Toussaint, G. T., "A historical note on convex hull finding algorithms," Pattern Recognition Letters, vol. 3, January 1985, pp.21-28.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....regions of a set are defined as the connected components of this difference image. The use of concavity trees for shape recognition is presented in [3] Computational geometry algorithms for computing the convex hull of a set of points of the Euclidean plane can be found in [15, chaps. 4 5] [20] and [14] Algorithms for extracting the convex hull or concavity regions of binary images are proposed in many papers (e.g. 21, 17, 2, 18, 1, 4] In this paper, we show that the convex hull of a set can be defined in terms of morphological transformations suited to the processing of grey ....

G. Toussaint. A historical note on convex hull finding algorithms. Pattern Recognition Letters, 3:21--28, 1985.

....sets is also a convex set, there exists a smallest convex set containing any given set. This smallest convex set is called the convex hull of the input set. Algorithms for computing the Euclidean convex hull of a finite set of points of the Euclidean plane can be found in [10] 19, chaps. 4 5] [31], 18] and [16] An extended bibliography on this topic is given in [20] In image analysis, convex hulls are at the basis of useful shape indices such as the concavity index, i.e. the ratio between the surface area of a 2 D connected bounded set and that of its convex hull. Alternatively, the ....

G. Toussaint. A historical note on convex hull finding algorithms. Pattern Recognition Letters, 3:21-28, 1985.

....Shrinking that rubber band produces a polygon of minimum perimeter, as shown in figure 2.4(b) 18 Chapter 2. Review of Shape Description Methods The convex hull of a finite point set was first presented by Graham [40] and later expanded among others by McQueen and Toussaint [65] and Toussaint [101]. It defines the minimum area convex polygon containing all points of the set. The set difference between the convex hull and the point set is called the convex deficiency of the set. However, as most digital objects tend to be irregular, convex deficiency has rather small scattered components. ....

G.T. Toussaint. A historical note on convex hull finding algorithms. Pattern Recognition Letters, 3:21--28, 1985.

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Toussaint, G. T., "A historical note on convex hull finding algorithms," Pattern Recognition Letters, vol. 3, January 1985, pp.21-28.

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Toussaint, G. T., "A historical note on convex hull finding algorithms," Pattern Recognition Letters, vol. 3, January 1985, pp. 21-28.

....j ,x i ] or [x i 1 ,x i 1 ] It suffices to realize that each diagonal can be inserted with a constant number of local angle tests. A similar procedure can be used to triangulate the exterior of a one mouth polygon. First we can use an O(n) time algorithm for finding the convex hull of P [To1]. This will identify the two vertices x i and x j that form the lid of the pocket K ij of CH(P) One of the two ears of K ij must occur at either x i or x j and can then be identified in a constant number of steps (i.e. independent of n) Triangulation of K ij can then proceed as ....

Toussaint, G. T., "A historical note on convex hull finding algorithms," Pattern Recognition Letters, vol. 3, January 1985, pp.21-28.

....lie outside it and (4) out of all such full circles select the smallest encountered. This algorithm has a time complexity of O(n 4 ) An improved adaptive algorithm for this problem was proposed by Bass and Schubert in 1967 [BS67] and although no complexity analysis is given it can be shown[To85a] that their algorithm runs in time O(h 4 n log n) where h is the number of points that are extreme points of the convex hull of C. In 1972 Elzinga and Hearn [EH72] proposed a more 2 efficient (in the worst case) algorithm that runs in O(n 2 ) time. About four years later the computer ....

....where the most efficient algorithms for 3 computing such structures can be found. 2. Some Basic Computational Geometric Tools 2. 1 Convex Hulls One of the most useful structures in computational geometry is the convex hull of a set of points C, i.e. the smallest convex set containing C [To85a]. In designing their facility location algorithm Bass and Schubert [BS67] used the fact that the smallest enclosing circle of a set of points C is determined only by points which are extreme on the convex hull of C. This fact together with an O(n log n) time algorithm which they propose for ....

Toussaint, G. T., "A historical note on convex hull finding algorithms," Pattern Recognition Letters, vol. 3, January 1985, pp. 21-28.

....intersection points on L i lie to one side of p ij . The intersection point p ij for i j and 1 i,j n is said to be critical if and only if p ij is extreme with respect to both lines L i and L j . It is known that the convex hull of n points in the plane can be computed in O(n log n) time [To85]. Straightforward application of such algorithms to all the points in I thus leads to O(n 2 log n) time algorithms. Surprisingly, Ching Lee [CL85] present an O(n log n) time algorithm for this problem and also establish an W(n log n) lower bound. Atallah [At86] independently rediscovered the ....

Toussaint, G. T., "A historical note on convex hull finding algorithms," Pattern Recognition Letters, vol. 3, January 1985, pp. 21-28. CH(n) L x L i L j Fig. 2 Illustrating the counterexample for the line deletion algorithm.

....compute CH(P ) A linear scan can then test all chains of segments (as computed above) to see if any are cutting chains. The convex Figure 4: hull of a set of n points can be computed in O(n log n) time [5] or O(n log h) time [8] and [9] where h is the cardinality of the convex hull points. See [16] for a historical account of the planar convex hull problem. We have given some preliminary methods to reject sets of segments that do not admit a simple circuit. However, a set of segments may pass the above tests and still not admit a simple circuit. A set of five CH connected segments that ....

G.T. Toussaint. A historical note on convex hull finding algorithms. Pattern Recognition Lett., 3:21--28, 1985.

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