Multidimensional Nonseparable Subband Coding

这项研究提供了整个多维完美重构子带编码滤波器组的参数化。这样做的动机在于图像(视频)类型信号的有效编码和压缩的实际应用。由此得到的多维滤波器族的参数描述正被用于设计具有期望的频率分离特性的完美重构滤波器组。由这些滤波器组的迭代产生的整个不可分多维光滑小波族也是以这种方式完全参数化的。所采用的技术实质上是来自多维系统理论的技术。通过将双正交问题表示为多维多项式环或稳定真有理函数环上的矩阵扩张或矩阵补全问题，以及将拟么正问题表示为描述或等价地综合多维无损系统的问题，来处理双正交问题。类似的策略也被用于非因果公式以及处理特殊类别的解决方案，例如线性相位解决方案。

This research is providing a parameterization of the entire family of multidimensional perfect reconstruction subband coding filter banks. The motivation for doing this lies in practical application of efficient coding and compression of image (video) type signals. The parametric description of the family of multidimensional filter banks so obtained is being used to design perfect reconstruction filter banks having desirable frequency separation properties. The entire family of nonseparable multidimensional smooth wavelets resulting from iterations of these filter banks is also being completely parameterized in this way. The techniques being adopted essentially consist of those from multidimensional system theory. The biorthogonal problems are being approached by formulating them as matrix extension or matrix completion problems in the ring of multidimensional polynomials or stable proper rational functions; and the paraunitary problems as problems of describing, or equivalently, synthesizing multidimensional lossless systems. Similar strategies are being used in the acausal formulation as well as in dealing with special classes of solutions such as the linear phase solutions.