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Solving geometric problems with the rotating calipers
, 1983
"... Shamos [1] recently showed that the diameter of a convex nsided polygon could be computed in O(n) time using a very elegant and simple procedure which resembles rotating a set of calipers around the polygon once. In this paper we show that this simple idea can be generalized in two ways: several se ..."
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Cited by 112 (14 self)
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Shamos [1] recently showed that the diameter of a convex nsided polygon could be computed in O(n) time using a very elegant and simple procedure which resembles rotating a set of calipers around the polygon once. In this paper we show that this simple idea can be generalized in two ways: several sets of calipers can be used simultaneously on one convex polygon, or one set of calipers can be used on several convex polygons simultaneously. We then show that these generalizations allow us to obtain simple O(n) algorithms for solving a variety of problems defined on convex polygons. Such problems include (1) finding the minimumarea rectangle enclosing a polygon, (2) computing the maximum distance between two polygons, (3) performing the vectorsum of two polygons, (4) merging polygons in a convex hull finding algorithms, and (5) finding the critical support lines between two polygons. Finding the critical support lines, in turn, leads to obtaining solutions to several additional problems concerned with visibility, collision, avoidance, range fitting, linear separability, and computing the Grenander distance between sets. 1.
Modeling Image Analysis Problems Using Markov Random Fields
, 2000
"... this article are addressed mainly from the computational viewpoint. The primary concerns are how to dene an objective function for the optimal solution for an image analysis problem and how to nd the optimal solution. The reason for dening the solution in an optimization sense is due to various unce ..."
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Cited by 6 (1 self)
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this article are addressed mainly from the computational viewpoint. The primary concerns are how to dene an objective function for the optimal solution for an image analysis problem and how to nd the optimal solution. The reason for dening the solution in an optimization sense is due to various uncertainties in imaging processes. It may be dicult to nd the perfect solution, so we usually look for an optimal one in the sense that an objective, into which constraints are encoded, is optimized
Expectations of Random Sets and Their Boundaries Using Oriented Distance Functions
, 2009
"... Shape estimation and object reconstruction are common problems in image analysis. Mathematically, viewing objects in the image plane as random sets reduces the problem of shape estimation to inference about sets. Currently existing definitions of the expected set rely on different criteria to constr ..."
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Cited by 1 (1 self)
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Shape estimation and object reconstruction are common problems in image analysis. Mathematically, viewing objects in the image plane as random sets reduces the problem of shape estimation to inference about sets. Currently existing definitions of the expected set rely on different criteria to construct the expectation. This paper introduces new definitions of the expected set and the expected boundary, based on oriented distance functions. The proposed expectations have a number of attractive properties, including inclusion relations, convexity preservation and equivariance with respect to rigid motions. The paper introduces a special class of separable oriented distance functions for parametric sets and gives the definition and properties of separable random closed sets. Further, the definitions of the empirical mean set and the empirical mean boundary are proposed and empirical evidence of the consistency of the boundary estimator is presented. In addition, the paper gives loss functions for set inference in frequentist framework and shows how some of the existing expectations arise naturally as optimal estimators. The proposed definitions of the set and boundary expectations are illustrated on theoretical examples and real data. 1