. We present the lifting scheme, a simple construction of second generation wavelets, wavelets that are not necessarily translates and dilates of one fixed function. Such wavelets can be adapted to intervals, domains, surfaces, weights, and irregular samples. We show how the lifting scheme leads to a faster, in-place calculation of the wavelet transform. Several examples are included. Key words. wavelet, multiresolution, second generation wavelet, lifting scheme AMS subject classifications. 42C15 1. Introduction. Wavelets form a versatile tool for representing general functions or data sets. Essentially we can think of them as data building blocks. Their fundamental property is that they allow for representations which are efficient and which can be computed fast. In other words, wavelets are capable of quickly capturing the essence of a data set with only a small set of coefficients. This is based on the fact that most data sets have correlation both in time (or space) and frequenc...

This paper constructs K-regular M-band orthonormal wavelet bases. K-regularity of the wavelet basis is known to be useful in numerical analysis applications and in image coding using wavelet techniques. Several characterizations of K-regularity and their importance are described. An explicit formula is obtained for all minimal length M-band scaling filters. A new state-space approach to constructing the wavelet filters from the scaling filters is also described. When M-band wavelets are constructed from unitary filter banks they give rise to wavelet tight frames in general (not orthonormal bases). Conditions on the scaling filter so that the wavelet bases obtained from it is orthonormal is also described.

Abstract. This note is a very basic introduction to wavelets. It starts with an orthogonal basis of piecewise constant functions, constructed by dilation and translation. The “wavelet transform ” maps each f(x) to its coefficients with respect to this basis. The mathematics is simple and the transform is fast (faster than the Fast Fourier Transform, which we briefly explain), but approximation by piecewise constants is poor. To improve this first wavelet, we are led to dilation equations and their unusual solutions. Higher-order wavelets are constructed, and it is surprisingly quick to compute with them — always indirectly and recursively. We comment informally on the contest between these transforms in signal processing, especially for video and image compression (including high-definition television). So far the Fourier Transform — or its 8 by 8 windowed version, the Discrete Cosine Transform — is often chosen. But wavelets are already competitive, and they are ahead for fingerprints. We present a sample of this developing theory. 1. The Haar wavelet To explain wavelets we start with an example. It has every property we hope for, except one. If that one defect is accepted, the construction is simple and the computations are fast. By trying to remove the defect, we are led to dilation equations and recursively defined functions and a small world of fascinating new problems — many still unsolved. A sensible person would stop after the first wavelet, but fortunately mathematics goes on. The basic example is easier to draw than to describe: Figure 1. Scaling function φ(x), wavelet W(x), and the next level of detail.

The wavelet representation using orthonormal wavelet bases has received widespread attention. Recently M-band orthonormal wavelet bases have been constructed and compactly supported M-band wavelets have been parameterized [15, 12, 32, 17]. This paper gives the theory and algorithms for obtaining the optimal wavelet multiresolution analysis for the representation of a given signal at a predetermined scale in a variety of error norms [23]. Moreover, for classes of signals, this paper gives the theory and algorithms for designing the robust wavelet multiresolution analysis that minimizes the worst case approximation error among all signals in the class. All results are derived for the general M-band multiresolution analysis. An efficient numerical scheme is also described for the design of the optimal wavelet multiresolution analysis when the least-squared error criterion is used. Wavelet theory introduces the concept of scale which is analogous to the concept of frequency in Fourier ana...

this article we explore the notion of the multiresolution operator, its spectral theory, and applications. This operator (also called the transition operator) has appeared as a fundamental tool in several aspects of wavelet theory, such as the Lawton-Cohen theorem on wavelet orthonormal bases [2, 18, 19], and the work of Eirola [8] and others [1, 3, 4, 5, 23] on the Sobolev smoothness of wavelet scaling functions. After introducing the multiresolution operator and observing its connection with the convolution and downsampling operations of multirate signal processing, we present a review of the work of Lawton and that of Eirola. Throughout the paper we work in the setting of rank m wavelet systems (m is the integer dilation factor, not necessarily 2). This involves some generalization of previous work, and yields initial results on the differentiability of wavelet scaling functions for rank m ? 2. In particular, we find that the minimal support rank 3 scaling functions with

In this paper, we consider the asymptotic regularity of Daubechies scaling functions and construct examples of M-band scaling functions which are both orthonormal and cardinal for M 3. Keywords Multiresolution, M-band scaling function, M-band wavelets, cardinal function, orthonormality. 1 The author is partially supported by the National Science Foundation of China and the Zhejiang Provincial Science Foundation of China. 1. Introduction Let M 2 be a fixed positive integer. A family of closed subspaces V j ; j 2 Z; of L 2 , the space of all square integrable functions on the real line, is said to be a multiresolution of L 2 if the following conditions hold: (i) V j ae V j+1 , and f 2 V j if and only if f(M \Delta) 2 V j+1 for all j 2 Z; (ii) [ j2Z V j is dense in L 2 and " j2Z V j = ;; (iii) There exists a function OE in V 0 such that fOE(\Delta \Gamma k); k 2 Zg is a Riesz basis of V 0 . Here we say that fOE(\Delta \Gamma k); k 2 Zg is a Riesz basis of V 0 if there exis...

Symmetric quadrature mirror filters (QMF's) offer several advantages for wavelet-based image coding. Symmetry and odd-length contribute to efficient boundary handling and preservation of edge detail. Symmetric QMF's can be obtained by mildly relaxing the filter bank orthogonality conditions. We describe a computational algorithm for these filter banks which is also symmetric in the sense that the analysis and synthesis operations have identical implementations, up to a delay. The essence of a wavelet transform is its multiresolution decomposition, obtained by iterating the lowpass filter. This allows one to introduce a new design criterion, smoothness (good behavior) of the lowpass filter under iteration. This design constraint can be expressed solely in terms of the lowpass filter tap values (via the eigenvalue decomposition of a certain finite-dimensional matrix). Our innovation is to design near-orthogonal QMF's with linear-phase symmetry which are optimized for smoothness under ite...

. It is known that paraunitary matrices can be factorized into shift products of orthogonal matrices or linear factors. When number of rows of such a matrix (i. e. the number of channels of a paraunitary filter bank) is even, the symmetry constraints corresponding to linear phase property of the filter bank can be expressed as restrictions on factors---except the very first one, all must be centrosymmetric. For odd number of rows the situation is more complicated. It turns out that paraunitary matrices comprising of an even number of square blocks do not exist and quadratic centrosymmetric factors have to be used in the 0-shift product factorization. The centrosymmetric linear and quadratic factors can be easily obtained from partitions of centrosymmetric orthogonal matrices. Their parameterizations are also described. The characterizations of paraunitary matrices obtained from these factorizations are complete; the question of number of free parameters is discussed. Furthermore, the p...

This tutorial reviews the theory, structure and design methods for M-channel perfect-reconstruction (PR) filter banks. Some of the filter banks being considered here are the two-channel orthogonal filter bank

. This paper explores the Sobolev regularity of rank M wavelets and refinement schemes. We find that the regularity of orthogonal wavelets with maximal vanishing moments grows at most logarithmically with filter length when M is odd, but linearly for even M . When M = 3 and M = 4, we show that the regularity does achieve these upper bounds for asymptotic growth, complementing earlier results for M = 2. A new class of wavelet filters is introduced, by asserting zeros of the wavelet symbol at preperiodic points of the mapping ø : ! !M! mod 2ß. While this class includes the generalized Daubechies wavelets, numerical experiments demonstrate that the class also includes wavelet families with greater smoothness for a given filter length. Finally, members of the class of wavelets that have maximal Sobolev regularity for a given filter length are found as the solution to an optimization problem. Key words. wavelets, refinement equations, Sobolev regularity, smoothness, filter design AMS subj...