Constraining Active Contour Evolution via Lie Groups of Transformation

We present a novel approach to constraining the evolution of active contours used in image analysis. The proposed approach constrains the final curve obtained at convergence of curve evolution to be related to the initial curve from which evolution begins through an element of a desired Lie group of plane transformations. Constraining curve evolution in such a way is important in numerous tracking applications where the contour being tracked in a certain frame is known to be related to the contour in the previous frame through a geometric transformation such as translation, rotation, or affine transformation, for example. It is also of importance in segmentation applications where the region to be segmented is known up to a geometric transformation. Our approach is based on suitably modifying the Euler-Lagrange descent equations by using the correspondence between Lie groups of plane actions and their Lie algebras of infinitesimal generators, and thereby ensures that curve evolution takes place on an orbit of the chosen transformation group while remaining a descent equation of the original functional. The main advantage of our approach is that it does not necessitate any knowledge of nor any modification to the original curve functional and is extremely straightforward to implement. Our approach therefore stands in sharp contrast to other approaches where the curve functional is modified by the addition of geometric penalty terms. We illustrate our algorithm on numerous real and synthetic examples.