The restoration of degraded images is a fundamental and challenging problem in image processing. This problem is ill-posed. The total-variation regularization and its variants are commonly used to convert to a well-posed problem. The resulting regularized model usually has a non-differentiable objective functional, which together with the large dimension of the underlying image makes the minimization theoretically and numerically difficult. Typical numerical treatments for this minimization are indirect in the sense that the methods are developed for a smoothed or dual model of the original model. With this project, the principal investigators use tools from convex analysis to find the solution of the image restoration models directly under a unified framework. The PIs address more general mathematical challenges and computational difficulties associated with the obtained fixed-point formulation. This project provides a fixed-point characterization for the solutions of models with least squares and max norm fidelity terms combined with the total variation regularization term. The study considers images corrupted by Gaussian noise, impulsive Gaussian noise and Poisson noise, which are all of relevance for different applications.
Restoring images from available data is required in a variety of applications including computer tomography; natural resources and pollution control via satellite imaging in environmental sciences; and fingerprint and face recognition in security identification. Advanced mathematical models and efficient computational algorithms for solving this problem are essential. The developed numerical schemes support improved automatic image restoration for these applications. Furthermore, interdisciplinary approaches resulting from the projects enrich upper level undergraduate and graduate curriculum development and teaching activities.
During the support period August 1 2011 to July 2014, the PIs developed mathematically sound and computational efficient algorithms for solving the non-smooth convex and non-convex optimization problems. These problems arise from applications of practical importance such as incomplete data recovery with a sparse representation and compressive sensing. Research results obtained from this project consist of two types. Results in Theoretical Development: Fixed-point Characterization of Solutions for Non-smooth Optimization Problems: A variety of optimization problems in image/signal processing are non-smooth in nature. This prevents us from directly using the optimization methods developed primarily for smooth functional and therefore, imposes algorithms challenges to develop efficient numerical algorithms for solving the non-smooth optimization problems. The PIs overcome this barrier by transferring a non-smooth optimization problem as a fixed-point problem which is formulated via the help of the proximity operator. Convergence Analysis of Fixed-point Algorithms: The mapping associated with the fixed-point equations arising from the optimization problem in image/signal processing is the composition of the proximity operator and an expansive affine transformation, and therefore is expansive. This imposes the algorithmic and mathematical challenges in finding the fixed-points of the mapping. The PIs introduced a notion of weakly firmly non-expansive mappings and established under certain conditions that the sequence generated from a weakly firmly non-expansive mapping is convergent via a multi-step iteration. Non-Convex Optimization in Sparse Optimization: Most interesting problems are non-convex. Non-convexity is recognized as a major obstacle in solving sparse optimization problems. Presently, most existing work uses the l1-norm (convex) as an approximate of the l0-norm (non-convex). The PIs use the l0-norm directly and analyzed mathematically the solutions of these problems and developed iterative schemes for finding the solutions. In the application of 1-bit compressive sensing, by exploiting the explicit form of the proximity operator and the geometric properties of the l0-norm, they proved that their proposed schemes can converge to a local minimizer of the optimization problem. Results in Mathematical Modeling and Applications: Mathematical Modeling:Solving an application problem in image/signal processing requires to identify intrinsic features embedded in the problem and then to build a model with mechanistic interpretations. Edges and textures in different orientations are important features in images. The PIs designed a discrete cosine transform based tight framelet system that can simultaneously capture the edges and textures of an image. The usefulness of this system have verified in various applications including image restoration and wavelet domain inpainting. In the 1-bit compressive sensing project, by applying the maximum a posterior a robust one-sided l0 objective model for the 1-bit compressive sensing was proposed. Accuracy of the results from the model measured in various quality metrics is significantly higher than that from existing models in the literature in their numerical experiments. Applications in Medical Images: The PIs have developed a preconditioned alternating projection algorithm (PAPA) with total variation (TV) regularizer for solving the penalized maximum likelihood optimization model for SPECT reconstruction.This algorithm belongs to a novel class of the fixed-point proximity methods. For high-noise simulated SPECT data PAPA-TV significantly outperforms the clinical EM-GPF and the conventional OSL-TV in terms of "hot" lesion detectability, noise suppression, MSE and computational efficiency. The proposed PAPA-TV might permit clinically useful reconstructions of low-dose SPECT data thus offering a promising approach to the important need of reducing radiation dose to patients in selected Nuclear Medicine studies.