熊刚强, 余建德, 熊昌镇, 齐东旭. U-正交变换的可逆实现及其图像无损编码[J]. 计算机研究与发展, 2012, 49(4): 856-863.
 引用本文: 熊刚强, 余建德, 熊昌镇, 齐东旭. U-正交变换的可逆实现及其图像无损编码[J]. 计算机研究与发展, 2012, 49(4): 856-863.
Xiong Gangqiang, Yu Jiande, Xiong Changzhen, Qi Dongxu. Reversible Factorization of U Orthogonal Transform and Image Lossless Coding[J]. Journal of Computer Research and Development, 2012, 49(4): 856-863.
 Citation: Xiong Gangqiang, Yu Jiande, Xiong Changzhen, Qi Dongxu. Reversible Factorization of U Orthogonal Transform and Image Lossless Coding[J]. Journal of Computer Research and Development, 2012, 49(4): 856-863.

## Reversible Factorization of U Orthogonal Transform and Image Lossless Coding

• 摘要: 把U-正交变换应用到图像无损编码中,研究U-正交矩阵的基本三角可逆矩阵(TERM)分解与单行基本可逆矩阵(SERM)分解.一个N阶U-正交矩阵的TERM分解由N－1个自由变量决定,用区间收缩方法可以搜索到TERM分解的局部近似最优解.如果用行交换方法搜索正交矩阵的SERM分解,那么一个8阶的正交矩阵最多只有40320种可能的SERM分解,用穷举法即能找到SERM的近似最优分解.最后,用U-正交矩阵的可逆分解对图像进行无损编码,实验表明可逆U-正交变换的无损编码的码率与浮点U-正交变换的近似无损编码的码率基本相同,SERM分解要比TERM分解更有效,三次U-正交变换的编码效果与离散余弦变换的编码效果几乎完全相同.因此,在图像无损编码中，可用三次U-正交变换代替DCT.

Abstract: U orthogonal transform is applied into the image lossless coding, and the factorizations of U orthogonal matrices into triangular elementary reversible matrices (TERMs) and single-row elementary reversible matrices (SERMs) are investigated. The TERM factorization of an N by N matrix is determined by N-1 free variables, and therefore, the local approximate optimal TERM factorization can be found by shrinking search-interval of the N-1 free variables. If row exchange is used, an 8×8 orthogonal matrix has only 40320 forms of SERM factorizations, and the approximate optimal SERM factorization can be found with the exhaustion search algorithm. At the end, image lossless coding is achieved by using reversible U matrices, and the experimental results show that the code-rate of lossless compression based on reversible U transform is comparable to that of near lossless compression based on float U orthogonal transform; the coding efficiency of SERM factorization outperforms that of TERM; the image coding performance of U orthogonal transform of degree 3 is approximate to that of DCT. As a result, the U orthogonal transformation of degree 3 can be used into the image lossless coding instead of DCT.

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