SIGMA 9 (2013), 031, 25 pages      arXiv:1209.4850      https://doi.org/10.3842/SIGMA.2013.031
Contribution to the Special Issue “Symmetries of Differential Equations: Frames, Invariants and Applications”

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

Mireille Boutin ^a and Shanshan Huang ^b
a) School of Electrical and Computer Engineering, Purdue University, USA
^b) Department of Mathematics, Purdue University, USA

Received September 24, 2012, in final form April 03, 2013; Published online April 11, 2013

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.

Key words: moments; symmetry detection; moving frame; shape recognition.

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