Patrick J. Van Fleet

Multiwavelets and Integer Transforms

In many applications of image processing, the given data are integer-valued. It is therefore desirable to construct transformations that map data of this type to an integer (or rational) ring. Calderbank, Daubechies, Sweldens, and Yeo [1] devised two methods for modifying orthogonal and biorthogonal wavelets so that they map integers to integers. The first method involves appropriately scaling the transform so that data that has been transformed and truncated can be recovered via the inverse wavelet transform. In developing this method, the authors of [1] created a useful factorization of the 4-tap Daubechies orthogonal wavelet transform [2]. We have observed that this factorization can be extended to 4-tap multiwavelets of arbitrary size. In this paper we will discuss this generalization and illustrate the factorization on two multiwavelets. In particular, the well-known Donovan, Geronimo, Hardin, and Massopust (DGHM) [3] multiwavelet transform can be scaled so that it maps integers to integers. Since this transform is (anti)symmetric in addition to orthogonal, regular, and compactly supported, the ability to modify it so that it maps integers to integers should be useful in image processing applications.In many applications of image processing, the given data are integer-valued. It is therefore desirable to construct transformations that map data of this type to an integer (or rational) ring. Calderbank, Daubechies, Sweldens, and Yeo [1] devised two methods for modifying orthogonal and biorthogonal wavelets so that they map integers to integers. The first method involves appropriately scaling the transform so that data that has been transformed and truncated can be recovered via the inverse wavelet transform. In developing this method, the authors of [1] created a useful factorization of the 4-tap Daubechies orthogonal wavelet transform [2]. We have observed that this factorization can be extended to 4-tap multiwavelets of arbitrary size. In this paper we will discuss this generalization and illustrate the factorization on two multiwavelets. In particular, the well-known Donovan, Geronimo, Hardin, and Massopust (DGHM) [3] multiwavelet transform can be scaled so that it maps integers to integers. Since this transform is (anti)symmetric in addition to orthogonal, regular, and compactly supported, the ability to modify it so that it maps integers to integers should be useful in image processing applications.

Some recurrence formulas for box splines and cone splines

A degree elevation formula for multivariate simplex splines was given by Micchelli[6] and extended to hold for multivariate Dirichlet splines in [8]. We report similar formulae for multivariate cone splines and box splines. To this end, we utilize a relation due to Dahmen and Micchelli[4] that connects box splines and cone splines and a degree reduction formula given by Cohen, Lyche, and Riesenfeld in [2].

Nonnegative Scaling Vectors on the Interval

In this paper, we outline a method for constructing nonnegative scaling vectors on the interval. Scaling vectors for the interval have been constructed in [1–3]. The approach here is different in that the we start with an existing scaling vector ϕ that generates a multi-resolution analysis for L2(R) to create a scaling vector for the interval. If desired, the scaling vector can be constructed so that its components are nonnegative. Our construction uses ideas from [4,5] and we give results for scaling vectors satisfying certain support and continuity properties. These results also show that less edge functions are required to build multi-resolution analyses for L2 ([a; b]) than the methods described in [5,6].