Signal and Image Denoising via Wavelet Thresholding: Orthogonal and Biorthogonal, Scalar and Multiple Wavelet Transforms

In this paper we discuss wavelet thresholding in the context of scalar orthogonal, scalar biorthogonal, multiple orthogonal and multiple biorthogonal wavelet transforms. Two types of multiwavelet thresholding are considered: scalar and vector. Both of them take into account the covariance structure of the transform. The form of the universal threshold is carefully formulated. The results of numerical simulations in signal and image denoising are presented. Multiwavelets outperform scalar wavelets for three out of four noisy 1D test signals, and the Chui-Lian scaling functions and wavelets combined with repeated row preprocessing appears to be a good general method. Vector thresholding does not always outperform scalar thresholding. Multiwavelets generally outperform scalar wavelets for image denoising for all four noisy 2D test images, and the results are visually very impressive. Only for `Lenna' and `fingerprints' with signal to noise ratios of 2 do scalar wavelets perform best. As f..

Construction of Multiscaling Functions with Approximation and Symmetry

. This paper presents a new and e#cient way to create multiscaling functions with given approximation order, regularity, symmetry, and short support. Previous techniques were operating in time domain and required the solution of large systems of nonlinear equations. By switching to the frequency domain and employing the latest results of the multiwavelet theory we are able to elaborate a simple and e#cient method of construction of multiscaling functions. Our algorithm is based on a recently found factorization of the refinement mask through the two-scale similarity transform (TST). Theoretical results and new examples are presented. Key words. approximation order, symmetry, multiscaling functions, multiwavelets AMS subject classifications. 41A25, 42A38, 39B62 PII. S0036141096297182 1. Introduction. This paper discusses the construction of multiscaling functions which generate a multiresolution analysis (MRA) and lead to multiwavelets. A standard (scalar) MRA assumes that there is ..

Construction of Multiscaling Functions with Approximation and Symmetry

. This paper presents a new and efficient way to create multi-scaling functions with given approximation order, regularity, symmetry and short support. Previous techniques were operating in time domain and required the solution of large systems of nonlinear equations. By switching to the frequency domain and employing the latest results of the multiwavelet theory we are able to elaborate a simple and efficient method of construction of multi-scaling functions. Our algorithm is based on a recently found factorization of the refinement mask through the two-scale similarity transform (TST). Theoretical results and new examples are presented. Key words. approximation order, symmetry, multi-scaling functions, multiwavelets AMS subject classifications. 41A25, 42A38, 39B62 1. Introduction. This paper discusses the construction of multi-scaling functions which generate a multiresolution analysis (MRA) and lead to multiwavelets. A standard (scalar) MRA assumes that there is only one scaling..

The Application of Multiwavelet Filter Banks to Image Processing

Multiwavelets are a new addition to the body of wavelet theory. Realizable as matrix-valued filter banks leading to wavelet bases, multi-wavelets offer simultaneous orthogonality, symmetry, and short support, which is not possible with scalar 2-channel wavelet systems. After reviewing this recently developed theory, we examine the use of multiwavelets in a filter bank setting for discrete-time signal and image processing. Multiwavelets di#er from scalar wavelet systems in requiring two or more input streams to the multiwavelet filter bank. We describe two methods (repeated row and approximation /deapproximation) for obtaining such a vector input stream from a one-dimensional signal. Algorithms for symmetric extension of signals at boundaries are then developed, and naturally integrated with approximation-based preprocessing. We describe an additional algorithm for multiwavelet processing of two-dimensional signals, two rows at a time, and develop a new family of multiwavelets (the constr..