From the inequality |P(z)|2 + |P(-z)|2 ≤1, assuming that both of the low-pass filters and high-pass filters are unknown, we design compactly supported wavelet tight frames. The unknowing of low-pass filters allows the design more freedom, and both the low-pass filters and high-pass filters have symmetries or anti-symmetries. We give the algorithm for filters with odd and even lengths separately, some concrete examples of wavelet tight frames with the length 4, 5, 6, 7, and at last we give the result of decomposing Lena image with them.
The concept of paraunitary two-scale similarity transform （PTST） is introduced. We discuss the property of PTST, and prove that PTST preserves the orthogonal, approximation order and smoothness of the given orthogonal multiscaling functions. What is more, by applying PTST, we present an algorithm of constructing high order balanced multiscaling functions by balancing the already existing orthogonal nonbalanced multiscaling functions. The corresponding transform matrix is given explicitly. In addition, we also investigate the symmetry of the balanced multiscaling functions. Finally, construction examples are given.
