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22
Detecting Commuting Patterns by Clustering Subtrajectories
, 2008
"... In this paper we consider the problem of detecting commuting patterns in a trajectory. For this we search for similar subtrajectories. To measure spatial similarity we choose the Fréchet distance and the discrete Fréchet distance between subtrajectories, which are invariant under differences in spee ..."
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Cited by 11 (6 self)
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In this paper we consider the problem of detecting commuting patterns in a trajectory. For this we search for similar subtrajectories. To measure spatial similarity we choose the Fréchet distance and the discrete Fréchet distance between subtrajectories, which are invariant under differences in speed. We give several approximation algorithms, and also show that the problem of finding the ‘longest’ subtrajectory cluster is as hard as MaxClique to compute and approximate.
Fréchet distance for curves, revisited
 In ESA
, 2006
"... Abstract. We revisit the problem of computing the Fréchet distance between polygonal curves, focusing on the discrete Fréchet distance, where only distance between vertices is considered. We develop efficient approximation algorithms for two natural classes of curves: κbounded curves and backbone c ..."
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Cited by 10 (5 self)
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Abstract. We revisit the problem of computing the Fréchet distance between polygonal curves, focusing on the discrete Fréchet distance, where only distance between vertices is considered. We develop efficient approximation algorithms for two natural classes of curves: κbounded curves and backbone curves, the latter of which are widely used to model molecular structures. We also propose a pseudo–outputsensitive algorithm for computing the discrete Fréchet distance exactly. The complexity of the algorithm is a function of the complexity of the freespace boundary, which is quadratic in the worst case, but tends to be lower in practice. 1
Stable Inverse Dynamic Curves
"... Figure 1: From left to right: The user sketches a smooth curve over the tail of a character. The curve is automatically converted into a dynamic rod model at stable equilibrium under gravity. The user can then animate the curve (e.g., pull then release it) with the guarantee thatthe choseninitial sh ..."
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Cited by 8 (2 self)
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Figure 1: From left to right: The user sketches a smooth curve over the tail of a character. The curve is automatically converted into a dynamic rod model at stable equilibrium under gravity. The user can then animate the curve (e.g., pull then release it) with the guarantee thatthe choseninitial shapewillbe preservedafter slight(orpossiblystrong) motion. See theaccompanying video forthe fullanimation. 2d animation is a traditional but fascinating domain that has recently regained popularity both in animated movies and video games. This paper introduces a method for automatically convertingasmoothsketchedcurveintoa2ddynamiccurveatstableequilibrium under gravity. The curve can then be physically animated to produce secondary motions in 2d animations or simple video games. Ourapproachproceedsintwosteps. Wefirstpresentanew techniquetofitasmoothpiecewisecirculararcscurvetoasketched curve. Then we show how to compute the physical parameters of a dynamicrodmodel(supercircle)sothatitsstablerestshapeunder gravity exactly matches the fitted circular arcs curve. We demonstratetheinteractivityandcontrollabilityofourapproachonvarious exampleswhereausercanintuitivelysetupefficientandprecise2d animations byspecifyingthe inputgeometry.
Protein structurestructure alignment with discrete Fréchet distance
 Proc. 5th AsiaPacific Bioinform. Conf. 131–141. Downloaded
, 2007
"... Matching two geometric objects in 2D and 3D spaces is a central problem in computer vision, pattern recognition and protein structure prediction. In particular, the problem of aligning two polygonal chains under translation and rotation to minimize their distance has been studied using various dista ..."
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Cited by 4 (3 self)
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Matching two geometric objects in 2D and 3D spaces is a central problem in computer vision, pattern recognition and protein structure prediction. In particular, the problem of aligning two polygonal chains under translation and rotation to minimize their distance has been studied using various distance measures. It is well known that the Hausdorff distance is useful for matching two point sets, and that the Fréchet distance is a superior measure for matching two polygonal chains. The discrete Fréchet distance closely approximates the (continuous) Fréchet distance, and is a natural measure for the geometric similarity of the folded 3D structures of biomolecules such as proteins. In this paper, we present new algorithms for matching two polygonal chains in 2D to minimize their discrete Fréchet distance under translation and rotation, and an effective heuristic for matching two polygonal chains in 3D. We also describe our empirical results on the application of the discrete Fréchet distance to the protein structurestructure alignment. 1.
Protein Local Structure Alignment Under the Discrete Fréchet Distance
"... Protein structure alignment is a fundamental problem in computational and structural biology. While there has been lots of experimental/heuristic methods and empirical results, very few results are known regarding the algorithmic/complexity aspects of the problem, especially on protein local structu ..."
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Cited by 4 (1 self)
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Protein structure alignment is a fundamental problem in computational and structural biology. While there has been lots of experimental/heuristic methods and empirical results, very few results are known regarding the algorithmic/complexity aspects of the problem, especially on protein local structure alignment. A wellknown measure to characterize the similarity of two polygonal chains is the famous Fréchet distance, and with the application of proteinrelated research, a related discrete Fréchet distance has been used recently. In this paper, following the recent work of Jiang et al. we investigate the protein local structural alignment problem using bounded discrete Fréchet distance. Given m proteins (or protein backbones, which are 3D polygonal chains), each of length O.n/, our main results are summarized as follows: If the number of proteins, m, is not part of the input, then the problem is NPcomplete; moreover, under bounded discrete Fréchet distance it is NPhard to approximate the maximum size common local structure within a factor of n 1. These results hold both when all the proteins are static and when translation/rotation are allowed. If the number of proteins, m, is a constant, then there is a polynomial time solution for the problem. Key words: approximation, discrete Fréchet distance, Fréchet distance, NPhardness, protein structure alignment.
Faster Retrieval with a TwoPass DynamicTimeWarping Lower Bound
, 2009
"... The Dynamic Time Warping (DTW) is a popular similarity measure between time series. The DTW fails to satisfy the triangle inequality and its computation requires quadratic time. Hence, to find closest neighbors quickly, we use bounding techniques. We can avoid most DTW computations with an inexpensi ..."
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Cited by 4 (0 self)
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The Dynamic Time Warping (DTW) is a popular similarity measure between time series. The DTW fails to satisfy the triangle inequality and its computation requires quadratic time. Hence, to find closest neighbors quickly, we use bounding techniques. We can avoid most DTW computations with an inexpensive lower bound (LB Keogh). We compare LB Keogh with a tighter lower bound (LB Improved). We find that LB Improvedbased search is faster. As an example, our approach is 2–3 times faster over randomwalk and shape time series.
HOMOTOPIC FRÉCHET DISTANCE BETWEEN CURVES OR, WALKING YOUR DOG IN THE WOODS IN POLYNOMIAL TIME
, 2008
"... The Fréchet distance between two curves in the plane is the minimum length of a leash that allows a dog and its owner to walk along their respective curves, from one end to the other, without backtracking. We propose a natural extension of Fréchet distance to more general metric spaces, which requi ..."
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Cited by 3 (0 self)
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The Fréchet distance between two curves in the plane is the minimum length of a leash that allows a dog and its owner to walk along their respective curves, from one end to the other, without backtracking. We propose a natural extension of Fréchet distance to more general metric spaces, which requires the leash itself to move continuously over time. For example, for curves in the punctured plane, the leash cannot pass through or jump over the obstacles (“trees”). We describe a polynomialtime algorithm to compute the homotopic Fréchet distance between two given polygonal curves in the plane minus a given set of polygonal obstacles.
Computing the discrete Fréchet distance with imprecise input
, 2010
"... We consider the problem of computing the discrete Fréchet distance between two polygonal curves when their vertices are imprecise. An imprecise point is given by a region and this point could lie anywhere within this region. By modelling imprecise points as balls in dimension d, we present an algori ..."
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Cited by 2 (1 self)
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We consider the problem of computing the discrete Fréchet distance between two polygonal curves when their vertices are imprecise. An imprecise point is given by a region and this point could lie anywhere within this region. By modelling imprecise points as balls in dimension d, we present an algorithm for this problem that returns in time 2 O(d2) m 2 n 2 log 2 (mn) the Fréchet distance lower bound between two imprecise polygonal curves with n and m vertices, respectively. We give an improved algorithm for the planar case with running time O(mn log 2 (mn)+(m 2 +n 2) log(mn)). In the ddimensional orthogonal case, where points are modelled as axisparallel boxes, and we use the L∞ distance, we give an O(dmn log(dmn))time algorithm. We also give efficient O(dmn)time algorithms to approximate the Fréchet distance upper bound, as well as the smallest possible Fréchet distance lower/upper bound that can be achieved between two imprecise point sequences when one is allowed to translate them. These algorithms achieve constant factor approximation ratios in “realistic ” settings (such as when the radii of the balls modelling the imprecise points are roughly of the same size).
Computing the Discrete Fréchet Distance in Subquadratic Time ∗
, 2012
"... The Fréchet distance is a similarity measure between two curves ..."
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Cited by 2 (1 self)
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The Fréchet distance is a similarity measure between two curves
Voronoi Diagram of Polygonal Chains Under the Discrete Fréchet Distance
"... Polygonal chains are fundamental objects in many applications like pattern recognition and protein structure alignment. A wellknown measure to characterize the similarity of two polygonal chains is the famous (continuous/discrete) Fréchet distance. In this paper, for the first time, we consider the ..."
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Cited by 1 (0 self)
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Polygonal chains are fundamental objects in many applications like pattern recognition and protein structure alignment. A wellknown measure to characterize the similarity of two polygonal chains is the famous (continuous/discrete) Fréchet distance. In this paper, for the first time, we consider the Voronoi diagram of polygonal chains in ddimension under the discrete Fréchet distance. Given a set C of n polygonal chains in ddimension, each with at most k vertices, we prove fundamental properties of such a Voronoi diagram VDF (C). Our main results are summarized as follows. • The combinatorial complexity of VDF (C) is at most O(ndk+ɛ). • The combinatorial complexity of VDF (C) is at least Ω(ndk) for dimension d = 1, 2; and Ω(nd(k−1)+2) for dimension d> 2.