A Fourier–Shannon approach to closed contours modelling

Abstract

This paper describes a modelling method for continuous closed contours. The initial input data set consists of two-dimensional (2-D) points, which may be represented as a discrete function in a polar coordinate system. The method uses the Shannon interpolation between these data points to obtain the global continuous contour model. A minimal description of the contour is obtained using the link between the Shannon interpolation kernel and the Fourier series of polar development (FSPD) for periodic functions. The Shannon interpolation kernel allows the direct interpretation of the contour smoothness in terms of both samples and Fourier frequency domains.

In order to deal with deformation point sources, often encountered in active modelling techniques, a method of local deformation is proposed. Each local deformation is performed in an angular sector centred on the deformation point source. All the neighbouring characteristic samples are displaced in order to minimize the oscillations of the newly created model outside the deformation sector. This deformation technique preserves the frequency characteristics of the contour, regardless of the number and the intensity of deformation sources. In this way, the technique induces a frequency modelling constraint, which may be subsequently used in an active detection and modelling environment.

Experiments on synthetic and real data prove the efficiency of the proposed technique. The method is currently used to model contours of the left ventricle of the heart obtained from ultrasound apical images. This work is part of a larger project, the aim of which is to analyse the space and time deformations of the left ventricle. The 2-D Fourier–Shannon model is used as a basis for more complex three-dimensional and four-dimensional Fourier models, able to recover automatically the movement and deformation of the left ventricle of the heart during a cardiac cycle.