Lossy to Lossless Image Coding Using Allpass Filters

基于小波的图像编码在国际标准JPEG2000中得到了广泛的研究和采用。在基于小波的图像编码中，两波段完全重构滤波器组起着非常重要的作用。分析和合成滤波器要求具有完全线性的相位响应（对应于对称小波基），允许我们使用对称扩展方法来精确处理图像的边界。小波滤波器组也应该是标准正交的，以避免子带图像之间的冗余。不幸的是，除了哈尔小波，FIR滤波器没有非平凡的标准正交对称小波。为了获得更好的压缩性能，小波基必须具有合理的正则性。因此，要获得比Haar小波更有规律性的结果，至少要放弃上述条件中的一个。例如，JPEG2000的基线编解码器支持的D-9/7和D-5/3小波是双正交的。另一方面，已知IIR小波滤波器可以同时满足正交性和对称性。利用全通滤波器提出了一类IIR正交对称小波。将基于全通的标准正交对称小波应用于图像压缩，提出了一种有效的小波滤波器组实现方法。首先，提出了不可逆实数到实数小波的实现方法，并利用全通滤波器对其进行分解。然后，利用全通滤波器的可逆实现实现可逆的整数到整数小波。最后，利用JPEG2000的参考软件，研究了基于allpass的标准正交对称小波的编码性能，并与D-9/7和D-5/3小波进行了比较。实验结果表明，基于allpass的标准正交对称小波比D-9/7和D-5/3小波具有更好的压缩性能。

Wavelet-based image coding has been extensively studied and adopted in the international standard JPEG2000. In the wavelet-based image coding, two-band PR (perfect reconstruction) filter banks play a very important role. The analysis and synthesis filters are required to have exactly linear phase responses (corresponding to symmetric wavelet bases), allowing us to use the symmetric extension method to accurately handle the boundaries of images. The wavelet filter banks should also be orthonormal to avoid redundancy between the subband images. Unfortunately, there are no nontrivial orthonormal symmetric wavelets with FIR filters, except for the Haar wavelet. To achieve better compression performance, a reasonable regularity is necessary for wavelet bases. Therefore, at least one of the above-mentioned conditions has to be given up to get more regularity than the Haar wavelet. For example, the D-9/7 and D-5/3 wavelets supported by the baseline codec of JPEG2000 are biorthogonal. On the other hand, it is known that IIR, wavelet filters can simultaneously satisfy both of the orthonormality and symmetry. A class of IIR orthonormal symmetric wavelets has been proposed by using allpass filters.We apply the allpass-based orthonormal symmetric wavelets to image compression, and propose an effective implementation of the wavelet filter banks. Firstly, we present the implementation of irreversible real-to-real wavelets and show its decomposition process by using allpass filters. Then, we make use of the invertible implementation of allpass filters to realize the reversible integer-to-integer wavelets. Finally, we investigate the coding performance of the allpass-based orthonormal symmetric wavelets by using the reference software of JPEG2000, and compare the performance with the D-9/7 and D-5/3 wavelets. It is shown from the experimental results that the allpass-based orthonormal symmetric wavelets can achieve better compression performance than the D-9/7 and D-5/3 wavelets.