Mathematical Sciences: A Common Framework for Curve Evolution and Image Segmentation

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9531293 Shah In recent years, curve evolution has developed into an important tool in Computer Vision and has been applied to a wide variety of problems such as smoothing of shapes, shape analysis and shape recovery. The underlying principle is the evolution of a simple closed curve whose points move in the direction of the normal with a prescribed velocity. A fundamental limitation of the method as it stands is that it cannot deal with important image features such as triple points. The method also requires a choice of an "edge-strength" function, defined over the image domain, indicating the likelihood of an object boundary being present at any point in the image domain. This implies a separate preprocessing step that is in essence precomputing approximate boundaries in the presence of noise. The initial curve also has to be preselected. The investigator demonstrates that the different versions of curve evolution used in Computer Vision together with the preprocessing step and the choice of the initial curve can be integrated in the form of a new segmentation functional that unifies and extends curve evolution models. Moreover, the numerical solutions obtained retain sharp discontinuities or "shocks," thus providing sharp demarcation of object boundaries. The new functional described provides a unified approach to the segmentation problem and shape analysis. The principal objective of the proposed research is to carry out a mathematical analysis of the new functional, apply it to real images, and study its possible extensions. This project is in the field of machine vision, with applications especially in medical imaging. The underlying mathematical theory proposed here is potentially applicable also to seemingly unrelated fields such as liquid crystal technology, metallurgy, flame propagation and combustion. A computer receives visual information through a camera in the form of millions of numbers per picture frame, each number describing th e intensity and the color of light at a point in the scene. The sheer volume of the visual information makes the task of its interpretation by the computer extremely difficult. The first processing task for the computer before it can begin to recognize an object in a scene is to find the object's outline. This is what is called the segmentation problem. The fundamental difficulty here is how to separate the object's outline from the rest of a cluttered scene. Even when the computer has successfully located the object outline, it is difficult to identify the object because its appearance changes depending on its orientation as the object and the camera move. Worse, the object might be partially obscured. Therefore, characteristic features that distinguish one object from another and that remain unchanged as the appearance of the object changes must be defined and calculated. This is the problem of shape analysis. This project formulates a new unified approach to the fundamental mathematical questions underlying the two problems. It develops new mathematical tools based upon this approach and applies them to specific classes of practical problems, in particular to analysis of biomedical images.