Fast Reverse Jacket Transform Based on a Simple Sparse Matrix Decomposition

This paper presents a simple decomposition of the Reverse Jacket [RJ] matrix which is used to develop a fast algorithm for the [RJ]N transform. The matrix decomposition is of the form of the matrix products of Hadamard matrices and successively lower order coefficient matrices. This decomposition leads very clearly to a block circulant sparse matrix factorization of the [RJ]N matrix. The main property of [RJ]N is that the inverse matrices of its elements can be obtained very easily and have a special structure. [RJ]N is derived using the weighted Hadamard Transform corresponding to the Hadamard matrix [H]N and a basic symmetric matrixΛ. Each element of [RJ]N is a generalized Quincunx and polygonal subsampling matrix. In this paper we represent in particular the systematical block-wise sparse matrix of extending method for [RJ]N .