The Pascal Triangle of a Discrete Image: Definition, Properties and Application to Shape Analysis

TL;DR

This paper introduces a novel Pascal triangle representation for discrete images using complex moments, revealing geometric properties and enabling symmetry detection through group action analysis.

Contribution

It defines the Pascal triangle of an image, links it to Fourier coefficients of the Radon transform, and develops methods for symmetry detection under group actions.

Findings

Entries of the triangle relate to Fourier coefficients of the Radon transform

Group actions can be extended to the Pascal triangle for symmetry analysis

Proposed tests effectively detect image symmetries and equivalences

Abstract

We define the Pascal triangle of a discrete (gray scale) image as a pyramidal arrangement of complex-valued moments and we explore its geometric significance. In particular, we show that the entries of row k of this triangle correspond to the Fourier series coefficients of the moment of order k of the Radon transform of the image. Group actions on the plane can be naturally prolonged onto the entries of the Pascal triangle. We study the prolongation of some common group actions, such as rotations and reflections, and we propose simple tests for detecting equivalences and self-equivalences under these group actions. The motivating application of this work is the problem of characterizing the geometry of objects on images, for example by detecting approximate symmetries.