3,940 research outputs found
A Linear Formulation for Disk Conformal Parameterization of Simply-Connected Open Surfaces
Surface parameterization is widely used in computer graphics and geometry
processing. It simplifies challenging tasks such as surface registrations,
morphing, remeshing and texture mapping. In this paper, we present an efficient
algorithm for computing the disk conformal parameterization of simply-connected
open surfaces. A double covering technique is used to turn a simply-connected
open surface into a genus-0 closed surface, and then a fast algorithm for
parameterization of genus-0 closed surfaces can be applied. The symmetry of the
double covered surface preserves the efficiency of the computation. A planar
parameterization can then be obtained with the aid of a M\"obius transformation
and the stereographic projection. After that, a normalization step is applied
to guarantee the circular boundary. Finally, we achieve a bijective disk
conformal parameterization by a composition of quasi-conformal mappings.
Experimental results demonstrate a significant improvement in the computational
time by over 60%. At the same time, our proposed method retains comparable
accuracy, bijectivity and robustness when compared with the state-of-the-art
approaches. Applications to texture mapping are presented for illustrating the
effectiveness of our proposed algorithm
A Face Fairness Framework for 3D Meshes
In this paper, we present a face fairness framework for 3D meshes that
preserves the regular shape of faces and is applicable to a variety of 3D mesh
restoration tasks. Specifically, we present a number of desirable properties
for any mesh restoration method and show that our framework satisfies them. We
then apply our framework to two different tasks --- mesh-denoising and
mesh-refinement, and present comparative results for these two tasks showing
improvement over other relevant methods in the literature.Comment: 15 page
Compressed Sensing and Reconstruction of Unstructured Mesh Datasets
Exascale computing promises quantities of data too large to efficiently store
and transfer across networks in order to be able to analyze and visualize the
results. We investigate Compressive Sensing (CS) as a way to reduce the size of
the data as it is being stored. CS works by sampling the data on the
computational cluster within an alternative function space such as wavelet
bases, and then reconstructing back to the original space on visualization
platforms. While much work has gone into exploring CS on structured data sets,
such as image data, we investigate its usefulness for point clouds such as
unstructured mesh datasets found in many finite element simulations. We sample
using second generation wavelets (SGW) and reconstruct using the Stagewise
Orthogonal Matching Pursuit (StOMP) algorithm. We analyze the compression
ratios achievable and quality of reconstructed results at each compression
rate. We are able to achieve compression ratios between 10 and 30 on moderate
size datasets with minimal visual deterioration as a result of the lossy
compression.Comment: 18 pages, 7 figure
Multi-Kernel Diffusion CNNs for Graph-Based Learning on Point Clouds
Graph convolutional networks are a new promising learning approach to deal
with data on irregular domains. They are predestined to overcome certain
limitations of conventional grid-based architectures and will enable efficient
handling of point clouds or related graphical data representations, e.g.
superpixel graphs. Learning feature extractors and classifiers on 3D point
clouds is still an underdeveloped area and has potential restrictions to equal
graph topologies. In this work, we derive a new architectural design that
combines rotationally and topologically invariant graph diffusion operators and
node-wise feature learning through 1x1 convolutions. By combining multiple
isotropic diffusion operations based on the Laplace-Beltrami operator, we can
learn an optimal linear combination of diffusion kernels for effective feature
propagation across nodes on an irregular graph. We validated our approach for
learning point descriptors as well as semantic classification on real 3D point
clouds of human poses and demonstrate an improvement from 85% to 95% in Dice
overlap with our multi-kernel approach.Comment: accepted for ECCV 2018 Workshop Geometry Meets Deep Learnin
MeshCNN: A Network with an Edge
Polygonal meshes provide an efficient representation for 3D shapes. They
explicitly capture both shape surface and topology, and leverage non-uniformity
to represent large flat regions as well as sharp, intricate features. This
non-uniformity and irregularity, however, inhibits mesh analysis efforts using
neural networks that combine convolution and pooling operations. In this paper,
we utilize the unique properties of the mesh for a direct analysis of 3D shapes
using MeshCNN, a convolutional neural network designed specifically for
triangular meshes. Analogous to classic CNNs, MeshCNN combines specialized
convolution and pooling layers that operate on the mesh edges, by leveraging
their intrinsic geodesic connections. Convolutions are applied on edges and the
four edges of their incident triangles, and pooling is applied via an edge
collapse operation that retains surface topology, thereby, generating new mesh
connectivity for the subsequent convolutions. MeshCNN learns which edges to
collapse, thus forming a task-driven process where the network exposes and
expands the important features while discarding the redundant ones. We
demonstrate the effectiveness of our task-driven pooling on various learning
tasks applied to 3D meshes.Comment: For a two-minute explanation video see https://bit.ly/meshcnnvide
Efficient Feature-based Image Registration by Mapping Sparsified Surfaces
With the advancement in the digital camera technology, the use of high
resolution images and videos has been widespread in the modern society. In
particular, image and video frame registration is frequently applied in
computer graphics and film production. However, conventional registration
approaches usually require long computational time for high resolution images
and video frames. This hinders the application of the registration approaches
in the modern industries. In this work, we first propose a new image
representation method to accelerate the registration process by triangulating
the images effectively. For each high resolution image or video frame, we
compute an optimal coarse triangulation which captures the important features
of the image. Then, we apply a surface registration algorithm to obtain a
registration map which is used to compute the registration of the high
resolution image. Experimental results suggest that our overall algorithm is
efficient and capable to achieve a high compression rate while the accuracy of
the registration is well retained when compared with the conventional
grid-based approach. Also, the computational time of the registration is
significantly reduced using our triangulation-based approach
Fast Disk Conformal Parameterization of Simply-connected Open Surfaces
Surface parameterizations have been widely used in computer graphics and
geometry processing. In particular, as simply-connected open surfaces are
conformally equivalent to the unit disk, it is desirable to compute the disk
conformal parameterizations of the surfaces. In this paper, we propose a novel
algorithm for the conformal parameterization of a simply-connected open surface
onto the unit disk, which significantly speeds up the computation, enhances the
conformality and stability, and guarantees the bijectivity. The conformality
distortions at the inner region and on the boundary are corrected by two steps,
with the aid of an iterative scheme using quasi-conformal theories.
Experimental results demonstrate the effectiveness of our proposed method
A Connectivity-Aware Multi-level Finite-Element System for Solving Laplace-Beltrami Equations
Recent work on octree-based finite-element systems has developed a multigrid
solver for Poisson equations on meshes. While the idea of defining a regularly
indexed function space has been successfully used in a number of applications,
it has also been noted that the richness of the function space is limited
because the function values can be coupled across locally disconnected regions.
In this work, we show how to enrich the function space by introducing functions
that resolve the coupling while still preserving the nesting hierarchy that
supports multigrid. A spectral analysis reveals the superior quality of the
resulting Laplace-Beltrami operator and applications to surface flow
demonstrate that our new solver more efficiently converges to the correct
solution.Comment: This work was done when the first author was a PhD student at Johns
Hopkins Universit
hp-finite-elements for simulating electromagnetic fields in optical devices with rough textures
The finite-element method is a preferred numerical method when
electromagnetic fields at high accuracy are to be computed in nano-optics
design. Here, we demonstrate a finite-element method using hp-adaptivity on
tetrahedral meshes for computation of electromagnetic fields in a device with
rough textures. The method allows for efficient computations on meshes with
strong variations in element sizes. This enables to use precise geometry
resolution of the rough textures. Convergence to highly accurate results is
observed.Comment: Proceedings article, SPIE conference "Optical Systems Design 2015:
Computational Optics
Solid Geometry Processing on Deconstructed Domains
Many tasks in geometry processing are modeled as variational problems solved
numerically using the finite element method. For solid shapes, this requires a
volumetric discretization, such as a boundary conforming tetrahedral mesh.
Unfortunately, tetrahedral meshing remains an open challenge and existing
methods either struggle to conform to complex boundary surfaces or require
manual intervention to prevent failure. Rather than create a single volumetric
mesh for the entire shape, we advocate for solid geometry processing on
deconstructed domains, where a large and complex shape is composed of
overlapping solid subdomains. As each smaller and simpler part is now easier to
tetrahedralize, the question becomes how to account for overlaps during problem
modeling and how to couple solutions on each subdomain together algebraically.
We explore how and why previous coupling methods fail, and propose a method
that couples solid domains only along their boundary surfaces. We demonstrate
the superiority of this method through empirical convergence tests and
qualitative applications to solid geometry processing on a variety of popular
second-order and fourth-order partial differential equations.Comment: submitted to Computer Graphics Foru
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