为了更有效地进行图像编码，先用U-正交函数系构造出一类新型的U-正交变换, 并以三次U-正交变换为例，研究了基于U-正交变换的图像编码算法。该编码算法首先通过离散U-正交函数系的基函数构造U-正交变换的变换矩阵,并根据U-正交矩阵的对称性给出了U-正交变换的快速算法; 然后应用三次U-正交变换对图像实施2维变换, 再用JPEG标准中的量化矩阵、Huffman码表与熵编码方法对图像的三次U-变换系数进行量化与编码, 实现了基于三次U-正交变换的图像编码算法。实验结果表明, 三次U-正交变换的编码增益、去相关效率与DCT基本相同,而编码效果却与JPEG编码效果非常接近, 且计算复杂度与基于FFT的快速DCT算法基本一致。由此可见, 应用U-正交变换对图像进行编码压缩是一类行之有效的方法,并有望在视频编码中得到应用。

Discrete cosine transform (DCT) has been applied extensively to the area of image compressing; in order to improve image encoding, this paper introduces a class of orthogonal complete piecewise k-degree polynomials in L²［0,1］ (so-called U-system). Firstly, a class of new U-orthogonal transform is constructed using U-orthogonal basis, and an algorithm of image coding based on U-orthogonal transform is presented by investigating 3-degree U-orthogonal transform (so-called U3). Secondly, two methods of calculating discrete U-orthogonal transform matrices are established, and the fast U-transform is derived from symmetrical characteristic of U-transform matrices. Thirdly, coding gain and de-correlation efficiency of U3 are studied, and then JPEG algorithm is realized using U3 instead of DCT. The experiments show coding gain and de-correlation efficiency of U3 are close to that of DCT, and the computational complexity of U-transform is approximate to that of DCT which computed using fast Fourier transform algorithm. Moreover the effect of reconstructed image from our scheme is comparable to that of decoded image from baseline JEPG. So it is effective to apply U-transform, which may be used extensively in the application of video coding, to the field of image compression.