Euclidean-distance-based canonical forms for non-rigid 3D shape retrieval

Abstract

Retrieval of 3D shapes is a challenging problem, especially for non-rigid shapes.One approach giving favourable results uses multidimensional scaling (MDS) to compute a canonical form for each mesh, after which rigid shape matching can be applied. However, a drawback of this method is that it requires geodesic distances to be computed between all pairs of mesh vertices. Due to the superquadratic computational complexity, canonical forms can only be computed for low-resolution meshes. We suggest a linear time complexity method for computing a canonical form, using Euclidean distances between pairs of a small subset of vertices. This approach has comparable retrieval accuracy but lower time complexity than using global geodesic distances, allowing it to be used on higher resolution meshes, or for more meshes to be considered within a time budget.

An evaluation of canonical forms for non-rigid 3D shape retrieval

“…The Euclidean distance based method by Pickup et al [32] computes a canonical form of a mesh by stretching out its limbs so that its extremities are distant from one another. This is achieved more efficiently than using multidimensional scaling (MDS) on the geodesic distances [11].…”

“…This aims to preserve the edge lengths of the mesh, to ensure isometric deformation. In order to maximise the distance between feature points, the value of δ ij for each pair of the N sampled vertices is set to 10, as suggested by Pickup et al [32]. As long as this value is large enough and the parameter β, discussed below, is optimised accordingly, any value can be chosen.…”

“…This in turn ensures that we can use the same value for the parameter β regardless of mesh resolution. Pickup et al [32] offer two different solutions to handling this issue. The first is to increase the number of feature points to N by adding extra randomly selected vertices as feature points.…”

Skeleton-based canonical forms for non-rigid 3D shape retrieval

“…Wang and Zha [20] speeded up the canonical form computation by only computing geodesic distances between all pairs of a set of detected feature points, and unbending the mesh by creating target axes used to align sets of geodesic contours. Pickup et al [21] also used feature points, but restricted their number to the square root of the number of mesh vertices. They maximised the distance between pairs of these feature points whilst preserving the mesh's edge lengths.…”

“…In some objects, the mesh is disconnected into two or three different components. The Euclidean distance based methods by Pickup et al [21] do not require any modification for this, but all the other methods fail on these meshes. For the MDS and GPS methods, we therefore delete all but the largest component.…”