Voronoi‐Based Similarity Distances between Arbitrary Crystal Lattices

Mosca, Marco M.; Kurlin, Vitaliy

doi:10.1002/crat.201900197

Cited by 26 publications

(14 citation statements)

References 11 publications

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“…Even if such a subset is sufficiently large to cover a big extended cell, its content is highly variable. In the simpler cases of lattices, two metric functions on arbitrary lattices were defined by using Voronoi domains [30], though their computation was only approximate.…”

Section: Periodic Point Sets and A Review Of Past Workmentioning

confidence: 99%

Average minimum distances of periodic point sets – foundational invariants for mapping periodic crystals

Widdowson¹,

Mosca²,

Pulido³

et al. 2021

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The fundamental model of any solid crystalline material (crystal) at the atomic scale is a periodic point set. The strongest natural equivalence of crystals is rigid motion or isometry that preserves all inter-atomic distances. Past comparisons of periodic structures often used manual thresholds, symmetry groups and reduced cells, which are discontinuous under perturbations or thermal vibrations of atoms. This work defines the infinite sequence of continuous isometry invariants (Average Minimum Distances) to progressively capture distances between neighbors. The asymptotic behaviour of the new invariants is theoretically proved in all dimensions for a wide class of sets including non-periodic. The proposed near linear time algorithm identified all different crystals in the world's largest Cambridge Structural Database within a few hours on a modest desktop. The ultra fast speed and proved continuity provide rigorous foundations to continuously parameterise the space of all periodic crystals as a high-dimensional extension of Mendeleev's table of elements.

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Section: Periodic Point Sets and A Review Of Past Workmentioning

confidence: 99%

Average minimum distances of periodic point sets – foundational invariants for mapping periodic crystals

Widdowson¹,

Mosca²,

Pulido³

et al. 2021

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“…The recent stronger invariants are Pointwise Distance Distributions [33]. Their generic completeness under isometry holds in a more challenging setting of periodic point sets [34][35][36][37][38].…”

Section: A Discussion Of Novel Contributions and Further Open Problemsmentioning

confidence: 99%

Isometry Invariant Shape Recognition of Projectively Perturbed Point Clouds by the Mergegram Extending 0D Persistence

Yury

Kurlin

2021

Mathematics

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Rigid shapes should be naturally compared up to rigid motion or isometry, which preserves all inter-point distances. The same rigid shape can be often represented by noisy point clouds of different sizes. Hence, the isometry shape recognition problem requires methods that are independent of a cloud size. This paper studies stable-under-noise isometry invariants for the recognition problem stated in the harder form when given clouds can be related by affine or projective transformations. The first contribution is the stability proof for the invariant mergegram, which completely determines a single-linkage dendrogram in general position. The second contribution is the experimental demonstration that the mergegram outperforms other invariants in recognizing isometry classes of point clouds extracted from perturbed shapes in images.

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“…Optimal geometric matching of Voronoi domains with a shared centre led [21] to two continuous metrics (up to orientation-preserving isometry and similarity) on lattices. The minimisation over infinitely many rotations was implemented in [21] by sampling and gave approximate algorithms for these metrics. The complete invariant isoset [7] for periodic point sets in R n has a continuous metric that can be approximated [6] with a factor O(n).…”

Section: Definition 23 (Voronoi Domain V (λ))mentioning

confidence: 99%

A complete isometry classification of 3-dimensional lattices

Kurlin¹

2022

Preprint

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A periodic lattice in Euclidean 3-space is the infinite set of all integer linear combinations of basis vectors. Any lattice can be generated by infinitely many different bases. This ambiguity was only partially resolved, but standard reductions remained discontinuous under perturbations modelling crystal vibrations. This paper completes a continuous classification of 3-dimensional lattices up to Euclidean isometry (or congruence) and similarity (with uniform scaling).The new homogeneous invariants are uniquely ordered square roots of scalar products of four vectors whose sum is zero and all pairwise angles are non-acute. These root invariants continuously change under perturbations of basis vectors. The geometric methods extend the past work of Delone, Conway and Sloane.1 The hard problem to continuously classify lattices up to isometry We extend the continuous isometry classification of 2-dimensional lattices [19] to dimension 3. A lattice Λ ⊂ R n consists of integer linear combinations of basis vectors v 1 , . . . , v n . This basis spans a parallelepiped called a unit cell U ⊂ R n .The problem to classify lattices up to isometry is motivated by periodic crystals whose structures are determined in a rigid form. Hence the most natural equivalence of crystals is rigid motion. We start from general isometries that also include mirror reflections because the sign of a lattice similar to [19, Definition 3.4] easily distinguishes mirror images. As in R 2 , the space of lattices up to rigid motion in R 3 is a 2-fold cover of the smaller Lattice Isometry Space LIS(R 3 ).The previous work [19, section 1] provided important motivations for a continuous classification problem, which we state below for 3-dimensional lattices.

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Voronoi‐Based Similarity Distances between Arbitrary Crystal Lattices

Cited by 26 publications

References 11 publications

Average minimum distances of periodic point sets – foundational invariants for mapping periodic crystals

Average minimum distances of periodic point sets – foundational invariants for mapping periodic crystals

Isometry Invariant Shape Recognition of Projectively Perturbed Point Clouds by the Mergegram Extending 0D Persistence

A complete isometry classification of 3-dimensional lattices

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