The Pascal Triangle of a Discrete Image: Definition, Properties and Application to Shape Analysis
This work addresses shape analysis in computer vision, providing a novel mathematical framework for detecting symmetries and equivalences in images, which is incremental in applying group theory to image moments.
The paper tackles the problem of characterizing object geometry in images, such as detecting approximate symmetries, by defining the Pascal triangle of a discrete image as a pyramidal arrangement of complex-valued moments and showing its geometric significance, including that row entries correspond to Fourier series coefficients of Radon transform moments.
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.