Tools for 3D-object retrieval: Karhunen-Loeve transform and spherical harmonics

Abstract

We present tools for 3D object retrieval in which a model, a polygonal mesh, serves as a query and similar objects are retrieved from a collection of 3D objects. Algorithms proceed first by a normalization step (pose estimation) in which models are transformed into a canonical coordinate frame. Second, feature vectors are extracted and compared with those derived from normalized models in the search space. Using a metric in the feature vector space nearest neighbors are computed and ranked. Objects thus retrieved are displayed for inspection, selection, and processing. For the pose estimation we introduce a modified Karhunen-Loeve transform that takes into account not only vertices or polygon centroids from the 3D models but all points in the polygons of the objects. Some feature vectors can be regarded as samples of functions on the 2-sphere. We use Fourier expansions of these functions as uniform representations allowing embedded multi-resolution feature vectors. Our implementation demonstrates and visualizes these tools.

Using Irreducible Group Representations for Invariant 3D Shape Description

“…There is already a huge amount of work concerning feature extraction for 3D surface models by the use of Spherical Harmonics (SH). Vranic et al [10] compute a so-called spherical extent function of the model-surface and make a spherical harmonic transform of this function, but the rotational invariance is obtained by normalization. Kazhdan et al [4] eliminate the rotational dependency by taking the magnitude of the invariant subspaces of the Spherical Harmonic transform.…”

Efficient 3D shape matching and retrieval using a concrete radialized spherical projection representation

“…Table 1 shows the number of categories and the total number of models for each dataset. We compare the proposed descriptor with the following shape retrieval methods which are considered state-of-the-art with respect to their discriminative power: (i) hybrid (DSR472) [5]; (ii) depth buffer-based (DB) [5]; (iii) silhouette-based (SIL) [5]; (iv) the ray-based using spherical harmonics (RSH) [27]; (v) light-field (LF) [11]; (vi) the spherical harmonic representation of the Gaussian Euclidean distance transform descriptor (SH-GEDT) [14].…”

“…The set of vertices of the polygonal model [3] or the centroids of the model's triangles [29] have been used as input to the PCA. In [5,27] the so-called CPCA is presented. Another approach to normalize a 3D model for rotation (similar to PCA), is the use of singular value decomposition (SVD) [28].…”

“…where M is the set of vertices of the polygonal model and is given in Appendix A; this is not included in [5,27] where CPCA was introduced. The eigenvectors of C are then computed, which represent the principal directions of the model's surface area and form an orthonormal basis for R 3 space.…”

Mesh Alignment using Grid based PCA

“…Nevertheless, several important shape descriptors, such as (Cantoni et al, 2013), shape histograms (Ankerst et al, 1999) and descriptors based on higher order moments (Elad et al, 2002), are not rotationally invariant and thus require alignment. Extensions to PCA to overcome its shortcomings include PCA performed on the normals of a surface (Papadakis et al, 2007), and a continuous version of PCA applied to whole mesh triangles rather than just their vertices (Vranic et al, 2001). The latter is independent from distribution of vertices within the mesh and thus, overcomes some of the limitations of PCA the same way our method does.…”