一种采用矩阵分解的正交4抽头多小波无损压缩算法

摘要

An algorithm for orthogonal 4-tap integer multiwavelet transforms is proposed. By the singular value decomposition (SVD), the transform matrix can be rewritten in a product of two block diagonal matrices and a permutation matrix, and the block matrix in block diagonal matrices is factorized into triangular elementary reversible matrices (TERMs), which map integers to integers by rounding arithmetic. The cost of factorizing block matrix into TERMs does not increase with the increasing dimension of transform matrix, and the proposed algorithm is in-place calculation without allocating auxiliary memory. The examples of integer multiwavelet transform for CL, DGHM, SA4, SA4-1, SA4-2, SA4-3 and OPTFR verify that the proposed algorithm outperforms the existing algorithm for orthogonal 4-tap integer multiwavelet transform. Compared with the Van Fleet algorithm, the weighted entropy of the proposed method is decreased by 1.9-2. 8 bits.%针对4抽头多小波整数变换算法在图像无损压缩中加权熵较大的问题,提出了一种利用矩阵奇异值分解和三角分解的4抽头正交多小波整数变换算法.首先把多小波系数矩阵组成的右循环变换矩阵分解为两个块对角阵与一个置换阵之积,并对块对角阵中的块矩阵进行基本三角分解,其次对输入图像中的每一列依次与基本三角阵相乘,并对每一次相乘的结果进行取整运算,最后在输入图像列变换结果的基础上对每一行再重复上述对图像的列运算.因为在整个变换过程中该算法实现了原位计算,所以减少了运算的存储空间和运算时间.对CL、DGHM、SA4、SA4-1、SA4-2、SA4-3、OPTFR多小波的实验结果表明:相对于4抽头多小波Van Fleet算法,该算法对图像压缩的加权熵减少了1.9～2.8 b.