The topological contour closure requirement for object extraction from 2D-digital images

Object contour extraction from given 2D-digital images is an essential processing step for object extraction, recognition and analysis. The exact execution of this contour extraction, which is known in the literature as a segmentation problem is extremely important as subsequent steps or algorithms applied are highly dependent on this object contour extraction step. 2D-objects have always closed contours. Scanning the exhaustive number of publications on object contour extraction will disclose that this basic topological property has not been taken into consideration. It has not even been explicitly required from segmentation algorithms. The presently published image processing methods for object contour extraction possess the severe disadvantage that the extracted contours are not necessarily closed. They exhibit discontinuous contours or bifurcated segments. This is also the reason for the obligation to use consecutive contour restoration algorithms requiring intensive computation time. The paper sets the central condition on object contour extraction methods to be topologically preserving. Thus, extracted contours of 2D-objects should necessarily be closed without discontinuous contours or bifurcation. By applying graph theoretical methods, it can be shown that preserving the object contour closure requirements implies that the resulting extracted contour graph is a directional Euler line. Contour segmentation algorithms or inadequate reduction of image resolution that lead to contour graphs other than the directed Euler line, should therefore not be acceptable. In the paper, the necessary and sufficient conditions that should be satisfied by an algorithm to produce only closed contours are given. The global requirement of contour closure is reduced to a set of local conditions for contour extractions. With these conditions one can examine whether a segmentation algorithm will produce closed contours. Further, an algorithm that satisfies these conditions is introduced and practical segmentation results with guaranteed object contour closure from 2 D-object images are shown