Completion Energies and Scale.

Dept. of Comput. Sci. & Appl. Math., Weizmann Inst. of Sci., Rehovot, Israel
IEEE Transactions on Pattern Analysis and Machine Intelligence (Impact Factor: 5.78). 11/2000; 22(10):1117-1131. DOI: 10.1109/34.879792
Source: DBLP


Abstract The detection of smooth curves in images and their completion over gaps are two important problems,in perceptual grouping. In this paper we examine,the notion of completion 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 energy. In addition, we introduce a fast numerical method to compute the curve of least energy, and show that our approximations 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 generalized 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.

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Available from: Ronen Basri, Sep 26, 2013
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    • "Following results in Psychophysics [39], the curvature has been introduced into various fields of computer vision. It was first seen in the context of shape completion through the celebrated Euler's elastica energy L(C) 0 |κ C (s)| 2 ds, see [57], [35], [48] and the subsequent developments in [25], [41], [32], [17]. Other applications of curvature are image segmentation [40], [3], [2], [51], [12], inpainting [4], [15], [45], [28], [30], image smoothing and denoising (see [56] and references in [11]), image analysis [52], [26] or surface interpolation and smoothing [36], [55], [24], [6], [29]. "
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    ABSTRACT: We present the first ratio-based image segmentation method that allows imposing curvature regularity of the region boundary. Our approach is a generalization of the ratio framework pioneered by Jermyn and Ishikawa so as to allow penalty functions that take into account the local curvature of the curve. The key idea is to cast the segmentation problem as one of finding cyclic paths of minimal ratio in a graph where each graph node represents a line segment. Among ratios whose discrete counterparts can be globally minimized with our approach, we focus in particular on the elastic ratio [Formula: see text] that depends, given an image I, on the oriented boundary C of the segmented region candidate. Minimizing this ratio amounts to finding a curve, neither small nor too curvy, through which the brightness flux is maximal. We prove the existence of minimizers for this criterion among continuous curves with mild regularity assumptions. We also prove that the discrete minimizers provided by our graph-based algorithm converge, as the resolution increases, to continuous minimizers. In contrast to most existing segmentation methods with computable and meaningful, i.e., nondegenerate, global optima, the proposed approach is fully unsupervised in the sense that it does not require any kind of user input such as seed nodes. Numerical experiments demonstrate that curvature regularity allows substantial improvement of the quality of segmentations. Furthermore, our results allow drawing conclusions about global optima of a parameterization-independent version of the snakes functional: the proposed algorithm allows determining parameter values where the functional has a meaningful solution and simultaneously provides the corresponding global solution.
    IEEE Transactions on Image Processing 02/2011; 20(9):2565-81. DOI:10.1109/TIP.2011.2118225 · 3.63 Impact Factor
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    • "We now turn to obtain the curves of shortest admissible path between two endpoints in T (I), as defined in Problem 2. At this point we limit the discussion to retinal curves α(·) which are functions over some coordinate system in the image plane, i.e., α(x) = [x, y(x)], as assumed in many previous studies of the curve completion (e.g.,[10] [18]). Consequently, all admissible curves in T (I) are 'lifted' functions of the form β(x) = [x, y(x), θ(x)] , where x becomes the curve parameter (as demonstrated in Figs. 1 and 2). "
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    ABSTRACT: The phenomenon of visual curve completion, where the visual system completes the missing part (e.g., due to occlusion) between two contour fragments, is a major problem in perceptual organization research. Previous computational approaches for the shape of the completed curve typically follow formal descriptions of desired, image-based perceptual properties (e.g, minimum total curvature, roundedness, etc.). Unfortunately, however, it is difficult to determine such desired properties psychophysically and indeed there is no consensus in the literature for what they should be. Instead, in this paper we suggest to exploit the fact that curve completion occurs in early vision in order to formalize the problem in a space that explicitly abstracts the primary visual cortex. We first argue that a suitable abstraction is the unit tangent bundle R<sup>2</sup> × S<sup>1</sup> and then we show that a basic principle of “minimum energy consumption” in this space, namely a minimum length completion, entails desired perceptual properties for the completion in the image plane. We present formal theoretical analysis and numerical solution methods, we show results on natural images and their advantage over existing popular approaches, and we discuss how our theory explains recent findings from the perceptual literature using basic principles only.
    Computer Vision and Pattern Recognition (CVPR), 2010 IEEE Conference on; 07/2010
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    • "In terms of geometry, we consider features derived from the euclidian distance between u and v, the corresponding segments' centers, the square of these two distances and min and max lengths of the segments. High level geometric features are considered as well, like the Elastica defined in [15], computed between extremities or between segments centers, and multiplied or not by the corresponding distances and square distances in order to introduce scale-invariant and scale-dependent curvature constraints. The photometric features are also considered to re introduce low-level information in the process: min and max mean intensity along the segments and terms insuring likely links: the means of the output of the detection along cubic splines linking extremities or centers of the segments (and tangents to segments) -multiplied or not by corresponding distances and square distances to take the length of the cubic curve into account or not. "
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    ABSTRACT: Fluroscopic imaging provides means to assess the motion of the internal structures and therefore is of great use during surgery. In this paper we propose a novel approach for the segmentation of curvilinear structures in these images. The main challenge to be addressed is the lack of visual support due to the low SNR where traditional edge-based methods fail. Our approach combines machine learning techniques, unsupervised clustering and linear programming. In particular, numerous invariant to position/rotation classifiers are combined to detect candidate pixels of curvilinear structure. These candidates are grouped into consistent geometric segments through the use of a state-of-the art unsupervised clustering algorithm. The complete curvilinear structure is obtained through an ordering of these segments using the elastica model in a linear programming framework. Very promising results were obtained on guide wire segmentation in fluoroscopic images.
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