Isometry invariant shape recognition of projectively perturbed point clouds by the mergegram extending 0D persistence

TL;DR

This paper introduces a stable isometry invariant called the mergegram for recognizing shapes from noisy point clouds under affine or projective transformations, outperforming existing methods.

Contribution

It proves the stability of the mergegram invariant and demonstrates its effectiveness in shape recognition tasks involving noisy, projectively perturbed point clouds.

Findings

Mergegram completely determines a single-linkage dendrogram in general position.

Mergegram outperforms other invariants in recognizing shape classes from perturbed point clouds.

The method is robust to noise and size variations in point cloud data.

Abstract

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.