正交变换在图像压缩中的应用

Orthogonal Transformation applied to Image Compression

用矩阵表示图像,构造正交均值差分变换矩阵,对原始图像进行正交变换,进一步取阈值,仅存储绝对值大于阈值的系数,获得数据压缩.解压缩过程只需作逆均值差分变换.最后将该算法分别应用于灰度和彩色图像的压缩处理,结果验证了算法的有效性.由于算法中所有变换都通过矩阵运算处理,且意义直观明了,故该算法是大学线性代数教学中一个非常好的应用案例.

A typical application of linear algebra to image compression is proposed. When the image is expressed by a matrix, the orthogonal averaging and differencing matrix is constructed first, and then the original matrix is decomposed to some coefficients. The coefficients which are bigger than the threshold are retained while the rest are replaced to zeros. As a result, the data were compressed. The inverse transformation was conveniently used for the decoding process. The decoding images show that the proposed algorithm is practical and effective. In addition, since the algorithm is easy to understand by undergraduates, it should be introduced to them as a good application case of the linear algebra.