Image Segmentation Based on the Poincaré Map Method

Active contour models (ACMs) integrated with various kinds of external force fields to pull the contours to the exact boundaries have shown their powerful abilities in object segmentation. However, local minimum problems still exist within these models, particularly the vector field's “equilibrium issues.” Different from traditional ACMs, within this paper, the task of object segmentation is achieved in a novel manner by the Poincaré map method in a defined vector field in view of dynamical systems. An interpolated swirling and attracting flow (ISAF) vector field is first generated for the observed image. Then, the states on the limit cycles of the ISAF are located by the convergence of Newton-Raphson sequences on the given Poincaré sections. Meanwhile, the periods of limit cycles are determined. Consequently, the objects' boundaries are represented by integral equations with the corresponding converged states and periods. Experiments and comparisons with some traditional external force field methods are done to exhibit the superiority of the proposed method in cases of complex concave boundary segmentation, multiple-object segmentation, and initialization flexibility. In addition, it is more computationally efficient than traditional ACMs by solving the problem in some lower dimensional subspace without using level-set methods.