Showing papers on "Multiresolution analysis published in 1991"

Using the refinement equations for the construction of Pre-Wavelets II: powers and two

Abstract: We study basic questions of wavelet decompositions associated with multiresolution analysis. A rather complete analysis of multiresolution associated with the solution of a refinement equation is presented. The notion of extensibility of a finite set of Laurent polynomials is shown to be central in the construction of wavelets by decomposition of spaces. Two examples of extensibility, first over the torus and then in complex space minus the coordinate axes are discussed. In each case we are led to a decomposition of the fine space in a multiresolution analysis as a sum of the adjacent coarse space plus an additional space spanned by the multiinteger translates of a finite number of pre-wavelets. Several examples are provided throughout to illustrate the general theory.

Using the refinement equation for the construction of pre-wavelets

Abstract: A variety of methods have been proposed for the construction of wavelets. Among others, notable contributions have been made by Battle, Daubechies, Lemarie, Mallat, Meyer, and Stromberg. This effort has led to the attractive mathematical setting of multiresolution analysis as the most appropriate framework for wavelet construction. The full power of multiresolution analysis led Daubechies to the construction ofcompactly supported orthonormal wavelets with arbitrarily high smoothness. On the other hand, at first sight, it seems some of the other proposed methods are tied to special constructions using cardinal spline functions of Schoenberg. Specifically, we mention that Battle raises some doubt that his block spin method “can produce only the Lemarie Ondelettes”. A major point of this paper is to extend the idea of Battle to the generality of multiresolution analysis setup and address the easier job of constructingpre-wavelets from multiresolution.

Multiresolution properties of the wavelet Galerkin operator

Abstract: This paper extends recent results by the author [J. Math. Phys. 31, 1898 (1990), J. Math. Phys. 32, 57 (1991)] that show a scaling parameter sequence h yields an orthonormal wavelet basis for L2(R) if and only if an associated operator Sh has eigenvalue 1 with multiplicity 1. The operator transforms a sequence a by Sh(a)(k)=2Σm,n∼(h(m))h(n)a(2k+m−n). A correspondence is derived between Sh and Galerkin projection operators related to the multiresolution analysis defined by the orthonormal wavelet basis. The spectrum of Sh is characterized in terms of the Fourier modulus of the (unique) scaling function φ that satisfies φ(x)=2Σnh(n)φ(2x−n). This characterization yields several results including a direct, alternate proof that the eigenvalue 1 of Sh has multiplicity 1.

Lossless systems in wavelet transforms

Abstract: A brief review of wavelet transforms and their relation to paraunitary filter banks is given. Proofs of these relations are presented, using standard multirate signal processing notations. >

Multiresolution analysis of remotely sensed imagery

Abstract: A multiresolution method for analysing remotely sensed images is described, in which correlation filters, based on two-point differences, are applied over a range of scales with octave separation. When applied to typical Earth backgrounds viewed from space, the measured probability distributions of filter outputs exhibit strongly non-Gaussian statistics and satisfy scaling laws which allow a representation of the imagery in terms of fractal geometry. The method may be used as a basis for image, or image-region, characterization and, using the tails of the resulting normalised distributions, for the identification of those localized image features which are most unusual; that is, have lowest relative probability.

Discrete Wavelet Transforms: The Relationship of the a Trous and Mallat Algorithms

Abstract: : In a general sense this paper represents an effort to clarify the relationship of discrete and continuous wavelet transforms. More narrowly, it focuses on bringing together two separately motivated implementations of the wavelet transform, the algorithm a trous and Mallat's multiresolution decomposition. It is observed that these algorithms are both special cases of a single filter bank structure, the discrete wavelet transform, the behavior of which is governed by one's choice of filters. In fact, the a trous algorithm, originally devised as a computationally efficient implementation, is more properly viewed as a nonorthogonal multiresolution algorithm for which the discrete wavelet transform is exact. A systemative framework for the discrete wavelet transform is provided, and conditions are derived under which it computes the continuous wavelet transform exactly.

Image Segmentation Using Affine Wavelets

Abstract: : This thesis discusses the use of the multiresolution representation and Radial Basis Function (RBF) neural networks to segment both FLIR and SAR imagery. The multiresolution approximation coefficients are used as features into the RBF network which learns to distinguish between different cultural and natural regions or objects. The wavelets used are Mallat's spline wavelet and Daubechies' compactly supported wavelets. Additionally, this thesis provides an explanation of wavelets in a tutorial manner. It introduces wavelet theory and discusses two different approaches to generating the multiresolution or wavelet representation.

Noisy Image Restoration Using Multiresolution Analysis And Markov Random Fields

Abstract: Noisy image restoration is a necessary stage in many image processing applications. In this paper, we will deal with the case of grey level images corrupted by additive gaussian white noise. The restoration methods derived from the one introduced by Geman and Geman [l], which uses Markov random fields with line processes, are known to give a high level of performance, but are also costly in terms of computation due to the non-convexity of the criteria. Amongst the relaxation methods, the deterministic GNC method (Graduated Non-Convexity) [2] gives goods results, and is quicker than the stochastic methods. Unfortunately, the volume of computation remains high. The purpose of this paper is to present a new method in which the image is decomposed at different resolutions and the restoration is carried out on each resulting image. The noisy image is first decomposed by 2D wavelet transform. The wavelets used are either separable biorthogonal [3], that is the decomposition base is different from the reconstruction base (the two bases being dual), or non separable, associated with a staggered subsampling of the transformed images, which leads to an isotropic decomposition. At a given resolution, the energy of the texture image is modelled so as to preserve contours. To achieve this, the energy is depending on a Boolean field referred to as the line process [l], which represents the presence or absence of an edge, and on an intensity process represented by a Markovian field. The minimisation of this energy gives the smoothed image (intensity process) and the image edges (line process). For the next step, two different methods have been developed: The first involves synthesizing an image at the finer resolution, by using the restored texture image and the noisy wavclet coefficients. A new restoration is carried out on this image, then the algorithm is iterated. In comparison to the GNC method [2] on the original image, the computation time can be reduced by factor 20. The second method uses the non-separable isotropic wavelet transform. The restoration is performed on the texture image as well as on the wavelet coefficients, on which the energy parameters are adapted (in particular the smoothing factor). The restored image is obtained by the synthesis of the restored images (texture + coefficients). Results, using these different techniques, are shown for some real images to which a 4.4dB synthetic white noise was added. These results are compared, from visual and computation time points of view, with a classical GNC restoration on the original image.