Geometric Flow Approach for Region-Based Image Segmentation

Geometric flows have been successfully used for surface modeling and designing, largely because they are inherently good at controlling geometric shape evolution. Variational image segmentation approaches, on the other hand, detect objects of interest by deforming certain given shapes. This motivates us to revisit the minimal partition problem for segmentation of images, and propose a new geometric flow-based formulation and solution to it. Our model intends to derive a mapping that will evolve given contours or piecewise-constant regions toward objects in the image. The mapping is approximated by B-spline basis functions, and the positions of the control points are to be determined. Starting with the energy functional based on intensity averaging, we derive a Euler-Lagrange equation and then a geometric evolution equation. The linearized system of equations is efficiently solved via a special matrix-vector multiplication technique. Furthermore, we extend the piecewise-constant model to a piecewise-smooth model which effectively handles images with intensity inhomogeneity.