Dept. of Electr. and Comput. Eng., Rice Univ., Houston, TX.
IEEE Trans Image Process. 1995;4(2):162-76. doi: 10.1109/83.342190.
Multiplicity M, K-regular, orthonormal wavelet bases (that have implications in transform coding applications) have previously been constructed by several authors. The paper describes and parameterizes the cosine-modulated class of multiplicity M wavelet tight frames (WTFs). In these WTFs, the scaling function uniquely determines the wavelets. This is in contrast to the general multiplicity M case, where one has to, for any given application, design the scaling function and the wavelets. Several design techniques for the design of K regular cosine-modulated WTFs are described and their relative merits discussed. Wavelets in K-regular WTFs may or may not be smooth, Since coding applications use WTFs with short length scaling and wavelet vectors (since long filters produce ringing artifacts, which is undesirable in, say, image coding), many smooth designs of K regular WTFs of short lengths are presented. In some cases, analytical formulas for the scaling and wavelet vectors are also given. In many applications, smoothness of the wavelets is more important than K regularity. The authors define smoothness of filter banks and WTFs using the concept of total variation and give several useful designs based on this smoothness criterion. Optimal design of cosine-modulated WTFs for signal representation is also described. All WTFs constructed in the paper are orthonormal bases.
多进制、K-正则、正交小波基（在变换编码应用中有重要意义）已经被多位作者构造出来。本文描述并参数化了余弦调制类的多进制小波紧框架（WTF）。在这些 WTF 中，尺度函数唯一地确定了小波。这与一般的多进制情况不同，在一般情况下，对于任何给定的应用，都需要设计尺度函数和小波。本文描述了几种设计 K-正则余弦调制 WTF 的技术，并讨论了它们的相对优点。K-正则 WTF 中的小波可能是也可能不是光滑的，因为编码应用使用具有短尺度和小波向量的 WTF（因为长滤波器会产生振铃伪影，这在图像编码等应用中是不希望出现的），所以本文提出了许多短长度的 K 正则 WTF 的平滑设计。在某些情况下，还给出了尺度和小波向量的解析公式。在许多应用中，小波的平滑度比 K 正则性更重要。作者使用总变差的概念定义了滤波器组和 WTF 的平滑度，并根据该平滑度标准给出了几个有用的设计。还描述了用于信号表示的余弦调制 WTF 的最优设计。本文构造的所有 WTF 都是正交基。
