Reconstructing polygons from x-rays

Meijer, Henk; Skiena, Steven

doi:10.1007/bf00151583

Cited by 7 publications

(6 citation statements)

References 19 publications

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“…Geometric probing was introduced by Cole and Yap [CY87] in 1983. Many probing tools, together with reconstruction algorithms, have been developed -finger probes [CY87], hyperplane (or line) probes [DEY86,Li88], diameter probes [RG94], x-ray probes [ES88,Gar92], histogram (or parallel x-ray) probes [MS96], half-plane probes [Ski91], composite probes [BL91,Li88,Ski88] among others. Another closely related area, called geometric testing, studies a verification problem: find a set of probes to determine if a given set of geometric objects contains one which is equivalent to a certain query object (see [Rom95] for a survey).…”

Section: Introductionmentioning

confidence: 99%

“…x-ray probes [ES88,Gar92], histogram (or parallel x-ray) probes [MS96], half-plane probes [Ski91], composite probes [BL91,Li88,Ski88] among others. Another closely related area, called geometric testing, studies a verification problem: find a set of probes to determine if a given set of geometric objects contains one which is equivalent to a certain query object (see [Rom95] for a survey).…”

Section: Introductionmentioning

confidence: 99%

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Probing convex polygons with a wedge

Bose¹,

Carufel²,

Shaikhet³

et al. 2016

Computational Geometry

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Abstract. Minimizing the number of probes is one of the main challenges in reconstructing geometric objects with probing devices. In this paper, we investigate the problem of using an ω-wedge probing tool to determine the exact shape and orientation of a convex polygon. An ω-wedge consists of two rays emanating from a point called the apex of the wedge and the two rays forming an angle ω. To probe with an ω-wedge, we set the direction that the apex of the probe has to follow, the line − → L , and the initial orientation of the two rays. A valid ω-probe of a convex polygon O contains O within the ω-wedge and its outcome consists of the coordinates of the apex, the orientation of both rays and the coordinates of the closest (to the apex) points of contact between O and each of the rays. We present algorithms minimizing the number of probes and prove their optimality. In particular, we show how to reconstruct a convex n-gon (with all internal angles of size larger than ω) using 2n − 2 ω-probes; if ω = π/2, the reconstruction uses 2n − 3 ω-probes. We show that both results are optimal. Let NB be the number of vertices of O whose internal angle is at most ω, (we show that 0 ≤ NB ≤ 3). We determine the shape and orientation of a general convex n-gon with NB = 1 (respectively NB = 2, NB = 3) using 2n − 1 (respectively 2n + 3, 2n + 5) ω-probes. We prove optimality for the first case. Assuming the algorithm knows the value of NB in advance, the reconstruction of O with NB = 2 or NB = 3 can be achieved with 2n + 2 probes,-which is optimal.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Probing convex polygons with a wedge

Bose¹,

Carufel²,

Shaikhet³

et al. 2016

Computational Geometry

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“…These include the finger probes previously studied by Cole and Yap; hyperplane probes, which consist of a hyperplane (whose angle is chosen by the algorithm) which sweeps over the whole space and stops when it collides with the polygon; and silhouette probes (also called projection probes [13]), which provide the projection of the polygon onto a chosen subspace. Other probes which have been studied for convex polygons include x-ray probes ( [1,8,12]), which measure the length of intersection between a chosen line and the unknown polygon, and half-plane probes [14], which measure the area of intersection between a chosen half-plane and the unknown polygon.…”

Section: Introductionmentioning

confidence: 99%

An efficient proximity probing algorithm for metrology

Panahi

Adler

Stappen

et al. 2013

2013 IEEE International Conference on Automation Science and Engineering (CASE)

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Metrology, the theoretical and practical study of measurement, has applications in automated manufacturing, inspection, robotics, surveying, and healthcare. An important problem within metrology is how to interactively use a measuring device, or probe, to determine some geometric property of an unknown object; this problem is known as geometric probing. In this paper, we study a type of proximity probe which, given a point, returns the distance to the boundary of the object in question. We consider the case where the object is a convex polygon P in the plane, and the goal of the algorithm is to minimize the upper bound on the number of measurements necessary to exactly determine P . We show an algorithm which has an upper bound of 3.5n + k + 2 measurements necessary, where n is the number of vertices and k ≤ 3 is the number of acute angles of P . Furthermore, we show that our algorithm requires O(1) computations per probe, and hence O(n) time to determine P .

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“…These include the finger probes previously studied by Cole and Yap; hyperplane probes, which consist of a hyperplane (whose angle is chosen by the algorithm) which sweeps over the whole space and stops when it collides with the polygon; and silhouette probes (also called projection probes [17]), which provide the projection of the polygon onto a chosen subspace. Other probes which have been studied for convex polygons include x-ray probes ( [1], [11], [16]), which measure the length of intersection between a chosen line and the unknown polygon, and half-plane probes [18], which measure the area of intersection between a chosen half-plane and the unknown polygon. A related problem, identifying a convex polygon from a known set, was considered by Goldberg and Rao in [15] using diameter probes.…”

mentioning

confidence: 99%

Efficient Proximity Probing Algorithms for Metrology

Adler

Panahi

Stappen

et al. 2015

IEEE Trans. Automat. Sci. Eng.

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Abstract-Metrology, the theoretical and practical study of measurement, has applications in automated manufacturing, inspection, robotics, surveying, and healthcare. The geometric probing problem considers how to optimally use a probe to measure geometric properties. In this paper, we consider a proximity probe which, given a point, returns the distance to the boundary of the nearest object. When there is an unknown convex polygon P in the plane, the goal is to minimize the number of probe measurement needed to exactly determine the shape and location of P . We present an algorithm with upper bound of 3.5n + k + 2 probes, where n is the number of vertices and k ≤ 3 is the number of acute angles of P . The algorithm requires constant time per probe, and hence O(n) time to determine P . We also address the related problem where the unknown polygon is a member of a known finite set Γ and the goal is to efficiently determine which polygon is present. When m is the size of Γ and n is the maximum number of vertices of any member of Γ, we present an O(n m) algorithm with an upper bound of 2n + 2 probes.Note to Practitioners: Abstract-This paper was inspired by the problem of using a sensor with low-dimensional output, such as a range sensor, to determine the shape of an object, which is typically too complex for a single measurement to characterize. Existing approaches to shape measurement generally use sensors with high-dimensional output, such as cameras, or use lowdimensional sensors to measure fixed points, such as with Scanning Probe Microscopy (SPM). It is shown here that by employing an algorithm that uses the results of previous measurements to determine how the next measurement is taken, a non-directional range sensor can be used to efficiently and exactly determine the shape of a convex polygon. This suggests an alternative approach to obtaining information on the shape of an object in cases where low-dimensional sensors are more accurate, faster, or cheaper than their counterparts.

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Reconstructing polygons from x-rays

Cited by 7 publications

References 19 publications

Probing convex polygons with a wedge

Probing convex polygons with a wedge

An efficient proximity probing algorithm for metrology

Efficient Proximity Probing Algorithms for Metrology

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