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

Symmetry Integrability and Geometry-Methods and Applications

article

Mireille Boutin¹  Shanshan Huang² 

[1] School of Electrical and Computer Engineering, Purdue University;Department of Mathematics, Purdue University

关键词: moments;    symmetry detection;    moving frame;    shape recognition;   

DOI  :  10.3842/SIGMA.2013.031

来源: National Academy of Science of Ukraine

【 摘 要 】

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.

【 授权许可】

Unknown