Interactive reconstruction via geometric probing

Skiena, Steven

doi:10.1109/5.163406

Cited by 37 publications

(18 citation statements)

References 69 publications

(114 reference statements)

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“…Efficient algorithms for probing convex polytopes have been the subject of several papers, starting with Cole and Yap [7], who studied the complexity (in terms of number of probes required) of Determining the Shape of an Unknown Convex Polygon by using probes which travel along a straight line chosen by the algorithm and stop when they collide with the polygon (later referred to as finger probes [1,8,9]). A number of probe types and algorithms were presented by Dobkin et al [9].…”

Section: Introductionmentioning

confidence: 99%

“…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%

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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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Section: Introductionmentioning

confidence: 99%

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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“…That 3n -3 x-ray probes are necessary follows from Lindenbaum and Bruckstein's [14] lower bound for determination where two opposing finger probes on a line I count as a single probe, since P fq I can be computed from the two contact points. Geometric probing problems were first introduced by Cole and Yap [1], and have since inspired a significant literature, as surveyed in [20] [23].…”

Section: Introductionmentioning

confidence: 99%

Reconstructing polygons from x-rays

Meijer

Skiena

1996

Geom Dedicata

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We present strategies for interactively reconstructing polygons from carefully chosen x-ray probes, generalizing previous results for convex polygons to a significantly larger class of objects. In particular, we show that n -4-h + 2 parallel x-ray probes are sufficient to determine an n-gon P with h vertices on its convex hull, provided no three vertices of P are collinear. If given an upper bound n / on the number of vertices of P, then 2n' + 2 parallel probes or 3n' origin probes suffice. Further, we show that [lg n -2J probes are necessary. Finally, we present verification strategies for arbitrary polygons. Interactive probing strategies have the potential to minimize radiation exposure in medical imaging.Mathematics Subject Classifications (1991): 51M20, 52C05, 51N20.

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“…A pioneering paper, by Cole and Yap [10], studied probes which travel along a straight line chosen by the algorithm and stop when they collide with the polygon (later referred to as finger probes [1], [11], [12], [13]). Different probe types and algorithms were presented by Dobkin et al [12], and generalized to higher-dimensional cases.…”

mentioning

confidence: 99%

“…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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Interactive reconstruction via geometric probing

Cited by 37 publications

References 69 publications

An efficient proximity probing algorithm for metrology

An efficient proximity probing algorithm for metrology

Reconstructing polygons from x-rays

Efficient Proximity Probing Algorithms for Metrology

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