A Fast Finite-State Algorithm for Generating RGB Palettes of Color Quantized Images

Short Paper

A Fast Finite-State Algorithm for Generating RGB Palettes of Color Quantized Images

YU-LEN HUANG AND RUEY-FENG CHANG⁺ Department of Computer Science and Information Engineering

Tunghai University Taichung, 407 Taiwan E-mail: [email protected]

+Department of Computer Science and Information Engineering National Chung Cheng University

Chiayi, 621 Taiwan E-mail: [email protected]

On the WWW, the transmission time for videos and images impacts the perform- ance of a web site. In order to reduce the bandwidth that is used to transmit images over the Internet, most image formats adopt only a limited number of colors used simultane- ously to display color images on a video monitor. Hence, generating a good color palette for a color digitized image is an important task for Internet applications. In general, the Linde-Buzo-Gray (LBG) algorithm can be used to cluster a color digitized image in which each pixel is considered as a 3-dimension vector in an RGB color space for gen- erating a color palette. The codebook generated by the LBG algorithm can be considered as the color palette for the color image. In order to obtain a good color palette, the LBG algorithm needs a large amount of computation time. In this paper, we propose a color finite-state LBG (CFSLBG) algorithm that reduces the computation time by exploiting the correlations of palette entries between the current and previous iterations. Instead of searching the whole color palette, the CFSLBG algorithm searches only a small number of colors that are very close to the training vector. Thus, the computation time for color quantization is reduced. The proposed approach generates RGB palettes efficiently with little sacrifice of quantized image quality. This paper describes the implementation of this work and simulation results.

Keywords: palettes generating, color quantization, LBG algorithm, finite-state algorithm, VQ

1. INTRODUCTION

Video monitors typically use the three primary color components, red, green, and blue, to specify the color of each pixel in a color image. Each primary component usually provides 8 bits for specifying the color of each pixel in a full-color digitized image.

Received January 21, 2002; revised January 21 & April 24, 2003; accepted May 8, 2003.

Communicated by Ming-Syan Chen.

* This work was supported by the National Science Council, Taiwan, R.O.C., under Grand NSC 91-2213-E-029-021.

There has been steady growth in the Internet resources utilization for the last decade. The World Wide Web (WWW), which combines multimedia and networking techniques, is widely used to access multimedia information on the Internet. Hence, the WWW has become the most widely used part of the Internet. The transmission time required for videos and images impacts the performance of a WWW site. In order to reduce the net- work bandwidth required to transmit the images over the Internet, most of the color im- age formats adopt only a limited number of colors to simultaneously display full-color images on display devices. Generally speaking, 256 colors are enough to display a color image. The set of these quantized 256 colors is called a color map or palette. The monitor uses the palette to display a color image. Each entry in a color palette contains a 24-bit value, which specifies the three color components. Thus, in order to achieve better net- work performance, a full-color digitized image is usually quantized using a 256-color palette.

Color quantization algorithms can be grouped into splitting algorithms and cluster- ing-based algorithms. Splitting algorithms iteratively split the color space of the original image into color subspaces [1, 2]. Then, the representative color of each subspace be- comes the quantized color. Generally, splitting algorithms are fast. However, they cannot always obtain the optimal solution because the splitting operations cannot be resumed.

On the other hand, clustering-based algorithms extract quantized colors by applying various clustering algorithms. The algorithms may generate an optimal color palette, but these approaches are very time consuming and complicated [3-8]. In this paper, we pro- pose an effortless and straightforward clustering algorithm, which is fast and excellent for generating palette for a color image.

Vector quantization (VQ) has been shown to be an efficient method of image coding [9-12]. The input vectors are individually quantized to the closest codeword in the code- book. The codebook is generated by using some clustering algorithms from a set of train- ing images. Image compression is achieved by transmitting the codeword indices for the information of the encoded images. Image decompression is done by utilizing the indices as addresses to the corresponding codewords in the decoder’s codebook. Generating a good codebook is the key step in VQ. The iterative clustering algorithm proposed by Linde, Buzo, and Gray (LBG) is usually used in VQ [3]. A number of methods for gener- ating color palettes using the VQ codebook design techniques were proposed in [13-15].

In view of color image coding, each pixel can be considered as a 3-dimension vector in the RGB color space. These vectors are the input vectors of the color VQ (CVQ) scheme.

The LBG algorithm is most popular method used to select a color palette with a limited number of colors from a full-color digitized image.

In each iteration of the LBG algorithm, it searches the whole color palette in order to find the corresponding palette entry for each training vector. That is, the LBG algo- rithm requires a large amount of computation for color quantization. This paper proposes a novel color finite-state LBG (CFSLBG) algorithm that reduces the computation time required to select palettes from color images. Instead of searching the whole color palette, the proposed algorithm searches only a small part of the palette to find the corresponding palette entry for each training vector. In the CFSLBG algorithm, the number of palette entries that need to be searched for a training vector in each iteration is always much smaller than the size of the whole palette. For this reason, the duration of each iteration is greatly reduced. The computation time of the CFSLBG algorithm is much smaller than

that of the LBG algorithm.

The rest of this paper is organized as follows. We discuss the VQ technique and LBG algorithm in section 2. Section 3 presents the structure of the proposed CFSLBG algorithm. Experimental results are given in section 4. Finally, conclusions are drawn in section 5.

2. LBG ALGORITHM FOR GENERATING COLOR PALETTES

A vector quantization is defined as a mapping from a k-dimension Euclidean space R^k to a finite subset of R^k. This finite set C={xˆi:i=1,...,N} is called a codebook, where N is the size of the codebook. Each vector xˆ_i =(xˆ₀ ,...,xˆk₋₁) in codebook C is called a codeword. The codebook used by the VQ encoder and decoder is generated by using an iterative clustering algorithm, such as the LBG algorithm.

In color digitized images, each pixel is considered as a 3-dimension vector in the RGB space. Each of these vectors contains three values for each of the primary compo- nents, red, green, and blue. In many applications, the codebook in VQ is considered as the color palette when a color digitized image is quantized. In general, a complete color VQ (CVQ) image coding includes three basic steps. First, an index i, which points to the closest color vector xˆ in the color palette, is assigned to each input vector x = (xR, xG, xB) by the CVQ encoder. Then, the index i is transmitted to the CVQ decoder. Finally, the CVQ decoder decompresses the image by using the transmitted indices to find the corre- sponding color from the color palette. Fig. 1 shows the basic structure of the CVQ scheme. The distortion between the input vector x and its corresponding palette entry xˆ is measured as the squared Euclidean distortion measure, i.e.,

(

) (

) (

)

) ˆ ,ˆ (

2 2

2 2

B B G G R

R x x x x x

x d

− +

− +

−

=

−

= x x x

x (1)

CVQ RGB Palette CVQ

Encoder

Codeword of x

Quantized Color Image Channel

Index of x

Input Vector x

Closest Codeword Searching

Table Look-Up

CVQ Decoder

R G

B

R G

B

Fig. 1. The basic structure of the CVQ scheme.

In the LBG algorithm, color palette generation is an iterative process which converts the previous palette into the current palette from the training color image. The algorithm begins with a predefined initial color palette and repeats the mapping until the achieved improvement of the average distortion between the current and previous iterations is small enough. The performance of the LBG algorithm is dependent on the initial color palette. A number of techniques for designing the initial codebook were proposed in [16].

Then, an LBG algorithm iteration clusters each training vector to the closest entry in the previous color palette. Each palette entry is replaced by the centroid of the set of training vectors that are closer to this palette entry than to the others in the color palette. Let

ˆ) ( x

CS be the set of these training vectors. The centroid of the set CS( xˆ) is defined as

( )

( ) ( ) ∑

∈

=

x x

x x x

ˆ ˆ

ˆ 1 centroid

CS CS

CS , (2)

where ||CS( xˆ) denotes the number of training vectors in CS( xˆ). The average distortion D_AVG of training vectors and their closest palette entries can be expressed as

D_AVG T d

x T

=

∑

1

b g

where |T| is the number of training vectors in training set T and xˆ is the closest palette entry for the training vector x. The algorithm terminates when the difference of the aver- age distortion between iteration i^th and (i − 1)^th is smaller than a predefined distortion threshold ε ⁽ε > 0). Let P_i be the color palette generated in the i^th iteration, and let xˆ be ⁱ a palette entry in P_i. The LBG algorithm for color image coding is concisely given as follows:

Step 1: Design an initial RGB color palette P0 and set i ← 1, DAVG₀← ∞.

Step 2: For each training vector x, find the closest palette entry xˆ^i-1 by searching the whole color palette P_i−1.

Step 3: Compute the average distortion DAVGi. If |D_AVGi−1− DAVG_i| / DAVG_i is smaller than ε, then stop.

Step 4: For each entry xˆ^i-1 in Pi−1, generate a new entry xˆ ⁱ ← centroid(CS(xˆ^i-1)) and add xˆ into ⁱ Pi. Set i← i + 1 and go to step 2.

In step 2 of the LBG algorithm, each training vector requires searching throughout the entire color palette. Because of the inefficiency of full searching, the LBG algorithm requires a large amount of computation to obtain a good color palette from the training image. We have observed that the current palette entry for a training vector x is usually very close to the palette entry for x in the previous iteration. Therefore, we propose a modified LBG algorithm in this paper that searches only a small number of palette en- tries for x. Because it does not search the whole color palette, the computation time re- quired by our algorithm for color palette generation is much reduced.

3. COLOR FSLBG ALGORITHM

Although the VQ scheme yields acceptable performance for image coding, the fi- nite-state vector quantization (FSVQ) schemes [16-21] improve performance for an or- dinary VQ. An FSVQ can be viewed as a finite collection of ordinary VQ’s, each with its own codebook associated with a state, which is called the state codebook. The encoding state of the current input vector is decided by a state function F(x). This coding state may be described by a state variable s∈ S = {si: i = 1, …, M}, where M is the total number of states. The FSVQ is defined as a mapping from R^k × S to a subset of a master code- book MC = {x_i: i = 1, …, N}. For each state si, the FSVQ encoder selects Nfcodewords by means of the state function from the master codebook MC as the state codebook SCs. For each input vector x, the encoder decides the current state s and then searches the state codebook SCsto find its corresponding codeword. In the decompression phase, the de- coder finds the same current state s and the corresponding codeword in the same state codebook SCs by means of the transmitted index. The codebook size of the state code- book is much smaller than that of the master codebook. Hence, the searching time can be reduced, and the image quality can be maintained. A fast finite-state algorithm that re- duces the computation time for vector quantizer design by exploiting FSVQ techniques was proposed in [22]. That paper shows that the fast finite-state algorithm can reserve the quality of encoded image. We notice that the finite-state technique was well suited to color image coding. Thus, this paper proposes a color finite-state LBG algorithm that modifies the training step of the ordinary LBG algorithm for generating a color palette.

Let the palette Pi−1 be generated before the i^th iteration in the LBG algorithm, and let the previous codeword xˆ^i-1 be the closest color vector of Pi−1 for a training vector x in the (i − 1)^th iteration. That is, the codeword xˆ^i-1 can be used as the state in the i^th itera- tion in our CFSLBG algorithm. Clearly, the state space in the i^th iteration of the CFSLBG algorithm is the palette Pi−1. For each state s, the state palette SPs is the subset of whole palette Pi, and the size of SPs is Nf. The Nf codewords in SPs are the closest codewords to s in the whole palette Pi−1. The block diagram of generating the state palette in the CFSLBG algorithm is shown as Fig. 2. In the first iteration, there is no previous informa- tion for the CFSLBG algorithm. Thus, the first iteration of the CFSLBG algorithm is the same as that of the ordinary LBG algorithm, in which the full search algorithm is used to select a codeword for each training vector. At the following iteration, i.e. i≥ 2, the in- formation in the previous iteration is used to determine the states of the training vectors.

The CFSLBG algorithm is described in the following steps.

Step 1: Design an initial RGB color palette P0 and set DAVG0← ∞.

Step 2: For each training vector x, find the closest palette entry x by searching the ˆ⁰ whole color palette P0. Compute the average distortion DAVG1.

Step 3: For each palette entry x in ˆ⁰ P0, generate a new entry ˆx ¹ ← centroid(CS(xˆ⁰)) and add x into Pˆ¹ 1. Set i ← 2.

Step 4: Set the state space S = Pi−1.

Step 5: For each state s in S, find the set of the Nf closest codewords in the whole color palette Pi−1 and define this set as the state palette SPs. For each training vector x, use the previous codeword xˆ^i-1 as the state s. Find the closest palette entry x ˆⁱ by searching the state palette SPs.

State Function F(x): s =
x^{i - 1}

Whole Color RGB Palette Pi -1

Find the Closest Codewords to state s

State Color Palette SPs to State s Closest Codeword

Searching

x^{i - 1} x
xⁱ

Fig. 2. Block diagram of generating the state palette in the iteration i (i ≥ 2) of the CFSLBG algo- rithm.

Step 6: Compute the average distortion DAVGi. If | DAVGi-1− DAVGi| / DAVGi is smaller than ε, then stop.

Step 7: For each entry xˆ^i-1 in Pi−1, generate a new entry x ˆⁱ ← centroid(CS(xˆ^i-1)) and add x into Pˆⁱ i. Set i ← i + 1 and go to step 4.

Note that the state palette size Nf is much smaller than the whole color palette size N.

The Nf codewords for each state are found by an insertion sorting algorithm applied to the entire palette. The size of the state palette Nf is very small, such as 4, 8, 12, or 16. In this situation, the insertion sorting algorithm is efficient. The CFSLBG algorithm is much faster than the LBG algorithm. Moreover, the image quality is very close to that of the LBG algorithm, based on our experimental results described in section 4. Therefore, the CFSLBG algorithm is an effective method for generating palettes of color digitized images.

4. SIMULATIONS AND RESULTS

Our CFSLBG algorithm and the LBG algorithm for generating color palettes in the RGB space were simulated on a SUN SPARC workstation IPX for several still 512 × 512 RGB color images with 24 bits per pixel (bpp). To evaluate the performance of the color palette numerically, the RGB peak signal-to-noise ratio (PSNR) between the original color image and the quantized color image was calculated, where the RGB PSNR is de- fined as

MSE dB.

log 255 10

= PSNR

RGB 2

10 (4)

Note that the mean squared error for an n × n RGB color image is defined as











 



 



= n

∑∑

2

RGB 1 ( ,ˆ )

3

MSE 1 x_ij x_ij , (5)

where x_ij and xˆ_ij denote the original and quantized primary color intensity levels, re- spectively.

We compared the ordinary LBG algorithm and the CFSLBG algorithm in the simu- lations. Two distortion thresholds, ε1 = 0.0001 and ε2 = 0.001, where used to generate a color palette that contained 256 colors from a color image. In the CFSLBG algorithm, the size of the whole palette is 256, and the size of state palette Nf is 4, 8, 12, or 16. Ta- bles 1-2 show the number of iterations and execution times at a distortion threshold of ε

= 0.0001 for the test color images. The number of iterations and execution times at larger distortion threshold, ε = 0.001, for the test color images are listed in Tables 4-5. In our experiments, the execution time of the CFSLBG algorithm was only about 8% of the time required by the LBG algorithm. Tables 3 and 6 show the PSNR values for the color images with the different distortion threshold values. It can be seen that when the size of the state palette is larger than 8, the performance of the CFSLBG algorithm in terms of the RGB PSNR is very close to that of the LBG algorithm.

Table 1. The numbers of iterations required by the CFSLBG and LBG algorithms (N = 256 and ε = 0.0001) for different images.

CFSLBG Images

Nf = 4 Nf = 8 Nf = 12 Nf = 16 LBG

Lena 46 48 53 50 41

Peppers 45 51 62 53 47

F-16 53 76 82 66 80

Sailboat 35 24 62 21 33 Tiffany 68 45 55 66 34

Table 2. The execution times (in seconds) required by the CFSLBG and LBG algorithms (N = 256 and ε = 0.0001) for different images.

CFSLBG Images

Nf = 4 Nf = 8 Nf = 12 Nf = 16 LBG

Lena 62.3 73.4 95.0 96.9 1501.6

Peppers 61.3 112.0 156.8 168.5 1820.3

F-16 62.4 110.9 157.6 166.4 3140.6

Sailboat 58.6 66.1 145.8 86.3 1310.9

Tiffany 75.7 85.7 125.8 178.3 1404.4

Table 3. The RGB PSNR of the CFSLBG and LBG algorithms (N = 256 and ε = 0.0001) for different images.

CFSLBG Images

Nf = 4 Nf = 8 Nf = 12 Nf = 16 LBG

Lena 35.006 35.286 35.303 35.321 35.336

Peppers 27.513 28.119 28.417 28.408 28.339

F-16 36.408 37.372 37.523 37.509 38.002

Sailboat 29.120 29.049 29.418 29.363 29.509 Tiffany 33.251 34.270 34.516 34.723 34.319

Table 4. The numbers of iterations required by the CFSLBG and LBG algorithms (N = 256 and ε = 0.001) for different images.

CFSLBG Images

Nf = 4 Nf = 8 Nf = 12 Nf = 16 LBG

Lena 26 30 33 25 36

Peppers 37 46 56 49 41

F-16 43 56 63 59 46

Sailboat 22 17 30 15 25 Tiffany 64 46 49 52 30

Table 5. The execution times (in seconds) required by the CFSLBG and LBG algorithms (N = 256 and ε = 0.001) for different images.

CFSLBG Images

Nf = 4 Nf = 8 Nf = 12 Nf = 16 LBG

Lena 50.5 66.7 87.1 87.4 1306.5

Peppers 58.9 91.3 137.4 154.9 1577.3

F-16 58.1 91.2 129.4 152.4 1812.1

Sailboat 50.7 57.3 90.0 72.4 1010.9

Tiffany 73.4 86.7 116.3 148.3 1241.6

Table 6. The RGB PSNR of the CFSLBG and LBG algorithms (N = 256 and ε ^{= 0.001)} for different images.

CFSLBG Images

Nf = 4 Nf = 8 Nf = 12 Nf = 16 LBG

Lena 34.832 35.215 35.252 35.152 35.311

Peppers 27.515 27.996 28.344 28.373 28.324

F-16 36.247 37.241 37.341 37.488 37.587

Sailboat 29.113 29.034 29.367 29.358 29.491 Tiffany 33.236 34.271 34.508 34.692 34.295

(a) (b)

(c) (d)

Fig. 3. Results for the image Peppers (N = 256 and ε = 0.001): (a) original image, (b) LBG algo- rithm, (c) CFSLBG algorithm (N_f = 4), and (d) CFSLBG algorithm (N_f = 8).

(a) (b)

Fig. 4. Results for the image Lena (N = 256 and ε = 0.001): (a) original image, (b) LBG algorithm, (c) CFSLBG algorithm (Nf = 4), and (d) CFSLBG algorithm (Nf = 8).

(c) (d)

Fig. 4. (Cont’d) Results for the image Lena (N = 256 and ε = 0.001): (a) original image, (b) LBG algorithm, (c) CFSLBG algorithm (N_f = 4), and (d) CFSLBG algorithm (N_f = 8).

The images Peppers and Lena shown in Figs. 3 (a) and 4 (a) are the original 512 × 512 color images with 24 bpp. Figs. 3 (b)-(d) show the quantized results for the image Peppers when the distortion threshold ε was 0.001. Fig. 3 (b) shows the quantized image obtained using the color palette that was generated by the LBG algorithm. Figs. 3 (c)-(d) show the quantized results of the CFSLBG algorithm with Nf = 4 and Nf = 8, respectively.

Figs. 4 (b)-(d) show the quantized results for the image Lena when ε was 0.001. Fig. 4 (b) shows the quantized image obtained using the color palette that was generated by the LBG algorithm. Figs. 4 (c)-(d) show the quantized results of the CFSLBG algorithm with Nf = 4 and Nf = 8, respectively.

5. CONCLUSIONS

In this paper, the CFSLBG algorithm for generating a RGB palette from a training color image has been proposed. The ordinary LBG algorithm ignores the correlation of corresponding palette entries between the current and previous iterations. In general, the current corresponding palette entry for a training vector is close to the previous corre- sponding palette entry. The CFSLBG algorithm searches only the palette entries that are close to the palette entry for the training vector in the previous iteration. However, the number of palette entries is much smaller than the size of the whole palette. This is the reason why our CFSLBG algorithm is faster than the ordinary LBG algorithm. In our experiment, the quality of the color quantized image obtained using the CFSLBG palette was very close to that of the image obtained using the LBG palette. Moreover, the com- putation time of the CFSLBG algorithm is only about 8% of the time required by the LBG algorithm. The experimental results demonstrate the excellent performance of the proposed approach in color quantization. We conclude that the CFSLBG algorithm is an effective method for designing RGB palettes of color digitized images.

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Yu-Len Huang (黃育仁) was born in Chiayi, Taiwan, on May 22, 1970. He re- ceived the B.S. degree in Computer Science from Tunghai University, Taiwan, R.O.C., in 1992, and the M.S. and Ph.D. degrees in Computer Science and Information Engi- neering from National Chung Cheng University, Taiwan, R.O.C., in 1994 and 1999. He is currently an Assistant Professor in the Department of Computer Science and Informa- tion Engineering, Tunghai University, Taiwan, R.O.C.. His research interests include digital image/video coding, neural networking, computer networking, and medical imag- ing. Dr. Huang is a member of IEEE and Phi Tau Phi.

Ruey-Feng Chang (張瑞峰) was born in Taichung, Taiwan, on August 25, 1962.

He received the B.S. degree in Electrical Engineering from National Cheng Kung Uni- versity, Tainan, Taiwan, R.O.C., in 1984, and the M.S. degree in Computer and Decision Sciences and the Ph.D. degree in Computer Science from National Tsing Hua University, Hsinchu, Taiwan, R.O.C., in 1988 and 1992, respectively. Since 1992, he has been with the Department of Computer Science and Information Engineering, National Chung Cheng University, Chiayi, Taiwan, R.O.C., and he is now a Professor. His research inter- ests include image/video processing and retrieval, medical computer-aided diagnosis system, and multimedia systems and communication. Dr. Chang is a member of IEEE, ACM, SPIE, and Phi Tau Phi.