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Completion Energies and Scale Eitan Sharon, Achi Brandt, Ronen Basri Dept. of Applied Math The Weizmann Inst. of Science Rehovot, 76100, Israel Abstract The detection of smooth curves in images and their com- pletion over gaps are two important problems in perceptual grouping. In this paper we examine the notion of com- pletion energy and introduce a fast method to compute the most likely completions in images. Specifically, we develop two novel analytic approximations to the curve of least en- ergy. In addition, we introduce a fast numerical method to compute the curve of least energy, and show that our ap- proximations are obtained at early stages of this numerical computation. We then use our newly developed energies to find the most likely completions in images through a gener- alized summation of induction fields. Since in practice edge elements are obtained by applying filters of certain widths and lengths to the image, we adjust our computation to take these parameters into account. Finally, we show that, due to the smoothness of the kernel of summation, the process of summing induction fields can be run in time that is linear in the number of different edge elements in the image, or in log where is the number of pixels in the image, using multigrid methods. 1. Introduction The smooth completion of fragmented curve segments is a skill of the human visual system that has been demon- strated throughmany compellingexamples. Due to this skill people often are able to perceive the boundaries of objects evenin the lack of sufficient contrast or in the presence of oc- clusions. A numberof computational studies have addressed the problem of curve completion in an attempt to both pro- vide a computational theory of the problem and as part of a process of extracting the smooth curves from images. These studies commonly obtain two or more edge elements (also referred to as edgels ) and find either the most likely comple- tions that connect the elements or the smoothest curves trav- Research supported in part by Israel Ministry of Science Grant 4135- 1-93 and by the Gauss Minerva Center for Scientific Computation. Research supported in part by the Unites States-Israel Binational Sci- ence Foundation, Grant No. 94-00100. eling through them. The methods proposed for this problem generally require massive computations, and their results strongly depend on the energy function used to evaluate the curves in the image. It is therefore important to develop methods which simplify the computation involved in these methods while providing results competitive with the exist- ing approaches. Below we present such a method that di- rectly relates to a numberof recent studies of completionand curve salience [9, 18, 5, 19, 12, 7] (see also [2, 6, 8, 14, 15]).
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