Differential geometry and image processing

Abstract

There are various ways of observing and interpreting images. At higher levels of abstraction, one looks at an image as collections of objects, while at the other end it is just a collection of curves and surfaces. We study this viewpoint, through tools of differential geometry. Differential Geometry is the study of geometry using calculus. We mainly concentrate on one of its applications, active contours. Active contours are primarily used for object segmentation and object boundary detection. Traditional edge detection algorithms fail to give closed contours as object boundary when the gradient along the object edge is varying, because one may not be able to set the proper threshold for the magnitude of gradient. But active contours always give closed boundary of objects. They are of three types, parametric active contours, level set active contours and geodesic active contours. We describe the Level set active contours in detail and give a brief account of the other two. Level set Active contours fail to detect the inner as well as outer boundary of objects with holes, using a single initial contour. This problem is stated in Weeratunga and Kamath. We have tried to solve this problem and are successful to some extent in doing so. By suitably modifying the level set contour evolution equation we have not only succeeded in detecting both boundaries of objects with holes, but also detected boundaries in case of objects with multiple holes and objects lying inside holes of other objects.