377 research outputs found
Gradient Young measures generated by quasiconformal maps in the plane
In this contribution, we completely and explicitly characterize Young
measures generated by gradients of quasiconformal maps in the plane. By doing
so, we generalize the results of Astala and Faraco \cite{AstalaFaraco} who
provided a similar result for quasiregular maps and Bene\v{s}ov\'a and
Kru\v{z}\'ik \cite{bbmk2013} who characterized Young measures generated by
gradients of bi-Lipschitz maps. Our results are motivated by non-linear
elasticity where injectivity of the functions in the generating sequence is
essential in order to assure non-interpenetration of matter
Bijective Mappings Of Meshes With Boundary And The Degree In Mesh Processing
This paper introduces three sets of sufficient conditions, for generating
bijective simplicial mappings of manifold meshes. A necessary condition for a
simplicial mapping of a mesh to be injective is that it either maintains the
orientation of all elements or flips all the elements. However, these
conditions are known to be insufficient for injectivity of a simplicial map. In
this paper we provide additional simple conditions that, together with the
above mentioned necessary conditions guarantee injectivity of the simplicial
map.
The first set of conditions generalizes classical global inversion theorems
to the mesh (piecewise-linear) case. That is, proves that in case the boundary
simplicial map is bijective and the necessary condition holds then the map is
injective and onto the target domain. The second set of conditions is concerned
with mapping of a mesh to a polytope and replaces the (often hard) requirement
of a bijective boundary map with a collection of linear constraints and
guarantees that the resulting map is injective over the interior of the mesh
and onto. These linear conditions provide a practical tool for optimizing a map
of the mesh onto a given polytope while allowing the boundary map to adjust
freely and keeping the injectivity property in the interior of the mesh. The
third set of conditions adds to the second set the requirement that the
boundary maps are orientation preserving as-well (with a proper definition of
boundary map orientation). This set of conditions guarantees that the map is
injective on the boundary of the mesh as-well as its interior. Several
experiments using the sufficient conditions are shown for mapping triangular
meshes.
A secondary goal of this paper is to advocate and develop the tool of degree
in the context of mesh processing
Lipschitz regularity for inner-variational equations
We obtain Lipschitz regularity results for a fairly general class of
nonlinear first-order PDEs. These equations arise from the inner variation of
certain energy integrals. Even in the simplest model case of the Dirichlet
energy the inner-stationary solutions need not be differentiable everywhere;
the Lipschitz continuity is the best possible. But the proofs, even in the
Dirichlet case, turn out to relay on topological arguments. The appeal to the
inner-stationary solutions in this context is motivated by the classical
problems of existence and regularity of the energy-minimal deformations in the
theory of harmonic mappings and certain mathematical models of nonlinear
elasticity; specifically, neo-Hookian type problems.Comment: No 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
VOLMAP: a Large Scale Benchmark for Volume Mappings to Simple Base Domains
Correspondences between geometric domains (mappings) are ubiquitous in computer graphics and engineering, both for a variety of downstream applications and as core building blocks for higher level algorithms. In particular, mapping a shape to a convex or star-shaped domain with simple geometry is a fundamental module in existing pipelines for mesh generation, solid texturing, generation of shape correspondences, advanced manufacturing etc. For the case of surfaces, computing such a mapping with guarantees of injectivity is a solved problem. Conversely, robust algorithms for the generation of injective volume mappings to simple polytopes are yet to be found, making this a fundamental open problem in volume mesh processing. VOLMAP is a large scale benchmark aimed to support ongoing research in volume mapping algorithms. The dataset contains 4.7K tetrahedral meshes, whose boundary vertices are mapped to a variety of simple domains, either convex or star-shaped. This data constitutes the input for candidate algorithms, which are then required to position interior vertices in the domain to obtain a volume map. Overall, this yields more than 22K alternative test cases. VOLMAP also comprises tools to process this data, analyze the resulting maps, and extend the dataset with new meshes, boundary maps and base domains. This article provides a brief overview of the field, discussing its importance and the lack of effective techniques. We then introduce both the dataset and its major features. An example of comparative analysis between two existing methods is also present
Quasiconformal mappings that highly distort dimensions of many parallel lines
We construct a quasiconformal mapping of -dimensional Euclidean space, , that simultaneously distorts the Hausdorff dimension of a nearly
maximal collection of parallel lines by a given amount. This answers a question
of Balogh, Monti, and Tyson.Comment: 12 page
On the optimality of gluing over scales
We show that for every , there exist -point metric spaces
(X,d) where every "scale" admits a Euclidean embedding with distortion at most
, but the whole space requires distortion at least . This shows that the scale-gluing lemma [Lee, SODA 2005] is tight,
and disproves a conjecture stated there. This matching upper bound was known to
be tight at both endpoints, i.e. when and , but nowhere in between.
More specifically, we exhibit -point spaces with doubling constant
requiring Euclidean distortion ,
which also shows that the technique of "measured descent" [Krauthgamer, et.
al., Geometric and Functional Analysis] is optimal. We extend this to obtain a
similar tight result for spaces with .Comment: minor revision
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