1,118 research outputs found
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
Multilevel Solvers for Unstructured Surface Meshes
Parameterization of unstructured surface meshes is of fundamental importance in many applications of digital geometry processing. Such parameterization approaches give rise to large and exceedingly ill-conditioned systems which are difficult or impossible to solve without the use of sophisticated multilevel preconditioning strategies. Since the underlying meshes are very fine to begin with, such multilevel preconditioners require mesh coarsening to build an appropriate hierarchy. In this paper we consider several strategies for the construction of hierarchies using ideas from mesh simplification algorithms used in the computer graphics literature. We introduce two novel hierarchy construction schemes and demonstrate their superior performance when used in conjunction with a multigrid preconditioner
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
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