Sampling and reconstruction of smooth three‐dimensional convex shapes
Kohhei Ohtake, Michio Kaneko · Electronics and Communications in Japan (Part III Fundamental Electronic Science) · 1992
Abstract In digitally transmitting or recording three‐dimensional shapes, to decrease the amount of transmitted information and memory volume, investigations of efficient shape description and sampling‐reconstruction methods are considered important. This paper investigates these problems with shapes restricted to smooth, bounded three‐dimensional convex shapes. First, as the representation methods of these shapes, the tangential plane polar coordinate representation is shown and it is explained that this is an efficient representation consistent with the curvature. Next, from this representation function, spectra and band of the shape are introduced by using surface harmonics as basic functions and their fundamental properties are described. Based on this, a sampling theorem of three‐dimensional shapes is shown which states that for band‐limited shapes, the shape can be reconstructed from a fixed number of sample values. Moreover, for band‐unlimited shapes, distortion occurs in shapes reconstructed from a finite number of sample values. A sampling method is shown which minimizes this shape distortion power and clarifies that the method is a variable density sampling which arranges sampling points with density proportional to the curvature of shape. Finally, simulation results of shape‐sampling and reconstruction system performed on ellipsoid to confirm the authors' theory are described.