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The functional maps framework has achieved remarkable success in non-rigid shape matching. However, the traditional functional map representations do not explicitly encode surface orientation, which can easily lead to orientation-reversing correspondence. The complex functional map addresses this issue by linking oriented tangent bundles to favor orientation-preserving correspondence. Nevertheless, the absence of effective restrictions on the complex functional maps hinders them from obtaining high-quality correspondences. To this end, we introduce novel and powerful constraints to determine complex functional maps by incorporating multiple complex spectral filter operator preservation constraints with a rigorous theoretical guarantee. Such constraints encode the surface orientation information and enforce the isometric property of the map. Based on these constraints, we propose a novel and efficient method to obtain orientation-preserving and accurate correspondences across shapes by alternatively updating the functional maps, complex functional maps, and pointwise maps. Extensive experiments demonstrate our significant improvements in correspondence quality and computing efficiency. In addition, our constraints can be easily adapted to other functional maps-based methods to enhance their performance. [ABSTRACT FROM AUTHOR]

We construct a novel descriptor called anisotropic wavelet energy decomposition descriptor (AWEDD) for non-rigid shape analysis, based on anisotropic diffusion geometry. We first extend the Dirichlet energy of the vertex coordinate function to an anisotropic version, then use multiscale anisotropic spectral manifold wavelets to decompose the Dirichlet energy to all vertices and collect local energy at each vertex to form AWEDD. AWEDD simultaneously encodes multiscale extrinsic and intrinsic shape features, which are more informative and robust than purely intrinsic or extrinsic descriptors. And the introduction of anisotropy endows AWEDD with stronger abilities of feature discrimination and intrinsic symmetry identification. Our results demonstrate that AWEDD is more discriminative than current state-of-the-art descriptors. In addition, we show that AWEDD is an excellent choice of the initial inputs for various shape analysis approaches, such as functional map pipelines and deep convolutional architectures. [ABSTRACT FROM AUTHOR]

Incorporating multiscale spectral manifold wavelets preservation into the functional map framework for shape correspondence achieves great results in terms of both efficiency and effectiveness. However, fixing the dimension of the spectral embedding strategy in iterations of optimization is troublesome, such as missing high-frequency information when the dimension is small or getting trapped in local minima at a high dimension. In this paper, we present a simple and efficient method for refining correspondences from low frequency to high frequency with a theoretical guarantee. We formulate a strong constraint where the multiscale spectral manifold wavelets should be preserved at each scale correspondingly in the case of the arbitrary dimension of spectral embeddings. To solve the formula, we deduce two relaxed optimization subproblems and propose an incremental alternating iterative algorithm between the spatial and spectral domains via spectral up-sampling and filtering, computing the functional maps and pointwise maps in turn. Our results demonstrate that our method is robust to noisy initialization and scalable with regard to shape resolutions. The deformable shape correspondence benchmark experiments show our method produces more accurate and smoother results than state of the arts. [ABSTRACT FROM AUTHOR]

RBF-based methods play a very important role in the point cloud reconstruction field. However, solving a linear system is the bottleneck of such methods, especially when there are a large number of points and lead the computing to be time-consuming and unstable. In this paper, we firstly construct a novel implicit progressive-iterative approximation framework based on RBFs, which could elegantly reconstruct curves and surfaces or even higher dimensional data in an approximation or interpolation way, avoiding expensive computational cost on solving linear systems. Then, we further accelerate the proposed method with a strategy inspired from the conjugate gradient algorithm. In our framework, using proper RBFs allows to simply transform the iteration matrix to be symmetrical and positive definite. Such a property contributes to reduce the computational cost greatly and produce high-quality reconstruction results. Plenty of numerical examples on various challenging data are provided to demonstrate our efficiency, effectiveness, and superiority to other methods. [ABSTRACT FROM AUTHOR]

This paper presents a method for generating implicit surfaces from polygon soups based on compactly supported radial basis functions (CSRBFs). The surface is represented as the zero level set of an implicit function which interpolates the polygonal data with their outward normal constraints. By specifying two parameters, the support size and the shape parameter, users can flexibly control the accuracy of the reconstructed surfaces. For determining coefficients of RBFs, our method uses a quasi-interpolation framework to avoid solving a large linear system, which allows processing large meshes efficiently and robustly. Moreover, a relationship between the shape parameter and the support radius is provided for the quasi-solution validity, and an error bound of the reconstructed surfaces approximating the original models is deduced through the rigorous theoretical analysis. [ABSTRACT FROM AUTHOR]