Multidimensional Nonseparable Subband Coding

本研究提供了一种多维完美重构子带编码滤波器组的参数化方法。这样做的动机在于对图像（视频）类型信号进行高效编码和压缩的实际应用。由此得到的多维滤波器组族的参数化描述可用于设计具有理想频分离性能的重构滤波器组。由这些滤波器组的迭代产生的整个不可分离的多维光滑小波族也被完全参数化。所采用的技术基本上由多维系统理论中的技术组成。将双正交问题表述为多维多项式环或稳定的固有有理函数中的矩阵扩展或矩阵补全问题；而准酉问题是描述或等价地，综合多维无损系统的问题。类似的策略也被用于因果公式以及处理特殊类型的解，如线性相位解。

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.