Multiresolution analysis

About: Multiresolution analysis is a research topic. Over the lifetime, 4032 publications have been published within this topic receiving 140743 citations. The topic is also known as: Multiresolution analysis, MRA.

Ripples in Mathematics: The Discrete Wavelet Transform

Abstract: 1. Introduction.- 1.1 Prerequisites.- 1.2 Guide to the Book.- 1.3 Background Information.- 2. A First Example.- 2.1 The Example.- 2.2 Generalizations.- Exercises.- 3. The Discrete Wavelet Transform via Lifting.- 3.1 The First Example Again.- 3.2 Definition of Lifting.- 3.3 A Second Example.- 3.4 Lifting in General.- 3.5 DWT in General.- 3.6 Further Examples.- Exercises.- 4. Analysis of Synthetic Signals.- 4.1 The Haar Transform.- 4.2 The CDF(2,2) Transform.- Exercises.- 5. Interpretation.- 5.1 The First Example.- 5.2 Further Results on the Haar Transform.- 5.3 Interpretation of General DWT.- Exercises.- 6. Two Dimensional Transforms.- 6.1 One Scale DWT in Two Dimensions.- 6.2 Interpretation and Examples.- 6.3 A 2D Transform Based on Lifting.- Exercises.- 7. Lifting and Filters I.- 7.1 Fourier Series and the z-Transform.- 7.2 Lifting in the z-Transform Representation.- 7.3 Two Channel Filter Banks.- 7.4 Orthonormal and Biorthogonal Bases.- 7.5 Two Channel Filter Banks in the Time Domain.- 7.6 Summary of Results on Lifting and Filters.- 7.7 Properties of Orthogonal Filters.- 7.8 Some Examples.- Exercises.- 8. Wavelet Packets.- 8.1 From Wavelets to Wavelet Packets.- 8.2 Choice of Basis.- 8.3 Cost Functions.- Exercises.- 9. The Time-Frequency Plane.- 9.1 Sampling and Frequency Contents.- 9.2 Definition of the Time-Frequency Plane.- 9.3 Wavelet Packets and Frequency Contents.- 9.4 More about Time-Frequency Planes.- 9.5 More Fourier Analysis. The Spectrogram.- Exercises.- 10. Finite Signals.- 10.1 The Extent of the Boundary Problem.- 10.2 DWT in Matrix Form.- 10.3 Gram-Schmidt Boundary Filters.- 10.4 Periodization.- 10.5 Moment Preserving Boundary Filters.- Exercises.- 11. Implementation.- 11.1 Introduction to Software.- 11.2 Implementing the Haar Transform Through Lifting.- 11.3 Implementing the DWT Through Lifting.- 11.4 The Real Time Method.- 11.5 Filter Bank Implementation.- 11.6 Construction of Boundary Filters.- 11.7 Wavelet Packet Decomposition.- 11.8 Wavelet Packet Bases.- 11.9 Cost Functions.- Exercises.- 12. Lifting and Filters II.- 12.1 The Three Basic Representations.- 12.2 From Matrix to Equation Form.- 12.3 From Equation to Filter Form.- 12.4 From Filters to Lifting Steps.- 12.5 Factoring Daubechies 4 into Lifting Steps.- 12.6 Factorizing Coiflet 12 into Lifting Steps.- Exercises.- 13. Wavelets in Matlab.- 13.1 Multiresolution Analysis.- 13.2 Frequency Properties of the Wavelet Transform.- 13.3 Wavelet Packets Used for Denoising.- 13.4 Best Basis Algorithm.- 13.5 Some Commands in Uvi_Wave.- Exercises.- 14. Applications and Outlook.- 14.1 Applications.- 14.2 Outlook.- 14.3 Some Web Sites.- References.

A Generalized Local Binary Pattern Operator for Multiresolution Gray Scale and Rotation Invariant Texture Classification

Abstract: This paper presents generalizations to the gray scale and rotation invariant texture classification method based on local binary patterns that we have recently introduced. We derive a generalized presentation that allows for realizing a gray scale and rotation invariant LBP operator for any quantization of the angular space and for any spatial resolution, and present a method for combining multiple operators for multiresolution analysis. The proposed approach is very robust in terms of gray scale variations, since the operator is by definition invariant against any monotonic transformation of the gray scale. Another advantage is computational simplicity, as the operator can be realized with a few operations in a small neighborhood and a lookup table. Excellent experimental results obtained in a true problem of rotation invariance, where the classifier is trained at one particular rotation angle and tested with samples from other rotation angles, demonstrate that good discrimination can be achieved with the occurrence statistics of simple rotation invariant local binary patterns. These operators characterize the spatial configuration of local image texture and the performance can be further improved by combining them with rotation invariant variance measures that characterize the contrast of local image texture. The joint distributions of these orthogonal measures are shown to be very powerful tools for rotation invariant texture analysis.

A sampling theorem for wavelet subspaces

Abstract: The classical Shannon sampling theorem is extended to the subspaces used in the multiresolution analysis in wavelet theory. Under weak hypotheses, these subspaces are first shown to have a Riesz basis formed from the reproducing kernels. These in turn are used to construct the sampling sequences. Examples are given. >

Multiresolution representation of data: a general framework

Abstract: In this paper we present a general framework for a multiresolution representation of data which is obtained by the discretization of mappings. This framework, which can be viewed as a generalization of the theory of wavelets, includes discretization corresponding to unstructured grids in several space dimensions, and thus is general enough to enable us to embed most numerical problems in a multiresolution setting. Furthermore, this framework allows for nonlinear (data-dependent) multiresolution representation schemes and thus enables us to design adaptive data-compression algorithms.In this paper we also study the stability of linear schemes for a multiresolution representation and derive sufficient conditions for existence of a multiresolution basis.

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.