Searching Aligned Groups of Objects with Fuzzy Criteria

The detection of aligned groups of objects is important for satellite image interpretation. This task can be challenging when objects have different sizes. In this paper, we propose a method for extracting aligned objects from a labeled image. In this method we construct a neighborhood graph of the objects of the image, and its dual graph where we incorporate information about the relative direction of the objects, evaluated using fuzzy measures of relative position. The groups of objects satisfying the fuzzy criterion of being locally aligned are extracted from the dual graph. These groups are the candidates for being (globally) aligned. The method was tested on synthetic images, and on objects extracted from real images demonstrating that the method extracts the aligned groups of objects even if the objects have different sizes. 1 Alignment and Related Work Alignment can be defined as the spatial property possessed by a group of objects arranged in a straight line. Determining the groups of aligned objects is crucial for image interpretation. According to the Gestalt theory, the human perceptual vision system groups objects together using certain rules. Among these rules there is one called continuity of direction which groups together objects in the same direction, and one particular case is the constancy of direction that refers to alignments [5]. An aligned group of objects has the characteristic that it should be seen as a whole, since if its elements are observed in an independent manner then the alignment property is lost. Having to look it as a whole makes alignment detection a difficult task. Identifying the aligned groups of objects in satellite images is important for several applications. Satellite images provide a huge amount of geographical information, and aligned groups of objects can be seen as a way to reduce this information in a pertinent way. For example in cartography, it is necessary to find groups of aligned buildings for map generalization [12]. Observing if a group of buildings is aligned can give information about the structure of their arrangement, and whether they belong to a urban, rural or residential area [6]. In object detection, complex semantic classes such as parking areas (car parkings, 1 Definition taken from ThinkMap Visual Thesaurus http://www.visualthesaurus.com/ E. Hüllermeier, R. Kruse, and F. Hoffmann (Eds.): IPMU 2010, LNAI 6178, pp. 605–613, 2010. c © Springer-Verlag Berlin Heidelberg 2010 606 M.C. Vanegas, I. Bloch, and J. Inglada ports, truck parkings or airports) comprise aligned groups of transport vehicles. Therefore, the identification of aligned groups of transport vehicles can be useful for detecting instantiations of these complex classes, and is meaningful for the description of this kind of scenes. Alignment extraction has been studied in image processing as a low level feature. For instance methods relying on the Hough transform [5] or the Radon transform [7] are used to find groups of points in digital images which fall into a line. Other examples are the identification of aligned segments which have the same orientations as the alignment [5,10,11,8]. However, alignment extraction as a high level feature has been less studied. One example is the work of [4], where an algorithm to detect aligned groups of buildings in maps is presented. In this algorithm buildings with aligned barycenters are extracted, and the quality of the alignments is evaluated based on the criteria of proximity and similarity laws of Gestalt theory. Nevertheless, when the groups are composed of objects of different sizes, it is not possible to detect the alignment by observing just their barycenters (see Fig. 1). Thus, when considering extended objects and not only points the notion of “falling into a line” becomes imprecise. Therefore it is necessary to consider a degree of satisfaction of the relation of alignment. In this work we propose a novel method to detect alignments of objects that can be applied to objects of different sizes, or to fuzzy objects. In our approach, we use the direction orientation between any two elements of the group to determine their degree of alignment. To measure the orientation between two objects we make use of what we call orientation histogram which is based on the angle histogram introduced by Mijama and Ralescu in [9] (Sec. 2). Our strategy consists in first determining the locally aligned groups which are the candidates to form an aligned group of objects. Then we measure the degree of alignment of each candidate group (Sec. 3) and solve conflicts. The results of the method are shown on synthetic and real images in Sec. 5. Fig. 1. Problems encountered when the group has objects of different sizes: an aligned group of objects with not aligned barycenters 2 Angle and Orientation Histograms Angle histograms have proved to be an adequate way for evaluating the directional spatial relation between two objects, since they take into account the shape of the regions [9]. They can be interpreted as a function that captures the directional position between two objects. Let a and b be two objects defined by two regions in the image space I, that we denote by a and b. The angle Searching Aligned Groups of Objects with Fuzzy Criteria 607 histogram from a to b is obtained by computing for each pair of points pa ∈ a and pb ∈ b the angle between the segment joining them and the horizontal axis, denoted by ∠(pa, pb). Angles are organized in a histogram, normalized by the largest frequency: H(b)(θ) = ∑ pa∈a,pb∈b|∠(pa,pb)=θ 1 maxφ∈[0,2π) ∑ pa∈a,pb∈b|∠(pa,pb)=φ 1 . (1) To determine if an object a is in a given direction with respect to an object b (for example “right of”), we can compute the angle histogram H(b) and compare it with a template for the relation “right of” by using for instance a conjunctive operator or the compatibility between the computed histogram and the template [9]. Angle histograms are easily extended to fuzzy objects. In addition, they are invariant to simultaneous translation, scaling and rotation of both objects. They are not symmetrical, but they satisfy: H(b)(θ) = H(a)(θ + π). Since we are interested in the orientation of two objects with respect to the horizontal axis, we introduce the notion of orientation histogram, which is simply an angle histogram where the angles are computed modulus π and its support has a length equal to π. For the case where a and b are fuzzy objects with membership function μa : I → [0, 1] and μb : I → [0, 1], respectively, the orientation histogram is given by: O(a, b)(θ) = ∑ pa,pb∈I|mod(∠(pa,pb),π)=θ μa(pa) ∧ μb(pb) maxφ∈[0,π) ∑ pa,pb∈I|mod(∠(pa,pb),π)=φ μa(pa) ∧ μb(pb) , (2) where ∧ is a t-norm. The orientation histogram is a fuzzy subset of [0, π[ that represents the orientation between two objects with respect to the horizontal axis, it preserves the same properties as the angle histogram, and in addition it is symmetrical. To compare if two orientation histograms are similar, it is important to consider the imprecision that is linked to the comparison of two angles that are approximately the same. When a fuzzy morphological dilation [3] is performed on an orientation histogram using a structuring element ν0, then the high values of the histogram will be propagated to the similar angle values according to ν0. The structuring element ν0 is designed such that ν0(θ− θ̃) represents the degree to which θ̃ and θ are “approximately” equal (modeled by a trapezoid function in our experiments). Then the similarity degree between two orientation histograms can be given by the maximum height of the intersection of the dilated histograms: sim(O(a, b), O(c, d)) = max θ∈[0,π) [Dν0(O(a, b)) ∧Dν0(O(c, d))] (θ), (3) where ∧ is a t norm, and the fuzzy morphological dilation is given byDν0(μ)(θ) = supθ̃∈[0,π[min(μ(θ̃), ν0(θ − θ̃)) [3]. 608 M.C. Vanegas, I. Bloch, and J. Inglada This degree of similarity can be extended to evaluate the similarity degree between several orientation histograms. Let {O(ai, bi)}i=0 be a set of orientation histograms. Then the degree of similarity between them is given by: sim (O(a0, b0), . . . , O(aN , bN)) = max θ∈[0,π[ N ∧ i=0 Dν0(O(ai, bi))(θ). (4) 3 Alignment Detection In this section we propose the definitions of globally aligned and locally aligned, both relations depend on a neighborhood relation. Let a, b be two objects. We define Nd(a) as the Voronoi neighborhood of a constrained by a distance d, and the binary relation Neigh(a, b) is satisfied if b ∩Nd(a) = ∅. A group S is said to be globally aligned if all its members are connected by the Neigh relation, and if there exists an angle θ such that every member of the group is able to see the other members of the group in a direction θ or θ + π with respect to the horizontal axis. Thus, it is possible to define the degree of global alignment as follows: Definition 1. Let S = {a0, . . . , aN}, with N ≥ 3, be a group of objects in I, connected by the Neigh relation. The degree of global alignment of S is given by: μALIG(S) = sim (O(a0, S \ {a0}), . . . , O(aN , S \ {aN})) . (5) A group S with μALIG(S) = β is called a globally aligned group to a degree β.A group S = {a0, . . . , aN} is said to be locally aligned to a degree β, if for every two pairs of neighboring objects, having one object in common, the orientations between the objects of each pair are similar to a degree β , and also if the group is connected by the neighbor relation. The latter can be summarized by saying that a group S with |S| ≥ 3 is locally aligned to a degree β if it satisfies the following relations: R1 : ∀x, y, z (Neigh(x, y) ∧Neigh(y, z)) ⇒ (sim(O(x, y), O(y, z)) ≥ β) R2 : ∀a, b ∃x0, . . . , xm for m > 1 such that x0 = a, xm = b and m−1 ∧