The 2D orientation is unique through principal moments analysis

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 not such a 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 further propose a new method to efficiently compute the shape orientation: Principal Moments Analysis. Besides the theoretical proof of correctness, we describe experiments demonstrating robustness to noise and illustrating with real images.