IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL. 29, NO. 5, MAY 2019 1559 Letters Efficient Chroma Sub-Sampling and Luma Modification for Color Image Compression

Shuyuan Zhu , Member, IEEE, Chang Cui, Ruiqin Xiong , Member, IEEE, Yuanfang Guo , Member, IEEE, and Bing Zeng , Fellow, IEEE

Abstract— In color image compression, the chroma components are by a factor of 2 in both the horizonal and vertical directions in the often sub-sampled before compression and up-sampled after compression. pixel domain, leading to the YUV 4:2:0 format. Although sub-sampling the chroma components saves the bit-cost for Unfortunately, sub-sampling of the chroma components usually compression, it often induces extra color distortions in the compressed images. In this letter, we propose two approaches to tackle this problem. induces extra distortions to the compressed RGB images. These First, we propose a sub-sampling method in the transform domain and distortions may be reduced by applying the interpolation dependent apply it to both chroma components. Then, based on this sub-sampling, sub-sampling [2]Ð[5] to each individual chroma component. Note we propose a novel method to modify the luma component. In our that these techniques [2]Ð[5] must be carried out in the YUV space. proposed luma modification algorithm, the distortions that occurred in the two chroma components can be coupled together and utilized to In practice, they cannot guarantee a low RGB distortion for the modify the luma component. With our proposed chroma sub-sampling compressed color image. To solve this problem, the RGB distortion and luma modification algorithms, we can achieve a low RGB distortion constrained chroma sub-sampling is proposed in [6], where sub- in practical image coding. The experimental results demonstrate that our sampling of the chroma components is optimized to achieve a lower proposed methods offer more significant coding gains compared with the RGB distortion. The solution in [6] has demonstrated some promising state-of-the-art methods for the compression of color images. results when it is employed in the 4:2:0 format coding. Index Terms— Color image, compression, distortion, sub-sampling, Besides of the optimization of the sub-sampling operation for modification. chroma components, the luma component may also be modified I. INTRODUCTION to achieve a high-quality color image compression [7]. In [7], OLOR image compression typically happens in the YUV space, the traditional spatial sub-sampling is performed on two chroma Cwhere the input RGB signal is converted into the YUV signal components and a low RGB distortion is guaranteed by modifying the which is composed by one luma (Y) component and two chroma luma component according to the sub-sampled chroma components. (U and V) components. The chroma component is usually more This method is named as the joint chroma sub-sampling and distor- smooth than the luma component, and the latter one contains plenty tion minimization-based luma modification (JCSLM). Unfortunately, of textures. After the color space conversion, the luma and chroma JCSLM has to be carried out before compression. When it is adopted components are divided into image blocks. Then, the discrete cosine in practical coding schemes, the compressed color image still suffers transform (DCT), quantization and entropy coding are accordingly from certain compression distortion, which accordingly limits the performed on these blocks to achieve the compression. coding efficiency. In general, the RGB-to-YUV space conversion removes the redun- In this letter, we firstly propose a transform domain sub-sampling dancies between color channels [1], and the compression of the scheme for both the chroma components. Our proposed method YUV signal can achieve a high compression ratio more easily. targets a low RGB distortion and yields a comparable sub-sampling In practical YUV based image compression, the chroma sub-sampling performance to the YUV 4:2:0 format. Specifically, this proposed is employed very frequently to reduce the image data-size. This sub- scheme converts each N × N chroma block into an (N/2) × (N/2) sampling saves a considerable bit-cost, and thus potentially improves coefficient block, which accordingly induces a low bit-cost for the coding efficiency. Currently, the most popular chroma sub- compression. Then, based on this sub-sampling scheme, we propose sampling scheme down-samples two chroma components separately a novel method to modify the luma component. When our proposed modification algorithm is adopted in practical image coding, the Manuscript received November 1, 2018; revised January 14, 2019; accepted distortions, including both the sub-sampling and compression distor- January 23, 2019. Date of publication January 29, 2019; date of cur- rent version May 3, 2019. This work was supported in part by the tions, occurred on two chroma components can be coupled together National Natural Science Foundation of China under Grant 61672134, and utilized to modify the luma component. At last, we can achieve Grant 61772041, Grant 61802391, and Grant 61720106004, in part by the state-of-the-art performance, i.e., a low RGB distortion as well the Fundamental Research Funds for Central Universities of China under as a high coding efficiency, for the compression of color images. Grant ZYGX2016J038, and in part by the 111 Projects under Grant B17008. This letter was recommended by Associate Editor S. Liu. (Corresponding authors: Yuanfang Guo; Bing Zeng.) II. RGB COLOR DISTORTION S. Zhu, C. Cui, and B. Zeng are with the School of Information and Com- The RGB-to-YUV space conversion is implemented by perform- munication Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China (e-mail: [email protected]). ing a transform on the red (R), green (G) and blue (B) pixels { , , } R. Xiong is with the Institute of Digital Media, School of Electronic Engi- xRi xGi xBi to produce the corresponding Y, U and V pixels { , , } neering and Computer Science, Peking University, Beijing 100871, China. xYi xUi xVi as Y. Guo is with the Laboratory of Intelligent Recognition and Image ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ Processing, School of Computer Science and Engineering, Beihang University, xYi xRi 16 Beijing 100072, China (e-mail: [email protected]). ⎣ ⎦ = ⎣ ⎦ + ⎣ ⎦, xUi A xGi 128 (1) Color versions of one or more of the figures in this letter are available xV xB 128 online at http://ieeexplore.ieee.org. i i Digital Object Identifier 10.1109/TCSVT.2019.2895840 where A is the color transform matrix. 1051-8215 © 2019 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. 1560 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL. 29, NO. 5, MAY 2019

2 2 2 If the degradation, such as the sub-sampling or the compression, where DL is an N × M matrix and DH is an N × (N − M) happens to the Y, U and V pixels, the reconstructed R, G and matrix. If we remove XH and only reserve XL for further processing, {ˆ , ˆ , ˆ } Bpixels xRi xGi xBi should be determined by the degraded the transform domain sub-sampling for X is accordingly achieved. {ˆ , ˆ , ˆ } ˆ = Y, U a nd V pi xe l s xYi xUi xVi based on the inverse transform With the sub-sampled X, i.e., XL , x can be rebuilt as x DL XL. { , , } of (1). Then, the color distortion measured on xRi xGi xBi is However, due to the removal of the high-frequency components, calculated as rebuilding x directly from XL tends to induce a high distortion. When ⎡ ⎤ ⎡ ⎤ T the compression is performed on X , the reconstruction of x from x −ˆx x −ˆx L Ri Ri Ri Ri the compressed X will suffer from a more serious distortion. d = ⎣x −ˆx ⎦ ⎣x −ˆx ⎦ L RGBi Gi Gi Gi Gi To achieve a high-quality reconstruction, we propose to preserve xB −ˆxB xB −ˆxB ⎡ i i ⎤ i i ⎡ ⎤ the high-frequency information as much as possible in the remaining −ˆ T −ˆ xYi xYi xYi xYi low-frequency coefficients. Specifically, we produce XL with the = ⎣ −ˆ ⎦ ( −1)T ( −1) ⎣ −ˆ ⎦. xUi xUi A A xUi xUi (2) minimal reconstruction distortion as xV −ˆxV xV −ˆxV i i i i X˜ = arg min x − D X 2. (6) =[ ] =[ ] =[ ] L L L 2 Let us use xY xYi K ×1, xU xUi K ×1 and xV xVi K ×1 XL to represent the K-point column-vectors for the Y, U and V compo- Then, the optimal XL is generated as nents, respectively. When the degradation happens, with the degraded − ˆ =[ˆ ] ˆ =[ˆ ] X˜ = (DT D ) 1(DT x). (7) luma and chroma components, i.e., xY xYi K ×1, xU xUi K ×1 L L L L and xˆ =[ˆx ] × , we calculate the distortions of the Y, U and V V Vi K 1 According to (7), a sub-sampled coefficient vector with M coef- components as e = x −ˆx , e = x −ˆx and e = x −ˆx , Y Y Y U U U V V V ficients is produced and these coefficients will be further exploited respectively. Then, based on e , e and e , the YUV error vector Y U V to reconstruct x, leading to our proposed upward-conversion guided can be constructed as e =[eT eT eT ]T . Similarly, in the RGB YUV Y U V transform domain sub-sampling (UGTS). space, the corresponding RGB error vector eRGB can be obtained. − The proposed UGTS can be applied to both the chroma compo- By assuming that λ , (m, n = 0, 1, 2) is the element of A 1, m n nents for color image compression. Specifically, we specify M = we can exploit λ , to form a color conversion matrix  such that m n N2/4 to yield a comparable sub-sampling performance to the YUV × eRGB = eYUV. (3) 4:2:0 format. With UGTS, the high-frequency information of an N N chroma block will be preserved in the N2/4 remaining coefficients Specifically,  is formed as ⎡ ⎤ as much as possible. These coefficients are further compressed by 00 01 02 quantization and entropy coding, and a considerable bit-cost can be ⎣ ⎦  = 10 11 12 , saved accordingly. 20 21 22 IV. OUR PROPOSED LUMA MODIFICATION METHOD where each m,n is the K × K diagonal matrix and m,n = diag(λm,n, ··· ,λm,n). Based on (3), the RGB distortion can be Our proposed UGTS algorithm can certainly be applied to the calculated as chroma components to reduce the color distortion caused by sub- samplings. Meanwhile, the color distortion can also be reduced by = T dRGB eRGBeRGB modifying the luma component. In this work, we propose a novel = T (T ) . eYUV eYUV (4) method based on UGTS to modify the luma component for a low RGB color distortion. When this luma modification method is applied III. UPWARD-CONVERSION GUIDED TRANSFORM to practical image coding, both the sub-sampling and the compression DOMAIN SUB-SAMPLING distortions of the two chroma components can be coupled together In the sub-sampling based image coding, sub-sampling is usually and utilized to modify the luma component. Then, a low RGB carried out in the pixel domain to reduce the resolution of the input distortion can be accordingly achieved in practical image coding. image. Besides, sub-sampling can also be performed in the transform domain by dropping some of the high-frequency coefficients from A. Decomposition of RGB Distortion the transformed block [8], [9]. Compared with the pixel domain sub- If we define W = T , (4) can be rewritten as sampling based coding scheme, image interpolation is not required in = T , the transform domain sub-sampling based scheme to reconstruct the dRGB eYUVWeYUV (8) image signal, which yields a lower computational complexity. In this where W is a 3N2 ×3N2 symmetric matrix. To make a much clearer work, we propose the upward-conversion guided transform domain analysis of the composition of dRGB, we perform the Cholesky sub-sampling for the compression of color images. factorization on W to factorize it as W = VT V,whereV is a × Based on the Kronecker product, the 2D transform of an N N 3N2 × 3N2 upper triangular matrix, i.e., image block can be implemented in a 1D manner as X = (C ⊗ C)x, ⎡ ⎤ where x is the N × N block in vector form, X contains N2 transform V0 V1 V2 ⎣ ⎦ coefficients, ⊗ denotes the Kronecker product and C is the N × N V = 0V3 V4 , (9) transform matrix. Meanwhile, if an N2 × N2 matrix D is defined as 00V5 − − D = C 1 ⊗ C 1, the inverse transform can be obtained via x = DX. and each sub-matrix Vi = diag(γi , ··· ,γi ){i = 0, ··· , 5} is an If we employ the first M coefficients of X to form a low-frequency N2 × N2 diagonal matrix. With the N2 × N2 identity matrix I, each vector X and utilize the other (N2 − M) coefficients to form a high- L Vi can be represented by Vi = γi I. frequency vector X , X can be partitioned as X =[XT XT ]T .Also, H L H Basedon(8)and(9),dRGB can be decomposed as we can partition D as D =[DL DH ] such that   2 XL dRGB = dk , (10) x =[DL DH ] , (5) XH k=0 ZHU et al.: EFFICIENT CHROMA SUB-SAMPLING AND LUMA MODIFICATION FOR COLOR IMAGE COMPRESSION 1561

where According to (18), if the minimal d0 is desired, we can modify T xY as d0 = (V0eY + V1eU + V2eV ) (V0eY + V1eU + V2eV ) 1 = + + 2 ˜ = + (γ + γ ) V0eY V1eU V2eV 2 xY xY 1eU 2eV γ0 =γ + γ + γ 2, γ γ 0eY 1eU 2eV 2 (11) = + 1 ( − ˜ ) + 2 ( − ˜ ). xY xU DL XUL xV DL XVL (19) d = (V e + V e )T (V e + V e ) γ0 γ0 1 3 U 4 V 3 U 4 V 2 = γ + γ , Based on the above modification, d0 = 0 can be achieved accordingly. 3eU 4eV 2 (12) At last, with the minimal d0, d1 and d2, a minimal dRGB can be and achieved. After obtaining the sub-sampled chroma coefficients, i.e., d = (V e )T (V e ) 2 5 V 5 V {X˜ , X˜ }, and the modified luma component, i.e., x˜ , we can =γ 2. UL VL Y 5eV 2 (13) further process them to achieve the compression.

According to the decomposition of dRGB above, we can minimize dRGB by minimizing each di independently. Moreover, this mini- C. Compression-Dependent Luma Modification mization provides a potential solution to simultaneously modify the In practical image coding, if the compression is performed on both luma component and sub-sample the chroma components to achieve the chroma components, we can also implement our proposed luma a low RGB distortion. modification based on the compressed U and V components, and thus propose the compression-dependent luma modification algorithm. ˇ ˜ B. Luma Modification With Transform Domain Sub-Sampling Let XVL denote the compressed XVL . When the compression is of the Chroma Components applied to the sub-sampled V component, the overall distortion of the V component, including both the sub-sampling and compression For convenience, we focus on the 1D case to describe how to = − ˇ distortions, is calculated as eV xV DLXVL . Based on eV , achieve the minimal dRGB by jointly modifying the luma component ˜ the optimal XU , i.e., XU , can be computed according to (16). and sub-sampling the chroma components. Let us use x , x and L L Y U Similarly, with the compressed X˜ , which is denoted as Xˇ , x to represent the N2-point pixel vectors converted from the UL UL V the overall distortion of the U component can be calculated as e = N × N Y, U and V blocks, respectively. Let X and X be U UL VL x −D Xˇ . Based on e and e , the modified luma component x˜ the (N2/4)-point vectors converted from the (N/2) × (N/2) sub- U L UL U V Y can be obtained according to (19). Then, we apply the compression to sampled U and V coefficient blocks, respectively. In our proposed x˜ and generate the compressed x˜ , which is denoted as xˇ .Atlast, method, the optimal modified luma pixels and the optimal sub- Y Y Y d0 is calculated by sampled chroma coefficients will be produced to minimize dRGB. Based on the decomposition of d , i.e., (10)Ð(13), both the = γ 2 ˜ −ˇ 2. RGB d0 0 xY xY 2 (20) optimal sub-sampled chroma coefficients and the luma component can be determined in an alternating iterative manner as follows. Comparing (20) with (11), after our proposed luma modification, ˜ According to our proposed UGTS algorithm, when the transform d0 is only determined by the compression distortion of xY .Note domain sub-sampling is applied to the V component, the opti- that this distortion is highly related to the quantization adopted in compression rather than the distortions occurred on the two chroma mal XVL , which minimizes d2, is determined by components. Therefore, a smaller dRGB can be achieved for practical ˜ = ( T )−1( T ). image coding. XVL DL DL DL xV (14) With X˜ , e is calculated via e = x − D X˜ . VL V V V L VL V. E XPERIMENTAL RESULTS If the transform domain sub-sampling is also applied to the U component, e should be calculated by e = x − D X and The conversion matrix employed in [7] is adopted in this work U U U L UL = (12) can be rewritten as to implement the RGB-to-YUV space conversion, and N 16 is specified for our proposed UGTS and luma modification algorithms. = γ ( − ) + γ 2. d1 3 xU DL XUL 4eV 2 (15) All the simulations are conducted not only on three popular image data sets, including the Kodak data set, the IMAX data set and the According to (15), the optimal X , which minimizes d , is deter- UL 1 SCI data set, which are utilized in [7], but also on eight classical mined via test images, including Lena, Airplane, Baboon, Barbara, Peppers, γ ˜ = ( T )−1 T ( + 4 ) Splash, Tiffany and Monarch, as shown in Fig. 1. The experiments are XUL DL DL DL xU eV γ3 performed on Matlab R2016b supported by the Intel Core i7-7500U γ = ( T )−1 T ( + 4 ( − ˜ )). CPU at 2.90 GHz with 8G memory and the Microsoft Windows DL DL DL xU xV DL XVL (16) γ3 10 operating system. The experimental results are presented in this As can be concluded from (16), the optimal sub-sampled coefficients section, where the color PSNR (CPSNR) [7], derived from the color- of the U component are determined not only by the original pixels mean-square-error (CMSE) over three color channels, is utilized as of the U component, but also by the distortions of the V component. the distortion metric. Specifically, CPSNR is calculated via ˜ ˜ Then, with the optimal XU and XV , d0 is computed by 2 L L = 255 . CPSNR 10log10 (21) = γ 2 + 1 (γ + γ ) 2. CMSE d0 0 eY 1eU 2eV 2 (17) γ0 A. Comparisons of Different Sub-Sampling Methods d If the luma component is modified, 0 should be determined by the and Luma Modification Methods modified xY , which is denoted as x˜Y ,as Firstly, we compare our proposed UGTS algorithm with the = γ 2 ( −˜ ) + 1 (γ + γ ) 2 traditional direct spatial down-sampling (DSDS) and the interpolation d0 0 xY xY 1eU 2eV 2 (18) γ0 1562 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL. 29, NO. 5, MAY 2019

Fig. 1. Eight classical test images: Lena, Airplane, Baboon, Barbara, Peppers, Splash, Tiffany, and Monarch.

TABLE I AVERAGE CPSNR (DB) RESULTS OF RECONSTRUCTED IMAGES

dependent image down-sampling (IDID) [4] by performing them on the U and V components of the converted YUV images. When UGTS is carried out, the full-resolution reconstruction is imple- Fig. 2. Compressed image partons for Peppers: From left to right and from mented based on (7). If DSDS or IDID is carried out, the bicubic top to bottom: JPEG, JCSLM, IDID, UGTDS, TS-LM, and CDTS-LM. interpolation is utilized to rebuild the full-resolution components. TABLE II After the YUV-to-RGB space conversion, the average CPSNR result AVERAGE BD-RATE REDUCTION FOR VARIOUS IMAGE SETS(%) over all the test images for each image data set is calculated to evaluate the performance of the sub-sampling method. The average CPSNR results for the three sub-sampling methods are presented in Table I. As can be observed from Table I, our proposed UGTS offers the best reconstruction results compared with the DSDS-based and IDID-based methods. These results also indicate that our proposed sub-sampling method can well preserve the necessary information in the sub-sampled images to recover high-quality images. Then, we compare our proposed transform domain sub-sampling based luma modification algorithm (TS-LM) with JCSLM [7]. The corresponding CPSNR results are also presented in Table I. One can see from Table I that our proposed TS-LM achieves a much better reconstruction quality than JCSLM. Besides, TS-LM also outperforms our UGTS algorithm. These results demonstrate that our proposed TS-LM can well preserve the necessary information in the sub-sampled image and effectively modify the luma component to three QF sets, {20, 30, 40, 50}, {40, 50, 60, 70} and {60, 70, achieve low RGB distortions. 80, 90}, are introduced to evaluate the performances of different coding scenarios. According to the results presented in Table II, our B. Compressing Color Images With Different Methods UGTS-based coding has outperformed the JCSLM-based coding and In this experiment, we integrate our proposed UGTS algorithm and IDID-based coding in all the coding scenarios. After our proposed luma modification algorithm into JPEG [10] to construct the corre- luma modification algorithm is carried out, our TS-LM-based coding sponding compression methods for color images. Specifically, UGTS and CDTS-LM-based coding offer more significant performance is applied to both the chroma components and then the sub-sampled gains compared with our UGTS-based coding. components are compressed accordingly. This method is denoted as For subjective comparisons, some compressed image portions the UGTS-based coding. Meanwhile, we utilize our proposed TS-LM of image Peppers are presented in Fig. 2. As can be observed algorithm to construct the TS-LM-based coding scheme. Besides, from Fig. 2, our proposed methods, especially the CDTS-LM-based we design the compression-dependent TS-LM (CDTS-LM), as men- coding, generate the compressed color images with less artifacts. tioned in Section IV-C, and employ it to form the CDTS-LM-based coding. We also apply IDID to both the chroma components to implement C. Discussion About the Computational Complexity the IDID-based coding and utilize the JCSLM algorithm to implement When the proposed UGTS and luma modification algorithms the JCSLM-based coding as the baseline methods. Note that all the are adopted in typical image coding scheme, they only introduce coding methods are implemented based on JPEG to guarantee fair additional operations at the encoder side. Moreover, our proposed comparisons. coding approaches do not require image interpolation to reconstruct Compared with the traditional JPEG coding, the average the complete image at the decoder side. On the contrary, in the BD-rate [11] saving over all the test images for each image data spatial sub-sampling based coding scheme, such as the traditional set is employed to evaluate the rate-distortion performance for each JPEG coding, the IDID-based coding and the JCSLM-based coding compression method, where CPSNR is adopted as the distortion schemes, image interpolation is always required at the decoder side metric. The average BD-rate savings for the five compression meth- to reconstruct the final color image, which may induce a higher ods are presented in Table II, in which the compression results of computational complexity. ZHU et al.: EFFICIENT CHROMA SUB-SAMPLING AND LUMA MODIFICATION FOR COLOR IMAGE COMPRESSION 1563

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