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Sparse Modeling and the Resolution of Inverse Problems in Biomedical Imaging

M. Unser

Keynote, Twelfth IEEE International Symposium on Biomedical Imaging: From Nano to Macro (ISBI'15), Brooklyn NY, USA, April 16-19, 2015.



Sparsity is a powerful paradigm for introducing prior constraints on signals in order to address ill-posed image reconstruction problems.

In this talk, we first present a continuous-domain statistical framework that supports the paradigm. We consider stochastic processes that are solutions of non-Gaussian stochastic differential equations driven by white Lévy noise. We show that this yields intrinsically sparse signals in the sense that they admit a concise representation in a matched wavelet basis.

We apply our formalism to the discretization of ill-conditioned linear inverse problems where both the statistical and physical measurement models are projected onto a linear reconstruction space. This leads to the specification of a general class of maximum a posteriori (MAP) signal estimators complemented with a practical iterative reconstruction scheme. While our family of estimators includes the traditional methods of Tikhonov and total-variation (TV) regularization as particular cases, it opens the door to a much broader class of potential functions that are inherently sparse and typically nonconvex. We apply our framework to the reconstruction of images in a variety of modalities including MRI, phase-contrast tomography, cryo-electron tomography, and deconvolution microscopy.

Finally, we investigate the possibility of specifying signal estimators that are optimal in the MSE sense. There, we consider the simpler denoising problem and present a direct solution for first-order processes based on message passing that serves as our gold standard. We also point out some of the pitfalls of the MAP paradigm (in the non-Gaussian setting) and indicate future directions of research.


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