1 Implementation of the Convex Edge Segment

Abstract | This report describes the implementation of convex line segment grouper proposed by David W. Jacobs 9]. The inputs of this algorithm are line segments obtained by edge detector and straight line nder. De nition of mutually convex lines(MCL) given by Daniel P. Huttenlocher and Peter C. Wayer 10] was adopted. Mathematical expressions of these de nitions will be give. All convex groups satisfying the length-gap ratio and the angle constraints will be found. Keywords | Perceptual grouping, convexity, proximity, non-accidental properties.
n n i i i i; i ;
The third condition was not included in Huttenlocher and Wayner's de nition. We added it to make the de ne compatible with Jacob's idea. It will prevent two distant lines from being considered as convex.
G2
ln,1 L1 l1,1 l1,2 ln,2 Gn
l2,1
Fig. 1. Convex group. The thick lines represent the lines in a group. The thin lines show the gaps between them.
The inputs of the system are line segments obtained by running a Canny edge detector 11] along with a straight line detector. Each line corresponds to two \oriented" lines, in which two endpoints are ordered. As shown in Figure 1, if l is an oriented line, then l 1 is its rst endpoint , and l 2 is its second. Therefore, n line segments in the image correspond to 2n oriented line segments. The output is convex groups and convex cycles de ned by Jacob.
C. Convex group and convex cycle We have described the properties a convex group or a convex cycle should have. Now, we will formally de ne a convex group in a recursive style, and then de ne a convex cycle. 1. S1 , i.e., any single line segment, is a convex group. 2. Line segment l +1 is added to convex group S only if (a) l +1 62 S (i.e., no line may appear more than once in a group). (b) l +1 is closely convex to l , which is the last line of sequence S . (c) l1 , which is the rst line to sequence S , is convex to l +1 . (d) Let S +1 = S fl +1 g. Then, the length-gap ratio of S +1 should be less than threshold k. That is, G < 1 +1(1? ) ? G1 ?1 , which is equivalent to 1 +1 stating that 1 +1 + 1 > k. (e) In S +1 , the sum of the absolute values of the angles turned is 2 when one traverses from l1 1 through all lines in S and then l +1 2 , returning nally to l1 1 . De nition 5 A convex group S +1 is called a convex cycle if
i i; i;
2
De nition 1 A set of oriented line segments is call a convex groups if for each oriented line segment, all the other line segments are on the same side of the oriented line segment as its normal, where the normal is de ned as lying to the right when we traverse the line segment from the rst endpoint to the second. Let S be a convex line segment group: (l1 ; l2 ; ; l ). De ne L to be the length of l , and G to be the gap between l 2 and l +1 1 . Let
II. Description of the algorithm
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I. Introduction
ERCEPTUAL grouping is an important method for abstracting meaningful organizations from observed visual data. It clusters low level image features to higher level geometric primitives and further to models of objects. The theoretical basis of perceptual grouping can be traced back to Gestalt psychologists' work 1] 2] 3]. They stated that it's not the separate parts of the image but their relative relationships that dramatically a ect human perception. Further explanations of Gestalt laws of perception and its applications in computer vision were provided individually by Witkin and Tenenbaum 4], Lowe 5], and Rock 6], known as the principle of non-accidentalness, which assert that certain causal organization almost certainly exists when a structure, whose relative likelihood is extremely small in randomly generated scenes, is observed, even if the prior likelihood of existence of this organization is very small. For this reason, it is very useful to look in the image for structures, especially complex structures, because they, if appear, are very likely to hint signi cant organizations. A lot of local geometric properties such as collinearity, proximity, parallelism 7], and more complex ones, such as continuous segments, parallels, and closed gures 8], have been used in a variety of recognition systems. In Jocob's paper, convexity, which frequently occurs in objects, was considered. Convexity are more distinctive than individual features, and are more readily matched between images, and, therefore, are more useful in various vision tasks such as solving point correspondence problem in stereo or motion. It is alos noticed that convexity preserves from di erent views. Jocob's convex grouper is guaranteed to nd all convex sets of line segments which satisfy convexity and length-