Region and Contour Identification of Physical Objects

The region occupied by and the contour of a physical object in 3-dimensional space are in a way dual or interchangeable characteristics of the object: the contour is the region’s boundary and the region is contained inside the contour. In the same way the characterization of the object’s contour by its Fourier descriptors, and the reconstruction of its region from the object’s multidimensional moments, are also dual problems. While both problems are well-understood in two dimensions, the complexity increases tremendously when moving to the three-dimensional world. In Section 2 we discuss how the latest techniques allow to reconstruct an object’s shape from the knowledge of its moments. For 2D significantly different techniques must be used, compared to the general 3D case. In Section 3, the parameterization of a 2D contour onto a unit circle and a 3D surface onto a unit sphere is described. Furthermore, the theory of Fourier descriptors for 2D shape representation and the extension to 3D shape analysis are discussed. The reader familiar with the use of either Fourier descriptors or moments as shape descriptors of physical objects may find the comparative discussion in the concluding section interesting.