Algebraic Approaches for Constructing Multi-D Wavelets

Wavelets have been a powerful tool in data representation and had a growing impact on various signal processing applications. As multi-dimensional (multi-D) wavelets are needed in multi-D data representation, the construction methods of multi-D wavelets are of great interest. Tensor product has been the most prevailing method in multi-D wavelet construction, however, there are many limitations of tensor product that make it insufficient in some cases. In this dissertation, we provide three non-tensor-based methods to construct multi-D wavelets. The first method is an alternative to tensor product, called coset sum, to construct multi-D wavelets from a pair of $1$-D biorthogonal refinement masks. Coset sum shares many important features of tensor product. It is associated with fast algorithms, which in certain cases, are faster than the tensor product fast algorithms. Moreover, it shows great potentials in image processing applications. The second method is a generalization of coset sum to non-dyadic dilation cases. In particular, we deal with the situations when the dilation matrix is \dil=p{\tt I}_\dm, where $p$ is a prime number and {\tt I}_\dm is the \dm-D identity matrix, thus we call it the prime coset sum method. Prime coset sum inherits many advantages from coset sum including that it is also associated with fast algorithms. The third method is a relatively more general recipe to construct multi-D wavelets. Different from the first two methods, we attempt to solve the wavelet construction problem as a matrix equation problem. By employing the Quillen-Suslin Theorem in Algebraic Geometry, we are able to build \dm-D wavelets from a single \dm-D refinement mask. This method is more general in the sense that it works for any dilation matrix and does not assume additional constraints on the refinement masks. This dissertation also includes one appendix on the topic of constructing directional wavelet filter banks

ON POLYNOMIAL SYMBOLS FOR SUBDIVISION SCHEMES

Abstract. Given a dilation matrix A: Z d → Z d, and G a complete set of coset representatives of 2π(A − ⊤ Z d /Z d), we consider polynomial solutions M to the equation � g∈G M(ξ + g) = 1 with the constraints that M ≥ 0 and M(0) = 1. We prove that the full class of such functions can be generated using polynomial convolution kernels. Trigonometric polynomials of this type play an important role as symbols for interpolatory subdivision schemes. For isotropic dilation matrices, we use the method introduced to construct symbols for interpolatory subdivision schemes satisfying Strang-Fix conditions of arbitrary order. 1