Biomedical Imaging GroupSTI
English only   BIG > Publications > Sparse Modeling

 Home Page
 News & Events
 Tutorials and Reviews
 Download Algorithms

 PDF not available
 PS not available
 All BibTeX References

Sparse Modeling and the Resolution of Inverse Problems in Biomedical Imaging

M. Unser

Keynote address, 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.

AUTHOR="Unser, M.",
TITLE="Sparse Modeling and the Resolution of Inverse Problems in
        Biomedical Imaging",
BOOKTITLE="Twelfth {IEEE} International Symposium on Biomedical Imaging:
        From Nano to Macro ({ISBI'15})",
address="Brooklyn NY, USA",
month="April 16-19,",
note="Keynote address")

© 2015 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from IEEE.
This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder.