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.

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...

In the last decade a number of perfect reconstruction filter bank structures have been proposed [11]. Of these the modulated filter banks are the easiest to design and implement [7, 6, 4]. However, the filters in a modulated filter bank cannot be linear phase. Recently, unitary FIR filter banks with linear-phase have been completely parameterized [10]. In that paper, the eigenstructure of the exchange matrix played an important role. In this correspondence we give a new characterization of the polyphase matrix of a filter bank with filters of various types of symmetry. Using matrix extensions of the well-known hyperbolic and orthogonal lattices, we give a alternative proof for the parameterization of linear-phase unitary filter banks. Moreover, a complete parameterization of FIR unitary filter banks with each of the different types of symmetries considered (not just linearphase) is also given. We also mention how the above results can be used to generate non-unitary filter banks with symmetries, though no completeness results are obtained. In some cases implicit, and in others explicit parameterization of wavelet tight frames associated with these filter banks are also given. In this paper, we only consider the case of M even. For odd M a similar theory can be developed.

Wavelet and short-time Fourier analysis is introduced in the context of frequency decompositions. Wavelet type frequency decompositions are associated with filter banks, and using this fact, filter bank theory is used to construct multiplicity M wavelet frames and tight frames. The way in which filter banks lead to decomposition and recomposition of arbitrary separable Hilbert spaces is also described. Efficient computational structures for both filter banks and wavelets are also discussed.

This paper derives relationships between the moments of the scaling function /0(t) associated with multiplicity M , K-regular, compactly supported, orthonormal wavelet bases [6, 5], that are extensions of the multiplicity 2, K-regular orthonormal wavelet bases constructed by Daubechies [2]. One such relationship is that the square of the first moment of the scaling function (/0(t)) is equal to its second moment. This relationship is used to show that uniform sample values of a function provides a third order approximation of its scaling function expansion coefficients. For the special case of M = 2, the results in this paper have been reported earlier [3]. 1. INTRODUCTION In this paper we derive relationships between the moments of the scaling function /0(t) associated with the compactly supported, multiplicity M , K regular, orthonormal wavelet bases. In particular, we show that the square of the first moment of /0 is the second moment of /0 . Hence samples of a function accurately ...

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