Linearly Converging Quasi Branch and Bound Algorithms for Global Rigid Registration

Dym, Nadav; Kovalsky, Shahar Z.

doi:10.1109/iccv.2019.00171

Cited by 15 publications

(15 citation statements)

References 32 publications

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“…Full separation with reasonable complexity for the joint action of rotations and permutations on d × n point clouds should be achievable when d is some fixed small number (recall that typically d = 3). See [14,15] for a discussion of the dependence of the complexity of point cloud tasks on the dimension d. Another interesting question is understanding whether the 2D M + 1 cardinality we require to obtain separating invariants is optimal. As mentioned above, in phase retrieval it is known that this number can be improved, and we believe this is the case for the invariant separation problems we discuss here as well.…”

Section: Discussionmentioning

confidence: 99%

Low Dimensional Invariant Embeddings for Universal Geometric Learning

Dym¹,

Gortler²

2022

Preprint

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This paper studies separating invariants: mappings on d-dimensional semi-algebraic subsets of D dimensional Euclidean domains which are invariant to semi-algebraic group actions and separate orbits. The motivation for this study comes from the usefulness of separating invariants in proving universality of equivariant neural network architectures.We observe that in several cases the cardinality of separating invariants proposed in the machine learning literature is much larger than the ambient dimension D. As a result, the theoretical universal constructions based on these separating invariants is unrealistically large. Our goal in this paper is to resolve this issue.We show that when a continuous family of semi-algebraic separating invariants is available, separation can be obtained by randomly selecting 2d+1 of these invariants. We apply this methodology to obtain an efficient scheme for computing separating invariants for several classical group actions which have been studied in the invariant learning literature. Examples include matrix multiplication actions on point clouds by permutations, rotations, and various other linear groups.

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Section: Discussionmentioning

confidence: 99%

Low Dimensional Invariant Embeddings for Universal Geometric Learning

Dym¹,

Gortler²

2022

Preprint

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“…The algorithm from [7] checks if two finite sets of m points are isometric in time O(m ⌈n/3⌉ log m). The latest advance is the O(m log m) algorithm in R 4 [20], see other significant results on matching bounded rigid shapes in [32,36,26,15]. The Euclidean Distance Geometry [24] studies the related problem of uniquely embedding (up to isometry of R n ) an abstract graph whose straight-line edges must have specified lengths.…”

Section: A Review Of the Past Work On Isometry Classifications And Me...mentioning

confidence: 99%

Exactly computable and continuous metrics on isometry classes of finite and 1-periodic sequences

Kurlin¹

2022

Preprint

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This paper introduces a metric that continuously quantifies the similarity between high-dimensional periodic sequences considered up to natural equivalences maintaining inter-point distances. This metric problem is motivated by periodic time series and point sets that model real periodic structures with noise. Most past advances focused on finite sets or simple periodic lattices. The key novelty is the continuity of the new metric under perturbations that can change the minimum period. For any sequences with at most m points within their periods, the metric is computed in time O(m 3 ).

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“…The latest advance for finite sets is the O(m log m) algorithm in R 4 [24]. Significant results on matching bounded rigid shapes and registration of finite point sets were obtained in [38,44,27,17]. The research on graph isomorphisms [40,1] can be potentially used for periodic graphs with fixed edges between points of a periodic set.…”

Section: Definitions and A Review Of Past Work On Comparing Periodic ...mentioning

confidence: 99%

Pointwise distance distributions of periodic point sets

Widdowson¹,

Kurlin²

2021

Preprint

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The fundamental model of a periodic structure is a periodic set of points considered up to rigid motion or isometry in Euclidean space. The recent work by Edelsbrunner et al defined the new isometry invariants (density functions), which are continuous under perturbations of points and complete for generic sets in dimension 3. This work introduces much faster invariants called higher order Pointwise Distance Distributions (PDD). The new PDD invariants are simpler represented by numerical matrices and are also continuous under perturbations important for applications.Completeness of PDD invariants is proved for distance-generic sets in any dimension, which was also confirmed by distinguishing all 229K known molecular organic structures from the world's largest Cambridge Structural Database. This huge experiment took only seven hours on a modest desktop due to the proposed algorithm with a near linear or small polynomial complexity in terms of key input sizes. Most importantly, the above completeness allows one to build a common map of all periodic structures, which are continuously parameterized by PDD and explicitly reconstructible from PDD. Appendices include first tree-based maps for several thousands of real structures.

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Linearly Converging Quasi Branch and Bound Algorithms for Global Rigid Registration

Cited by 15 publications

References 32 publications

Low Dimensional Invariant Embeddings for Universal Geometric Learning

Low Dimensional Invariant Embeddings for Universal Geometric Learning

Exactly computable and continuous metrics on isometry classes of finite and 1-periodic sequences

Pointwise distance distributions of periodic point sets

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