多小波图象变换的统计分析

多小波是一种新的小波，多小波的应用更是近几年才日见兴起，因此，有关多小波图象变换的一些基本统计数据，如均值、方差、量化后零系数的比例等等，尚未见诸学术刊物，而这些数据又是从事多小波图象编码研究的基本依据。从学术刊物和互联网上收集了5种不同性质的多小波，对这些多小波图象变换特性进行了详尽的统计分析。通过统计分析发现：（1）图象经过CL多小波变换后，能量不但汇聚在最低分辨率的子图象上，而且还进一步汇聚在最低分辨率子图象的第一个分量上，因此，CL多小波最适合图象编码；（2）图象经过CARDBAL多小波变换后，能量不但汇聚在最低分辨率的子图象上，而且还平均分摊在最低分辨率子图象的4个分量上，因此，通过相关性编码可以大幅度提高CARDBAL多小波图象编码的压缩比；（3）图象经过GHM多小波变换后，最低分辨率子图象的能量既不是集中在一个分量上，也不是平均分配在4个分量上，因此，尽管GHM是最早发现的多小波，且是目前最为常用的多小波，但它其实并不是图象编码的首选。

Statistical Analysis of Multiwavelet Image Transform

Multiwavelet is a new kind of wavelets and application of multiwavelet to signal processing is also a new practice these years. Perhaps this is why we hardly find the statistical data such as mean, variance and the proportion of zero valued quantized coefficients of multiwavelet transforms in literatures. In this paper, we collect five multiwavelets from literatures and Internet and make a statistical analysis of their performance in multiwavelet transform. From our analysis we can conclude that (1) After CL multiwavelet transform, the energy of an image will converge not only to the lowest resolution subimage, but further to the first component of the subimage. Therefore, CL multiwavelet is most qualified for image coding. (2) After CARDBAL multiwavelet transform, the energy of the lowest resolution subimage of an image will spread in average among its four components. Therefore, CARDBAL multiwavelet image coding has to appeal to correlative coding between the components of subimage to improve its compression ratio. (3) After GHM multiwavelet transform, the distribution of the lowest resolution subimage's energy is not concentrated on one of its components, not averaged among its components. Therefore, GHM multiwavelet is not particularly suitable to image coding, even though it is the first multiwavelet to be discovered and now widely used in applications.