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

Yury Elkin,Vitaliy Kurlin

doi:10.3390/math9172121

Yury Elkin, Vitaliy Kurlin

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https://doi.org/10.3390/math9172121

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Journal: Mathematics	Publication Date: Sep 1, 2021
Citations: 3	License type: CC BY 4.0

Affiliation: University of Liverpool

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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.

Highlights

Motivations, Shape Recognition Problem, and Overview of ResultsReal-life objects are often represented by unstructured point clouds obtained by laser range scanning or by selecting salient or feature points in images [1]
The recognition of point clouds of the same number of points is practically solved by the histogram of all pairwise distances, which is a complete isometry invariant in general position [3]
In dimension 0 the persistence diagram PD(A) for distance-based filtrations of a point cloud A consists of the pairs (0, s) ∈ R2, where values of s are distance scales at which subsets of A merge by the single-linkage clustering

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Motivations, Shape Recognition Problem, and Overview of ResultsReal-life objects are often represented by unstructured point clouds obtained by laser range scanning or by selecting salient or feature points in images [1]. The recognition of point clouds of the same number of points is practically solved by the histogram of all pairwise distances, which is a complete isometry invariant in general position [3]. One of the first approaches to recognize nearly identical point clouds A, B of different sizes in the same metric space, for example, in Rm, is to use the Hausdorff distance

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