Edge-Weighted Centroidal Voronoi Tessellation Based Algorithms for Image Segmentation

Centroidal Voronoi tessellations (CVTs) are special Voronoi tessellations whose generators are also the centers of mass (centroids) of the Voronoi regions with respect to a given density function. CVT-based algorithms have been proved very useful in the context of image processing. However when dealing with the image segmentation problems, classic CVT algorithms are sensitive to noise. In order to overcome this limitation, we develop an edge-weighted centroidal Voronoi Tessellation (EWCVT) model by introducing a new energy term related to the boundary length which is called "edge energy". The incorporation of the edge energy is equivalent to add certain form of compactness constraint in the physical space. With this compactness constraint, we can effectively control the smoothness of the clusters' boundaries. We will provide some numerical examples to demonstrate the effectiveness, efficiency, flexibility and robustness of EWCVT. Because of its simplicity and flexibility, we can easily embed other mechanisms with EWCVT to tackle more sophisticated problems. Two models based on EWCVT are developed and discussed. The first one is "local variation and edge-weighted centroidal Voronoi Tessellation" (LVEWCVT) model by encoding the information of local variation of colors. For the classic CVTs or its generalizations (like EWCVT), pixels inside a cluster share the same centroid. Therefore the set of centroids can be viewed as a piecewise constant function over the computational domain. And the resulting segmentation have to be roughly the same with respect to the corresponding centroids. Inspired by this observation, we propose to calculate the centroids for each pixel separately and locally. This scheme greatly improves the algorithms' tolerance of within-cluster feature variations. By extensive numerical examples and quantitative evaluations, we demonstrate the excellent performance of LVEWCVT method compared with several state-of-art algorithms. LVEWCVT model is especially suitable for detection of inhomogeneous targets with distinct color distributions and textures. Based on EWCVT, we build another model for "Super-pixels" which is in fact a "regularization" of highly inhomogeneous images. We call our algorithm for super-pixels as "VCells" which is the abbreviation of "Voronoi cells". For a wide range of images, VCells is capable to generate roughly uniform sub-regions and meanwhile nicely preserves local image boundaries. The under-segmentation error is effectively limited in a controllable manner. Moreover, VCells is very efficient. The computational cost is roughly linear in image size with small constant coefficient. For megapixel sized images, VCells is able to generate very dense superpixels in a matter of seconds. We demonstrate that VCells outperforms several state-of-art algorithms through extensive qualitative and quantitative results on a wide range of complex images. Another important contribution of this work is the "Detecting-Segment-Breaking" (DSB) algorithm which can be used to guarantee the spatial connectedness of resulting segments generated by CVT based algorithms. Since the metric is usually defined on the color space, the resulting segments by CVT based algorithms are not necessarily spatially connected. For some applications, this feature is useful and conceptually meaningful, e.g., the foreground objects are not spatially connected. But for some other applications, like the superpixel problem, this "good" feature becomes unacceptable. By simple "extracting-connected-component" and "relabeling" schemes, DSB successfully overcomes the above difficulty. Moreover, the computational cost of DSB is roughly linear in image size with a small constant coefficient. From the theoretical perspective, the innovative idea of EWCVT greatly enriches the methodology of CVTs. (The idea of EWCVT has already been used for variational curve smoothing and reconstruction problems.) For applications, this work shows the great power of EWCVT for image segmentation related problems.