A tabu search based maximum descent algorithm for VQ codebook design

753

Short Paper_________________________________________________

A Tabu Search Based Maximum Descent Algorithm

for VQ Codebook Design

HSIANG-CHEHHUANG, SHU-CHUANCHU*_{,JENG-SHYANGPAN}**

ANDZHE-MINGLU***

Department of Electronics Engineering National Chiao Tung University

Hsinchu, 300 Taiwan

*

School of Informatics and Engineering Flinders University of South Australia

Australia

**

Department of Electronic Engineering National Kaohsiung University of Applied Sciences

Kaohsiung, 807 Taiwan E-mail: [email protected]

***

Department of Automatic Test and Control Harbin Institute of Technology

Harbin, China

A maximum descent (MD) method has been proposed for vector quantization (VQ) codebook design. Compared with the traditional generalized Lloyd algorithm (GLA), the MD algorithm achieves better codebook performance with far less computation time. However, searching for the optimal partitioning hyperplane of a multidimensional cluster is a difficult problem in the MD algorithm. Three partition techniques have been pro-posed for the MD method in the literature. In this paper, a new partition technique based on the tabu search (TS) approach is presented for the MD algorithm. Experimental re-sults show that the tabu search based MD algorithm can produce a better codebook than can the conventional MD algorithms.

Keywords: VQ, codebook design, maximum descent method, tabu search, GLA

1. INTRODUCTION

Vector quantization (VQ) has been extensively and successfully used in speech en-coding and image compression [1]. The k-dimensional, N-level vector quantizer is de-fined as a mapping from a k-dimensional Euclidean space Rkinto a certain finite set C of

Rk. This finite set C = {Y1, Y2, …, YN} is called a VQ codebook. Each Yi

∈

Rkin codebook

C is called a codeword, i = 1, 2, …, N. The VQ quantizer consists of two procedures: an

encoder and a decoder. The encoder assigns each input vector X to an index i, which

Received November 24, 1999; revised April 17, 2000; accepted July 19, 2000. Communicated by Yung-Nien Sun.

points to the closest codeword Yiin the codebook. The decoder uses the index i to look

up the codeword Yiin the codebook. The squared Euclidean distortion measure is often

used to measure the distortion between the input vector X = {x1, x2, …, xk} and the

codeword Yi= {yi1, yi2, …, yik}, i.e.,

. (1)

The key problem with VQ is to generate a good codebook from a number of training vectors. The codebook design problem is essentially a clustering problem where a train-ing set S = {X1, X2, …, XM} is clustered into N subsets S1, S2, …, SN, which satisfy

(2)

and

if i≠ j . (3)

The centriod of the subset Siis the codeword Yi, and if

)

,

(

min

)

,

(

1 j N l j i l

Y

d

X

Y

X

d

≤ ≤

=

, (4)

then Xl

∈

Si. The aim of the codebook design algorithm is to minimize the total distortion

defined as follow:

. (5)

A generalized Lloyd clustering algorithm [2] proposed by Linde, Buzo, and Gray, called the LBG algorithm or GLA, is typically used to generate VQ codebooks. However, this iterative algorithm depends on the initial codebook, often obtains the local optimal codebook and requires intensive computation. Thus, the two prominent problems in codebook design are how to approach the global optimum and how to reduce the compu-tational complexity. In order to obtain better codebooks, Vaisey and Gersho [3] com-bined the simulated annealing (SA) technique and the LBG iteration in VQ codebook design. In their SA-GLA algorithm, they took the codebook design as a combinatorial problem by perturbing the encoder of a vector quantizer in order to let the LBG algo-rithm converge again and again so as to obtain lower overall distortion. This method can improve codebook performance; however, the computational time used by this algorithm is at least several times that used by the LBG algorithm. The stochastic relaxation (SR) approach [4] has also been proposed to improve codebook performance. The basic idea of the SR approach is to perturb the solution by adding some values to the codewords during each iteration. Many methods designed to reduce the computational time have appeared in the literature. The subspace distortion method [5] attempts to reduce the computation burden by reducing the dimension of the distortion measure in the LBG algorithm. The pairwise nearest neighbor (PNN) algorithm [6] is a bottom-up clustering algorithm that generates a codebook by merging nearest training vector clusters until the desired number of codewords is obtained. The codebooks generated by both methods are not as good as the codebook generated by the LBG algorithm although the computational time is reduced by several times.

∑

= − = k l il l i x y Y , X d 1 2 ) ( ) (

S

S

N i i

=

=

U

1

∑

₌ ≤≤ = M l j l N j d X ,Y D 11 ) ( min

I

j=Φ i S S

Designed to generate better codebooks while reducing the computational time, the maximum descent (MD) algorithm [7] has been proposed for VQ codebook design. This algorithm begins by treating the training vector set S = {X1, X2, …, XM} as a global

clus-ter. Then, the algorithm generates the required number of clusters one by one, subject to the maximum distortion reduction criterion, until the desired number of codewords is obtained. Compared with the LBG algorithm, the codebook performance is improved, and the computational time is substantially reduced. However, searching for the optimal partitioning hyperplane in a multidimensional cluster is a difficult problem with the MD algorithm. Three techniques [8] have been presented for searching for the optimal parti-tioning hyperplane with the MD method. However, these techniques all restrict the searching range to hyperplanes that are perpendicular to the basis vectors of the vector space that can be obtained using discrete cosine transform (DCT), so it is difficult for them to find the global optimal partitioning hyperplane. In order to find the global or nearly global optimal partitioning hyperplane, the tabu search (TS) approach first pro-posed by Glover [9] is introduced in this paper. The tabu search approach is a global op-timization technique with short-term memory, that can be used to solve many difficult combinatorial optimization problems. In previous works [10-11], the tabu search ap-proach has been successfully applied to codebook index assignment over channel noise and texture segmentation. The main problem with MD is determining how to split one cluster into two clusters with minimal mean squared error. This is a typical combinatorial optimization problem, so the tabu search approach is suitable for solving it.

Section 2 presents the main idea of the maximum descent algorithm and three con-ventional methods used to search for the optimal partitioning hyperplane. Section 3 pre-sents the proposed tabu search based maximum descent algorithm. Section 4 describes experiments carried out to compare the codebook performance of the proposed algorithm with that of the LBG algorithm, the simulated annealing method and the basic MD algo-rithms with the three partition techniques presented in [8]. Section 5 concludes the paper.

2. THE MAXIMUM DESCENT ALGORITHM

Let us consider the design of an N-level codebook C = {Y1, Y2, …, YN} for training

set S = {X1, X2, …, XM}, where Yi

∈

Rk, i = 1, 2, …, N, Xm

∈

Rk, m = 1, 2, …, M, M >> N.

The maximum descent algorithm first views the training vector set S as a global cluster. This cluster is split into two new clusters by an optimal partitioning hyperplane. One of these two clusters is then further partitioned into two clusters so as to generate three new clusters based on a maximum distortion reduction criterion. For the general case of

L(L>2) clusters, the MD method attempts to generate L+1 clusters by splitting one of the

former L clusters into two new clusters and keeping the other L-1 clusters unchanged such that the distortion can be reduced as much as possible. This procedure is performed until the desired number of clusters is obtained. Finally, the centriods of these clusters are taken as codewords.

Now, we will to investigate the distortion reduction resulting from splitting a cluster into two new clusters. Without loss of generality, we consider the case in which the original training set S has been partitioned into L clusters, i.e., S = {S1, S2, …, SL}.

As-sume that a hyperplane Hi(U, v) = {Z

∈

Rk: UTZ = v } further partitions the cluster Siinto

} : {Z S U Z v S T i ia = ∈ < , } : {Z S U Z v S T i ib = ∈ ≥ , (6)

where U

∈

Rk, v

∈

R, 1≤ i ≤ L, and T denotes the transpose.

The centroid of cluster Sican be defined as

, (7)

where niis the number of input vectors that belong to cluster Si. When the centriod Yiis

used to quantize all the vectors inside Si, the induced total squared Euclidean distortion

D(Si) is

, (8)

where||Xj − Yi||2denotes the squared Euclidean distortion between Xj and Yi. Then the

reduction in distortion induced by partitioning the cluster Siinto two clusters using the

partition plane Hican be expressed as

ri= D(Si)− [D(Sia) + D(Sib)]. (9)

Direct computation of Eq. (9) requires many calculations of squared distances. In order to reduce the computational complexity of Eq. (9), we can easily rewrite Eq. (9) as fol-lows (the proof for this can be found in [8]):

(10)

or

, (11)

where ni, niaand nibare the number of vectors in Si, Siaand Sib, respectively. Yi, Yiaand Yib

are the centroids of clusters Si, Siaand Sib, respectively.

From equation 10 we can easily prove that riis always greater than zero, i.e., that

the reduction in distortion is never less than zero. In order to obtain the maximum reduc-tion funcreduc-tion for cluster Si, we need to use some techniques to search for the optimal

partitioning hyperplane for cluster Si. To form L+1 clusters from the L clusters with

maximum distortion reduction, the maximum distortion reduction functions of all the clusters need to be calculated (Fig. 1a); then, one of the L clusters Spwhich satisfies

p i L i {rˆ} rˆ

max

1≤≤ = is split into two new clusters while the remaining L-1 clusters stay un-changed. In the step that obtains L+2 clusters from the L+1 clusters, only two maximum distortion reduction functions, i.e.,

rˆ

_pa and

rˆ

_pb, of the newly formed clusters Spa and

Spb(Fig. 1b) need to be computed since the

rˆ

_is of all other clusters have already been

i S X : j j i n X Y j i

∑

_∈ =

∑

_∈ − = i j S X : j i j i) X Y S ( D 2 2 ia i ib ia i i Y Y n n n r = ⋅ ⋅ − 2 ib i ia ib i i Y Y n n n r = ⋅ ⋅ − i

rˆ

i Hˆ

(a)

(b)

Fig. 1. The optimal formation of (L+1) clusters that provides maximum reduction of overall distor-tion.

computed in the previous step. Thus, in each step in forming an additional cluster, only two optimal partitioning hyperplanes need to be found. Based on this maximum descent criterion, the clusters are split one by one until the required N clusters are obtained. We can easily prove that only 2N-3 optimal partition searches are required to design an

N-level vector quantizer, so the number of optimal partition searches is proportional to

the size of the codebook.

It is noted that the MD algorithm has an advantage in that it tends to partition clus-ters that are densely populated and never generates a cluster that includes only one vector, which sometimes happens with the conventional LBG algorithm. However, exhaustive searching for the optimal partitioning hyperplane of a multidimensional cluster is a dif-ficult and computationally intensive problem, and it is very hard to carry out in practice. In order to reduce the complexity, three methods of searching for the optimal partitioning hyperplane were presented in [8] and are geiven below.

2.1 Constrained Exhaustive Search

This method searches for the optimal hyperplane among the hyperplanes that are perpendicular to the basis vectors of the vector space that can be obtained by means of the discrete cosine transform of the input image. For a cluster Si, the lower and upper

bounds of the vectors on one of the DCT basis axes are found. The algorithm first assigns a splitting threshold T to the lower bound and places the training vectors whose projec-tions on this axis are smaller than T into cluster Siaand puts the remaining ones into Sib. ri

for threshold T is computed. The threshold T is then increased by means of a predeter-mined incremental step. The procedures are repeated until T reaches the upper bound. The optimal partitioning hyperplane and the corresponding

rˆ

_i on this axis are recorded. Based on the same procedure, all other optimal partitioning hyperplanes for other basis axes are found. By finding the maximum

rˆ

_i among all the

rˆ

_is obtained, the constrained global optimal partitioning hyperplane and the corresponding distortion reduction func-tion are obtained for cluster Si.

2.2 Successive Search

Constrained exhaustive search is a computationally intensive technique. Its com-plexity can be reduced by using a successive search method based on the idea that the maximum reduction can often be obtained when D(Sia) is equal to D(Sib). This method

starts by assigning a threshold T half way between the upper and lower bounds on an axis. Then D(Sia) and D(Sib) are evaluated. If D(Sia) > D(Sib), the threshold is set to T1, which is

half way between T and the lower bound. If D(Sia)≤ D(Sib), T1is set to a value half way

between T and the upper bound. D(Sia) and D(Sib) are computed for T1, and the next

threshold T2can be determined using the same method described above. This procedure

is repeated until the difference between D(Sia) and D(Sib) is smaller than a pre-defined

value. The optimal hyperplane and the distortion reduction value are thus obtained. Similarly, successive search needs to be performed over all the basis axes in order to find the maximum distortion reduction value and the optimal hyperplane.

2.3 Fast LBG Search

If the optimal partition hyperplane is restricted to being perpendicular to only one of the basis vectors of the transformed vector space, then we can use the following fast LBG algorithm to perform binary splitting of a cluster Siinto Siaand Sibso as to reduce the

amount of computation. In this method, the difference in the Euclidean distances d(X, Yia,

Yib) between the input vector X

∈

Siand the two centroids Yiaand Yibis defined as

fol-lows:

∑

∑

₌ − + ₌ − − = k j j j j k j j j b x a b a 1 2 2 1 ) ( ) ( 2 , (12) where X = (x1, x2, …, xk), Yia= (a1, a2, …, ak) and Yib= (b1, b2, …, bk). Based on Eq. (12), if

∑

∑

₌ − > k₌ − j j j j k j j j b x a b a 1 2 2 1 ) ( 2 1 ) ( , (13)

then the vector X is closer to the cluster Sia, so the vector X can be placed in Sia.

Other-wise, it is placed in Sib. Since (aj − bj) and do not change throughout the

partitioning process and can be pcalculated, the amount of computation is greatly re-duced.

3. TABU SEARCH BASED MD ALGORITHM

The above three methods restrict the searching range to hyperplanes that are per-pendicular to the basis vectors of the vector space that can be obtained by means of dis-crete cosine transform of the input image, so it is difficult for them to find the global op-timal hyperplane. In this section, we will introduce the tabu search technique which can be used to search for the optimal hyperplane.

∑

∑

₌ − − ₌ − = k j j j k j j j ib ia,Y x a x b Y , X d 1 2 1 2 _{( )} ) ( ) (

∑

= − k j j j b a 1 2 2 ) (

The basic idea of tabu search is to explore the search space of all feasible solutions by means of a series of moves and to forbid some search directions at the present itera-tion in order to avoid cycling and jump off local optima. The elements of moving from the current solution to its selected neighbor are partially or completely recorded in the tabu list for the purpose of forbidding reversal of replacement in a number of future it-erations. The tabu search approach begins with test solutions generated randomly and evaluates the objective function for these solutions. If the best of these solutions is not tabu or if it is tabu but satisfies the aspiration criterion, then this solution is selected as the new current solution to be used to generate test solutions for the next iteration. It is called the aspiration criterion if the test solution is a tabu solution but the objective value is better than the best value of all the iterations.

The basic task of the MD algorithm is to find the optimal partitioning of a certain cluster S into two clusters Saand Sbwhich maximizes the following objective function

(14) or

. (15)

Here n, naand nbare the number of vectors in S, Saand Sbrespectively, where n = na+ nb.

Y, Yaand Yb
are

centroids of clusters S, Sa, and Sb, respectively.

In order to describe the proposed algorithm, the corresponding indices of the train-ing vectors that belong to cluster S are used to form a solution. Every solution can be divided into two parts. The corresponding training vectors in the first part belong to cluster Sa, and the corresponding training vectors in the second part belong to cluster Sb.

Thus the optimal partitioning of S into Saand Sbcan be found using the following

algo-rithm.

Let Pt, pcand pbbe the test solutions, the best solution of the current iteration and

the best solution of all the iterations, respectively, where Pt= {p1, p2, …, pNs}, pi= {pia,

pib} is one of test solutions, piaincludes the indices of the training vectors that belong to

Sa, piaincludes the indices of the training vectors that belong to Sb, 1≤ i ≤ NS, and NSis

the number of test solutions, pc= {pca, pcb} and pb= {pba, pbb}.

Let Rt, rcand rbdenote the objective function values for the test solutions, the

ob-jective function value for the best solution of the current iteration, and the obob-jective function value for the best solution of all the iterations, respectively, where Rt= {r1,

r2, …, rNs}, riis the objective function of pi, 1≤ i ≤ NS, and NS is the number of test

solutions. The initail test solutions are generated randomly. After the first iteration, the test solutions are generated from the best solution of the current iteration by means of some moves. The tabu list memory stores the moved indices only. It is a tabu condition if the moved indices used to generate the test solution from the best solution of the current iteration are the same as any records in the tabu list memory. The algorithm is given as follows:

Step 1. Set the tabu list size be TS, the number of test solutions be NSand the maximum

number of iterations be Im. Set the iteration counter i = 1, and let the insertion

point of the tabu list tl= 1. Generate NS initail solutions Pt= {p1, p2, …, pNs}

2 a b a _{Y Y} n n n r= ⋅ ⋅ − 2 b a b _{Y Y} n n n r= ⋅ ⋅ −

randomly, calculate the corresponding objective values Rt= {r1, r2, …, rNs}

according to Eq. (14) and find the current best solution pc= pj, j = arg maxlrl, 1

≤ i ≤ NS. Set pb= pcand rb= rc.

Step 2. Copy the current best solution pcto each test solution pi, 1≤ i ≤ NS. For each test

solution pi= {pia, pib}, 1 ≤ i ≤ NS, generate a new solution by means of the

following three substeps.

Step 2.1. If nia≥ 2 and nib≥ 2, select piaor pibrandomly; else if nia= 1 and nib≥ 2,

select pib; else if nib= 1 and nia≥ 2, select pia; else if nia= 1 and nib= 1,

select neither of them.

Step 2.2. If piais selected, then select an index in piarandomly and move it to pib;

else if pibis selected, then select an index in pibrandomly and move it to

pia. If neither of them is selected, the solution remains unchanged.

Step 2.3. Calculate the corresponding objective value rifor the new test solution.

Step 3. Sort r1, r2 ,…, rNs in decreasing order. From the best new test solution to the

worst new test solution,if the new test solution is a non-tabu solution or if it is a tabu solution but its objective value is better than the best value of all iterations

rb, then choose this new solution as the current best solution pc, choose its

objective value as the current best objective value rc, and go to step 4; otherwise,

try the next new test solution. If all new test solutions are tabu solutions, then go to step 2.

Step 4. If rb< rc, set pb= pcand rb= rc. Insert the moved index of the current best solution

pcinto the tabu memory list. Set the insertion point of the tabu list tl= tl+ 1. If tl

> TS, set tl= 1. If i < Im, set i = i + 1 and go to step 2; otherwise, record the best

solution and terminate the algorithm.

4. PERFORMANCE COMPARISONS

In order to demonstrate the efficiency of the proposed algorithm, the LBG algorithm, the SA algorithm, the conventional MD algorithms and the tabu search based MD algo-rithm were all implemented to generate codebooks. For the SA algoalgo-rithm, the initial tem-perature T0was 35, and the temperature was decreased by 1% after each iteration step

until the number of iterations reached 30. Because each iteration of SA included the LBG algorithm, the CPU time of the SA was about 25 times that of the LBG algorithm in the experiments. Here, the parameters of tabu search were TS= 20, NS= 20, and Im= 200. All

the tests were run on a Pentium II PC running at 233MHz. Three images, LENA, PEP-PERS and F-16, with a resolution of 512× 512 pixels, 8bits/pixel, were used. The image LENA was used to generate codebooks of size 256 with dimension 16(4× 4), and the images PEPPERS and F-16 were used to test the coding performance of the codebooks. Results are shown in Tables 1 and 2. Table 1 compares the six algorithms based on CPU time and PSNR when generating codebooks of size 256 were generated. From Table 1
we can see that the PSNR of the tabu search based MD algorithm was improved by 0.5dB compared with the constrained exhaustive MD algorithm and by 1.4dB compared with the LBG algorithm although it needed more CPU time compared with the con-strained exhaustive MD algorithm. For the images outside the training set, Table 2 shows the performance of codebooks of size 256 that were generated by different algorithms. From Table 2, we can see that the PSNR of the tabu search based MD algorithm was

Table 1. Performance comparison of LBG, SA and various MD algorithms for the image within the training set.

Algorithm CPU time

(Sec) PSNR(dB)

LBG 360.5 30.23

SA 9035.2 31.48

Constrained Exhaustive Search MD 120.2 31.14 Successive Search MD 15.3 30.96 Fast LBG Search MD 3.8 31.08 Tabu Search Based MD 1023.1 31.64

Table 2. Performance comparison of LBG, SA and various MD algorithms for images outside the training set.

PSNR of the coded images (dB) Algorithm

Peppers F-16

LBG 28.45 26.37

SA 29.01 26.93

Constrained Exhaustive Search MD 28.94 26.82 Successive Search MD 28.85 26.75 Fast LBG Search MD 28.87 26.79 Tabu Search Based MD 29.14 27.08

improved by 0.2dB compared with the constrained exhaustive MD algorithm and by 0.7dB compared with the LBG algorithm. Although the proposed algorithm needs more CPU time than the conventional MD algorithms, it needs less time than the SA algorithm and has better performance than the SA algorithm and the other algorithms. In conclu-sion, the proposed algorithm can obtain better reconstructed image quality than the LBG algorithm and all the conventional MD methods not only for an image in the training set, but also images outside the training set.

5. CONCLUSIONS

In this paper, a tabu search based maximum descent method for generating books for vector quantization has been proposed. This method generates better code-books than does the LBG algorithm or the conventional MD algorithms although it needs more CPU time than the LBG algorithm and the conventional MD algorithms. Not only is the performance improved, but the computation time is also reduced based on a com-parison of the proposed algorithm with the simulated annealing method. Since codebook design is performed off line for most applications, the proposed tabu search based maxi-mum descent algorithm can be viewed as a good approach to codebook design.

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9. F. Glover and M. Laguna, Tabu Search, Boston, Kluwer Academic Publishers, 1997. 10. J. S. Pan and S. C. Chu, “Non-redundant VQ channel coding using tabu search

strat-egy,” Electronics Letters, Vol. 32, No. 17, 1996, pp. 1545-1546.

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H. C. Huang (
) got the Ph.D. in Electronics Engineering from National

Chiao Tung University in 2001. Currently he is the research associate at Department of Electronics Engineering, National Chiao Tung University. His current research interests include pattern recognition, image compression and watermarking.

S. C. Chu () received the B. M. degree in Industrial Engineering and

Man-agement from the National Taiwan Institute of Technology, Taiwan, in 1988. She is cur-rently a Ph.D. candidate at the Flinders University of South Australia, Adelaide, Austra-lia. Her current research interests include data mining and clustering.

J. S. Pan () received the B. S. degree in Electronic Engineering from the

National Taiwan Institute of Technology, Taiwan, in 1986, the M. S. degree in Communication Engineering from National Chiao Tung University, Taiwan, in 1988, and the Ph.D. degree in Electronic Engineering from the University of Edinburgh, U.K., in 1996. Currently, he is a Professor in the Department of Electronic Engineering, National Kaohsiung University of Applied Sciences, Taiwan. His current research interests include pattern recognition, speech coding and image processing.

Z. M. Lu ( ) received the B. S. degree, M. S. degree and Ph.D. all in

Elec-tronic Engineering from the Harbin Institute of Technology, Harbin, China, in 1995, 1997 and 2001, respectively. He is currently an Associate Professor at the Harbin Insti-tute of Technology. His current research interests include speech coding and image proc-essing.