Geometric modeling of rigid and non-rigid 3D shapes using the global geodesic function

Abstract

In this paper, we present a novel intrinsic geometric representation of 3D objects. We add the proposed modeling of objects to their topological graphs to ensure a full and compact description necessary for shape-based retrieval, recognition and analysis of 3D models. In our approach, we address the challenges due to pose variability, computational complexity and noisy data by intrinsically and simply describing a 3D object by a global geodesic function. We exploit the geometric features contained in the dense set of iso-levels of this function. Using Whitney easy embedding theorem, we embed the manifold of the geodesic iso-levels in Ropf <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> and obtain a single space curve as our geometry descriptor. 3D shape comparison is then reduced to comparing the resulting modeling curves. To quantify the dissimilarities between them we simply compute an L <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> distance between classical Euclidian invariants applied to space curves. The experimental results show that in addition to being straightforward and easy to compute, our modeling technique achieves a high level of discrimination, and appears to be robust to both noise and decimation.