Toussaint, G. T. and Avis, D., "On a convex hull algorithm for polygons and its application to triangulation problems," Pattern Recognition, vol. 15, No. 1, 1982, pp.23-29.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....for restricted classes of polygons. As noted earlier, convex or monotone polygons can be triangulated in linear time. A star shaped polygon can also be triangulated in linear time using an ear cutting procedure. PS81] shows how to test if a given simple polygon is monotonic in O(n) time. TA82] show how to triangulate a class of polygons called edge visible polygons in linear time. An edge visible polygon contains at least one edge such that every point inside the polygon is visible from some point on that edge. Given a general polygon, testing if it is simple does no seem to be all ....

G. T. Toussaint and D. Avis. On a convex hull algorithm for polygons and its application to triangulation problems. Pattern Recogn., 15:23--29, 1982.

....an easy exercise for the reader. The linearity follows from the fact that at each step, which takes constant time P s contains one less vertex. Note that other linear time algorithms could be used for triangulating P s . For example P s is edge visible from the mast and thus the algorithm of [13] can be used. Alternately, P s is monotonic in the direction perpendicular to the mast and therefore the algorithm of Garey et al. 3] applies. The advantages of the algorithm presented here are that, first, unlike those of [13] and [3] it does not incorporate backtracking and is thus simpler, ....

....P s is edge visible from the mast and thus the algorithm of [13] can be used. Alternately, P s is monotonic in the direction perpendicular to the mast and therefore the algorithm of Garey et al. 3] applies. The advantages of the algorithm presented here are that, first, unlike those of [13] and [3] it does not incorporate backtracking and is thus simpler, and second, the last diagonal to be added is . This latter property is crucial for solving the polygon intersection problem, The ear cutting algorithm is in essence a trimmed version of the algorithm of Garey et al. 3] that ....

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Toussaint GT, Avis D (1982) On a convex hull algorithm for polygons and its application to triangulation problems. Pattern Recognition. 15:23-29

.... C P ) where C P is a shape parameter no bigger than n that depends on the polygon P to be triangulated (for instance the number of reflex vertices [HM] FM] or the sinuosity of P [CI] On a different front an ever increasing class of polygons were shown to be triangulatable in linear time [TA82], T88] After a false start, Tarjan and Van Wyk [TV] made a major breakthrough with an O(n log log n) algorithm in 1986. This time bound was matched by a different but simpler algorithm by Kirkpatrick et al. KKT] three years later. In the mean time Clarkson et al. had published a randomized ....

Toussaint, G. and Avis, D. On a convex hull algorithm for polygons and its applications to triangulation problems, Pattern Recognition 15,1 (1982), 23--29.

....Let P be an n vertex simple polygon. For a point p and an object C in P , p is said to be weakly visible from C iff p is visible from some point on C (depending on p) Polygon P is said to be weakly visible from C iff every point p 2 P is weakly visible from C. Many sequential algorithms [1, 2, 3, 4, 8, 12, 15, 16, 17, 18, 20, 23, 24, 26, 29, 31, 33, 34, 35, 36] and parallel algorithms [9, 10, 11, 12, 13, 22, 25] for solving various weak visibility problems on simple polygons have been discovered. We consider the problem of computing the shortest weakly visible chain of a simple polygon (called it the SWVC problem) and the problem of computing a chain on ....

G. T. Toussaint and D. Avis. "On a convex hull algorithm for polygons and its applications to triangulation problems," Pattern Recognition, 15 (1) (1982), pp. 23--29.

....on the bay with f as the goal. However, we must first fix up the triangulation at the mouth of the bay. We need to triangulate the polygon defined by f and its left chain (or right chain) This polygon is weakly visible from f , and hence it can be triangulated in linear time by a simple algorithm [34]. Note that all the points in the triangles cut by f can reach f . At the next step, these triangles are strictly inside the non directional backprojection of f and thus can never be cut again. We run the algorithm of Section 3 on each bay, perhaps forming smaller bays. We repeat the process until ....

G. Toussaint and D. Avis, On a convex hull algorithm for polygons and its applications to triangulation problems, Pattern Recognition, 15 (1982), pp. 23--29.

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Toussaint, G. T. and Avis, D., "On a convex hull algorithm for polygons and its application to triangulation problems," Pattern Recognition, vol. 15, No. 1, 1982, pp.23-29.

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Toussaint, G.T. and Avis, D., "On a convex hull algorithm for polygons and its application to triangulation problems," Pattern Recognition, vol. 15, No. 1, 1982, pp. 23-29.

....edges, we are left with a planar subdivision of the interior of the convex hull of P consisting of triangles and at most two regions R 1 and R 2 on either side of s. The regions R 1 and R 2 are simple and weakly visible from s and can therefore be triangulated by the algorithm presented in [15]. A polygon P is weakly visible from an edge e if 8x 2 P; 9y 2 e such that the line segment xy is in P . R 1 R 2 T(L ) Insertion of s into T(L ) Re triangulation with s Figure 4: Illustration for proof of Lemma 3.4. A triangulated graph is a planar graph where every face is a triangle ....

....of the interior of the convex hull of P consisting of triangles and at most two regions R 1 and R 2 on either side of s. The regions R 1 and R 2 are simple and weakly visible from s. The regions can be triangulated by the algorithm presented below which is a modification of the algorithm TR POL in [15]. Given three points a; b; c in the plane, a; b; c) is a left turn if c lies to the left of ray(a; b) a right turn if c lies to the right of ray(a; b) a zero turn if c lies on ray(a; b) past b, and a 180 turn if c lies on ray(a; b) before b. We use the following notation in the algorithm. q ....

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G. TOUSSAINT AND D. AVIS. On a convex hull algorithm for polygons and its application to triangulation problems, Pattern Recognition, 15, 23--29, 1982.

....mouth and two ears. In fact, these notions suggest some interesting families of simple polygons. Recall that no O(n) time algorithm exists for triangulating an arbitrary simple polygon. However certain special classes of simple polygons such as star shaped ones do admit O(n) time triangulation [To2]. We now define another such class of polygons. Definition: A simple non convex polygon P is called a one mouth polygon provided it contains no more than one mouth. Definition: A simple polygon P is called a two ear polygon provided it contains no more than two ears. Definition: A simple ....

Toussaint, G. T. and Avis, D., "On a convex hull algorithm for polygons and its application to triangulation problems," Pattern Recognition, vol. 15, No. 1, 1982, pp.23-29.

....A simple polygon P is edge visible from an edge e of P if for every point x in P, there is a point y on e such that x and y are visible. Property (1) above of the unbounded region D is similar to edge visibility from a line at infinity. Edge visibility was introduced by Toussaint and Avis in [TA82], where they show that a Graham scan gives an easy algorithm to triangulate edge visible polygons; our algorithm is closely related to theirs. We will use some of their results in our proof of correctness. The proof of correctness of the sorting procedure will follow from lemmata 3.5 and 3.7 which ....

....i . r 1 r m 1 r 1 r m 1 e i lies on the chain from f i to r m 1 . Figure 3.3. Illustrating the statements of lemmata 3.3. and 3.4. 12of D can be joined to the line y = by a segment that lies entirely inside D. The statement of the lemma therefore follows directly from lemma 1 in [TA82]. Corollary 3.6.1. The content of the stack forms a convex chain of edges sorted in increasing order of orientation after each loop. Proof: Follows directly from the fact that the stack chain is made of left turns and that it is monotone with respect to the y axis. Corollary 3.6.2. If edge e i ....

Toussaint G.T., Avis D., "On a convex hull algorithm for polygons and its application to triangulation problems," Pattern Recognition, Vol. 15, No. 1, 1982, pp. 23-29.

....the algorithm runs in time O(nc) where c is the number of vertices on the convex hull found. However, as pointed out in [To83] the fact that a histogram is a very special type of polygon, namely a monotonic polygon allows us to compute the convex hull with a very simple O(n) time algorithm [TA82]. 2.1.2 Cluster Analysis One of the most powerful approaches to image segmentation that lends itself to the application of complicated images such as those of magazine or newspaper documents that contain textured pictures and diagrams as well as text blocks is the method of clustering and this ....

Toussaint, G. T. and Avis, D., "On a convex hull algorithm for polygons and its application to triangulation problems," Pattern Recognition, vol. 15, 1982, pp. 23-29.

....S he shows that this question can be answered in O(log n) time after the polygon is preprocessed in O(n log n) time using O(n) space. In this paper we focus on weak external visibility of a polygon. This topic is as yet quite unexplored compared to its internal counterpart. Toussaint and Avis [TA] considered the problem of determining if a polygon is weakly externally visible. Since we will restrict ourselves hereafter to notions of visibility that are both weak and external we will drop these adjectives for the sake of less cumbersome terminology) A polygon P is edge visible if for each ....

....adjectives for the sake of less cumbersome terminology) A polygon P is edge visible if for each point x P there exists a ray that supports P at x. This is equivalent to saying that P is visible from a circle at infinity (or, in fact, any circle that properly encloses P) Toussaint and Avis [TA], using related results of [AT] show that edge visibility of polygons can be recognized in O(n) time. This result is proved by showing that the edge visibility problem is equivalent to the somewhat less constrained vertex visibility problem: determine, for each vertex v P, if there exists a ray ....

[Article contains additional citation context not shown here]

Toussaint, G.T. and Avis, D., "On a convex hull algorithm for polygons and its application to triangulation problems," Pattern Recognition, vol. 15, No. 1, 1982, pp. 23-29.

....edges, we are left with a planar subdivision of the interior of the convex hull of P consisting of triangles and at most two regions R 1 and R 2 on either side of s. The regions R 1 and R 2 are simple and weakly visible from s and can therefore be triangulated by the algorithm presented in [15]. A polygon P is weakly visible from an edge e if 8x 2 P; 9y 2 e such that the line segment xy is in P . A triangulated graph is a planar graph where every face is a triangle except possibly the outer face. In [4] it is shown how to triangulate a set of disjoint line segments in O(n log n) ....

....the interior of the convex hull of P consisting of triangles and at most two regions R 1 and R 2 on either side of s. The regions R 1 and R 2 are simple and weakly visible from s. The regions can be triangulated by the algorithm presented below which is a modification of the algorithm TR POL in [15]. Given three points a; b; c in the plane, a; b; c) is a left turn if c lies to the left of ray(a; b) a right turn if c lies to the right of ray(a; b) a zero turn if c lies on ray(a; b) past b, and a 180 turn if c lies on ray(a; b) before b. We use the following notation in the algorithm. q ....

[Article contains additional citation context not shown here]

G. Toussaint and D. Avis. On a convex hull algorithm for polygons and its application to triangulation problems, Pattern Recognition, 15, 23--29, 1982.

....an algorithm of Rutovitz [Ru75] which runs in time O(n 2 ) where n is the number of grey levels. However, as pointed out in [To83] the fact that a histogram is a very special type of polygon, namely a monotonic polygon allows us to compute the convex hull with a very simple O(n) time algorithm [TA82]. 2.2 Cluster Analysis One of the most powerful approaches to image segmentation that lends itself to the application of complicated images such as those of outdoor scenes is the method of clustering and this is an area where a great deal of computational geometry can be readily applied. In this ....

Toussaint, G. T. and Avis, D., "On a convex hull algorithm for polygons and its application to triangulation problems," Pattern Recognition, vol. 15, 1982, pp. 23-29.

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