Open Access Publications from the University of California

## Nonconvex Sparse Recovery Methods

Critical to accurate reconstruction of sparse signals from low-dimensional observations is the solution of nonlinear optimization problems that promote sparse solutions. Sparse signal recovery is a common problem of many different applications ranging from photography to tomography and from radiology to biology. Within the compressive imaging community, minimizing the $\ell_1$-norm or the total variation (TV) seminorm penalized least-squares problem is the most conventional approach for sparse signal recovery. The least-squares data-fidelity term assumes a Gaussian noise model. However, there are many real-world applications that do not follow Gaussian noise statistics. For an instance, when the number of observed photon counts is relatively low at the camera detector, they are corrupted by Poisson noise. This phenomenon can be seen in a variety of different applications including astronomy, night vision, and medical imaging. Therefore, the contribution of the dissertation to sparse signal recovery is two-fold. First, we propose several nonconvex algorithms operate on Poisson statistics to promote sparsity. Second, we will present methods based on trust-region and alternating minimization techniques for sparse signal recovery under Gaussian statistics.
While convex optimization for low-light imaging has received some attention by the imaging community, non-convex optimization techniques for photon-limited imaging are still in their nascent stages. Theoretically, the non-convex $\ell_p$-norm regularization $(0 \leq p < 1)$ would lead to more accurate reconstruction than the convex $\ell_1$-norm relaxation. In this dissertation, we explore sparse Poisson intensity reconstruction methods using gradient based optimization approach with the nonconvex regularization techniques: $\ell_p$-norm, TV$_p$-seminorm, and the \textit{generalized} Shannon entropy. The proposed methods lead to more accurate and high strength reconstructions in medical imaging and computational genomics. In particular, we developed a stage-based nonconvex approach to solve time dependent bioluminescence and fluorescence lifetime imaging problems in the Poisson noise context.
In Gaussian noise context, we solve the $\ell_2$-$\ell_1$ and $\ell_2$-$\ell_p$ sparse recovery problems by transforming the objective function into an unconstrained differentiable function and apply a limited-memory trust-region method. Unlike gradient projection-type methods, which uses only the current gradient, our approach uses gradients from previous iterations to obtain a more accurate Hessian approximation. Numerical experiments with simulated compressive sensing 1D and 2D data are provided to illustrate that our proposed approach eliminates spurious solutions more effectively while improving the computational time to converge in comparison to standard approaches. Moreover, we employ nonconvex $\ell_p$-norm regularization for better recovery and demixing of sparse signals arise in image inpainting and source separation applications.