Efficient Global Minimization Methods for Sparsity and Regularity Promoting Optimization Problems in Imaging and Vision, 2014

The main goals of “Efficient Global Minimization Methods for Sparsity and Regularity Promoting Optimization Problems in Imaging and Vision, 2014” are design and analysis of regularity/sparsity promoting energy minimization models in image processing, computer vision and compressed sensing, and design and analysis of efficient numerical methods that can produce global or nearly global solutions of the resulting optimization problems. Imaging and vision are some of the core emerging technologies that are shaping our society. This can partly be explained by progressively better physical sensing devices for acquiring image data. Another important reason is the development of image processing and computer vision software for restoring, analyzing, simplifying and interpreting the image information. Energy minimization has been established as one of the most important paradigms for formulating problems in image processing and computer vision in a mathematical language. The problems can elegantly be formulated as finding the minimal state of some energy potential, which typically encodes the underlying assumption that the image data is regular/sparse, i.e. values at different image pixels are correlated. More recently, it has been realized that the expensive acquisition process can be greatly improved by incorporating the same assumption in an energy minimization framework, a field known as compressed sensing. A major challenge is to solve the resulting optimization problems efficiently. The most desirable models are often the most difficult to handle from an optimization perspective. Continuous optimization problems may be non-convex and contain many inferior local minima. Discrete optimization problems may be NP-hard, which means an algorithm is unlikely to exist which can always compute an exact solution without an unreasonable amount of effort. Even though the underlying problems are NP-hard, computational methods will be constructed that produce global solutions for practical input data, and otherwise close approximations.

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