Variational harmonic maps for space deformation

We propose 3D Geometric Mesh Network (3DGeoMeshNet), a method for 3D mesh reconstruction and completion. 3DGeoMeshNet is a novel Graph Convolutional Network (GCN)-based framework that leverages anisotropic convolution layers to effectively learn multi-scale global and local features directly in the spatial domain. Unlike traditional approaches that convert meshes into voxel grids or point clouds, our method operates directly on the polygonal mesh structure, preserving geometric fidelity throughout the learning and reconstruction process. In addition to 3D mesh reconstruction, we extend our framework to tackle 3D mesh completion task, where missing or incomplete regions of the mesh are first accurately recovered. The completion results are further refined through a set of pre- and post-processing steps. We extensively evaluate our approach on two benchmark datasets, COMA and DFAUST, and achieve SOTA results for 3D mesh reconstruction on both datasets. Additionally, our mesh completion experiments on the COMA dataset demonstrate the promising capability of 3DGeoMeshNet in recovering incomplete geometries. We further showcase the versatility of our method through additional applications, including mesh denoising, interpolation, and extrapolation, highlighting the robustness and generalization ability of our framework across various 3D mesh processing tasks.

This chapter takes up several independent topics in Euclidean Fourier analysis, all having some bearing on the subject of partial differential equations. Section 1 elaborates on the relationship between the Fourier transform and the Schwartz space, the subspace of $L^{1}(\mathbb{R}^{N})$ consisting of smooth functions with the property that the product of any iterated partial derivative of the function with any polynomial is bounded. It is possible to make the Schwartz space into a metric space, and then one can consider the space of continuous linear functionals; these continuous linear functionals are called “tempered distributions.” The Fourier transform carries the space of tempered distributions in one-one fashion onto itself. Section 2 concerns weak derivatives, and the main result is Sobolev’s Theorem, which tells how to recover information about ordinary derivatives from information about weak derivatives. Weak derivatives are easy to manipulate, and Sobolev’s Theorem is therefore a helpful tool for handling derivatives without continually having to check the validity of interchanges of limits. Sections 3–4 concern harmonic functions, those functions on open sets in Euclidean space that are annihilated by the Laplacian. The main results of Section 3 are a characterization of harmonic functions in terms of a mean-value property, a reflection principle that allows the extension to all of Euclidean space of any harmonic function in a half space that vanishes at the boundary, and a result of Liouville that the only bounded harmonic functions in all of Euclidean space are the constants. The main result of Section 4 is a converse to properties of Poisson integrals for half spaces, showing that harmonic functions in a half space are given as Poisson integrals of functions or of finite complex measures if their $L^{p}$ norms over translates of the bounding Euclidean space are bounded. Sections 5–6 concern the Calderon–Zygmund Theorem, a far-reaching generalization of the theorem concerning the boundedness of the Hilbert transform. Section 5 gives the statement and proof, and two applications are the subject of Section 6. One of the applications is to Riesz transforms, and the other is to the Beltrami equation, whose solutions are “quasiconformal mappings.” Sections 7–8 concern multiple Fourier series for smooth periodic functions. The theory is established in Section 7, and an application to traces of integral operators is given in Section 8.

I want to present some results on the regularity for harmonic maps between a surface of dimension two and a Riemannian manifold. First of all, we recall what harmonic maps between Riemannian manifolds are. Let (M, g) and (N , g) be two Riemannian manifolds of dimension m and n respectively, and assume that N is compact and is isometrically embedded in some Euclidean space R (which is always possible thanks to the Nash–Moser theorem). We introduce the Dirichlet functional on the set of maps between M and N . To do this we define the energy density of a map u from M into N at any point x of M by e(u)(x) = 1 2hij [u(x)]g (x)uαu j β

Click to increase image sizeClick to decrease image size Additional informationNotes on contributorsShif BerhanuSHIF BERHANU graduated with a B.Sc. degree from Addis Ababa University, Ethiopia, in 1980. In 1987 he received his Ph.D. from Rutgers University under the direction of Francois Trevès. He is a professor of mathematics at Temple University. Berhanu's main area of research involves complex vector fields, that is, firstorder partial differential operators with complex-valued coefficients. Currently, he, P. Cordaro, and J. Hounie are collaborating to write a book on the subject of complex vector fields.Ahmed MohammedAHMED MOHAMMED received his B.Sc. in 1983 and M.Sc. in 1986 from Addis Ababa University, Ethiopia. He obtained his Ph.D. in 1995 from Temple University under the guidance of Professor Cristian E. Gutierrez. Between 1995 and 2000, he taught in the Department of Mathematics and Computer Science at Kuwait University, Kuwait. He held a visiting position at Temple University from 2000 to 2002. Currently he is an associate professor in the Department of Mathematical Sciences at Ball State University. If he is not doing mathematics in his office or teaching a class, chances are you'll find him at the Starbucks in the university book store.

Given a harmonic function U in a domain Ω in Euclidean space, the problem of finding a harmonic conjugate V, generalizing the well-known case of the complex plane, was considered in [4] in the framework of Clifford analysis. By the nature of the given construction, which is genuinely cartesian, this approach lead to geometric constraints on the domain Ω. In this paper we consider the problem in a larger class of domains, by a spherical approach. Starting from a real-valued function u, and singling out the radial direction, we explicitly construct a harmonic function of the form w = erv, with v ∈ span(eθ1,…, eθm−1), such that u+w is monogenic, i.e. a null solution of the Dirac operator. As an illustration, the construction is applied to important classes of homogeneous monogenic polynomials and functions. Finally, it is investigated to which extent the approach also applies to the complex plane case.

In this paper, we formulate a nonlinear elasticity theory in which the ambient space is evolving. For a continuum moving in an evolving ambient space, we model time dependency of the metric by a time-dependent embedding of the ambient space in a larger manifold with a fixed background metric. We derive both the tangential and the normal governing equations. We then reduce the standard energy balance written in the larger ambient space to that in the evolving ambient space. We consider quasi-static deformations of the ambient space and show that a quasi-static deformation of the ambient space results in stresses, in general. We linearize the nonlinear theory about a reference motion and show that variation of the spatial metric corresponds to an effective field of body forces.

Harmonic approximation and Sarason's-type theorem

The possibility of reconstructing two‐dimensional electron‐density profiles in the ionosphere with ionospheric tomography is significant. However, due to the nature of the imaging system, there are several resolution degradation parameters. In order to compensate for these degradation parameters, a priori information must be used. This article introduces the orthogonal decomposition algorithm for image reconstruction, which uses the a priori information to generate a set of orthogonal basis functions for the source domain. This algorithm consists of two simple steps: orthogonal decomposition and recombination. In the development of the algorithm, it is shown that the degradation parameters of the imaging system result in correlations among projections of orthogonal functions. Gram–Schmidt orthogonalization is used to compensate for these correlations, producing a matrix that measures the degradation of the system. Any set of basis functions can be used, and depending upon this choice, the nature of the algorithm varies greatly. Choosing the basis functions of the source domain to be the Fourier kernels produces an algorithm capable of isolating individual frequency components of individual projections. This particular choice of basis functions also results in an algorithm that strongly resembles the direct Fourier method, but without requiring the use of inverse Fourier transforms.

Direct immersogeometric fluid flow and heat transfer analysis of objects represented by point clouds

This paper presents an analytical and computational method to describe natural frequencies of a spherical bubble residing near a solid sphere of an arbitrary size in an otherwise unbounded fluid. Under low capillary and Reynolds number limits, the relevant hydrodynamic fields are converted into time-invariant but frequency-dependent quantities by temporal Fourier transform. Then, the spatial variations in the velocity and the pressure can be expressed in terms of two sets of harmonic basis functions involving spherical coordinates centered around the particle and the bubble. A subsequent derivation of transformation coefficients between the aforementioned two sets allows a matrix equation relating the unknown amplitudes to the boundary conditions at all interfaces. Finally, natural frequencies corresponding to different modes of pulsation are obtained from the eigenvalues of the constructed matrix. The results show fast convergence of the computed frequencies with the increasing number of basis functions. These values change significantly with the distance of the bubble from the particle and even decay to zero for some modes when their surface-to-surface separation vanishes. Furthermore, bubble oscillation near a solid plate is also discussed when the radius of the solid sphere is increased to an infinitely large dimension. Thus, this article renders a comprehensive study of naturally pulsating submerged bubbles in the presence of a nearby solid surface of various kinds.

In this paper, we study linear biharmonic maps from Euclidean space into Sol, Nil, and Heisenberg spaces using the linear structure of the target manifolds. We show that a linear map from a Euclidean space into Sol, Nil, or Heisenberg space is biharmonic if and only if it is a harmonic map, and give a complete description of such maps.

We study maps u\in W^{1,2}(\Omega;\overline M) that minimize the Alt–Caffarelli energy functional \int_{\Omega} (|Du|^{2} + q^{2} \chi_{u^{-1}(M)})\,dx under the condition that the image u(\Omega) is confined within \overline M . Here, \Omega denotes a bounded domain in the ambient space \mathbb{R}^{n} (with n\geq 1 ), and M represents a smooth domain in the target space \mathbb{R}^{m} (where m\geq 2 ). Since our minimizing constraint maps coincide with harmonic maps in the interior of the coincidence set, {\rm int}(u^{-1}(\partial M)) , such maps are prone to developing discontinuities due to their inherent nature. This research marks the commencement of an in-depth analysis of potential singularities that might arise within and around the free boundary. Our first significant contribution is an \varepsilon -regularity theorem, founded on a novel method of Lipschitz approximation near points exhibiting low energy. Utilizing this approximation and extending the analysis through a bootstrapping approach, we show Lipschitz continuity of our maps whenever the energy is small. Our subsequent key finding reveals that, whenever the complement of M is uniformly convex and of class C^{3} , the maps minimizing the Alt–Caffarelli energy with a positive parameter q exhibit Lipschitz continuity within a universally defined neighborhood of the noncoincidence set u^{-1}(M) . In particular, this Lipschitz continuity extends to the free boundary. A noteworthy consequence of our findings is the smoothness of flat free boundaries and of the resulting image maps.

3-Dimensional reconstruction (3D reconstruction) generates a 3D computer model of a real object or scene from data such as images, it involves many stages and open problems. Existing methods focus on point clouds and reconstructed polygonal mesh within Manhattan-world constrains in urban scenes reconstruction. However, when dealing with structures like steel truss cable-stayed bridges with complex topology (i.e., connectivity and genus), existing methods fail to recover an appealing polygonal mesh from highly unstructured and noisy point clouds. A topology-aware 3D reconstruction method which can obtain high-level structures and low-level shapes is proposed in this paper. A convolutional neural network and point cloud network is designed to encode multi-view images and 3D point cloud into a compact code, which is then decoded into structure layouts (i.e., a hierarchical binary structural parsing tree) and 3D shapes (i.e., leaf nodes on the binary tree) by designing a recursive neural network and a distance field network respectively. These high-level structures and low-level shapes constitute a 3D digital model.

Much effort is invested in generating natural deformations of three-dimensional shapes. Deformation transfer simplifies this process by allowing to infer deformations of a new shape from existing deformations of a similar shape. Current deformation transfer methods can be applied only to shapes which are represented as a single component manifold mesh, hence their applicability to real-life 3D models is somewhat limited. We propose a novel deformation transfer method, which can be applied to a variety of shape representations -- tet-meshes, polygon soups and multiple-component meshes. Our key technique is deformation of the space in which the shape is embedded. We approximate the given source deformation by a harmonic map using a set of harmonic basis functions. Then, given a sparse set of user-selected correspondence points between the source and target shapes, we generate a deformation of the target shape which has differential properties similar to those of the source deformation. Our method requires only the solution of linear systems of equations, and hence is very robust and efficient. We demonstrate its applicability on a wide range of deformations, for different shape representations.

Regular maps on Cartesian products and disjoint unions of manifolds