Generalized stochastic processes as prior models for compressed sensing and sparse signal recovery
Event details
| Date | 30.03.2012 |
| Hour | 10:15 › 11:15 |
| Speaker | Michael Unser |
| Location | |
| Category | Conferences - Seminars |
We present a general, non-Gaussian statistical framework for the reconstruction of signals from a limited set of noisy linear measurements and compressed sensing. The underlying signals are continuously-defined and modeled as sparse stochastic processes (SSP).
Part I: Theory. SSP are generalized stochastic processes (in the sense Gelfand and Vilenkin) that are solutions of (potentially unstable) linear stochastic differential equations. They are described by a general innovation model that is specified by: 1) a whitening operator L, which shapes their Fourier spectrum, and 2) a Lévy exponent f, which controls the sparsity of the (non-Gaussian) innovations (white Lévy noise). We give sufficient conditions on (L,f) for these processes to be welldefined and derive their characteristic form which provides a complete statistical description. We also substantiate the claim that the observations of these processes are intrinsically sparse (with heavy tailed statistics).
Part II: Application. Using those results, we derive an extended family of MAP estimators that are directly applicable to biomedical image reconstruction. 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 (associated with infinitely divisible priors) that are inherently sparse and typically nonconvex. By introducing a discrete counterpart of the
innovation variables, we are able to develop an alternating minimization scheme that can handle arbitrary potential functions. We apply our framework to the reconstruction of magnetic resonance images and phase-contrast tomograms; we also present simulation examples where the proposed scheme outperforms the more traditional convex optimization techniques (in particular, TV).
Part I: Theory. SSP are generalized stochastic processes (in the sense Gelfand and Vilenkin) that are solutions of (potentially unstable) linear stochastic differential equations. They are described by a general innovation model that is specified by: 1) a whitening operator L, which shapes their Fourier spectrum, and 2) a Lévy exponent f, which controls the sparsity of the (non-Gaussian) innovations (white Lévy noise). We give sufficient conditions on (L,f) for these processes to be welldefined and derive their characteristic form which provides a complete statistical description. We also substantiate the claim that the observations of these processes are intrinsically sparse (with heavy tailed statistics).
Part II: Application. Using those results, we derive an extended family of MAP estimators that are directly applicable to biomedical image reconstruction. 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 (associated with infinitely divisible priors) that are inherently sparse and typically nonconvex. By introducing a discrete counterpart of the
innovation variables, we are able to develop an alternating minimization scheme that can handle arbitrary potential functions. We apply our framework to the reconstruction of magnetic resonance images and phase-contrast tomograms; we also present simulation examples where the proposed scheme outperforms the more traditional convex optimization techniques (in particular, TV).
Practical information
- General public
- Free
Organizer
- CIB
Contact
- Isabelle Derivaz-Rabii