,as a function of total entropy rate. From Figs. 5 and 6,it can be seen that the expected distortion of the3DLVQ is smaller than that of the 2DLVQ in most casesexcept for low description loss rate and low-total entropyrate. As the total entropy rate or description-loss rateincreases, 3DLVQ outperforms 2DLVQ more remarkablywith a lager difference between the two expecteddistortions.

4.2. Uniformly distributed source

A two-dimensional uniformly distributed source isconsidered. We keep the per-description entropy rate R

fixed and the central distortions (denoted as D0) of thetwo systems are equal. Fig. 7 plots the operational pointsobtained by the two MDLVQ schemes. The normalizedexpected side distortion for 3DLVQ is calculated asplDs1+(1�pl)Ds2. The side distortion D̄s1 for the 2DLVQ isindependent of description loss rate pl. As mentioned in[10], for either 2DLVQ or 3DLVQ, optimal N is independentof the total entropy Rtotal. Therefore, the relative differ-ences among the operational points on each curve in thefigure remain unchanged irrespective of Rtotal. In otherwords, the curves only make translational shift as Rtotal

changes. It can be seen that the 3DLVQ exhibits moregraceful stairs even at the high description loss rate up to90%.

ARTICLE IN PRESS

M. Liu, C. Zhu / Signal Processing 88 (2008) 2754–2763 2761

4.3. Image coding in spacial domain

The image ‘‘Lena’’ (512�512) is encoded and decodedin the spatial domain by the two MDLVQ systems. Foreach row in the image, two neighboring pixels form aninput vector. Fig. 8 depicts the reconstructed ‘‘Lena’’images using 2DLVQ and 3DLVQ. The results are obtainedwith the quantization scale Qp ¼ 8, and the indexnumbers for the two systems are N2D ¼ N3D ¼ N ¼ 13. Qp

is used to scale the cell of the lattice, thus different ratescan be obtained as Qp changes. From the figure, it can beseen that the side-reconstructed image quality in 3DLVQis much better than that in 2DLVQ (3.3 dB higher), whenthere is only 1-description available for both 3DLVQ and

Fig. 7. 2DLVQ and 3DLVQ for uni

6 8 10 12 14 16 18 20-2

0

2

4

6

8

10

12

14

Total entropy rate (bit per sample)

Diff

eren

ce: D

2D -

D3D

(dB

)

2% description loss rate10% description loss rate20% description loss rate

Fig. 6. Difference of expected overall distortions between 2DLVQ and

3DLVQ vs. total entropy rate for zero-mean and unit variance Gaussian

source.

2DLVQ. The central distortions obtained from bothsystems are the same. Note that in Fig. 8, the total bitrates for 3DLVQ and 2DLVQ systems are 4.14 and 3.12 bpp,respectively.

4.4. Image coding in wavelet domain

We consider the rate-distortion performance for imagecoding in the wavelet domain with lower bit rates. Four-level wavelet decomposition using the 9

7-tap filter isapplied to ‘‘Lena’’ image. Two neighbor wavelet coeffi-cients in a sub band are considered as an input vector.After MDLVQ encoding, the produced indexes are codedlosslessly by adaptive arithmetic coding in each sub band.

Fig. 9 compares the expected distortions of 2DLVQ and3DLVQ. By ignoring the last term in (6) and (8) thatrepresents the distortion when no description is received,the expected distortions for 3DLVQ and 2DLVQ for theimage coding are calculated as

D2D ¼ð1� plÞ

2D̄0 þ 2 � ð1� plÞplD̄s1

1� p2l

D3D ¼ð1� plÞ

3D0 þ 3 � ð1� plÞp2l Ds1 þ 3 � ð1� plÞ

2plDs2

1� p3l

8>>>><>>>>:

(9)

Description loss rates pl of 10% and 20% are tested,respectively. It can be seen from Fig. 9 that the rate-distortion performance of our proposed 3DLVQ systemappreciably outperforms that of 2DLVQ system at lowerbit rates.

It is known that MDC is more suitable for theapplications, where one or more descriptions are mostlikely to be lost. Therefore, a lower expected sidedistortion is highly desirable in MDC. We consider theexpected side distortion by excluding the central-decoded

formly distributed source.

ARTICLE IN PRESS

Fig. 8. Reconstructed ‘‘Lena’’, with N2D ¼ N3D ¼ N ¼ 13, Qp ¼ 8. (a) Original image, (b) 2DLVQ central-decoded image, 41.58 dB, (c) 2DLVQ side-decoded

image, 24.71 dB, (d) 3DLVQ central-decoded image, 41.58 dB, (e) 3DLVQ side-decoded image with 2-descriptions received, 32.83 dB and (f) 3DLVQ side-

decoded image with 1-description received, 28.14 dB.

0 0.5 1

Exp

ecte

d di

stor

tion

per s

ampl

e

Total bit rate (bit per sample)

Exp

ecte

d di

stor

tion

per s

ampl

e

Total bit rate (bit per sample)1.5 2 2.5 3 3.5

0

10

20

30

40

50

603DLVQ2DLVQ

3DLVQ2DLVQ

0 0.5 1 1.5 2 2.5 3 3.50

10

20

30

40

50

60

70

80

Fig. 9. Comparison of expected distortions for 2DLVQ and 3DLVQ in coding image ‘‘Lena’’. (a) N2D ¼ N3D ¼ 7 with 10% description loss rate and (b)

N2D ¼ N3D ¼ 7 with 20% description loss rate.

M. Liu, C. Zhu / Signal Processing 88 (2008) 2754–27632762

ARTICLE IN PRESS

0 0.5 1 1.5 2 2.5 3 3.5

3DLVQ2DLVQ

3DLVQ2DLVQ

0 0.5 1 1.5 2 2.5 3 3.50

20

40

60

80

100

120

140

160

180

0

20

40

60

80

100

120

140

160

180

Exp

ecte

d si

de d

isto

rtion

per

sam

ple

Exp

ecte

d si

de d

isto

rtion

per

sam

ple

Total bit rate (bit per sample) Total bit rate (bit per sample)

Fig. 10. Comparison of expected side distortions for 2DLVQ and 3DLVQ in coding image ‘‘Lena’’. (a) N2D ¼ N3D ¼ 7 with 10% description loss rate and

(b) N2D ¼ N3D ¼ 7 with 20% description loss rate.

M. Liu, C. Zhu / Signal Processing 88 (2008) 2754–2763 2763

distortion term in (9), which is obtained by

Ds2D ¼

2 � ð1� plÞplD̄s1

1� p2l � ð1� plÞ

2¼ D̄s1

Ds3D ¼

3 � ð1� plÞp2l Ds1 þ 3 � ð1� plÞ

2plDs2

1� p3l � ð1� plÞ

3¼ plDs1 þ ð1� plÞDs2

8>>>><>>>>:

(10)

Fig. 10 plots the expected side distortions of 2DLVQ and3DLVQ for different total bit rates. It can be seen that withthe same total bit rate, our proposed 3DLVQ systemachieves significantly smaller expected side distortionthan that in 2DLVQ, especially at lower bit rates.

5. Conclusion

We have addressed the issue of 3DLVQ index assign-ment design in a different way, compared with theexisting 3DLVQ index assignment schemes. Firstly, guid-ing principles for 3DLVQ index assignment have beendeveloped, which have been shown to be optimal in termsof minimizing 1-description and 2-description side dis-tortion. More importantly, based on the principle, aconcrete three-step design scheme has been proposed totackle the complicated assignment. The 3DLVQ with ourproposed index assignment has been tested for codingdifferent sources. Numerical and experimental resultshave demonstrated the effectiveness of our design.

Acknowledgments

The authors would like to thank Dr. Xiaolin Wu forsome constructive discussion and for providing of Mr.Xiang Huang’s thesis. We would also like to thank theeditor and the anonymous reviewers for their valuableand constructive comments that greatly improved thispaper.

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