IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 7, JULY 2007 3659 Error Inhomogeneity in Wavelet-Based Compression Nai-Xiang Lian, Student Member, IEEE, Vitali Zagorodnov, Member, IEEE,and Yap-Peng Tan, Senior Member, IEEE Abstract—Despite the popularity of wavelet-based image com- pression, its shortcoming of having error inhomogeneity (EI), namely the error that is different for even and odd pixel location, has not been previously analyzed and formally addressed. The difference can be substantial, up to 3.4-dB peak signal-to-noise ratio (PSNR) for some images and compression ratios. In this paper, we show that the EI is caused by asymmetrical filtering of quantization errors after the upsampling step in wavelet synthesis process. Nonuniformity and correlation of quantization errors can also contribute to this EI, albeit to a smaller degree. In addition to explaining the source of EI, the model we developed in this paper also allows predicting its amount for a given wavelet. Further- more, we show how to redesign wavelet filters to reduce this EI at a cost of a small reduction in the overall PSNR performance. For applications that are sensitive to PSNR degradation, we also show how to design wavelet filters that can gradually tradeoff PSNR performance for reduced EI. Index Terms—Coding gain (CG), compression, compression error, error inhomogeneity (EI), quantization error, wavelet transform. I. INTRODUCTION B ECAUSE of its excellent energy compaction capability on piecewise smooth signals such as images, wavelet transform has become a powerful alternative to discrete cosine transform (DCT) for image compression. Shapiro’s pioneering embedded zerotree wavelet (EZW) compression algorithm [1] and Said and Pearlman’s set partitioning of images in hier- archical trees (SPIHT) algorithm [2] have both demonstrated that wavelet-based compression can achieve much better peak signal-to-noise ratio (PSNR) performance and visual quality compared to DCT-based JPEG compression. Many improve- ments that followed [3]–[9] have finally led to a new still image compression standard JPEG2000 [10], which is set to replace JPEG in the near future. With much literature written on this topic, to our best knowl- edge, no work has ever been reported saying that existing Manuscript received November 15, 2005; revised October 26, 2006. The as- sociate editor coordinating the review of this manuscript and approving it for publication was Dr. Ta-Hsin Lin. N.-X. Lian is with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798, Singapore (e-mail: pg03806980@ntu.edu.sg). V. Zagorodnov is with the School of Computer Engineering, Nanyang Tech- nological University, Singapore 639798, Singapore (e-mail: zvitali@ntu.edu. sg). Y.-P. Tan is with the Division of Information Engineering, School of Elec- trical and Electronic Engineering, Nanyang Technological University, Singa- pore 639798, Singapore (e-mail: eyptan@ntu.edu.sg). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TSP.2007.894250 Fig. 1. Decomposing an image into four subimages according to the pixel indexes. wavelet-based image compression can introduce spatially varying error. Specifically, we have found that some pixels in an image reconstructed after wavelet compression have greater or smaller distortion than others. For example, consider decomposition of the reconstructed image into four subim- ages, as shown in Fig. 1. Here, pixels with even indexes compose subimage 0; pixels with even and odd compose subimage 1, and so on. Note that these four subimages have slightly different contents with shift of one row and/or column of pixels. Simple empirical tests then show that regardless of the image content, subimage 0 has a consistently higher PSNR than other subimages, and subimage 3 has the lowest. We conducted this experiment on 24 test images (Fig. 2), and the results are shown in Tables I–III. Here, by error inhomo- geneity (EI), we mean the difference in PSNR performance be- tween different subimages, especially between subimages 0 and 3. The inhomogeneity can be observed in different compression methods (Tables I and II), wavelet filters (Tables I and II), and compression bit rates (Table III), with a very high statistical sig- nificance (statistical P-value 0.00001, the probability that EI does not hold), ruling out the possibility of random occurrence. Furthermore, the amount of EI is sufficiently large to be visu- ally perceptible, as shown on the examples of “BARBARA” and “LENA” in Fig. 3. For both images, some parts of subimage 3 lack textures that are well preserved in subimage 0. As the main contribution of the paper, we show that the EI is caused by asymmetrical filtering of quantization errors after the upsampling step of wavelet synthesis process (Section II). In particular, we have discovered that error variances of even and odd samples of the reconstructed signal depend on different syn- thesis filter coefficients, and hence, in general, are not equal to each other. Moreover, for popular wavelets such as Daubechies 9-7 and 5-3, the error variance of the even samples is consis- tently larger than that of the odd samples, leading to the ob- served EI. We have also found that nonuniformity and corre- lation of quantization errors can also affect the EI, albeit to 1053-587X/$25.00 © 2007 IEEE