REPRESENTING EXTRATED EDGES FROM IMAGES BY USING CONSTRAINED DELAUNAY TRIANGULATION

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1 REPRESENTING EXTRATED EDGES FROM IMAGES BY USING CONSTRAINED DELAUNAY TRIANGULATION Simena Dinas Escuela de Ingeniería de Sistemas y Computación, Universidad del Valle simena.dinas@correounivalle.edu.co José María Bañón Pinar Escuela de Ingeniería de Sistemas y Computación, Universidad del Valle jose.banon@correounivalle.edu.co * Only for Final Camera-Ready Submission ABSTRACT Delaunay Triangulation has been used in computational geometry and mathematics; it is a net formed by the union of non-overlapping triangles, where there are no points inside the circumcircle of each triangle. It is popular for meshing algorithms; however, in real world some important edges can be ignored to preserve the Delaunayhood of the structure. Constrained Delaunay Triangulation is an extension of Delaunay which forces required edges in the Triangulation; moreover, the resulting triangulation is not always a Delaunay Triangulation. Due to the continuously increasing images volume; there are some researches on extracting information from images. An application combining extracted data from images and Constrained Delaunay Triangulation is shown. KEYWORDS Delaunay Triangulation, Constrained Delaunay Triangulation, Delaunay Constrained Triangulation, Edge Extraction, Corner Extraction. 1. INTRODUCTION Delaunay Triangulation (DT) was created in 1934 by Boris Nikolaevich Delone ( ); it is a triangle net in which every triangle satisfies a Delaunay condition: the circumcircle of a triangle includes only the vertex of the triangle. In other words, the circumcircle does not contain any vertex of other triangles. (de Berg et al, 2008). The most common methods to construct a DT are: Lawson method (Lawson, 1977), Bowyer method (Bowyer, 1981) and Watson method (Watson, 1981). In addition, every simplex (triangle or edge) satisfying the circumcircle property is usually called a Delaunay simplex; moreover, the union of all simplices produces the convex hull. Two extension of DT including constraints have been widely studied: Constrained Delaunay Triangulation (CDT) and Delaunay Constrained Triangulation (DCT). Constraints are used to modeling edges for two-dimensional approaches, whereas facets are used for three-dimensional approaches. The first in propose an algorithm with optimal time to compute CDT was (Chew, 1987). It is an unstable method to computing a triangulation, where to preserves the constrained edges is more important than guarantees the Delaunay condition. In contrast, DCT is a stable method to compute a DT, where is more important is to preserve the Delaunay condition rather than guarantees constrained edges (Rognant et al., 1999). 2. BACKGROUND 2.1 Delaunay Triangulation

2 DT is a structure widely used in computational geometry and extended to others multipurpose areas. Some applications of DT include: video compression (Vomáčka and Puncman, 2009); networking (Wu et al., 2007); protein folding (Ouyang and Liang, 2008); spatial clustering (Yang and Cui, 2010); boundary detection (Liu et al., 2008); image representation (Kohout, 2007); among others. The goal of a triangulation is to produce a mesh. Meshes are a common way to represent continuous surfaces. The input of a Triangulation is a set of points and the result is a set of triangles or linked edges without overlapping (Zimmer, 2005). Building a Triangulation has different problems, but the starting point and the dispersion should not be important. Several proposals of triangulation have been developed (Preparata and Shamos, 1985; Shojaee et al., 2006); Greedy Triangulation (Dickerson et al., 1994), Triangulation of Garey (Garey et al., 1978), Radial Sweep (Hjalle and Dæhlen, 2006); (Mirante and Weingarten, 1982) and Delaunay Triangulation (de Berg et al., 2008). 2.1 Delaunay Stable and Unstable Methods When it is necessary to define fixed edges in DT, these edges are constraints, thus DT has to be called Constrained Delaunay Triangulation -CDT. If it is the most important is to preserve constrained edges rather than the Delaunay nature the result is a CDT, which is a triangulation including the constrained edges but with as close as possible Delaunay Triangulation (Chew, 1987). On the other hand, in a Delaunay Constrained Triangulation -DCT: it is most important is to conserve the Delaunay nature, but there is a poor work with the constraints (Rognant et al., 1999). The result of DCT is DT with as many as possible constrained edges. Figure 1, depicts the difference between stable and unstable DTs. (a) (b) (c) (d) Figure 1. (a) DT. (b) CDT. (c) DT & CDT. (d) DCT including surrounded virtual points Constrained Delaunay Triangulation (CDT) Independently, (Lee et al., 1986) and (Chew, 1987) introduced CDTs. However, they used different names for each approach; Generalized Delaunay Triangulation (Lee et al., 1986) and obstacle triangulation (Chew, 1987). A triangle is Constrained Delaunay if i) its interior does not intersect any input segment, and ii) its circumcircle does not enclose vertices visible from interior of the triangle. Nevertheless, an edge is Constrained Delaunay if i) it does not cross any input segment, and ii) it has a circumcircle, which does not enclose vertices visible from interior of the edge. CDT is an unstable method due to the importance constraints rather than the Delaunayhood of the triangulation. However, in the literature several stable methods can be found to construct a DT respecting poorly the constrained edges called DCT (Rognant et al., 1999). Nevertheless, some techniques can combine both: they are DT and they respect the constraints Delaunay Constrained Triangulation (DCT) A DCT is usually called a stable method because the final triangulation guarantees the Delaunay Structure at the same time respects as constrained edges as possible. Stable methods can be constructed by inserting new points in the triangulation called Virtual Points. Most common approaches can be found in figure 2: (a) (b) (c)

3 Figure 2. (a) DT. (b) CDT. (c) Densification. (d) Dichotomy. (e) Orthogonal. (f) Intersection. Delaunay Triangulation. It is depicted in figure 2(a) for a set of four points. Original configuration: It is an CDT (See figure 2 (b)); the constrained edge (C) is the horizontal line. Densification method: It is an DCT. The first step is to calculate the distance d of the C to the closest point; C is divided into small constrained edges with maximum length d. This method is a reliable and easy to implement; however, the high computational cost and the amount of virtual points required are notable shortcomings. (See figure 2(c)). Dichotomy method: It is DCT created by splitting the constrained edges into small constrained edges. The convergence can be guaranteed by the densification method. (See figure 2(d)). It is equally reliable, computationally less expensive and requires less virtual points than densifcation. The shortcoming of this technique is related to the random division of the segment. Orthogonal projection method: It is an DCT. For each vertex around the C, an edge is orthogonally projected. Due to an orthogonal projection on an edge produces a perpendicular edge; the edge is created by using the orthogonal projection of a vertex and the crossing point. (See figure 2(e)). It is reliable, and calculating the orthogonal projection is not complex, while the number of new virtual points can be calculated easily. Intersection method: It is a DCT resulting of overlapping a DT and constrained edges. Virtual points are created in the intersections. This method is useful when a new constraint is inserted in a DT. When the DT is unknown, can be less expensive using another method tan construct the DT, however, it is reliable. (See figure 2(f)) Virtual Points and Intersections (d) (e) (f) A Virtual Point is a point, which do not belong to the triangulation points; however, it is created for the cross of two constrained edges; thus, their creation depends on the applications. For instance, in X intersections, it is necessary to split the constraint into four new constraints where each constraint includes the virtual point (crossing point) and one of the four endpoints. When the intersection point is include in both lines, but it coincides with and endpoint of one of them (T intersection) is required to split only one of the lines into two lines, but is not required to create a new virtual point. The other instance is an L intersection; in this instance it is not necessary to create a virtual point because the crossing point coincides (one for each constrained edge). Virtual points can be used in the triangulation, but if the constraint is deleted, the virtual point is deleted too, see figure 1 (d). 3. PROPOSAL In general terms, the DT can be proved with cloud of points; however, data can be easily understandable and difficult to trace. For this experiment were selected two images: Lena picture is an image with curves, whereas Tsukuba image has straight lines and strong angles. Extracted features from the image were corners or interesting points (figure 3(b)) and edges (see figure 3(c)). Using interesting points from figure 3(b); a DT was created (see figure 3(d)) whereas figure 3(e) was created by using edges from figure 3(c). In order to obtain information from images, extracted edges from Lena picture were defined as constrained for CDT; they are flattening as shown in figure 3(d). The process of flattening is done by replacing curves with polylines (or set of lines that approximate the curve). Each flattened edge is defined as a constrained edge to do a CDT, as depicted in figure 3(d). The points of the flattening process enrich the set of points used to construct the CDT. Even though figures 3(d) and 3(e) were constructed by different set of data they are similar. Not all sets of data have the same similarity; however this picture is similar because of it is formed by small constrained edges because the flattening process fragments the curves in order to obtaining straight lines.

4 (a) (b) (c) (d) (e) Figure 3. (a) Lena Picture. (b) Cloud of points extracted from Lena Picture. (c) Set of edges extracted from Lena Picture and flattened. (d) DT from the cloud of points. (e) CDT from set of flattened edges. Tsukuba image is depicted in figure 4(a). Figures 4(b) and 4(c) are created by extracting interesting points and edges from the Tsukuba image, respectively. A DT is constructed by using the cloud of points formed by interesting points; it is depicted in figure 4(d). Finally, the CDT (see figure 4(e)) is enriched by using the extracted corners and new points created in the flattening process. In contrast, with figures 3(d) and 3(e), figures 4(d) and 4(e) are enriched by means of larger edges. In these pictures there are lower quantities of curved lines; however, there are upper quantities of angles, then the flattening process divides the lines with angles. (a) (b) (c) (d) Figure 4. (a) Tsukuba Picture. (b) Cloud of Points extracted from Tsukuba Picture. (c) Set of edges extracted from Lena Picture and flattened. (d) DT from the cloud of points. (e) CDT from set of flattened edges. (e) 4. RESULTS The main results is that comparing the triangles in figures 3(d) and (e) there are 90% equals; it is due to the small constrained edges. In contrast, the Tsukuba image has large constraints (see figures 3(d) and (e)); the triangles are 25% equals. For the Tsukuba image, was evaluated the interior of CDT triangles related to the real image and was found that 70% of the triangles has similar information about color with a scale of colors and 5 of color range variation for each channel. 5. CONCLUSIONS

5 The main contribution of this paper was focused on showing the relevance of modeling real world information from images by using CDT. Although the features modeled were the corner and edges, other features can be modeled, but the meaning of the feature depends on the applications. For instance, regions can be modeled by using CDT; however, it is necessary to define the meaning and the representation of a region. Another contribution was the state of the art on CDT; even though the use of this technique is increasing, there are no enough contributions oriented to condensate the literature information about this technique. CDT can be widely used to model computer vision problems; particularly; it can be used to tracking features; improving the dense disparity map estimation; estimating block matching; among others. This document presents an initial contribution to obtaining information about distinguishable features from images; however, it is open to finding more application and to proving this technique on computer vision. REFERENCES Bowyer, A Computing Dirichlet tessellations. The Computer Journal, 24(2), p Chew, L. P Constrained Delaunay triangulations. p of: Proc. of the 3rd annual symposium on Computational geometry. SCG '87. New York, NY, USA: ACM. de Berg, M., Cheong, O., van Kreveld, M., & Overmars, M Computational Geometry: Algorithms and Applications. 3rd edn. Dickerson, M. T., Drysdale, R. L. S., McElfresh, S. A., & Welzl, E. Fast greedy triangulation algorithms. In Proceedings of the tenth annual symposium on Computational geometry, SCG 94, pages , New York, NY, USA, ACM. Garey, M. R., Johnson, D. S., Preparata, F. P., & Tarjan, R. E. Triangulating a simple polygon. Information Processing Letters, 7: , Hjelle, Ø., & Dæhlen, M. Triangulations and Applications (Mathematics and Visualization). Springer-Verlag New York, Inc., Secaucus, NJ, USA, Kohout, J. On digital image representation by the delaunay triangulation. In Domingo Mery and Luis Rueda, editors, Advances in Image and Video Technology, Lecture Notes in Computer Science. Springer Berlin Heidelberg Lawson, Ch. L Software for C1 surface interpolation. p of: Mathematical Software III, J. Rice ed. Academic Press, New York. Lee, D-T., & Lin, A. K. Generalized Delaunay Triangulation for planar graphs, Discrete and Computational Geometry, vol. 1, pp , Liu, D., Nosovskiy, G. V., & Sourina, O. Effective clustering and boundary detection algorithm based on delaunay triangulation. Pattern Recogn. Lett., 29: , July Mirante, A., & Weingarten, N. The radial sweep algorithm for constructing triangulated irregular networks. IEEE Comput. Graph. Appl., 2:11 21, March Ouyang, Z., & Liang, J. Predicting protein folding rates from geometric contact and amino acid sequence. Protein Science, 17(7): , Preparata, F. P., & Shamos, M. I. Computational Geometry: An Introduction. Springer-Verlag New York, Inc., New York, NY, USA, Rognant, L., Chassery, J. M., Goze, S., & Planès, J.G The Delaunay Constrained Triangulation: The Delaunay Stable Algorithms. p. 147 of: Proc. of the 1999 International Conference on Information Visualisation. Washington, DC, USA. Shojaee, D., Alesheikh, A. A., & Helali, H. Triangulation for surface modelling, Vomáčka, T., & Puncman, P. A novel video compression scheme based on kinetic delaunay triangulation. In Algoritmy 2009: 18th Conference on Scientific Computing, pages , Bratislava, Slovak University of Technology. Watson, D. F Computing the n-dimensional Delaunay Tessellation with Application to Voronoi Polytopes. The Computer Journal, 24(2), pp Wu, Ch.-H., Lee, K.-Ch., & Chung, Y.-Ch. A delaunay triangulation based method for wireless sensor network deployment. Comput. Commun., 30: , October Yang, X., & Cui, W. A novel spatial clustering algorithm based on delaunay triangulation. Journal of Software Engineering and Applications, JSEA, 3(2): , Zimmer, H. Voronoi and delaunay techniques. Lecture Notes in Computer Science VIII, 2005.