Surface Registration with Eigenvalues and Eigenvectors

Abstract

This paper presents a novel surface registration technique using the spectrum of the shapes, which can facilitate accurate localization and visualization of non-isometric deformations of the surfaces. In order to register two surfaces, we map both eigenvalues and eigenvectors of the Laplace-Beltrami of the shapes through optimizing an energy function. The function is defined by the integration of a smoothness term to align the eigenvalues and a distance term between the eigenvectors at feature points to align the eigenvectors. The feature points are generated using the static points of certain eigenvectors of the surfaces. By using both the eigenvalues and the eigenvectors on these feature points, the computational efficiency is improved considerably without losing the accuracy in comparison to the approaches that use the eigenvectors for all vertices. In our technique, the variation of the shape is expressed using a scale function defined at each vertex. Consequently, the total energy function to align the two given surfaces can be defined using the linear interpolation of the scale function derivatives. Through the optimization of the energy function, the scale function can be solved and the alignment is achieved. After the alignment, the eigenvectors can be employed to calculate the point-to-point correspondence of the surfaces. Therefore, the proposed method can accurately define the displacement of the vertices. We evaluate our method by conducting experiments on synthetic and real data using hippocampus, heart, and hand models. We also compare our method with non-rigid Iterative Closest Point (ICP) and a similar spectrum-based methods. These experiments demonstrate the advantages and accuracy of our method.

Registration-free localization of defects in 3-D parts from mesh metrology data using functional maps

“…We point out that despite the notation, A + does not necessarily correspond to the same region as B + , since the sign of the eigenvectors is always ambiguous. To correctly match the components of A (namely A + and A − ) with the components of B (namely B + and B − ), we compare the number of points each component contains and switch the notations between B + and B − if necessary, an idea due to Hamidian et al (2019). Now suppose we obtain four sub-meshes, with A + corresponding to B + and A − corresponding to B − , respectively.…”

“…The LB operator is therefore the negative divergence of the gradient of the function, and it encodes not only the curvature of f , but also the curvature of the manifold M itself. For this reason, this operator is widely used in computer graphics and in manifold learning (e.g., see Belkin and Niyogi (2008), Hamidian et al (2019), Reuter (2006)). The LB operator appears in the heat diffusion partial differential equation (Evans 2010, Zhao andDel Castillo 2021a):…”

Multiscale Mesh Deformation Component Analysis With Attention-Based Autoencoders

“…In Finite Element Methods, multiscale analysis is widely used [11], [12], [13]. In the spectral geometry field, research works apply multiscale technology on the deformation representation [14], physics-based simulation of deformable objects [15], and surface registration [16] by analyzing the nonisometric global and local (multiscale) deformation. Moreover, for shape editing, multiscale technology also enables modeling rich facial expressions on human faces [17].…”

Non-Cooperative Target Pose Estimation based on Improved Iterative Closest Point Algorithm

“…In medium-distance detection, the iterative closest point (ICP) algorithm was utilized for registration and a 3D reconstruction was completed. In fact, the ICP algorithm shows efficiency and accuracy for rigid registration, but it needs good initial parameters [20,21] and tends to fall into local extremes [22]. Wang et al [23] weighed the corresponding points of the point cloud and introduced M-estimation to the objective function.…”