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...se boundary has been drawn in black with a pencil on a white sheet is perceived by connectedness (the boundary is a black line), constant width (of the stroke), convexity and closure (of the black pencil stroke), parallelism (between opposite sides), orthogonality (between adjacent sides), finally, equidistance (of both pairs of opposite sides). The square is a global gestalt, and the result of concurring geometric qualities that we shall call partial gestalts. Many Computer Vision methods attempt to compute the (very diverse in nature) partial gestalts. To take an instance, the snakes method [10] attempts to capture the closed smooth curves, a combination of the “closure” and “good continuation” gestalts. Some more recent works try to define grouping rules for combining local information into organized contours [8], [13]. In [2], we have treated alignments (straight edges) and in [3] general boundaries and edges. In this paper, we treat four more examples of partial gestalts, namely, the object alignments, the clusters, and quality grouping by color, orientation or size. In [1], a vanishing point detector is treated by a clever extension of the same method. As a first evidence of the ...

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... (between opposite sides), orthogonality (between adjacent sides), finally, equidistance (of both pairs of opposite sides). The square is a global gestalt, and the result of concurring geometric qualities that we shall call partial gestalts. Many Computer Vision methods attempt to compute the (very diverse in nature) partial gestalts. To take an instance, the snakes method [10] attempts to capture the closed smooth curves, a combination of the “closure” and “good continuation” gestalts. Some more recent works try to define grouping rules for combining local information into organized contours [8], [13]. In [2], we have treated alignments (straight edges) and in [3] general boundaries and edges. In this paper, we treat four more examples of partial gestalts, namely, the object alignments, the clusters, and quality grouping by color, orientation or size. In [1], a vanishing point detector is treated by a clever extension of the same method. As a first evidence of the recursive character of gestalt laws, we shall push one of the experiments to prove that the partial gestalt recursive building up can be led up to the third level (gestalts built by three successive partial gestalt grouping...

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...ween opposite sides), orthogonality (between adjacent sides), finally, equidistance (of both pairs of opposite sides). The square is a global gestalt, and the result of concurring geometric qualities that we shall call partial gestalts. Many Computer Vision methods attempt to compute the (very diverse in nature) partial gestalts. To take an instance, the snakes method [10] attempts to capture the closed smooth curves, a combination of the “closure” and “good continuation” gestalts. Some more recent works try to define grouping rules for combining local information into organized contours [8], [13]. In [2], we have treated alignments (straight edges) and in [3] general boundaries and edges. In this paper, we treat four more examples of partial gestalts, namely, the object alignments, the clusters, and quality grouping by color, orientation or size. In [1], a vanishing point detector is treated by a clever extension of the same method. As a first evidence of the recursive character of gestalt laws, we shall push one of the experiments to prove that the partial gestalt recursive building up can be led up to the third level (gestalts built by three successive partial gestalt grouping princ...

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...When two testing methods are available, perception must obviously choose the one giving the smaller test number. Indeed, there is perceptual evidence that grouping processes may depend on density, and that different methods could be relevant for dense and for sparse patterns [17]. Hence, the second testing method we present here should be preferred for sparse distributions of points, whereas the initial model based on density would give a smaller number of tests when the number of points is large. This economy principle in the number of tests is being developed in recent works of Geman et al. [5], [6]. We compared both definitions of object alignments in the examples of Fig. 1. When we use the larger Ns corresponding to all widths (from 3 to 12 pixels) and all segments of the image, we simply do not detect any alignment. This is due to the testing overdose: by doing so, we have tested many times the same alignments, and have also tested many strips which contained no dots at all. The second definition ofNs happens to give a perceptually correct result. This result is displayed in Fig. 1b where we see the only detected strip. This same alignment is no more detectable on the right. The ...

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...ost cases, we shall consider in the next sections, a considered event will be defined as "-meaningful if NTBðp; n; kÞ ": We call in the following the left-hand member of this inequality the “number of false alarms” (NFA). When " 1, we talk about meaningful events. This seems to contradict the necessary notion of a parameter-less theory. Now, it does not since the -dependency of meaningfulness is in fact a log -dependency. We refer to [2] for a complete discussion of this definition. The general method we have just outlined can be viewed as a systematization of Stewart’s “MINPRAN” method [15]. It was also proposed in the early Lowe work [12], but, to the best of our knowledge, not systematically developed. 2 OBJECT ALIGNMENTS The first partial gestalt we shall consider is a direct application of the above definition. We consider the case of objects whose barycenters are aligned. Assume that we observe M objects of a certain kind in an image. Our a contrario assumption for the application of 508 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 25, NO. 4, APRIL 2003 . A. Desolneux is with UFR Math-Info, 45, rue des Saints-Peres, 75270 Paris cedex 06, France. E-m...

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...ally, equidistance (of both pairs of opposite sides). The square is a global gestalt, and the result of concurring geometric qualities that we shall call partial gestalts. Many Computer Vision methods attempt to compute the (very diverse in nature) partial gestalts. To take an instance, the snakes method [10] attempts to capture the closed smooth curves, a combination of the “closure” and “good continuation” gestalts. Some more recent works try to define grouping rules for combining local information into organized contours [8], [13]. In [2], we have treated alignments (straight edges) and in [3] general boundaries and edges. In this paper, we treat four more examples of partial gestalts, namely, the object alignments, the clusters, and quality grouping by color, orientation or size. In [1], a vanishing point detector is treated by a clever extension of the same method. As a first evidence of the recursive character of gestalt laws, we shall push one of the experiments to prove that the partial gestalt recursive building up can be led up to the third level (gestalts built by three successive partial gestalt grouping principles). 1.1 Helmholtz Principle In [2], we outlined a computatio...

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... a considered event will be defined as "-meaningful if NTBðp; n; kÞ ": We call in the following the left-hand member of this inequality the “number of false alarms” (NFA). When " 1, we talk about meaningful events. This seems to contradict the necessary notion of a parameter-less theory. Now, it does not since the -dependency of meaningfulness is in fact a log -dependency. We refer to [2] for a complete discussion of this definition. The general method we have just outlined can be viewed as a systematization of Stewart’s “MINPRAN” method [15]. It was also proposed in the early Lowe work [12], but, to the best of our knowledge, not systematically developed. 2 OBJECT ALIGNMENTS The first partial gestalt we shall consider is a direct application of the above definition. We consider the case of objects whose barycenters are aligned. Assume that we observe M objects of a certain kind in an image. Our a contrario assumption for the application of 508 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 25, NO. 4, APRIL 2003 . A. Desolneux is with UFR Math-Info, 45, rue des Saints-Peres, 75270 Paris cedex 06, France. E-mail: desolneux@math-info.univ-paris5.fr. . L. Mois...

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...iversity of Minnesota. Downloaded on February 15,2010 at 11:29:42 EST from IEEE Xplore. Restrictions apply. object’s quality QðOÞ falls in a; b. With the same generic argument as in Section 1, we have Definition 3. Definition 3. An interval a; b is -meaningful if NFAða; bÞ Ni Bðpða; bÞ;M; kða; bÞÞ ; where Ni is the number of considered intervals (Ni LðLþ 1Þ=2). An interval a; b is said to be maximal meaningful if it is meaningful and if it does not contain, or is not contained in, a more meaningful interval. It can be proven in the same way as for orientation alignments [2], [4] that maximal meaningful intervals do not intersect. Thus, we get an operational definition of meaningful modes as disjoint subintervals of 1; L. 3.2 Size of Objects The preceding arguments are easily adapted to Helmholtz type assumptions on nonuniform histograms. A very generic way to group objects in an image is their similarity of size. Now, it would be a total nonsense to assume any uniform law on the objects sizes. There are several powerful arguments in favor of a statistical decreasing law for size. These arguments derive from perspective laws, or from the occlusion dead leaves model,...

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...2. Recommended for acceptance by D. Jacobs and M. Lindenbaum. For information on obtaining reprints of this article, please send e-mail to: tpami@computer.org, and reference IEEECS Log Number 117717. 0162-8828/03/$17.00 ß 2003 IEEE Published by the IEEE Computer Society Authorized licensed use limited to: University of Minnesota. Downloaded on February 15,2010 at 11:29:42 EST from IEEE Xplore. Restrictions apply. Helmholtz principle is that the M barycenters ðxi; yiÞ are i.i.d. uniformly on a domain . A meaningful alignment of points must be a meaningful peak in the Hough Transform (see [11], [14] for a very similar approach). Now, the accuracy matter must be addressed. Points are supposed to be aligned if they all fall into a strip thin enough, in sufficient number. Let S be a strip of width a. Let pðSÞ denote the prior probability for a point to fall in S, and let kðSÞ denote the number of points (among the M) which are in S. Definition 2. A strip S is -meaningful if NFAðSÞ Ns BðpðSÞ;M; kðSÞÞ ; where Ns is the number of considered strips. 2.1 The Number of Tests We now have to discuss what the considered strips will be. In [2], we considered the relatively close problem of or...

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... get an operational definition of meaningful modes as disjoint subintervals of 1; L. 3.2 Size of Objects The preceding arguments are easily adapted to Helmholtz type assumptions on nonuniform histograms. A very generic way to group objects in an image is their similarity of size. Now, it would be a total nonsense to assume any uniform law on the objects sizes. There are several powerful arguments in favor of a statistical decreasing law for size. These arguments derive from perspective laws, or from the occlusion dead leaves model, or directly from statistical observations of natural images [7]. Our Helmholtz qualitative hypothesis is then: the prior distribution of the size of objects is decreasing. Definition 4. An interval a; b is -meaningful (for the decreasing assumption) if NFAða; bÞ Ni max p2D Bðpða; bÞ;M; kða; bÞÞ ; where Pd is the set of decreasing probability distributions ðpiÞ on f1; 2; . . . ; Lg, and pða; bÞ Pb ia pi. In the same way as in the flat histogram assumption, one can define maximal meaningful intervals and prove that maximal meaningful intervals do not intersect. 4 MEANINGFUL GROUPS OR CLUSTERS 4.1 Model The cluster example is the seminal one in...

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...two testing methods are available, perception must obviously choose the one giving the smaller test number. Indeed, there is perceptual evidence that grouping processes may depend on density, and that different methods could be relevant for dense and for sparse patterns [17]. Hence, the second testing method we present here should be preferred for sparse distributions of points, whereas the initial model based on density would give a smaller number of tests when the number of points is large. This economy principle in the number of tests is being developed in recent works of Geman et al. [5], [6]. We compared both definitions of object alignments in the examples of Fig. 1. When we use the larger Ns corresponding to all widths (from 3 to 12 pixels) and all segments of the image, we simply do not detect any alignment. This is due to the testing overdose: by doing so, we have tested many times the same alignments, and have also tested many strips which contained no dots at all. The second definition ofNs happens to give a perceptually correct result. This result is displayed in Fig. 1b where we see the only detected strip. This same alignment is no more detectable on the right. The teste...

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...ds attempt to compute the (very diverse in nature) partial gestalts. To take an instance, the snakes method [10] attempts to capture the closed smooth curves, a combination of the “closure” and “good continuation” gestalts. Some more recent works try to define grouping rules for combining local information into organized contours [8], [13]. In [2], we have treated alignments (straight edges) and in [3] general boundaries and edges. In this paper, we treat four more examples of partial gestalts, namely, the object alignments, the clusters, and quality grouping by color, orientation or size. In [1], a vanishing point detector is treated by a clever extension of the same method. As a first evidence of the recursive character of gestalt laws, we shall push one of the experiments to prove that the partial gestalt recursive building up can be led up to the third level (gestalts built by three successive partial gestalt grouping principles). 1.1 Helmholtz Principle In [2], we outlined a computational method to decide whether a given partial gestalt (computed by any segmentation or grouping method) is reliable or not. We treated the detection of straight segments, as one of the most basic ges...

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... this paper, we show that the method is fully general and can be extended to a grouping by any quality. We treat as an illustration the alignments of objects, their grouping by color and by size, and the vicinity gestalt (clusters). Collaboration of the gestalt grouping laws and their pyramidal structure are illustrated in a case study. Index Terms—Gestalt grouping laws, a contrario probabilistic model, binomial law, number of false alarms, histogram modes, clusters, alignments. æ 1 WHAT IS A PARTIAL GESTALT? ACCORDING to Gestalt theory, “grouping” is the main process in our visual perception [9], [16]. Whenever points (or previously formed visual objects) have one or several characteristics in common, they get grouped and form a new, larger visual object, a “Gestalt.” Some of the main grouping characteristics are proximity (clustering), color constancy (connectedness), “good continuation” (differentiability of boundaries), alignment (presence of straight lines or objects of a same kind aligned), parallelism (between lines, oriented objects, etc.), similarity of shape (between objects), common orientation (between points or oriented objects), convexity (of boundaries, of a group), clo...

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... paper, we show that the method is fully general and can be extended to a grouping by any quality. We treat as an illustration the alignments of objects, their grouping by color and by size, and the vicinity gestalt (clusters). Collaboration of the gestalt grouping laws and their pyramidal structure are illustrated in a case study. Index Terms—Gestalt grouping laws, a contrario probabilistic model, binomial law, number of false alarms, histogram modes, clusters, alignments. æ 1 WHAT IS A PARTIAL GESTALT? ACCORDING to Gestalt theory, “grouping” is the main process in our visual perception [9], [16]. Whenever points (or previously formed visual objects) have one or several characteristics in common, they get grouped and form a new, larger visual object, a “Gestalt.” Some of the main grouping characteristics are proximity (clustering), color constancy (connectedness), “good continuation” (differentiability of boundaries), alignment (presence of straight lines or objects of a same kind aligned), parallelism (between lines, oriented objects, etc.), similarity of shape (between objects), common orientation (between points or oriented objects), convexity (of boundaries, of a group), closure (...