Color image compression using quantization, thresholding, and edge detection techniques all based on the moment-preserving principle

Color image compression using quantization, thresholding, and

edge detection techniques all based on the moment-preserving

principle

1

Chen-Kuei Yang

a,2

, Wen-Hsiang Tsai

b,)

a_{Department of Information Management, Ming Chuan UniÕersity, 250, Sec. 5, Chung Shan N. Rd., Shih-Lin, Taipei 111, Taiwan, ROC} b_{Department of Computer and Information Science, National Chiao Tung UniÕersity, HsinChu 300, Taiwan, ROC}

Received 14 May 1997; revised 27 October 1997

Abstract

A new approach to color image compression with high compression ratios and good quality of reconstructed images using quantization, thresholding, and edge detection all based on the moment-preserving principle is proposed. An input image with 24 bits per pixel is quantized into 8 bits per pixel using a new color quantization method based on the moment-preserving principle. The quantized image is then divided into n=n non-overlapping square blocks. Two representative colors for each block are computed by moment-preserving thresholding. A bit-map is then generated, consisting of 0s and 1s indicating whether the block pixels are assigned to the first color or the second according to the Euclidean distance measure. A moment-based edge detector is performed further on the bit-map of each non-uniform block. The two parameters l and u of a line edge with the equation of x cos uqy sin usl are obtained. The image is finally coded with a codebook of a 256-color palette; a 1-bit indicator for each block which specifies whether the block is uniform or not; an 8-bit color index for a uniform block, or two 8-bit color indices, a 3-bit index foru, and a 2-bit or 3-bit index for l for a non-uniform block. An average compression ratio of 22.49 or 33.32 can be obtained for 4=4 or 5=5 image blocks, respectively. Experimental results show the feasibility and efficiency of the proposed approach for color image compression. q 1998 Elsevier Science B.V. All rights reserved.

Keywords: Color image compression; Color quantization; Bit-map; Moment-preserving principle

1. Introduction

The objective of an image compression technique is to remove as much redundant information as

)_{Corresponding author. E-mail: [email protected].} 1_{This work was supported partially by National Science} Coun-cil, Republic of China under Grant NSC83-0408-E009-010.

2_{E-mail: [email protected].}

possible without destroying the image integrity. These two goals, however, are mutually conflicting. In a digital true-color image, each color component is quantized with 8 bits, and so a color is specified with 24 bits. As a result, there are 224 possible colors for the image. However, the human vision system cannot differentiate so many colors. Further-more, a color image usually contains a lot of data redundancy and requires a large amount of storage 0167-8655r98r$19.00q 1998 Elsevier Science B.V. All rights reserved.

Ž . PII S 0 1 6 7 - 8 6 5 5 9 7 0 0 1 6 6 - 9

space. In order to lower the transmission and storage cost, image compression is desired, in which color quantization to reduce the number of possible colors is usually included. In this study, the moment-pre-serving principle is used in the image compression procedure to keep the quality of the resulting image good and acceptable for the human vision system.

Ž .

The standard block truncation coding BTC algo-rithm is a simple block-based image compression algorithm first developed by Delp and Mitchell Ž1979 that preserves the block mean and the block. standard deviation. The absolute moments BTC ŽLema and Mitchell, 1984 , upper and lower mean.

Ž .

BTC Udpikar and Raina, 1985 , and adaptive BTC ŽHui, 1990 were suggested to improve the quality of. coded images. Some other modified or hybrid BTC algorithms have since been proposed to further re-duce the bit rate either concerning the mean vector Ža, b. ŽDelp and Mitchell, 1991 , or the bit-map. ŽArce and Gallagher, 1983; Zeng et al., 1992;

Ud-. Ž

pikar and Raina, 1987 , or both Zeng and Neuvo, . 1993; Wu and Coll, 1991; Efrati et al., 1991 , or using the variable block size with hierarchical

tech-Ž

nique Oshri et al., 1993; Roy and Nasrabadi, 1991; .

Wu and Coll, 1993 . All of them use the moment-preserving principle first and then apply other tech-niques to truncate the image data. These algorithms were originally designed for gray-scale images. However, we can apply these methods separately on each color plane of color images, and then merge the resulting triple set mean vectors and bit-maps in some way. Therefore, reducing the three planes of bit-maps becomes very important in compressing

Ž .

color images. Wu and Coll 1992 used a single bit-map to quantize all the three color planes. This means that only one out of three bit-maps need be

Ž .

preserved. Kurita and Otsu 1993 used the mean vector and the covariance matrix of color vectors to compute the principal score for the pixels in an image block, and classified the pixels in the block into two classes. Two mean vectors and a bit-map are preserved.

In this study, a color image compression approach yielding high compression ratios and good recon-structed image quality is proposed. The approach consists of three steps, namely, quantization, thresh-olding, and edge detection, which are all based on the moment-preserving principle. First, an input color

image with 24 bits per pixel is quantized into 8 bits per pixel. The quantization process is based on the

Ž moment-preserving thresholding technique Tsai,

.

1985 . The color histogram is repeatedly sub-divided into smaller and smaller boxes, and two color values are computed automatically as two representative palette colors for every two separated boxes. A 3D lookup index table is constructed for use in pixel mapping. As a result, the computation in the quanti-zation process is fast. The output is a limited-color image as is desired. Second, the quantized image is divided into n=n non-overlapping square blocks. Each block is requantized into two representative

Ž . Ž .

colors R ,G , B1 1 1 and R ,G , B2 2 2 by applying the moment-preserving thresholding technique to each color component. A bit-map is then generated, con-sisting of 0s and 1s indicating whether the block pixels are assigned to X or X where X s R, G or1 2

B. The criterion of pixel assignment is based on the

Euclidean distance between the original color vector and the two representative color vectors. Third, a moment-based subpixel edge detection technique ŽLyvers et al., 1989 is performed on the bit-map of. each non-uniform block. The purpose is to divide the block into two regions separated by a line edge. This is reasonable, as demonstrated by our experimental results, because the block is small. The line edge is described by the equation x cosuqy sin usl which can be computed using analytic formulas, where l andu are the intercept and the angle of the line. The values of l and u are assumed to take certain specified values because the block size is small so that the number of possible edge orientations is limited. Thus, a total of no more than 6 bits of indices are found sufficient to code the two parame-ters l and u. Furthermore, the values of l and u need not be included in the transmission cost be-cause these values are predefined and fixed if the block size is given. The image is finally coded with a codebook of 256 color palettes, one 1-bit indicator for each block which specifies whether the block is uniform or not, and one or two 8-bit indices of the color palette according to the block’s uniformity. When the block is not uniform, additionally required are one 3-bit index foru, and one 2-bit or 3-bit index for l for 4=4 or 5=5 window size, respectively. An average compression ratio of 22.49 or 33.32 can be obtained for a 4=4 or 5=5 image block,

respec-tively. Experimental results show the feasibility and efficiency of the proposed approach for color image compression.

The remainder of this paper is organized as fol-lows. A brief review of moment-preserving thresh-olding and moment-base subpixel edge detection is first given in Section 2. In Section 3, the proposed color image quantization and compression methods are presented. Several experimental results showing the feasibility of the proposed approach are de-scribed in Section 4. Finally, some conclusions are given in Section 5.

2. Review of moment-preserving thresholding and subpixel edge detection

2.1. Moment-preserÕing thresholding

Given a gray-scale image f with T pixels whose

Ž . Ž .

gray value at pixel x, y is f x, y , the ith gray

moment of f is 1

i

m si

Ý Ý

f

Ž

x , y ,

.

i s 0,1,2, . . . .

Ž .

1

T x y

Let O be an operator applied to the input image

f, and g be the output image. If the ith moment of g

is set equal to that of f, then O is said to preserve the ith moment of the input data in the output data. By definition, we see that if an operator preserves more moments, then more information of the input image will be retained in the output image.

Suppose that it is desired to threshold f into one Ž

with two levels h and h_{1 2} with p and p being the_{1 2} . fractions of pixels with h and h , respectively . The_{1 2}

Ž .

algorithm by Tsai 1985 selects a threshold value in such a way that if all the pixels with below-threshold gray value in f are replaced by gray value h and all₁ the pixels with above-threshold gray value in f are replaced by gray value h , then the first three mo-2

ments of image f are preserved in the resulting bi-level image g. The concept can be expressed as follows: p h q p h s m , p h2_{q p h}2_{s m ,} 1 1 2 2 1 1 1 2 2 2 p h3_{q p h}3_{s m , p q p s 1,}

Ž .

₂ 1 1 2 2 3 1 2 Ž . where the left-hand terms in each equality of 2 is a moment of g and the corresponding right-hand term is a moment of f. To find the desired threshold value

Ž .

t, the equalities of 2 are first solved to obtain p ,1

p , h and h . Then, t is chosen as the gray value2 1 2

closest to the p -tile of the histogram of f. After t,1

h and h are computed, f can be bi-levelly thresh-_{1 2}

olded into g as follows:

h₁ if f x , y

Ž

.

-t,

g x , y s

Ž

.

½

Ž .

3

h₂ if f x , y

Ž

.

0t.

2.2. Subpixel edge detection

The subpixel edge detector proposed by Lyvers et

Ž .

al. 1989 is also based on the mass-moment preserv-ing principle and is reviewed here. A continuous two-dimensional edge specified by h, k, l and u is shown in Fig. 1. The edge is defined to lie within a unit circle. The mass moments of a two-dimensional

Ž .

function f x, y are given by p q

M_{p q}s

HH

x y f x , y d y d x.

Ž

.

Ž .

4 A rotation of the circular window by yu will align all moments containing edge information along the x-axis. The parameters h, k and l can be solved along the x-axis independently of u. A rotation of

Fig. 1. Two-dimensional ideal edge model characterized by four parameters h, k, l andu.

the window by an arbitrary angle f transforms the original mass moments into

p q qy s p q X M_{p q}s

Ý Ý

ž /ž /

Ž

y1

.

r s rs0 ss0 = py rqs qqrys cosf sinf M .

Ž

.

Ž

.

pq qyr , rqs 5

Ž .

The parameters h, k and l can be obtained as follows: 4 MX20y M00X l s ,

Ž .

6 3 M10 3 M10 k s ,

Ž .

7 3 2

(

2

Ž

1 y l

.

1 _y1

'

h s 2 M y k00

Ž

pysin ly2l 1yl ,

.

Ž .

8 2p

where the values of the mass moments can be ob-Ž .

tained from 4 , the rotated mass moments also can Ž .

be obtained from 5 and their relationship can be expressed in the following equations. The detail

Ž .

process can be seen in Lyvers et al., 1989 .

M00X s M ,00

Ž .

9 X

(

2 2 M s M q M_{10 01 10},

Ž .

10 M102M q 2 M M M q M20 01 10 11 012M02 X M s20 2 2 ,

Ž .

11 M q M01 10 M2_{M y 2 M M M q M}2_M 10 02 01 10 11 01 20 X M s02 _{M q M}2 2 .

Ž .

12 01 10

3. Proposed approach to color image compression 3.1. Proposed color quantization method

A color image can be defined as follows:

f : M=M™C:C ,

Ž .

13

Ž . < 4

where Cs r, g,b 0(r, g,b(255 is the RGB

Ž .

color space, x, y g M=M are the spatial

coordi-nates of a pixel, M is the integer set, and C s c , c , . . . ,c1 2 N4 is a set of colors used in the image.

There are 256=256=256 combinations of red, green and blue components. It puts heavy burden to deal with so many colors! By color quantization, not only the complexity can be reduced but the compres-sion rate is also increased. A quantized image may be regarded as a mapping defined by

q : M=M™R:C ,

Ž .

14

 4

where R s r , r , . . . ,r1 2 k is a set of representative colors used in the quantized image.

According to the above definition, the process of color image quantization basically can be divided into two major steps. The first step, called color palette design, selects the best match set of colors for a specific image. The second step, called pixel map-ping, associates each pixel of the image with a best match color in the color palette to yield an image with the highest quality. The optimization goal is to make the perceived difference between the original image and its quantized version as small as possible. It is very difficult to formulate a definitive solution to meet this goal in terms of perceived image quality. In fact, there is no good objective criterion available for measuring the perceived image quality. In this study, the mean absolute error is used to measure the difference between the original image f and its quantized reproduction q. Accordingly, the average quantization error is defined as follows:

1 < < d_M

Ž

f , q s

.

Ý

f x , y y q x , y ,

Ž

.

Ž

.

T _{Žx , y gM. =M} 15

Ž .

where T is the total number of pixels of the image.

3.1.1. Subsampling for histogram creation

A histogram gives the relative frequency of occur-rences of colors in an image. The original color image has 24 bits per pixel. There are thus 224

possible colors. To save memory space and computa-tion time, it is desirable to reduce the histogram

Ž .

resolution. Heckbert 1982 has suggested that the required resolution be at least 15 bits per pixel or 5 bits for each color component.

In our approach, to speed up histogram creation, not only the histogram resolution is reduced but also only the even-row-odd-column pixels are taken into

account for the histogram. Hence, only a quarter of the original image pixels are subsampled. The corre-sponding histogram was found still a good approxi-mation of the color distribution of the original image.

3.1.2. Design of color palette

Many algorithms have been proposed for the de-sign of color palettes. The popularity algorithm ŽHeckbert, 1982 selects the N most frequently oc-. curring colors in the image as the representative colors according to the image color distribution. In order to prevent the concentration of too many colors in one neighborhood, an improved popularity

algo-Ž .

rithm was proposed by Braudaway 1986 . When a most frequently occurring color is selected, the fre-quency count of its neighboring colors is reduced.

Ž .

The median cut algorithm Heckbert, 1982 uses a splitting technique to repeatedly sub-divide the color histogram into smaller and smaller boxes which con-tain approximately equal numbers of color occur-rences, and pick the mean values of the boxes as the representative palette colors. The mean-split

algo-Ž .

rithm Gentile et al., 1990 splits the box at the mean value instead of at the median, and recomputes the low mean and high mean for the two separated boxes as the representative palette colors. The

variance-Ž .

based algorithm Wan et al., 1990 is the same as the median cut algorithm except that it splits the box whose color distribution has the largest values of variances, and the centroids of the sub-boxes are chosen as the representative palette colors. The

Ž .

maxmin algorithm Houle and Dubois, 1986 tries to minimize the distances between the selected and the original image colors. The initial P palette colors are selected by the frequencies of occurrences which exceed a predetermined threshold. The choice of the

K representative colors is recursively made where K-P. One choice for the initial color value could

be the most frequently occurring color found in the image to be quantized. The new representative color value, selected from the unused colors, is the one whose minimum distance to the representative colors selected thus far is maximum. There are still a lot of

Ž

other algorithms Balasubramanian et al., 1994; Dixit, 1991; Orchard and Bouman, 1991; Goldberg, 1991;

.

Wu and Witten, 1985; Wu, 1992 for the design of color palettes, but they are less relevant to this study and so are not described here.

Table 1

The comparative quantization errors in terms of MAE values of the proposed method, the median cut method, and the mean split method

Proposed method Median cut Mean split

Lena 4.0595 4.4219 4.0472 Pepper 4.5827 5.7426 4.5178 Jet 4.0192 4.5178 4.2434 House 4.1007 4.7357 4.4269 Mandrill 6.5468 6.7324 6.5123 Candy 3.9946 4.9447 4.1294

In our approach, the entire color space is regarded initially as a single cube whose representative color is the mean vector of the R, G and B components. The histogram resolution is reduced to 5 bits per color component and the even rows and odd columns of the image pixels are subsampled. The variances of the color components are computed. The color plane with the largest variance is split and the color space becomes two separated boxes. Moment-preserving

Ž . Ž .

thresholding using Eqs. 2 and 3 is employed to compute two representative color values h and h1 2

for the two boxes and to choose as the cut point the value of the distribution closest to the p -tile of the1

corresponding histogram. Splitting of the color planes is repeated until the desired number of representative colors are achieved. K y 1 times of splitting are required if K representative palette colors are needed. After the splitting process is completed, all the color coordinates in each box are regarded as identical and are represented by the representative color value computed by the moment-preserving thresholding method. Finally, all the representative colors are

Table 2

Ž .

The comparative computation times in seconds of the methods mentioned in Table 1 with 4, 8, 16, 32, 64, 128 and 256 colors, respectively

Colors Proposed method Median cut Mean split

4 4.81 5.38 5.45 8 4.93 5.49 5.51 16 5.22 5.70 5.79 32 5.78 6.13 6.67 64 6.52 6.92 7.74 128 7.98 8.48 9.19 256 10.19 11.07 11.93

collected to form a color palette and are stored in a one-dimensional array as a color map. The computa-tion time of the proposed method is less than that of the maxmin, the mean-split, and the variance-based algorithms because some steps of the proposed method are performed using analytic formulas. The quantization error is less than that of the median-cut algorithm. These phenomena can be seen from Ta-bles 1 and 2.

3.1.3. Pixel mapping

When a palette has been designed, the remaining step is to map the original color of each pixel in the input image to their best match in the color palette. Several techniques for this purpose have been

intro-Ž

duced in the literature Heckbert, 1982; Balasubra-.

manian et al., 1994; Goldberg, 1991 . The simplest way is to compute the distance between the original color vector and all of the representative color vec-tors, and then choose the one with the minimum distance. This exhaustive search method was

pro-Ž .

posed by Heckbert 1982 . But the computation is slow. A locally sorted search method was also

re-Ž .

ported in Heckbert, 1982 . The computation time is reduced but it needs additional time to create a list of color vectors and sort the list by the distance key. The list is made by eliminating the representative color vectors whose distances to the centre of the box is above a threshold. Other faster methods

in-Ž .

clude binary tree search Orchard and Bouman, 1991

Ž .

and k y d tree search Friedman et al., 1977 . These algorithms are based on the arrangement of the K

representative colors in a binary tree structure. By a proper arrangement of the tree, a search time of

Ž .

O log K can be achieved.

In our approach, after the color palette is de-signed, each box’s region in the 3D histogram con-tains a set of color values which are represented by a single color. Each representative color is included in the color map mentioned previously. Regarding each color value in the box as a cell, we fill up all the cells in each box with identical pointers which all point to the corresponding representative color in the color map. After the cells in all boxes in the 3D histogram are filled with pointers, the histogram may be regarded as a lookup index table. The color value of each input pixel then can be mapped to the corresponding cell in the 3D lookup index table, and in turn the pointer in the lookup table can be re-trieved and used to find the corresponding represen-tative color in the color map. This is very convenient for pixel mapping because no search is needed. This direct pixel mapping method is shown in Fig. 2. Some experimental results and performance evalua-tions of the proposed method are given in Section 4.

3.2. Color image compression

After the input color image is quantized into 256 colors using the proposed quantization method, the image is divided into n=n non-overlapping blocks. Each block is requantized in this study into two colors by applying the moment-preserving threshold-ing to each color component again. Each image

Table 3

The predefined values of l andu for 4=4 and 5=5 window size

4=4 5=5

l y0.75 0.25 0.25 0.75 y0.8 y0.4 0 0.4 0.8 4=4 and 5=5

u ypr2 ypr3 ypr6 0 pr6 pr3 pr2

block can thus be coded with two indices of colors and one n=n bits bit-map. However, the two result-ing color vectors may not be found in the 256-color palette. Thus, the minimum value of the Euclidean distance is used as a criterion to select the best match color in the 256-color palette. The resulting bit-maps still occupy more than 50% of the output codes. In order to increase the compression ratio, the

Ž .

moment-based edge detector Lyvers et al., 1989 reviewed in Section 2.2 is performed on the bit-map. Hence, a non-uniform image block can be coded with two indices of colors and two edge parameters instead of the whole bit-map. And a uniform block can be coded with just a single index of color. Furthermore, since the block size is small, the range of the possible orientationsu of an edge in an image block is limited and so is the range of the intercept value l. The values of these two edge parameters can thus be fixed to certain specified values. Then, an edge block can be coded with two indices of colors and two indices of specified edge parameter values. From our experimental experience, in general there are very few cases in which all the values of the bits are one or zero in the bit-map. However, the variation of the bit values in a 4=4 or 5=5 block will become noticeable when there exist four or more bits which form a cluster and show a shape of

line and edge. According to this concept of visual discontinuity, a block may be assumed uniform when the value of the sum of all the bit values in the bit-map is greater than n=ny4 or less than 4. Note that as the number of uniform blocks increases, the compression ratio is improved.

The difference between two edges with very close directions is not perceivable when the image resolu-tion is high and the window size is small. Also, the number of the possible orientations of an edge in a small image block is limited. Therefore, the values of u and l can be predefined and fixed to be some specified values. The values of l are predefined to be the distances from the center of the block to those pixels along the x-axis in the block, and the values of u are selected to be those with the increments of pr6. These values of u and l in a 4=4 and 5=5 circular window are shown in Table 3. Because these values are predefined and fixed, it is not necessary to count them in the coding cost.

A 256-color palette codebook needs 3=5=256 bits. Each uniform block is coded with 1=8q1s9 bits and each edge block is coded with 2=8q1q2 q 3s22 bits or 2=8q1q2=3s23 bits for 4= 4 or 5=5 window size, respectively. the overall compression ratio so can be computed as follows:

M=N=3=8 r s , M=N 256=3=5qb =9q₀

ž

y b =b₀

/

n=n 16

Ž .

where M=N is the image size, b the number of0

uniform blocks, n=n the block size, and bs5 or 6

Table 4

The values of MAE and the compression ratios of the resulting images mentioned in Table 1 with 4=4 and 5=5 block sizes, respectively

Images 4=4 block size 5=5 block size

MAE values Compression ratios MAE values Compression ratios

Lena 7.9226 22.06 8.4477 31.09 Pepper 8.9569 21.74 10.1498 30.30 Jet 8.1365 23.70 8.8268 32.79 House 5.2712 24.28 6.0135 37.69 Mandrill 16.4548 19.18 19.3235 28.07 Candy 4.9882 23.98 5.4691 39.98 AÕerage 22.49 33.32

for the indices of the edge parameters with 4=4 or 5=5 window size, respectively.

4. Experimental results

The proposed algorithms have been tested on an IRIS Indigo workstation on several color images. Each color image has 24 bitsrpixel and is 512=512

Ž .

in size. We use the mean absolute error MAE as the criterion for the performance evaluation of image quantization and compression. Table 1 shows the comparative quantization errors in terms of the MAE values of the proposed method, the median cut method, and the mean split method, respectively. for each method, the images ‘‘Lena’’, ‘‘pepper’’, ‘‘jet’’, ‘‘house’’, ‘‘mandrill’’ and ‘‘candy’’ are tested and are quantized into 256 colors. Table 2 shows the comparative computation times of the methods men-tioned in Table 1. From Tables 1 and 2, several facts can be observed. First, the quantization results of the proposed method are better than that of the median cut method. Second, the computation time of the proposed method is the least among the methods mentioned above. Table 4 shows the MAE values of the reconstructed images and the compression ratios of the images mentioned above with 4=4 and 5=5 window sizes, respectively. Fig. 3 shows the stan-dard images of ‘‘Lena’’, ‘‘pepper’’ and ‘‘mandrill’’ of size 512=512 in top, middle and bottom, respec-tively. Fig. 4 shows the results of the proposed quantization method with 256 colors using the im-ages shown in Fig. 3. Figs. 5 and 6 show the reconstructed results of the proposed compression approach using the images shown in Fig. 3 with 4=4 and 5=5 window sizes, respectively. From Fig. 4, we can find that the image quality is almost the same as the original. It is difficult to find differ-ences between the original images and the repro-duced results by the human vision system. However, little differences can be distinguished by the

com-Ž .

puted MAE values see Table 1 . It can be seen from Figs. 5 and 6 and Table 4 that the reconstructed image quality is still good and acceptable. Only some slightly zigzag edges and a litter color differ-ence in regions with gradual color changes can be seen. This is due to the uses of BTC, fewer bits for

Fig. 3. The original images of ‘‘Lena’’, ‘‘pepper’’ and ‘‘mandrill’’ in top, middle and bottom, respectively.

Fig. 4. The quantization results with 256 colors of the images in Fig. 3.

Fig. 5. The reconstructed results of the images in Fig. 3 with 4=4 windows.

Fig. 6. The reconstructed results of the images in Fig. 3 with 5=5 windows.

the indices of l and u, as well as the detected line edges only in the blocks.

5. Conclusions

A new color image compression algorithm using quantization, thresholding, and edge detection all based on the moment-preserving principle has been proposed. The major contributions of this study

in-Ž .

clude the following. 1 A new approach to the Ž .

design of color palettes is proposed. 2 A fast pixel mapping method for color quantization and

repro-Ž .

duction is presented. 3 An edge operator is per-formed on the bit-map to avoid using the whole

Ž .

bit-map. 4 An efficient coding of the edge detec-Ž .

tion results has been proposed. 5 Reconstructed images with good quality and reasonable

compres-Ž .

sion ratios have been obtained. 6 The computation is fast and thus suitable for real-time applications due to the use of analytic formulas for some of the quantization and compression steps. Some possible improvements and further research topics are as fol-lows.

The human visual system is more sensitive to details in the luminance component than to details in the chrominance component. Hence, the RGB com-ponents can be first linearly transformed into the YIQ components, and the luminance and the chromi-nance components can be treated differently at the time of resampling the histogram with resolution reduction to save the memory space of storing the histogram. But this requires additional time for color model transformation. The size of an image block also need not be fixed. The use of variable block sizes in the compression scheme will yield generally higher compression ratios. The zigzag effect and image distortion in the compressed image can be improved by increasing the number of allowed speci-fied orientations of the detected edges, or by detect-ing more complicated features in the bit-map such as curves and lines.

References

Arce, G.R., Gallagher, N.C., 1983. BTC image coding using Ž .

median filter roots. IEEE Trans. Commun. 31 6 , 784–793. Balasubramanian, R., Bouman, C.A., Allebach, J.P., 1994.

Se-quential scalar quantization of color images. J. Electronic Ž .

Imaging 3 1 , 45–59.

Braudaway, G., 1986. A procedure for optimum choice of a small number of colors from a large color palette for color imaging. In: Proc. Electronic Imaging ’86, San Francisco, CA, pp. 75–79.

Delp, E.J., Mitchell, O.R., 1979. Image compression using block truncation coding. IEEE Trans. Commun. 27, 1142–1335. Delp, E.J., Mitchell, O.R., 1991. The use of block truncation

coding in DPCM image coding. IEEE Trans. Signal Process. Ž .

39 4 , 967–971.

Dixit, S.S., 1991. Quantization of color images for displayrprint-Ž . ing on limited color output device. Comput. Graphics 4 1 , 561–567.

Efrati, N., Liciztin, H., Mitchell, H.B., 1991. Classified block truncation coding–vector quantization: An edge sensitive im-age compression algorithm. Signal Processing: Imim-age Commu-nication 3, 275–283.

Friedman, J.J., Bentley, J.L., Finkel, R.A., 1977. An algorithm for finding best matches in logarithmic expected time. ACM Trans. Math. Software 3, 209–226.

Gentile, R.S., Allebach, J.P., Walowit, E., 1990. Quantization of color images based on uniform color spaces. J. Imaging

Ž . Technol. 16 1 , 12–21.

Goldberg, N., 1991. Color image quantization for high resolution Ž .

graphics display. Image and Vision Computing 9 5 , 303–312. Heckbert, P., 1982. Color image quantization for frame buffer

Ž .

display. Comput. Graphics 16 3 , 297–307.

Houle, H., Dubois, E., 1986. Quantization of color images for display on graphics terminals. In: Proc. IEEE Global Telecom-mun. Conf., pp. 1138–1142.

Hui, L., 1990. An adaptive block truncation coding algorithm for image compression. In: Proc. ICASSP 90, Vol. 4, pp. 2233– 2236

Kurita, T., Otsu, N., 1993. A method of block truncation coding Ž . for color image compression. IEEE Trans. Commun. 41 9 , 1270–1274.

Lema, M.D., Mitchell, O.R., 1984. Absolute moment block trun-cation coding and its applitrun-cation to color images. IEEE Trans. Commun. 32, 1148–1157.

Lyvers, E.P., Mitchell, O.R., Akey, M.L., Reeves, A.P., 1989. Subpixel measurement using a moment-based edge operator.

Ž .

IEEE Trans. Pattern Anal. Machine Intell. 11 12 , 1293–1309. Orchard, M.T., Bouman, C.A., 1991. Color quantization of

im-Ž .

ages. IEEE Trans. Signal Process. 39 12 , 2677–2690. Oshri, E., Shelly, N., Mitchell, H.B., 1993. Interpolative three-level

Ž . block truncation coding algorithm. Electron. Lett. 29 14 , 1267–1268.

Roy, J.U., Nasrabadi, N.H., 1991. Hierarchical block truncation Ž .

coding. Opt. Engrg. 30 5 , 551–556.

Tsai, W.H., 1985. Moment-preserving thresholding: a new ap-proach. CVGIP 29, 377–393.

Udpikar, V.R., Raina, J.P., 1985. A modified algorithm for block truncation coding of monochrome images. Electron. Lett. 21 Ž .20 , 900–902.

Udpikar, V.R., Raina, J.P., 1987. BTC image coding using vector Ž .

quantization. IEEE Trans. Commun. 35 3 , 352–356. Wan, S.J., Prusinkiewicz, P., Wong, S.K.M., 1990. Variance-based

color image quantization for frame buffer display. Color Res. Ž .

Appl. 15 1 , 52–58.

Wu, X., 1992. Color quantization by dynamic programming and Ž .

principal analysis. ACM Trans. Graphics 11 4 , 348–372. Wu, X.L., Witten, I.H., 1985. A fast k-means type clustering

algorithm. Technical Report 85r197r10, Dept. of CS, U. of Calgary, Calgary, Canada.

Wu, Y., Coll, D.C., 1991. BTC-VQ-DCT hybrid coding of digital Ž .

images. IEEEE Trans. Commun. 39 9 , 1283–1287. Wu, Y., Coll, D.C., 1992. Single bit-map Block Truncation

Coding of color images. IEEE J. Selected Areas Commun. 10 Ž .5 , 952–959.

Wu, Y., Coll, D.C., 1993. Multilevel block truncation coding using a minimax error criterion for high fidelity compression

Ž .

of digital images. IEEE Trans. Commun. 41 8 , 1179–1191. Zeng, B., Neuvo, Y., 1993. Interpolative BTC image coding with

Ž . vector quantization. IEEE Trans. Commun. 41 10 , 1436– 1437.

Zeng, B., Neuvo, Y., Venetsanopoulos, A.N., 1992. Interpolative BTC image coding. In: Proc. IEEE ICASSP 92, Vol 3, pp. 493–496.