Similarity Detection Algorithms for CAD Models

This document presents two spectral methods for shape matching and recognition. Our first method computes the Laplace-Beltrami spectra on a domain to describe its shape. Since the spectrum is an isometry invariant and contains geometrical information, it is optimally suited for shape analysis and shape matching of geometric data. We have recently focused on global shape comparison with an application to medical imaging. In our second approach, a wavelet-based spectral method has been studied. A new form of wavelet transform is employed to greatly reduce the sensitivity to translation and rotation of a model. We also take advantage of the multi-resolution attribute to operate on reduced data sets at lower resolutions, where we are able to rapidly detect the similarity 1. Laplace Spectra for Shape Recognition: This section describes our study on global shape analysis based on the Laplace-Beltrami spectrum of a Riemannian manifold [1,2,3,4,5] with an application in the field of medical imaging [6,7]. Previous approaches for global shape analysis in medical imaging include the use of invariant moments [8], the shape index [9], and global shape descriptors based on spherical harmonics [10]. Our methodology based on the Laplace-Beltrami spectrum differs in the following ways from these previous approaches: • It works for any Riemannian manifold, whereas spherical harmonics based methods are restricted to surfaces with spherical topology, and invariant moments do not easily generalize to arbitrary Riemannian manifolds. It may thus be used to analyze surface, solids, non-spherical objects, etc. • In the 2D case the only preprocessing requirement is the extraction of a surface approximation from the manually segmented binary volume. In 3D the method runs directly on the binary volume representation (voxel) without any preprocessing. No registration, remeshing, or additional mappings are necessary. Furthermore, the description is invariant to translations, rotations, isometries, and surface meshing. Given the Laplace-Beltrami operator Δ of a real-valued function f, with 2 C f ∈ , defined on a Riemannian manifold ) ( : : f grad div f M = Δ then the Helmholtz equation (also known as the Laplacian eigenvalue problem) is stated as f f λ − = Δ . If M is bounded we employ either the Dirichlet ) 0 ( ≡ f or the Neumann boundary condition ) 0 ( ≡ ∂ ∂ n f on the boundary. The solutions of this equation represent the spatial part of the solutions of the wave equation, with an infinite number of eigenvalue λ and eigenfunction f pairs. The possibly normalized beginning sequence of the Dirichlet or Neumann spectrum (the first n eigenvalues – called ShapeDNA) can be used as a fingerprint of the object’s intrinsic geometry (independent of the dimension of M – e.g. 2D surface or 3D solid). The ShapeDNA can be successfully applied for database retrieval. See Figure 1 for a plot presenting the first two principal components of the high dimensional ShapeDNA of a few closed surfaces. It can be seen how the ShapeDNA clusters the objects into meaningful groups (considering that the helmet is in fact a deformed ellipsoid where one cap has been flipped to the inside). Recently the ShapeDNA was applied to medical data representing two populations of brain parts (caudate nucleus obtained from MRI scans, see Figure 2) of female subjects diagnosed with Schizotypal Personality Disorder (SPD) and normal control (NC) subjects [6,7]. It could be demonstrated that the ShapeDNA can pick up statistically significant differences of the two populations not only in size, but also in shape, confirming the influence of SPD on the brain part NSF GRANT # 0629332 NSF PROGRAM NAME: CMMI (DMI) / Engineering Design