Curvatures of Smooth and Discrete Surfaces

Practical Computation of the Cut Locus on Discrete Surfaces

Computer Graphics Forum ◽

10.1111/cgf.14372 ◽

2021 ◽

Vol 40 (5) ◽

pp. 261-273

Author(s):

C. Mancinelli ◽

M. Livesu ◽

E. Puppo

Keyword(s):

Cut Locus ◽

Practical Computation ◽

Discrete Surfaces

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

Journal of Cerebral Blood Flow & Metabolism ◽

10.1038/jcbfm.1994.96 ◽

1994 ◽

Vol 14 (5) ◽

pp. 749-762 ◽

Cited By ~ 66

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.

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A curvature theory for discrete surfaces based on mesh parallelity

Mathematische Annalen ◽

10.1007/s00208-009-0467-9 ◽

2009 ◽

Vol 348 (1) ◽

pp. 1-24 ◽

Cited By ~ 40

Author(s):

Alexander I. Bobenko ◽

Helmut Pottmann ◽

Johannes Wallner

Keyword(s):

Discrete Surfaces ◽

Curvature Theory

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On Restriction Estimates for Discrete Quadratic Surfaces

International Mathematics Research Notices ◽

10.1093/imrn/rnx255 ◽

2018 ◽

Vol 2019 (23) ◽

pp. 7139-7159 ◽

Cited By ~ 1

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.

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