Practical Computation of the Cut Locus on Discrete Surfaces

2021 ◽  
Vol 40 (5) ◽  
pp. 261-273
Author(s):  
C. Mancinelli ◽  
M. Livesu ◽  
E. Puppo
Keyword(s):  
Cut Locus ◽  
Practical Computation ◽  
Discrete Surfaces

scholarly journals Remarks on Manifolds with Two-Sided Curvature Bounds

2021 ◽  
Vol 9 (1) ◽  
pp. 53-64
Author(s):  
Vitali Kapovitch ◽  
Alexander Lytchak
Keyword(s):  
Distance Functions ◽  
Cut Locus ◽  
Bounded Curvature ◽  
Curvature Bounds ◽  
Positive Reach

Abstract We discuss folklore statements about distance functions in manifolds with two-sided bounded curvature. The topics include regularity, subsets of positive reach and the cut locus.

scholarly journals Fast Nonsupervised 3D Registration of PET and MR Images of the Brain

1994 ◽  
Vol 14 (5) ◽  
pp. 749-762 ◽  
Author(s):  
Jean-François Mangin ◽  
Vincent Frouin ◽  
Isabelle Bloch ◽  
Bernard Bendriem ◽  
Jaime Lopez-Krahe
Keyword(s):  
Three Dimensional ◽  
Magnetic Resonance Images ◽  
Surface Matching ◽  
3D Registration ◽  
Matching Algorithm ◽  
Distance Map ◽  
Brain Surface ◽  
Discrete Surfaces ◽  
Positron Emission ◽  
The Brain

We propose a fully nonsupervised methodology dedicated to the fast registration of positron emission tomography (PET) and magnetic resonance images of the brain. First, discrete representations of the surfaces of interest (head or brain surface) are automatically extracted from both images. Then, a shape-independent surface-matching algorithm gives a rigid body transformation, which allows the transfer of information between both modalities. A three-dimensional (3D) extension of the chamfer-matching principle makes up the core of this surface-matching algorithm. The optimal transformation is inferred from the minimization of a quadratic generalized distance between discrete surfaces, taking into account between-modality differences in the localization of the segmented surfaces. The minimization process is efficiently performed via the precomputation of a 3D distance map. Validation studies using a dedicated brain-shaped phantom have shown that the maximum registration error was of the order of the PET pixel size (2 mm) for the wide variety of tested configurations. The software is routinely used today in a clinical context by the physicians of the Service Hospitalier Frédéric Joliot (>150 registrations performed). The entire registration process requires ∼5 min on a conventional workstation.

scholarly journals A curvature theory for discrete surfaces based on mesh parallelity

2009 ◽  
Vol 348 (1) ◽  
pp. 1-24 ◽  
Author(s):  
Alexander I. Bobenko ◽  
Helmut Pottmann ◽  
Johannes Wallner
Keyword(s):  
Discrete Surfaces ◽  
Curvature Theory

scholarly journals On Restriction Estimates for Discrete Quadratic Surfaces

2018 ◽  
Vol 2019 (23) ◽  
pp. 7139-7159 ◽  
Author(s):  
Kevin Henriot ◽  
Kevin Hughes
Keyword(s):  
Quadratic Form ◽  
Integer Matrix ◽  
Discrete Surfaces ◽  
Indefinite Quadratic Form ◽  
Indefinite Quadratic ◽  
Quadratic Surfaces

Abstract We obtain truncated restriction estimates of an unexpected form for discrete surfaces $$\begin{align*}S_N = \{\, ( n_1 , \dots , n_d , R( n_1 , \dots, n_d ) ) \,,\, n_i \in [-N,N] \cap {\mathbb{Z}} \,\},\end{align*}$$ where $R$ is an indefinite quadratic form with integer matrix.

scholarly journals Practical computation of the mixed μ problem

1992 ◽  
Author(s):  
Peter M. Young ◽  
Matthew P. Newlin ◽  
John C. Doyle
Keyword(s):  
Practical Computation

The Lorentzian Cut Locus

2017 ◽  
pp. 295-322
Author(s):  
John K. Beem ◽  
Paul E. Ehrlich ◽  
Kevin L. Easley
Keyword(s):  
Cut Locus
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