Evaluation of 2d Shape Likeness for Surface Reconstruction

Surface or shape reconstruction from 3D digitizations performed in planar samplings are frequent in product design, reverse engineering, rapid prototyping, medical and artistic applications, etc. The planar slicing of the object offers an opportunity to recover part of the neighborhood information essential to reconstruct the topological 2-manifold embedded in R3 that approximates the object surface. Next stages of the algorithms find formidable obstacles that are classified in this investigation by the following taxonomy: (i) Although real objects have manifold boundaries, in objects with thin sections or walls, the manifold property remains in the data sample only at the price of very small sampling intervals and large data sets. For relaxed sampling rates nonmanifold situations are likely. (ii) The position of the planar slices may produce an associated level function which is non – Morse. This means, the set of critical points of the associated level function is isomorphic to compact subsets of R1 or R2. The fact that the Hessian matrix at critical points is non-singular is the Morse condition (as a consequence critical points are isolated), and allows for the algorithms presented here. (iii) For Morse condition, the slicing interval may be such that several critical points occur between immediate slices (nonsimple condition). This article presents the degenerate cases arising from points (i)-(iii) and discusses a shape reconstruction algorithm for digitizations holding the Morse – Simple condition. It presents the results of applying the prescribed algorithms to data sets, and discusses future actions that enlarge the mentioned scope.