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Nonlinear Solution of Linear Inverse Problems by Wavelet–Vaguelette Decomposition

David L. Donoho

1995
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Applied and Computational Harmonic Analysis
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We describe the Wavelet-Vaguelette Decomposition (WVD) of a linear inverse problem. It is a substitute for the singular value decomposition (SVD) of an inverse problem, and it exists for a class of special inverse problems of homogeneous type { such a s n umerical dierentiation, inversion of Abel-type transforms, certain convolution transforms, and the Radon Transform. We propose to solve ill-posed linear inverse problems by nonlinearly \shrinking" the WVD coecients of the noisy, indirect data.
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... Our approach oers signicant advantages over traditional SVD inversion in the case of recovering spatially inhomogeneous objects. We suppose that observations are contaminated by white noise and that the object is an unknown element of a Besov space. We prove that nonlinear WVD shrinkage can be tuned to attain the minimax rate of convergence, for L 2 loss, over the entire Besov scale. The important case of Besov spaces B p;q , p < 2, which model spatial inhomogeneity, is included. In comparison, linear procedures { SVD included { cannot attain optimal rates of convergence over such classes in the case p < 2. For example, our methods achieve faster rates of convergence, for objects known to lie in the Bump Algebra or in Bounded Variation, than any linear procedure. 1 of unpublished manuscripts. singular system of band and time limiting operators, and of diraction-limiting operators; Spherical Harmonics weighted by special radial functions supply the singular functions for problems of whole-earth geomagnetic inversion Shure, Parker, and Backus (1982); Gegenbauer polynomials supply singular functions in problems of tomography (Davisson, 1982). Tchebyche polynomials weighted by appropriate radial functions supply singular functions in problems of limited angle tomography (Louis, 1986). Jacobi Polynomials and Tchebyshev polynomials supply the (e ), (h ) sequences for singular value decomposition of the Abel transform (Johnstone and Silverman, 1991). Bertero and coworkers Pike, De Mol, Boccaci, and others have published a series of elegant SVDderivations [4, 5, 7, 8] for operators arising in microscopy, such as timelimited Laplace Transform and the Poisson transform. See also Gori and Guattari for applications in signal processing [27].

doi:10.1006/acha.1995.1008
fatcat:g3xyd23uyzg5jk4oit7v34oxxa