Embedded Image Coding Based on Hierarchical Discrete Cosine Transform

 In this paper, we illustrate that the zerotree quantizer developed originally for wavelet compression can be effectively applied to Discrete Cosine Transform (DCT) in a hierarchical way. In this Hierarchical DCT (HDCT), the input image is partitioned into a number of 8× 8 blocks and a first level DCT is used to each of these blocks individually. As DC coefficients of DCT neighboring blocks are highly correlated and particularly pronounced to obtain the compression results at low bit rates, another level DCT is applied to only DC coefficients re-organized as 8× 8 blocks. This procedure is repeated until the last step is reached. All the HDCT coefficients within a DCT block are then rearranged into a subband structure in which the zerotree quantizer can be employed. The proposed algorithm yields a fully embedded, low-complexity coder with competitive PSNR performance. For example, when compared with the baseline JPEG on the 512 512 × standard image “Lena”, it gains 0.8 dB ~ 1.7 dB. In order to remove the blocking effects in the reconstructed images, especially at low bit rates, a simple and efficient method based on Sobel operators is also developed. Experimental results show that the proposed deblocking method works well and enhances decoding for decompressed images both objectively and subjectively. I. INTRODUTION Transform coding has been widely used in many practical image/video compression systems because its quantized coefficients after linear transform can yield better compression results than direct coding of image intensity in the spatial domain. KL transform can be found to be optimal under certain conditions, however, signal-independent sub-optimal approximations like DCT are often used for computational efficiency. In recent years, most of the research activities in image coding have shifted from DCT to wavelet transforms, especially after Shapiro introduced his famous embedded zerotree wavelet coder (EZW) [1] and Said and Pearlman developed an algorithm by set partition in hierarchical trees (SPIHT) [2]. Their methods provide very high compression efficiency in terms of Peak Signal-to-Noise Ratio (PSNR) versus required bits-per-pixel (bpp). But, it should be noted that the high compression efficiency in the rate/distortion sense quantitatively does not correspond to the same degree of compression efficiency in terms of visual quality versus bit rate. This has been pointed out in [3] by comparing visual results based on wavelet and DCT. Although wavelets are capable of providing more flexible space-frequency resolution tradeoffs than DCT, DCT still can produce very high compression efficiency when coupled with a zerotree quantizer instead of the traditional methods used in JPEG. Some better results using such techniques have been reported in [4,5], where EZDCT (Embedded Zerotree DCT coding) with arithmetic coding outperforms any other DCT-based coder published in the literature [4] and STQ (Significance Tree Quantization) without using arithmetic coding is superior to a version of EZDCT which also doesn’t use arithmetic coder [5]. We observe that DC coefficients in EZDCT are still highly correlated and particularly pronounced to obtain the compression results at low bit rates. We also notice that a maximum absolute coefficient is outputted first for all DCT coefficients in EZDCT and STQ, actually the maximum absolute coefficients in each block maybe greatly differ from each other, so it is not reasonable to output only one n for all blocks. So we present here an embedded zerotree image coder based on HDCT (EZHDCT) with improvements on using DCT in a hierarchical way and an additional step of outputting the magnitude of each block. EZHDCT achieves competitive PSNR performance when comparing to other DCT-based coders, such as the baseline JPEG [6], EZDCT and STQ. Another component of the work presented here relates to deblocking of DCT artifacts. As block-based image coding schemes always cause annoying blocking effects in the decoded images especially at low bit rates, a simple and efficient deblocking method is proposed to deal with this problem. The parameter of proposed deblocking method can be adaptively adjusted according to the complexity of the input image and the specified bit rate. II. EMBEDDED IMAGE CODING METHOD A. HDCT Structure An input image ( N M × ) is first divided into n n × blocks, where L n 2 = , 0 > L . Each block is then transformed by a first level DCT. Each n n × DCT block can be treated as a L -scale tree of coefficients labeled from 0 to 1 − ×n n . Fig. 1(a) is an example of 8 8 × DCT ( 8 = n , 3 = L ) coefficients labeled from 0 to 63. The tree structure for 8 8 × DCT coefficients in Fig. 1(a) can be viewed as a 64-subband decomposition. Because DC coefficients of neighboring blocks are highly correlated and particularly pronounced at low bit rates, a second level DCT transform is applied to each This procedure is repeated on all the DC coefficients ( ) / ( ) / ( n N n M × ) with DCT block size of m m × until the last step is reached. This scheme is called HDCT (Hierarchical DCT). All DCT coefficients with each block can be reorganized when an image is taken as a single entity. In order to obtain the dyadic decomposition such as in EZW and SPIHT. For example, we can take each 8 8 × DCT block as a 10-subband decomposition as was shown in Fig. 1(b). In Fig. 1(b), the ten subbands are composed of {0},{1},{2},{3},{4-7},{8-11},{12-15},{16-31},{32-47} and {48-63} coefficients respectively. One important step of HDCT (Hierarchical DCT) involves the grouping of the same subbands for all DCT blocks. We call this grouping of DCT coefficients into a single DCT clustering entity. This important step is illustrated as in Fig. 2. In Fig. 2, Go0 means Group of subband 0,..., Go9 means Group of subband 9. Fig. 3 is the diagrammatic illustration of the proposed organization strategy. This procedure is repeated on all the DC coefficients ( ) / ( ) / ( n N n M × ) with DCT block size of m m × until the last step is reached. Fig. 4 is an illustration of two-level organized DCT coefficients on the 512 512 × Lena image, where 8 = = n m . The gray values in Fig. 4 are obtained by ) ( * 4 255 t Coefficien ABS − with the exception of DC’s for better visual presentation. It can be seen that 1) signal energy is compacted mostly into DC coefficients and small number of AC coefficients related to the edges in spatial domain; 2) cross-subband similarity and decay of magnitude across subband; and 3) within subband clustering of significant coefficients. These DCT characteristics can be further utilized to DCT-based coders in order to get better compression performance and will widen DCT applications, such as applications on image compression, image retrieval, image recognition and so on. B. Magnitude Output Definition For all coefficients, j i c , , in EZDCT, STQ, EZW and SPIHT, a value, n , can be obtained and outputted in their initialization step, ( )   |} {| max log , ) , ( 2 j i j i c n = . (1) We observed that the maximum absolute coefficients in each block maybe greatly differ from each other, so it is not reasonable to only use one n for all blocks. In EZHDCT, we define k n for each Block k, ( )   |} {| max log , ) , ( 2 n m n m k b n = , (2) where n m b , is the coefficients in Block k. As there is much redundancy existed between k n and 1 − k n , a simple DPCM ( 1 − − = k k k n n d ) is used to decorrelate them. In order to obtain k d directly, such a mapping function is described as follows:    < − = . , 2 0 , 1 | | 2