Revisiting Complex Moments For 2D Shape Representation and Image Normalization

TL;DR

This paper introduces Principal Moment Analysis, a novel method for uniquely determining the orientation of arbitrary 2D shapes and normalizing images, addressing limitations of previous approaches especially in noisy conditions.

Contribution

It proposes a new shape orientation definition using Principal Moments, along with an efficient computation method and applications to image normalization, backed by theoretical and experimental validation.

Findings

The method accurately determines shape orientation for various shapes.

It demonstrates robustness to noise in shape and image normalization.

Experimental results validate theoretical correctness and practical effectiveness.

Abstract

When comparing 2D shapes, a key issue is their normalization. Translation and scale are easily taken care of by removing the mean and normalizing the energy. However, defining and computing the orientation of a 2D shape is not so simple. In fact, although for elongated shapes the principal axis can be used to define one of two possible orientations, there is no such tool for general shapes. As we show in the paper, previous approaches fail to compute the orientation of even noiseless observations of simple shapes. We address this problem. In the paper, we show how to uniquely define the orientation of an arbitrary 2D shape, in terms of what we call its Principal Moments. We show that a small subset of these moments suffice to represent the underlying 2D shape and propose a new method to efficiently compute the shape orientation: Principal Moment Analysis. Finally, we discuss how this method can further be applied to normalize grey-level images. Besides the theoretical proof of correctness, we describe experiments demonstrating robustness to noise and illustrating the method with real images.