Optimal Transform in Perceptually Uniform Color Space and Its Application in Image Coding *

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1 Optimal ransform in Perceptually Uniform Color Space and Its Application in Image Coding * Ying Chen 1,, Pengwei Hao 1,, and Anrong Dang 3 1 Center for Information Science, Peking University, Beijing, 0871, China phao@cis.pku.edu.cn Department of Computer Science, Queen Mary, University of ondon, E1 4S, UK {ying, phao}@dcs.qmul.ac.uk 3 Center for Science of Human Settlements, singhua University, Beijing, 0084, China danrong@tsinghua.edu.cn Abstract. o find an appropriate color transform is necessary and helpful for the applications of color images. In this paper, we proposed a new scheme to find color transforms close to the optimal transform and agree with human vision system for comparison. We first apply the perceptually uniform color space transform to convert RGB components into uniform CIE AB components, and then use principal components analysis (PCA) in the uniform space to find the image-dependent optimal color transforms (K) for each test image group and for all the images in all the groups. With the Ks, an approximate but image-independent transform in CIE AB space is presented, namely AR, which is just the AB space rotated and has an elegant and simple form. Finally, we apply it to image compression, and our experiment shows that AR performs better (PSR about 4dB higher) than the default color transform in JPEG Introduction Color images are everywhere in science, technology, medicine and industry. Color images are acquired and reproduced based on tristimulus values whose spectral composition is carefully chosen according to the principles of color science. Color space transform is critical for color feature extraction and data redundancy reduction. o find an appropriate color transform is necessary and very helpful in many color image applications, such as image display, processing, retrieval, recognition, and compression. In 1931, Commission Internationale de l'eclairage (CIE) defined three standard primaries, called X, Y and Z, to replace red, green and blue (RGB) and with positive weights to match all the colors we see. It took CIE a decade or more to find a transformation of CIE XYZ into a reasonable perceptually uniform space. So far, CIE standardized two perceptually uniform systems, CIE UV and CIE AB. Besides, many color standards enable users to have the freedom to choose the color space in * his work was supported by the Foundation for the Authors of ational Excellent Doctoral Dissertation of China, under Grant A. Campilho, M. Kamel (Eds.): ICIAR 004, CS 311, pp , 004. Springer-Verlag Berlin Heidelberg 004

2 70 Y. Chen, P. Hao, and A. Dang which to represent their data. RGB, CMYK, YIQ, HSV, CIE 1931 XYZ, CIE UV, CIE AB, YES, CCIR 601- YCbCr, and SMPE-C RGB are proposed for diverse requirements [1]. However, in many applications, we need some appropriate color transforms and we also wish the transformed or the inverse-transformed components are inter-comparable and the comparison done by computers agrees with that we do by our human visual system. herefore, we need compare the results in a perceptually uniform color space after applying inverse of our specific color space transforms. Our idea is to find some optimal color transforms in the uniform space. In this paper, a new scheme to find an optimal color transform is proposed. We transform color images into three components in the uniform space CIE AB, and then use principal components analysis (PCA) to find image-dependent optimal color transforms, Karhunen-òeve ransform (K- transform, or K). Finally, we take the optimal transform obtained from all the analyzed images as an image-independent color transform and apply it to image compression of some other test images with JPEG 000. Principal Components Analysis in CIE AB Color Space CIE AB color space is used in this paper as a perceptually uniform space and then the K- transforms are found in this space. he work is implemented with five different groups of color images and by three main steps: Step1. ransform all the images from the original image color space into CIE AB color space. Step. Compute the covariance matrix in CIE AB space with all the pixels of all the images in a group. Step3. Find the three eigenvalues and their corresponding eigenvectors of the covariance matrix and then make the K- transform with the three eigenvectors. Among the steps, Step and Step3 are actually principal components analysis..1 RGB to CIE AB he conversion from RGB color values into CIE AB components is implemented by converting from nonlinear RGB space into linear RGB space first, then into CIE XYZ space, and finally into CIE AB space. he original color space of the images we generally come across is nonlinear RGB space. onlinear RGB, also called gamma-corrected RGB, may be stored in a file as three 8-bit integers ranging from 0 to 55. A simple scaling conversion is required to transform the three integer values from 0 to 55 into floating-point numbers in [0, 1]. here are several possible choices for the conversion between nonlinear RGB and linear RGB, the transform and its inverse used in this paper are: From the gamma-corrected RGB values to a linear RGB values: For R, G, B 0.0, R = R, G = G, B = B (1) For R, G, B < 0. 0, R = R, G = G, B = B

3 Optimal ransform in Perceptually Uniform Color Space and Its Application 71 From the linear RGB values to gamma-corrected nonlinear values: For R, G, B 0.0, 1 / R = R, 1 / G = G, B 1/ = B (3) For R, G, B < 0. 0, R = R, G = G, B 1 / = B (4) he following is a linear transform from linear RGB to CIE XYZ (D50). X D50 R Y D50 G = A where A = (5) ZD50 B where the transform matrix A is derived by the matrix multiplication of two matrices: B = D = (6) where A=DB, B is the matrix to transform from linear RGB into XYZ (D65) [], D is the matrix to transform from XYZ (D65) into XYZ (D50) [1]. he conversion from CIE XYZ (D50) into CIE AB is computed as follows, which can be found in [1]: 3 * = 116( Y / Yn) 1 / 16,( Y / Yn) > * = ( Y / Yn),( Y / Yn) < ) (7) a* = 500[ f ( X / Xn) f ( Y / Yn)] b* = 00[ f ( Z / Zn) f ( Y / Yn)] 1 / 3 where f ( t) = 7.787t + 16/116 if t < , and f ( t) = t if t > In this paper, we use CIE XYZ (D50) as our XYZ space, and at the white point, Xn, Yn and Zn have the values of 0.964, 1.0, and 0.849, respectively. here are ways to speed up the transform, such as to approximate the cube root [4]. here are also some methods using the nonlinear functions in conversion between CIE XYZ and CIE AB [5] and in RGB gamma-correction, simpler and less timeconsuming. A system using linear interpolation to transform RGB to CIE AB can be found in [6].. Principal Components Analysis Having been converted into the CIE AB color space, the images are analyzed with principal components analysis (PCA) method [3]. Consider X={x 1, x,, x n } is a collection of pixel samples in the CIE AB color space. Each sample, x i =[x i1, x i, x i3 ], has 3 elements, which are 3 components *, a* and b* respectively. We use these samples to calculate the covariance matrix of X, denoted by S. S = n k = 1 ( x m)( x m) n where m is the mean vector of the n sample pixels, m = ( 1/ n) k = x. 1 k k k (8)

4 7 Y. Chen, P. Hao, and A. Dang S is a 3 3 symmetric matrix, and the solutions of the principal components analysis satisfy the equation: Sv= λ v, where λ is one of the 3 eigenvalues of matrix S and v is the corresponding eigenvector. For each covariance matrix, we can obtain three eigenvalues λ 1, λ and λ 3, and their eigenvectors v 1, v and v 3. hen, the K- transform is defined as y=vx, where x is a pixel in the CIE AB space, y is the resultant pixel after transformation, and V=[v 1, v, v 3 ]. he samples to make matrix X may be all the pixels of an image or all the pixels of all the images in a group. he computation of the covariance matrix is the most complex work and it has a temporal complexity of O(n ). A covariance matrix from an image gives a K- transform that is optimal for the single image. he matrix generated from all the images in a group provides a K- transform that is optimal for the group images as a whole. he three eigenvalues are useful if we want to know whether a component is more significant than another. he larger an eigenvalue is, the more significant its corresponding component is. In this paper, the eigenvector that corresponds to the maximum eigenvalue is called principal eigenvector. By using the principal eigenvector in the K- transform, the most important component can be found. his is helpful for many color image applications, such as color image coding and color image retrieval. 3 PCA Experiments In this paper, a database of 5 image groups is used for our experiments to obtain the optimal transforms. 1. Architectures: 56 images in JPEG format of Asian architectures in Japan, Iran and Indonesia.. Sports: 53 JPEG images of football and soccer games. 3. Animals: 50 JPEG images of feral horses. 4. Plants: 15 JPEG images of many diverse plants. 5. andscapes: 99 images in bitmap format. andscape pictures of Greenland. he images in Group 1, and 5 are from the image database of the Department of Computer Science & Engineering, University of Washington ( /research/imagedatabase). hose in Group 3 and 4 are from the collections of the digital library of the Department of Electrical Engineering and Computer Sciences, University of California, Berkeley ( For the JPEG files [7], each pixel is converted from YCbCr color space back into RGB color space first during decoding process. he principal components analysis takes place in the CIE AB color space, so the eigenvectors and the K- transform are also in the CIE AB space. he K- transform matrix V and three eigenvectors have a relation of V=[v 1, v, v 3 ]. he eigenvalues are sorted in the magnitude-descending order, and v 1 is the principal vector. In order to better understand the proportions between the eigenvalues, we use a rough eigpercent to stand for the percentages of eigenvalues relative to the sum of all three.

5 Optimal ransform in Perceptually Uniform Color Space and Its Application he Optimal ransforms of the 5 Individual Groups he K- transform matrix (transpose of eigenvectors matrix) and eigenvalues of image group 1 (architectures) are: , v, (9) [ λ, λ, λ ] = [ ,0.3070,0.17] = [ 74,15,11] 1 3 eigpercent () he K- transform matrix and eigenvalues of image group (sports) are: , v, (11) [ λ, λ, λ ] = [ 1.033,0.603,0.1647] = [ 71,18,11] 1 3 eigpercent (1) he K- transform matrix and eigenvalues of image group 3 (animals) are: , v, (13) [ λ, λ, λ ] = [ ,0.5858,0.406 ] = [ 54,7,19] 1 3 eigpercent (14) he K- transform matrix and eigenvalues of image group 4 (plants) are: , v, (15) [ λ, λ, λ ] = [.8,1.671,0.9036] = [ 49,30,1] 1 3 eigpercent (16) he K- transform matrix and eigenvalues of image group 5 (landscapes) are: , v, (17) [ λ, λ, λ ] = [ 4.005,1.391,0.746] = [ 67,,11] 1 3 eigpercent (18) It is very interesting that the principal eigenvectors of the 5 image groups are all close to (1,0,0). It implies that, in the CIE AB space, the principal component of the images is in the direction of *. 3. he Optimal ransform of All the Pictures in All the Groups We use all the test images to find the general K- transform for all the groups. he K- transform matrix (transpose of eigenvectors matrix), eigenvalues and their proportional percentages from all the images in all the groups are:

6 74 Y. Chen, P. Hao, and A. Dang V = [ v v, v ] 1, = [ λ, λ, λ ] = [ ,0.5694,0.5060] = [ 53,5,] 1 3 (19) eigpercent (0) he K- transform for all the images in our database also implies that the principal vector in CIE AB space is very close to the vector (1,0,0), and the other components are primarily in the AOB plane of the CIE AB coordinate system, where O is the origin in CIE AB space. Both vectors have a rotation angle of about π / 4 to CIE AB axes except a sign. herefore, through our experiments, an approximate optimal transform can be given as: V = 0 cos( π 4) sin( π 4) (1) 0 sin( π 4) cos( π 4) It looks so elegant and simple. More importantly, the color space after this transform is still perceptually uniform, and we believe that it is a better choice for perceptually uniform color space standard. Since the new space is just AB space rotated, we name the transformed space as AR. From the proportional percentages between the eignevalues, we know that the significance between the components in CIE AB space is roughly :1:1. his also looks very nice and will be very useful in image applications. he percentages of the three eigenvalues give the information of significance of components, which can be used in image retrieval or image compression during quantization. 4 Application in Image Coding he optimal transform in CIE AB has many applications. In this paper, we test lossy compression of color images in the image coding framework of the new international standard JPEG 000. Our transform gives better performance than the standardized color transform. 4.1 Intercomponent ransforms We use our optimal transform in CIE AB space presented above as an intercomponent transform in the new image coding standard framework, JPEG 000 [8]. We assume RGB components are in a cube of , and each component can be any integer from 0 to 55. For all possible RGB colors, the ranges of our transformed components AR are: [0, 0.004], A [ , 6.96], and R [ 114., ]. herefore, we use the new bounding cube to normalize

7 Optimal ransform in Perceptually Uniform Color Space and Its Application 75 AR components into [0, 55] by a simple linear mapping: f(x)=55*(x-x min )/(x max - x min ). In the standard JPEG 000, two color transforms are employed. he one for lossy image coding is in YCbCr space. he forward and the inverse of the transform are defined as: Y R R Y = Cb G G = Cb () Cr B B Cr he other color transform is integer reversible for lossless compression, and is a rough approximation of YCbCr color transform. 4. Compression Evaluation he quality of a compressed image is generally evaluated by peak signal-to-noise ratio (PSR) objectively and human visual system subjectively. In order to make our objective comparison agree with the subjective evaluation, PSR is calculated with the difference between the original image and the reconstructed image, and the difference is found in CIE AB: ( max min ) + ( Amax Amin ) + ( Bmax Bmin ) PSR = log (3) MSE 1 MSE = ( 1 ( x, ( x, ) + ( A1 ( x, A ( x, ) + ( B1 ( x, B( x, ) (4) M x, y Since our optimal color transform is just a 45-degree rotation in AOB plane, and is the same as * in AB space, we can estimate PSR in our AR space directly. 4.3 Evaluation Experiments In our evaluation experiments, we use 6 popular RGB color test images, baboon, barbara, goldhill, lena, peppers and airplane. he bit rates we tested are 0.5, 0.5, 0.15, 0.065, and bits per pixel (bpp). All the PSRs of our experiments are listed in able 1. able 1. Performance of lossy image compression in JPEG 000 framework (PSR, db) ame 0.5 bpp 0. 5 bpp 0.15 bpp bpp bpp YcbCr AR YcbCr AR YcbCr AR YcbCr AR YcbCr AR lena baboon barbara goldhill peppers airplane Average

8 76 Y. Chen, P. Hao, and A. Dang able 1 shows that our color space transform AR performs better for all the images at all the bit rates, average 4.3dB higher, minimum 3dB and maximum 6dB better than the default color transform of JPEG 000. By subjective comparison, the compression with our color transform also looks better than that with the default YCbCr color transform. 5 Conclusions Principal components analysis is applied to obtain the optimal color transforms for the images of 5 groups in the perceptually uniform color space, CIE AB. he eigenvectors of the 5 image groups are close. he principal eigenvectors corresponding to the maximum eigenvalues are all approximately in the direction of the *. he other two eigenvectors are almost perpendicular to the * direction. he color space K- transform for an image group is useful for image retrieval and image coding as pre-processing. When an image is given in practice, an appropriate fixed optimal color transform can be applied after we simply recognize what class the image belongs to. With hundreds of test images of diverse content, a general optimal K- transform is obtained and can be used as a fixed optimal color transform for all the images. Our approximately optimal K- transform AR is actually elegant and simple. It is just 45-degree rotated AB, and can be used as an image-independent color transform. As in our experiments with JPEG 000, AR performs better for lossy color image compression, and the average PSR is about 4 db higher than the default YCbCr. With our AR and a faster color space transform between RGB and CIE AB, an approximate linear transform from RGB to AR can be found for real-time applications and is more efficient and applicable, so it is our next investigation. References 1. Kasson, J.M., Plouffe, W.: An Analysis of Selected Computer Interchange Color Spaces. ACM ransactions on Graphics, Vol. 11, o. 4 (199) Poynton, C.: Frequently Asked Questions about Color. (1999) 3. Duda, R.O., Hart, P.E., Stork, D.G.: Pattern Classification. Wiley InterScience (1999) 4. Pratt, W.K.: Spatial transform coding of color images. IEEE ransactions on Communication echnology, Vol. 19 (1971) Connolly, C., Fliess,.: A Study of Efficiency and Accuracy in the ransformation from RGB to CIEAB Color Space. IEEE ransactions on Image Processing, Vol. 6, o. 7 (1997) Asakawa, K., Sugiura, H.: High-precision color transformation system. IEEE ransactions on Consumer Electronics, Vo.41, o. (1995) Wallace, G.K.: he JPEG Still Picture Compression Standard. Communications of the ACM, Vol. 34, o. 4 (1991) Christopoulos, C., Skodras, A., Ebrahimi,.: he JPEG000 still image coding system: An overview. IEEE rans. Consumer Electronics, Vol. 46, o. 4 (000)