 
Summary: IEEE TRANSACTIONS ON IMAGE PROCESSING 1
Revisiting Complex Moments For 2D Shape
Representation and Image Normalization
Jo~ao B. F. P. Crespo and Pedro M. Q. Aguiar, Senior Member, IEEE
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 and show how to uniquely define the orientation of an arbitrary 2D
shape, in terms of what we call its Principal Moments. We start by showing that a small subset of these
moments suffices to describe the underlying 2D shape, i.e., that they form a compact representation, which
is particularly relevant when dealing with large databases. Then, we 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 greylevel images. Besides the theoretical proof of correctness, we describe
experiments demonstrating robustness to noise and illustrating with real images.
Index Terms
Shape representation, Shape orientation, Image normalization, Complex moments, Invariants.
EDICS Category: ARSRBS
