Minimax geometric fitting of two corresponding sets of points

Imai, Keisuke; Sumino, S.; Imai, Hitoshi

doi:10.1145/73833.73863

Cited by 23 publications

(19 citation statements)

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“…However, this particular problem, the optimal matching of two point-sets with a fixed mapping between them, was addressed in [22]. To explain the solution, observe firstly that applying a rotation ρ around the coordinate-center with an angle θ to the point qj(xqj, yqj) will transform it to: ρ θ (qj) = (xqj cos θ − yqj sin θ, xqj sin θ + yqj cos θ).…”

Section: Optimal Similarity Matchingmentioning

confidence: 99%

“…Subsequently, translating it by a vector τ = (xτ , yτ ), brings it to the point τ (ρ θ (qj)) = ((xqj cos θ − yqj sin θ) + xτ , (xqj sin θ + yqj cos θ) + yτ ) (c.f. [22]). Hence, our original problem of obtaining the RT SD(T r1…”

Section: Optimal Similarity Matchingmentioning

confidence: 99%

“…The problem of the minimum enclosing circle can be reduced to finding the minimum value on the envelope of a family of functions fj(θ , xτ , yτ ). Its relationship with the dynamic furthest Voronoi diagram [8] of the points (xqj cos θ − yqj sin θ) + xτ , (xqj sin θ + yqj cos θ) + yτ is presented in details in [22]. Essentially, the algorithm fixes a pair of indices s and l and solves the constrained problem of finding the θ which yields the minimum enclosing circle having ρ θ (qj), subject to having a pair of points ρ θ (qs) and ρ θ (q l ) on its boundary.…”

Section: Optimal Similarity Matchingmentioning

confidence: 99%

“…For a fixed pair l and s (c.f. Lemma 4.1 in [22]) the complexity of obtaining the maximum diagram of u functions of the kind fj(θ , xτ , yτ (u) log u). Repeating the search for the minimum enclosing circle for all the possible pairs s and l on its boundary (and picking the smallest one among them) yields a complexity of O(u 2 λ7(u) log u).…”

Section: Optimal Similarity Matchingmentioning

confidence: 99%

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Dynamics-aware similarity of moving objects trajectories

Trajcevski¹,

Scheuermann²,

Tamassia

et al. 2007

Proceedings of the 15th Annual ACM International Symposium on Advances in Geographic Information Systems

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This work addresses the problem of obtaining the degree of similarity between trajectories of moving objects. Typically, a Moving Objects Database (MOD) contains sequences of (location,time) points describing the motion of individual objects, however, they also have an implicit information about the velocity, which is an important attribute describing the dynamics of a particular object. Our main goal is to extend the MOD functionalities with the capability of reasoning about how similar are the trajectories of objects that, possibly, move along geographically different routes. Towards this, we use a distance function which balances the lack of temporal-awareness of the Hausdorff distance with the generality (and complexity of calculation) of the Fréchet distance. Based on the observation that, as a firstapproximation in practice, the individual segments of trajectories are assumed to have constant speed, we provide efficient algorithms for: (1) optimal matching between trajectories; and (2) approximate matching between trajectories, both under translations and rotations, where the approximate algorithm guarantees a bounded error-quality with respect to the optimal one.

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Section: Optimal Similarity Matchingmentioning

confidence: 99%

Section: Optimal Similarity Matchingmentioning

confidence: 99%

Section: Optimal Similarity Matchingmentioning

confidence: 99%

Section: Optimal Similarity Matchingmentioning

confidence: 99%

See 2 more Smart Citations

Dynamics-aware similarity of moving objects trajectories

Trajcevski¹,

Scheuermann²,

Tamassia

et al. 2007

Proceedings of the 15th Annual ACM International Symposium on Advances in Geographic Information Systems

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show abstract

“…Imai et al [27] show that these bounds can be reduced somewhat if an assignment of points in A to points in B is given. Similarly, Arkin et al [6] show that one can improve the running times in the approximate case if the "noise regions" are disjoint.…”

Section: Previous Workmentioning

confidence: 99%

Approximate geometric pattern matching under rigid motions

Goodrich

Mitchell

Orletsky

1999

IEEE Trans. Pattern Anal. Machine Intell.

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Extraction of geometrically similar substructures: Least-squares and Chebyshev fitting and the difference distance matrix

Lesk

1998

Proteins

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In analysis, comparison and classification of conformations of proteins, a common computational task involves extractions of similar substructures. Structural comparisons are usually based on either of two measures of similarity: the root-mean-square (r.m.s.) deviation upon optimal superposition, or the maximal element of the difference distance matrix. The analysis presented here clarifies the relationships between different measures of structural similarity, and can provide a basis for developing algorithms and software to extract all maximal common well-fitting substructures from proteins. Given atomic coordinates of two proteins, many methods have been described for extracting some substantial (if not provably maximal) common substructure with low r.m.s. deviation. This is a relatively easy task compared with the problem addressed here, i.e., that of finding all common substructures with r.m.s. deviation less than a prespecified threshold. The combinatorial problems associated with similar subset extraction are more tractable if expressed in terms of the maximal element of the difference distance matrix than in terms of the r.m.s. deviation. However, it has been difficult to correlate these alternative measures of structural similarity. The purpose of this article is to make this connection. We first introduce a third measure of structural similarity: the maximum distance between corresponding pairs of points after superposition to minimize this value. This corresponds to fitting in the Chebyshev norm. Properties of Chebyshev superposition are derived. We describe relationships between the r.m.s. and minimax (Chebyshev) deviations upon optimal superposition, and between the Chebyshev deviation and the maximal element of the difference distance matrix. Combining these produces a relationship between the r.m.s. deviation upon optimal superposition and the maximal element of the difference distance matrix. Based on these results, we can apply algorithms and software for finding subsets of the difference distance matrix for which all elements are less than a specified bound, either to select only subsets for which the r.m.s.deviation is less than or equal to a specified threshold, or to select subsets that include all subsets for which the r.m.s. deviation is less than or equal to a threshold.

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Minimax geometric fitting of two corresponding sets of points

Cited by 23 publications

References 0 publications

Dynamics-aware similarity of moving objects trajectories

Dynamics-aware similarity of moving objects trajectories

Approximate geometric pattern matching under rigid motions

Extraction of geometrically similar substructures: Least-squares and Chebyshev fitting and the difference distance matrix

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