## Nonlinear solution of linear inverse problems by wavelet-vaguelette decomposition (1992)

Citations: | 189 - 12 self |

### BibTeX

@MISC{Donoho92nonlinearsolution,

author = {David L. Donoho},

title = {Nonlinear solution of linear inverse problems by wavelet-vaguelette decomposition},

year = {1992}

}

### Years of Citing Articles

### OpenURL

### Abstract

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 asnumerical di erentiation, 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 coe cients of the noisy, indirect data. Our approach o ers signi cant 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 Bp;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.

### Citations

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Citation Context ...f wavelets to represent functions, and many important advantages of wavelet bases have been discovered, particularly as regards sparse representation of objects. There are many possible wavelet bases =-=[12, 35, 36]-=-. We start with bases of L2 (IR). Using the construction of Daubechies (1988), we obtain a function of compact support, having M vanishing moments and M continuous derivatives and unit norm. This func... |

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62 |
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Citation Context ...nt SV D derivations [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]-=-. Finally, there are optimality results for SVD inversion; we quote an example. Suppose that the singular system of K admits a differential operator D m of m-th order, with the esas eigenfunctions. Le... |

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