In this paper, we propose a segmentation algorithm which combines the ideas from local watershed transforms and the region based deformable models. Traditionally, watersheds are computed in the whole image and then some region merging techniques are applied on them to reach the segmentation of structures. We propose that watershed regions can be used as operators in region-based deformable models. These regions are computed only when the deformable models reach them. Then, they are added to (or subtracted from) the deformable models via a measure computed from two terms: (i) statistical fit of regions to the models, region competition; (ii) smoothness of such fits, smoothness constraint. The proposed algorithm is computationally efficient because it operates on regions instead of pixels. In addition, this algorithm allows better boundary localization due to the edge information brought by watersheds. Moreover, the proposed algorithm can handle topological changes, e.g., split or merge, during the evolutions without an additional embedded surface as in the case of level set formulation. Furthermore, structure-based smoothness of segmented objects is obtained by using the smoothness term computed from the alignment of regions. We illustrate the efficiency and accuracy of the proposed technique on several medical data such as MRA and CTA data.

Variations in the projection of objects on a 2D image, e.g., due to occlusion and articulation, lead to edge maps which are noisy, contain gaps and spurious elements, and which are deformed. These variations in turn cause variations in the edge map which are typically regularized by the use of a salient measure for each edge element. The use of edge salience, however, typically faced with two drawback. First, salience measures take advantage of boundary continuity, but not of shape continuity, which includes continuity of the interior. Second, while each edge element can only belong to one object boundary, in the computation of salience measures, it often freely contributes to the salience of edges in competing grouping hypotheses as well. We identify both drawbacks with the lack of an explicit intermediate representation between the edge map and grouped object boundaries. We propose that (i) a symmetry map can fully represent the initial edge map so that both boundary and regional continuities can be represented via skeletal/shock continuity; (ii) a re-organization of the edge map in the form of completing gaps, discarding spurious elements, smoothing, and partitioning a contour (grouped set of edge elements) can be represented by transformations on the symmetry map; (iii) the optimal grouping corresponds to the least action path consisting of a sequence of symmetry transforms. The focus of this paper is to define transformations on the symmetry map and illustrate results for them. Specifically, we illustrate how spurious elements can be removed, gaps completed, and parts computed despite significant noise.

This paper presents an approach for computing the symmetries (skeletons) of an edge map consisting of a collection of curve segments. This approach is a combination of analytic computations in the style of computational geometry and discrete propagations on a grid in the style of the numerical solutions of PDE's. Specifically, waves from each of the initial curve segments are initialized and propagated as a discrete wavefront along discrete directions. In addition, to avoid error built up due to the discrete nature of propagation, shockwaves are detected and explicitly propagated along a secondary dynamic grid. The propagation of shockwaves, integrated with the propagation of the wavefront along discrete directions, leads to an exact simulation of propagation by the Eikonal equation. The resulting symmetries are simply the collection of shockwaves formed in this process which can be manipulated locally, exactly, and efficiently under local changes in an edge map (gap completion, removal of spurious elements, etc.). The ability to express grouping operations in the language of symmetry maps makes it an appropriate intermediate representation between low-level edge maps and high level object hypotheses.

The extraction of figure symmetry from image contours faces a number of fundamental difficulties: object symmetries are distorted due to (i) gaps in the bounding contour of a shape due to figure-ground blending, weak contrast edges, highlights, noise, etc.; (ii) an introduction of parts and occluders, and (iii) spurious edge elements due to surface markings, texture, etc. A framework for extracting such symmetries from real images is proposed based on the propagation of contour orientation information and the detection of four types of singularities (shocks) arising from the collision of propagating elements. In this paper, we show that an additional labeling of shocks based on whether the colliding wavefronts carry true orientation information (regular vs. rarefaction waves) allows a division of shocks into three sets: regular shocks are the partial shocks of partial contours as they remain invariant to the completion of the contour; semi-degenerate and degenerate shocks depict potential parts and gaps. Finally, shocks altered due to spurious edges, occlusion, and gaps are recovered via a simulation of inter-penetrating waves generated at select shock groups which with the aid of the above shock labels leads to second and further generations of shocks.

Figure-ground segmentation is a fundamental problem in computer vision. The main difficulty is the integration of low-level, pixel-based local image features to obtain global object-based descriptions. Active contours in the form of snakes, balloons, and level-set modeling techniques have been proposed that satisfactorily address this question for certain applications. However, these methods require manual initialization, do not always perform well near sharp protrusions or indentations, or often cross gaps. We propose an approach inspired by these methods and a shock-based representation of shape in terms of parts, protrusions, and bends. Since initially it is not clear where the objects or their parts are, parts are hypothesized in the form of fourth order shocks randomly initialized in homogeneous areas of images. These shocks then form evolving contours, or bubbles, which grow, shrink, merge, split and disappear to capture the objects in the image. In the homogeneous areas of the image bubbles deform by a reaction-diffusion process. In the inhomogeneous areas, indicated by differential properties computed from low-level processes such as edge-detection, texture, optical-flow and stereo, etc., bubbles do not deform. As such, the randomly initialized bubbles integrate low-level information and, in the process, segment the figures from the ground.>