Wavelet Method for Partial Differential Equations and Image Processing
In this chapter, applications of wavelet theory to partial differential equations and image processing are discussed.
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Wavelet Method for Partial Differential Equations and Image Processing
Abstract In this chapter, applications of wavelet theory to partial differential equations and image processing are discussed. Keywords Wavelet-based Galerkin method · Parabolic problems · Viscous Burger equations · Korteweg–de Vries equation · Hilbert transform and wavelets · Error estimation using wavelet basis · Representation of signals by frames · Iterative reconstruction · Frame algorithm · Noise removal from signals · Threshold operator · Model and algorithm · Wavelet method for image compression · Linear compression · Nonlinear compression
13.1 Introduction There has been a lot of research papers on applications of wavelet methods; see, for example, [2, 14–16, 20, 30, 33, 34, 52, 55, 56, 60, 65, 70, 80, 84, 99, 128, 132, 134, 145]. Wavelet analysis and methods have been applied to diverse fields like signal and image processing, remote sensing, meteorology, computer vision, turbulence, biomedical engineering, prediction of natural calamities, stock market analysis, and numerical solution of partial differential equations. Applications of wavelet methods to partial differential equations (PDEs) and signal processing will be discussed in this chapter. We require trial spaces of very large dimension for numerical treatment of PDEs by Galerkin methods which means we have to solve large systems of equations. Wavelets provide remedy for removing obstructions in applying Galerkin methods. It has been observed that the stiffness matrix relative to wavelet bases is quite close to sparse matrix. Therefore, efficient sparse solvers can be used without loss of accuracy. These are obtained from the following results: 1. Weighted sequence norm of wavelet expansion coefficients is equivalent to Sobolev norms in a certain range, depending on the regularity of the wavelets. 2. For a large class of operators, in the wavelet basis are nearly diagonal. 3. Smooth part of a function is removed by vanishing moments of wavelets.
© Springer Nature Singapore Pte Ltd. 2018 A. H. Siddiqi, Functional Analysis and Applications, Industrial and Applied Mathematics, https://doi.org/10.1007/978-981-10-3725-2_13
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13 Wavelet Method for Partial Differential Equations and Image Processing
As discussed in Sect. 12.3.4, a signal is a function of one variable belonging to L 2 (R) and wavelet analysis is very useful for signal processing. An image is treated as a function f defined on the unit square Q = [0, 1] × [0, 1]. We shall see here that image processing is closely linked with wavelet analysis. The concept of wavelets in dimension 2 is relevant for this discussion. Let ϕ be a scaling function and ψ(x) be the corresponding mother wavelet, then the three functions ψ1 (x, y) = ψ(x)ψ(y) ψ2 (x, y) = ψ(x)ϕ(y) ψ3 (x, y) = ϕ(x)ψ(y) form, by translation and dilation, an orthonormal basis for L 2 (R 2 ); that is {2 j/2 ψm (2 j x − k1 , 2 j y − k2 )}, j ∈ Z , k = (k1 , k2 ) ∈ Z 2 m = 1, 2, 3 is an orthonormal basis for L 2 (R 2 ). Therefore, each f ∈ L 2 (R 2 ) can be
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