30 research outputs found
Steklov Spectral Geometry for Extrinsic Shape Analysis
We propose using the Dirichlet-to-Neumann operator as an extrinsic
alternative to the Laplacian for spectral geometry processing and shape
analysis. Intrinsic approaches, usually based on the Laplace-Beltrami operator,
cannot capture the spatial embedding of a shape up to rigid motion, and many
previous extrinsic methods lack theoretical justification. Instead, we consider
the Steklov eigenvalue problem, computing the spectrum of the
Dirichlet-to-Neumann operator of a surface bounding a volume. A remarkable
property of this operator is that it completely encodes volumetric geometry. We
use the boundary element method (BEM) to discretize the operator, accelerated
by hierarchical numerical schemes and preconditioning; this pipeline allows us
to solve eigenvalue and linear problems on large-scale meshes despite the
density of the Dirichlet-to-Neumann discretization. We further demonstrate that
our operators naturally fit into existing frameworks for geometry processing,
making a shift from intrinsic to extrinsic geometry as simple as substituting
the Laplace-Beltrami operator with the Dirichlet-to-Neumann operator.Comment: Additional experiments adde
DA Wand: Distortion-Aware Selection using Neural Mesh Parameterization
We present a neural technique for learning to select a local sub-region
around a point which can be used for mesh parameterization. The motivation for
our framework is driven by interactive workflows used for decaling, texturing,
or painting on surfaces. Our key idea is to incorporate segmentation
probabilities as weights of a classical parameterization method, implemented as
a novel differentiable parameterization layer within a neural network
framework. We train a segmentation network to select 3D regions that are
parameterized into 2D and penalized by the resulting distortion, giving rise to
segmentations which are distortion-aware. Following training, a user can use
our system to interactively select a point on the mesh and obtain a large,
meaningful region around the selection which induces a low-distortion
parameterization. Our code and project page are currently available.Comment: Project page: https://threedle.github.io/DA-Wand/ Code:
https://github.com/threedle/DA-Wan
A mixed regularization approach for sparse simultaneous approximation of parameterized PDEs
We present and analyze a novel sparse polynomial technique for the
simultaneous approximation of parameterized partial differential equations
(PDEs) with deterministic and stochastic inputs. Our approach treats the
numerical solution as a jointly sparse reconstruction problem through the
reformulation of the standard basis pursuit denoising, where the set of jointly
sparse vectors is infinite. To achieve global reconstruction of sparse
solutions to parameterized elliptic PDEs over both physical and parametric
domains, we combine the standard measurement scheme developed for compressed
sensing in the context of bounded orthonormal systems with a novel mixed-norm
based regularization method that exploits both energy and sparsity. In
addition, we are able to prove that, with minimal sample complexity, error
estimates comparable to the best -term and quasi-optimal approximations are
achievable, while requiring only a priori bounds on polynomial truncation error
with respect to the energy norm. Finally, we perform extensive numerical
experiments on several high-dimensional parameterized elliptic PDE models to
demonstrate the superior recovery properties of the proposed approach.Comment: 23 pages, 4 figure
Unwind: Interactive Fish Straightening
The ScanAllFish project is a large-scale effort to scan all the world's
33,100 known species of fishes. It has already generated thousands of
volumetric CT scans of fish species which are available on open access
platforms such as the Open Science Framework. To achieve a scanning rate
required for a project of this magnitude, many specimens are grouped together
into a single tube and scanned all at once. The resulting data contain many
fish which are often bent and twisted to fit into the scanner. Our system,
Unwind, is a novel interactive visualization and processing tool which
extracts, unbends, and untwists volumetric images of fish with minimal user
interaction. Our approach enables scientists to interactively unwarp these
volumes to remove the undesired torque and bending using a piecewise-linear
skeleton extracted by averaging isosurfaces of a harmonic function connecting
the head and tail of each fish. The result is a volumetric dataset of a
individual, straight fish in a canonical pose defined by the marine biologist
expert user. We have developed Unwind in collaboration with a team of marine
biologists: Our system has been deployed in their labs, and is presently being
used for dataset construction, biomechanical analysis, and the generation of
figures for scientific publication
Surface Deformation Potentials on Meshes for Computer Graphics and Visualization
Shape deformation models have been used in computer graphics primarily to describe the dynamics of physical deformations like cloth draping, collisions of elastic bodies, fracture, or animation of hair. Less frequent is their application to problems not directly related to a physical process. In this thesis we apply deformations to three problems in computer graphics that do not correspond to physical deformations. To this end, we generalize the physical model by modifying the energy potential. Originally, the energy potential amounts to the physical work needed to deform a body from its rest state into a given configuration and relates material strain to internal restoring forces that act to restore the original shape. For each of the three problems considered, this potential is adapted to reflect an application specific notion of shape. Under the influence of further constraints, our generalized deformation results in shapes that balance preservation of certain shape properties and application specific objectives similar to physical equilibrium states. The applications discussed in this thesis are surface parameterization, interactive shape editing and automatic design of panorama maps. For surface parameterization, we interpret parameterizations over a planar domain as deformations from a flat initial configuration onto a given surface. In this setting, we review existing parameterization methods by analyzing properties of their potential functions and derive potentials accounting for distortion of geometric properties. Interactive shape editing allows an untrained user to modify complex surfaces, be simply grabbing and moving parts of interest. A deformation model interactively extrapolates the transformation from those parts to the rest of the surface. This thesis proposes a differential shape representation for triangle meshes leading to a potential that can be optimized interactively with a simple, tailored algorithm. Although the potential is not physically accurate, it results in intuitive deformation behavior and can be parameterized to account for different material properties. Panorama maps are blends between landscape illustrations and geographic maps that are traditionally painted by an artist to convey geographic surveyknowledge on public places like ski resorts or national parks. While panorama maps are not drawn to scale, the shown landscape remains recognizable and the observer can easily recover details necessary for self location and orientation. At the same time, important features as trails or ski slopes appear not occluded and well visible. This thesis proposes the first automatic panorama generation method. Its basis is again a surface deformation, that establishes the necessary compromise between shape preservation and feature visibility.Potentiale zur Flächendeformation auf Dreiecksnetzen für Anwendungen in der Computergrafik und Visualisierung Deformationsmodelle werden in der Computergrafik bislang hauptsächlich eingesetzt, um die Dynamik physikalischer Deformationsprozesse zu modellieren. Gängige Beispiele sind Bekleidungssimulationen, Kollisionen elastischer Körper oder Animation von Haaren und Frisuren. Deutlich seltener ist ihre Anwendung auf Probleme, die nicht direkt physikalischen Prozessen entsprechen. In der vorliegenden Arbeit werden Deformationsmodelle auf drei Probleme der Computergrafik angewandt, die nicht unmittelbar einem physikalischen Deformationsprozess entsprechen. Zu diesem Zweck wird das physikalische Modell durch eine passende Änderung der potentiellen Energie verallgemeinert. Die potentielle Energie entspricht normalerweise der physikalischen Arbeit, die aufgewendet werden muss, um einen Körper aus dem Ruhezustand in eine bestimmte Konfiguration zu verformen. Darüber hinaus setzt sie die aktuelle Verformung in Beziehung zu internen Spannungskräften, die wirken um die ursprüngliche Form wiederherzustellen. In dieser Arbeit passen wir für jedes der drei betrachteten Problemfelder die potentielle Energie jeweils so an, dass sie eine anwendungsspezifische Definition von Form widerspiegelt. Unter dem Einfluss weiterer Randbedingungen führt die so verallgemeinerte Deformation zu einer Fläche, die eine Balance zwischen der Erhaltung gewisser Formeigenschaften und Zielvorgaben der Anwendung findet. Diese Balance entspricht dem Equilibrium einer physikalischen Deformation. Die drei in dieser Arbeit diskutierten Anwendungen sind Oberflächenparameterisierung, interaktives Bearbeiten von Flächen und das vollautomatische Erzeugen von Panoramakarten im Stile von Heinrich Berann. Zur Oberflächenparameterisierung interpretieren wir Parameterisierungen über einem flachen Parametergebiet als Deformationen, die ein ursprünglich ebenes Flächenstück in eine gegebene Oberfläche verformen. Innerhalb dieses Szenarios vergleichen wir dann existierende Methoden zur planaren Parameterisierung, indem wir die resultierenden potentiellen Energien analysieren, und leiten weitere Potentiale her, die die Störung geometrischer Eigenschaften wie Fläche und Winkel erfassen. Verfahren zur interaktiven Flächenbearbeitung ermöglichen schnelle und intuitive Änderungen an einer komplexen Oberfläche. Dazu wählt der Benutzer Teile der Fläche und bewegt diese durch den Raum. Ein Deformationsmodell extrapoliert interaktiv die Transformation der gewählten Teile auf die restliche Fläche. Diese Arbeit stellt eine neue differentielle Flächenrepräsentation für diskrete Flächen vor, die zu einem einfach und interaktiv zu optimierendem Potential führt. Obwohl das vorgeschlagene Potential nicht physikalisch korrekt ist, sind die resultierenden Deformationen intuitiv. Mittels eines Parameters lassen sich außerdem bestimmte Materialeigenschaften einstellen. Panoramakarten im Stile von Heinrich Berann sind eine Verschmelzung von Landschaftsillustration und geographischer Karte. Traditionell werden sie so von Hand gezeichnet, dass bestimmt Merkmale wie beispielsweise Skipisten oder Wanderwege in einem Gebiet unverdeckt und gut sichtbar bleiben, was große Kunstfertigkeit verlangt. Obwohl diese Art der Darstellung nicht maßstabsgetreu ist, sind Abweichungen auf den ersten Blick meistens nicht zu erkennen. Dadurch kann der Betrachter markante Details schnell wiederfinden und sich so innerhalb des Gebietes orientieren. Diese Arbeit stellt das erste, vollautomatische Verfahren zur Erzeugung von Panoramakarten vor. Grundlage ist wiederum eine verallgemeinerte Oberflächendeformation, die sowohl auf Formerhaltung als auch auf die Sichtbarkeit vorgegebener geographischer Merkmale abzielt
Grid-based Finite Elements System for Solving Laplace-Beltrami Equations on 2-Manifolds
Solving the Poisson equation has numerous important applications. On a Riemannian 2-manifold, the task is most often formulated in terms of finite elements and two challenges commonly arise: discretizing the space of functions and solving the resulting system of equations. In this work, we describe a finite elements system that simultaneously addresses both aspects. The idea is to define a space of functions in 3D and then restrict the 3D functions to the mesh. Unlike traditional approaches, our method is tessellation-independent and has a direct control over system complexity. More importantly, the resulting function space comes with a multi-resolution structure supporting an efficient multigrid solver, and the regularity of the function space can be leveraged in parallelizing/streaming the computation. We evaluate our framework by conducting several experiments. These include a spectral analysis that reveals the embedding-invariant robustness of our discretization, and a benchmark for solver convergence/performance that reveals the competitiveness of our approach against other state-of-the-art methods. We apply our work to several geometry-processing applications. Using curvature flows, we show that we can support efficient surface evolution where the embedding changes with time. Formulating surface filtering as a solution to the screened-Poisson equation, we demonstrate that we can support an anisotropic surface editing system that processes high resolution meshes in real time