Parameterizing Arbitrary Shapes via Fourier Descriptors for Evidence-Gathering Extraction

According to the formulation of the Hough Transform, it is possible to extract any shape that can be represented by an analytic equation with a number of free parameters. Nevertheless, the extraction of arbitrary shapes has centered on nonanalytic representations based on a table which specifies the position of edge points relative to a fixed reference point. In this paper we develop a novel approach for arbitrary shape extraction which combines the analytic representation of shapes with the generality of the characterization by Fourier descriptors. The formulation is based on a definition of the Hough Transform obtained by considering the parametric representation of shapes and extends the descriptional power of the Hough Transform beyond simple shapes, thus avoiding the use of tables. Since we use an analytic representation of shapes, the developed technique inherits the robustness of the original formulation of the Hough Transform. Based on the developed formulation, and by using different strategies of parameter space decomposition, various methods of shape extraction are presented. In these methods the parameter space is reduced by using gradient direction information as well as the positions of grouped edge points. Different methods represent a compromise between speed, noise sensitivity, simplicity, and generality. Some examples of the extraction process on a selection of synthetic and real images are presented, showing the successful extraction of target shapes from noisy data.