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4Points Congruent Sets for Robust Pairwise Surface Registration
 INTERNATIONAL CONFERENCE ON COMPUTER GRAPHICS AND INTERACTIVE TECHNIQUES
, 2008
"... We introduce 4PCS, a fast and robust alignment scheme for 3D point sets that uses wide bases, which are known to be resilient to noise and outliers. The algorithm allows registering raw noisy data, possibly contaminated with outliers, without prefiltering or denoising the data. Further, the method ..."
Abstract

Cited by 16 (1 self)
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We introduce 4PCS, a fast and robust alignment scheme for 3D point sets that uses wide bases, which are known to be resilient to noise and outliers. The algorithm allows registering raw noisy data, possibly contaminated with outliers, without prefiltering or denoising the data. Further, the method significantly reduces the number of trials required to establish a reliable registration between the underlying surfaces in the presence of noise, without any assumptions about starting alignment. Our method is based on a novel technique to extract all coplanar 4points sets from a 3D point set that are approximately congruent, under rigid transformation, to a given set of coplanar 4points. This extraction procedure runs in roughly O(n2 + k) time, where n is the number of candidate points and k is the number of reported 4points sets. In practice, when noise level is low and there is sufficient overlap, using local descriptors the time complexity reduces to O(n + k). We also propose an extension to handle similarity and affine transforms. Our technique achieves an order of magnitude asymptotic acceleration compared to common randomized alignment techniques. We demonstrate the robustness of our algorithm on several sets of multiple range scans with varying degree of noise, outliers, and extent of overlap.
Incidences between points and circles in three and higher dimensions
 Geom
, 2002
"... We show that the number of incidences between m distinct points and n distinct circles in R d, for any d ≥ 3, is O(m 6/11 n 9/11 κ(m 3 /n)+m 2/3 n 2/3 +m+n), where κ(n) = (log n) O(α2 (n)) and where α(n) is the inverse Ackermann function. The bound coincides with the recent bound of Aronov and Shar ..."
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Cited by 11 (7 self)
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We show that the number of incidences between m distinct points and n distinct circles in R d, for any d ≥ 3, is O(m 6/11 n 9/11 κ(m 3 /n)+m 2/3 n 2/3 +m+n), where κ(n) = (log n) O(α2 (n)) and where α(n) is the inverse Ackermann function. The bound coincides with the recent bound of Aronov and Sharir, or rather with its slight improvement by Agarwal et al., for the planar case. We also show that the number of incidences between m points and n unrestricted convex (or boundeddegree algebraic) plane curves, no two in a common plane, is O(m 4/7 n 17/21 + m 2/3 n 2/3 + m + n), in any dimension d ≥ 3. Our results improve the upper bound on the number of congruent copies of a fixed tetrahedron in a set of n points in 4space and the lower bound for the number of distinct distances in a set of n points in 3space. Another application is an improved bound for the number of incidences (or, rather, containments) between lines and reguli in three dimensions. The latter result has already been applied by Feldman and Sharir to obtain a new bound on the number of joints in an arrangement of lines in three dimensions.
Similar Simplices in a dDimensional Point Set
, 2007
"... We consider the problem of bounding the maximum possible number fk,d(n) of ksimplices that are spanned by a set of n points in R d and are similar to a given simplex. We first show that f2,3(n) = O(n 13/6), and then tackle the general case, and show that fd−2,d(n) = O(n d−8/5) and 1 fd−1,d(n) = O ..."
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Cited by 4 (2 self)
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We consider the problem of bounding the maximum possible number fk,d(n) of ksimplices that are spanned by a set of n points in R d and are similar to a given simplex. We first show that f2,3(n) = O(n 13/6), and then tackle the general case, and show that fd−2,d(n) = O(n d−8/5) and 1 fd−1,d(n) = O ∗ (n d−72/55), for any d. Our technique extends to derive bounds for other values of k and d, and we illustrate this by showing that f2,5(n) = O(n 8/3).
An Efficient Approximation Algorithm for Point Pattern Matching Under Noise
 The 7th International Symposium, Latin American Theoretical Informatics (LATIN 2006). Lecture
"... Point pattern matching problems are of fundamental importance in various areas including computer vision and structural bioinformatics. In this paper, we study one of the more general problems, known as LCP (largest common point set problem): Let P and Q be two point sets in R 3, and let ɛ ≥ 0 be a ..."
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Cited by 3 (1 self)
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Point pattern matching problems are of fundamental importance in various areas including computer vision and structural bioinformatics. In this paper, we study one of the more general problems, known as LCP (largest common point set problem): Let P and Q be two point sets in R 3, and let ɛ ≥ 0 be a tolerance parameter, the problem is to find a rigid motion µ that maximizes the cardinality of subset I of Q, such that the Hausdorff distance dist(P, µ(I)) ≤ ɛ. We denote the size of the optimal solution to the above problem by LCP(P, Q). The problem is called exactLCP for ɛ = 0, and tolerantLCP when ɛ> 0 and the minimum interpoint distance is greater than 2ɛ. A βdistanceapproximation algorithm for tolerantLCP finds a subset I ⊆ Q such that I  ≥ LCP(P, Q) and dist(P, µ(I)) ≤ βɛ for some β ≥ 1. This paper has three main contributions. (1) We introduce a new algorithm, called DIHEDA, which gives the fastest known deterministic 4distanceapproximation algorithm for tolerantLCP. (2) For the exactLCP, when the matched set is required to be large, we give a simple sampling strategy that improves the running times of all known deterministic algorithms, yielding the fastest known deterministic algorithm for this problem. (3) We use expander graphs to speedup the DIHEDA algorithm for tolerantLCP when the size of the matched set is required to be large, at the expense of approximation in the matched set size. Our algorithms also work when the transformation µ is allowed to be scaling transformation.
Combinatorial Geometry Problems in Pattern Recognition
"... In the following I consider combinatorial geometry problems motivated by point pattern matching algorithms, and discuss the classical exact matching situation and several variants. ..."
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Cited by 1 (0 self)
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In the following I consider combinatorial geometry problems motivated by point pattern matching algorithms, and discuss the classical exact matching situation and several variants.
How many unit equilateral triangles can be generated by n points in convex position?
 Amer. Math. Monthly
"... What is the maximum number of times that the unit distance can occur among the distances between n points in the plane? This more than ftyyearold question of Paul Erdös, published in this MONTHLY [5], opened a whole new area of research in combinatorial geometry [10]. ..."
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Cited by 1 (1 self)
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What is the maximum number of times that the unit distance can occur among the distances between n points in the plane? This more than ftyyearold question of Paul Erdös, published in this MONTHLY [5], opened a whole new area of research in combinatorial geometry [10].
How Many Unit Equilateral Triangles Can Be Induced by n Points in Convex Position?
, 2001
"... Any set of n points in strictly convex position in the plane has at most b c triples that induce equilateral triangles of side length one. This bound cannot be improved. The case of general triangles is also discussed. ..."
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Any set of n points in strictly convex position in the plane has at most b c triples that induce equilateral triangles of side length one. This bound cannot be improved. The case of general triangles is also discussed.
Convex polyhedra in R³ spanning . . .
"... We construct nvertex convex polyhedra with the property stated in the title ..."
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We construct nvertex convex polyhedra with the property stated in the title
IIT Delhi
"... Figure 1: Reconstruction from raw scans using 4points congruent sets. Reconstruction results from nine input scans of a shinny water jug. Neighboring scans have 40 % overlap or less, and required an average of 16 seconds for fully automatic alignment starting from arbitrary initial poses. Pairwise ..."
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Figure 1: Reconstruction from raw scans using 4points congruent sets. Reconstruction results from nine input scans of a shinny water jug. Neighboring scans have 40 % overlap or less, and required an average of 16 seconds for fully automatic alignment starting from arbitrary initial poses. Pairwise alignment results are robust even with low overlap. Typical pairwise alignments are shown in (a) and (b), where for visualization we roughly mark the overlap regions in blue. The final alignment result, (c) and (d), is obtained without any data smoothing, outlier removal, local ICP refinement, global error distribution, or any assumption about starting alignment. We introduce 4PCS, a fast and robust alignment scheme for 3D point sets that uses wide bases, which are known to be resilient to noise and outliers. The algorithm allows registering raw noisy data, possibly contaminated with outliers, without prefiltering or denoising the data. Further, the method significantly reduces the number of trials required to establish a reliable registration between the underlying surfaces in the presence of noise, without any assumptions about starting alignment. Our method is based on a novel technique to extract all coplanar 4points sets from a 3D point set that are approximately congruent, under rigid transformation, to a given set of coplanar 4points. This extraction procedure runs in roughly O(n 2 + k) time, where n is the number of candidate points and k is the number of reported 4points sets. In practice, when noise level is low and there is sufficient overlap, using local descriptors the time complexity reduces to O(n + k). We also propose an extension to handle similarity and affine transforms. Our technique achieves an order of magnitude asymptotic acceleration compared to common randomized alignment techniques. We demonstrate the robustness of our algorithm on several sets of multiple range scans with varying degree of noise, outliers, and extent of overlap.
Thashing: An Efficient Approximation Algorithm for Point Pattern Matching Under Noise
, 2005
"... Point pattern matching problems are of fundamental importance in various areas including computer vision and structural bioinformatics. In this paper, we study one of the more general problems, known as tolerantLCP (largest common point set problem): given two point sets M and Q in R 3, and a toler ..."
Abstract
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Point pattern matching problems are of fundamental importance in various areas including computer vision and structural bioinformatics. In this paper, we study one of the more general problems, known as tolerantLCP (largest common point set problem): given two point sets M and Q in R 3, and a tolerance parameter ǫ> 0, the problem is to find a rigid motion µ such that the cardinality of subset I = LCP(M, Q) of Q, for which the Hausdorff distance dist(µ(I), M) ≤ ǫ, is maximized. A distanceapproximation algorithm for tolerantLCP finds a subset I such that I > LCP(M, Q)  and dist(µ(I), M) ≤ ǫ ′ for some ǫ ′ ≥ ǫ. In this paper, we introduce a new algorithm, called Thashing, which gives the fastest known deterministic distanceapproximation algorithm for tolerantLCP. Further, we use expander graphs to speedup the Thashing algorithm for tolerantLCP with large matched set at the expense of approximation in the matched set size.