A FOURIER-SHANNON APPROACH TO CLOSED CONTOURS MODELING

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 descri ption of the contour is obtained using the link between the Shannon in terpolation kernel and the Fourier series of polar development (FSPD) for periodic functions. The Shannon interpolation kernel allows the di rect interpretation of the contour smoothness in terms of both samples and Fourier frequency domains. In order to deal with deformation poin t 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 s ector. This deformation technique preserves the frequency characterist ics of the contour, regardless of the number and the intensity of defo rmation sources. In this way, the technique induces a frequency modell ing constraint, which may be subsequently used in an active detection and modelling environment. Experiments on synthetic and real data prov e the efficiency of the proposed technique. The method is currently us ed to model contours of the left ventricle of the heart obtained from ultrasound apical images. This work is part of a larger project, the a im 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 c omplex three-dimensional and four-dimensional Fourier models, able to recover automatically the movement and deformation of the left ventric le of the heart during a cardiac cycle.