The investigator develops extensions to the Mumford-Shah model to deal with certain problems in image analysis for which the model is not entirely satisfactory. Despite its successful use in a variety of important areas including computer vision, image processing, biomedical imaging, cognitive perception, computational neural science, and general data mining with clustering and categorization, the classical Mumford-Shah model has gradually revealed its insufficiency in modeling and computing more general classes of image and visual signals. The approach here is to advance the Mumford-Shah model in three distinct directions of stochastic, functional, and geometric modeling and analysis. The new models can effectively and faithfully analyze and compute more complex signal patterns that demand innate randomness, oscillations, or high-order geometric regularities and are beyond the scope of the original Mumford-Shah model. The main mathematical tools essential for the project involve stochastic analysis, functional analysis, differential geometry, nonlinear partial differential equations, non-convex optimizations, and scientific computing.
A fundamental problem common in many nationally important areas involving vision and images is to determine whether some specific pattern of interest is present, be it an enemy's disguised tank, a long knife in a passenger's carry-on, or a tumor block in a brain scan. Behind the split-second decision by human intelligence is an ensemble of sophisticated (but often subconscious) and clever decision rules, balance principles, and information blocks. To reveal and emulate such remarkable human intelligence in object identification and extraction is the major goal of the segmentation problem, and it has never been more urgent than in today's information age when massive sets of image/visual data are encountered everywhere. Automation, accuracy, genericity, adaptivity, and speed are key qualities that have been driving all the research efforts in the past few decades. Based on the classic method of Mumford-Shah, by incorporating several modern mathematical tools the investigator develops newer models that are more faithful in detecting real complex patterns, more broadly applicable and flexible, and computationally more efficient and tractable. The project helps train graduate students in these crucial areas. Results are valuable for scientific, engineering, medical, and industrial applications where pattern classification and data analysis methods are widely used.
