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A Recursive Algorithm for Calculating the Relative Convex Hull Gisela Klette AUT University Computing & Mathematical Sciences Auckland, New Zealand
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Motivation The calculation of relative convex hulls of simple polygons is –a special subject in computational geometry (shortest paths), –in image analysis (MLP: calculation of features), –in robotics (shortest path of a robot in a constrained environment),...
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Motivation: Image Analysis MLP is a digital length estimator of the circumference of a digital object that is multigrid convergent MLP is characteristic for the digital convexity of the shape MLP is a tangent estimator
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Motivation: Robotics Path Planning Informally: The relative convex hull of A relative to B is the shortest path between A and B. A B
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Definition 1 A subset S in R 2 of points is convex iff S is equal to the intersection of all half planes containing S. The convex hull CH(S) of a set of points S is the smallest (by area) convex polygon P that contains S. S CH(S)
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A cavity of a polygon A is the topological closure of any connected component of CH(A) \ A. Definition 2 Polygon with 4 cavities CAV 1 CAV 2 CAV 3 CAV 4
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Definition 3 A cover is a straight line segment in the frontier of CH(A) that is not part of the frontier of A. cover 4
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Definition 4 A polygon A is B-convex iff any straight line segment in B that has both end points in A, is also contained in A. The convex hull of A relatively to B is the intersection of all B-convex polygons containing A.
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Definition 5 The minimum length polygon (MLP) of a 2D digital object coincides with the relative convex hull of an inner grid polygon relatively to an outer grid polygon, normally defined in a way like simulating an inner and outer Jordan digitization.
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Properties of MLP - The inner and the outer polygon of a Jordan digitization have the constraint that they are at Hausdorff distance 1. -Mappings exist between vertices and between cavities in A and in B. -Convex vertices of the inner polygon and concave vertices of the outer polygon are candidates for the MLP. -MLP is uniquely defined for a given digitized object
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p Algorithm, general case [Toussaint, G.T.: An optimal algorithm for computing the relative convex hull of a set of points in a polygon. In: EURASIP, Signal processing lll: Theories and Applications, Part 2, pages 853–856, North-Holland, 1986.] 1.Find an extreme vertex p in A 2.Construct a polygon B\A with one newly created double-oriented edge between A and B 3.Triangulate the new polygon 4.Find the shortest path from p to p Time Complexity: O(n log log n) Computation of the relative convex hull:
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Algorithm, MLP [Klette, R., Kovalevsky, V.V., Yip., B.: Length estimation of digital curves. In: Vision Geometry, SPIE 3811, pages 117–129, 1999.] [Klette, R., Rosenfeld, A.: Digital Geometry. Morgan Kaufmann, San Francisco, 2004.] 1.Trace frontier of A and include concave vertices of B 2.Start at extreme vertex 3.Compute positive and negative sides 4.As long as next vertex is between positive and negative sides update the sides 5.Otherwise a new vertex for CH B (A) has been found Time Complexity: O(n) Computation of MLP:
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Tangential cover Convex hull of one zone Arithmetic MLP (AMLP) [Provencal, X., Lachaud, J.-O.: Two linear-time algorithms for computing the minimum length polygon of a digital contour. In: DGCI 2009, LNCS 5810, pages 104–117, Springer, Heidelberg, 2009.] 1.Input: contour words (Freeman code) 2.Compute tangential cover 3.Decompose into zones (convex, concave, inflexion) 4.Compute convex hull for each polyline per zone Time Complexity: O(n), only for polyominos Computation of MLP:
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[Provencal, X., Lachaud, J.-O.: Two linear-time algorithms for computing the minimum length polygon of a digital contour. In: DGCI 2009, LNCS 5810, pages 104–117, Springer, Heidelberg, 2009.] Combinatorial MLP (CMLP) CMLP Digital contour Time Complexity: O(n), only for polyominos
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Result 1 The B-convex hull of a simple polygon A is equal to the convex hull of A iff the convex hull of A is completely contained in B. A B A B B is concave B is convex
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Result 2 All vertices of the convex hull of a simple polygon A inside a simple polygon B are vertices of the B-convex hull of A.
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Result 3 All the vertices of the convex hull of I new belong to the relative convex hull of A between p s and p e. I new = {p s, q 3, q 4,…q 11, p e }
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New algorithm The relative convex hull CHB(A) for simple polygons A and B is only different from CH(A) if there is at least one cavity in A and one in B such that the intersection of those cavities is not empty.
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New recursive algorithm 1.Compute the convex hulls for the inner and the outer polygon (Melkman algorithm for example) 2.Copy vertices of the inner polygon one by one to the CH B (A) until it finds a cavity 3.Check the outer polygon for a cavity 4.Construct the new polygons and find the convex hulls 5.Copy all vertices of the inner polygon to the CH B (A) 6.Stop if the base case of the recursion is reached.
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New recursive algorithm Relative convex hull Covers First cavity Second cavity
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Discussion 1.Recursive algorithm works for arbitrary simple polygons A and B in O(n 2 ), worst case. 2.It is a recursive procedure that is very simple and of low time complexity. 3.The algorithm runs in linear time if the maximum depth of stacked cavities of A is limited by a constant. We continue to study the expected time complexity of the algorithm under some general assumptions of variations (i.e., distribution) for possible input polygons A and B. 4.The depth of stacked cavities could be used as a shape descriptor.
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