Axioms

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1015 KiB

Open AccessArticle

Heat Kernel Embeddings, Differential Geometry and Graph Structure

by Hewayda ElGhawalby and Edwin R. Hancock

Axioms 2015, 4(3), 275-293; https://doi.org/10.3390/axioms4030275 - 21 Jul 2015

Cited by 7 | Viewed by 5106

In this paper, we investigate the heat kernel embedding as a route to graph representation. The heat kernel of the graph encapsulates information concerning the distribution of path lengths and, hence, node affinities on the graph; and is found by exponentiating the Laplacian [...] Read more.

In this paper, we investigate the heat kernel embedding as a route to graph representation. The heat kernel of the graph encapsulates information concerning the distribution of path lengths and, hence, node affinities on the graph; and is found by exponentiating the Laplacian eigen-system over time. A Young–Householder decomposition is performed on the heat kernel to obtain the matrix of the embedded coordinates for the nodes of the graph. With the embeddings at hand, we establish a graph characterization based on differential geometry by computing sets of curvatures associated with the graph edges and triangular faces. A sectional curvature computed from the difference between geodesic and Euclidean distances between nodes is associated with the edges of the graph. Furthermore, we use the Gauss–Bonnet theorem to compute the Gaussian curvatures associated with triangular faces of the graph. Full article

(This article belongs to the Special Issue Discrete Differential Geometry and Its Applications to Imaging and Graphics)

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3194 KiB

Open AccessArticle

A Simplified Algorithm for Inverting Higher Order Diffusion Tensors

by Laura Astola, Neda Sepasian, Tom Dela Haije, Andrea Fuster and Luc Florack

Axioms 2014, 3(4), 369-379; https://doi.org/10.3390/axioms3040369 - 14 Nov 2014

Cited by 1 | Viewed by 5071

Abstract

In Riemannian geometry, a distance function is determined by an inner product on the tangent space. In Riemann–Finsler geometry, this distance function can be determined by a norm. This gives more freedom on the form of the so-called indicatrix or the set of [...] Read more.

In Riemannian geometry, a distance function is determined by an inner product on the tangent space. In Riemann–Finsler geometry, this distance function can be determined by a norm. This gives more freedom on the form of the so-called indicatrix or the set of unit vectors. This has some interesting applications, e.g., in medical image analysis, especially in diffusion weighted imaging (DWI). An important application of DWI is in the inference of the local architecture of the tissue, typically consisting of thin elongated structures, such as axons or muscle fibers, by measuring the constrained diffusion of water within the tissue. From high angular resolution diffusion imaging (HARDI) data, one can estimate the diffusion orientation distribution function (dODF), which indicates the relative diffusivity in all directions and can be represented by a spherical polynomial. We express this dODF as an equivalent spherical monomial (higher order tensor) to directly generalize the (second order) diffusion tensor approach. To enable efficient computation of Riemann–Finslerian quantities on diffusion weighted (DW)-images, such as the metric/norm tensor, we present a simple and efficient algorithm to invert even order spherical monomials, which extends the familiar inversion of diffusion tensors, i.e., symmetric matrices. Full article

(This article belongs to the Special Issue Discrete Differential Geometry and Its Applications to Imaging and Graphics)

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213 KiB

Open AccessArticle

The Gromov–Wasserstein Distance: A Brief Overview

by Facundo Mémoli

Axioms 2014, 3(3), 335-341; https://doi.org/10.3390/axioms3030335 - 2 Sep 2014

Cited by 17 | Viewed by 11068

Abstract

We recall the construction of the Gromov–Wasserstein distance and concentrate on quantitative aspects of the definition. Full article

(This article belongs to the Special Issue Discrete Differential Geometry and Its Applications to Imaging and Graphics)

7059 KiB

Open AccessArticle

Matching the LBO Eigenspace of Non-Rigid Shapes via High Order Statistics

by Alon Shtern and Ron Kimmel

Axioms 2014, 3(3), 300-319; https://doi.org/10.3390/axioms3030300 - 15 Jul 2014

Cited by 13 | Viewed by 5632

Abstract

A fundamental tool in shape analysis is the virtual embedding of the Riemannian manifold describing the geometry of a shape into Euclidean space. Several methods have been proposed to embed isometric shapes into flat domains, while preserving the distances measured on the manifold. [...] Read more.

A fundamental tool in shape analysis is the virtual embedding of the Riemannian manifold describing the geometry of a shape into Euclidean space. Several methods have been proposed to embed isometric shapes into flat domains, while preserving the distances measured on the manifold. Recently, attention has been given to embedding shapes into the eigenspace of the Laplace–Beltrami operator. The Laplace–Beltrami eigenspace preserves the diffusion distance and is invariant under isometric transformations. However, Laplace–Beltrami eigenfunctions computed independently for different shapes are often incompatible with each other. Applications involving multiple shapes, such as pointwise correspondence, would greatly benefit if their respective eigenfunctions were somehow matched. Here, we introduce a statistical approach for matching eigenfunctions. We consider the values of the eigenfunctions over the manifold as the sampling of random variables and try to match their multivariate distributions. Comparing distributions is done indirectly, using high order statistics. We show that the permutation and sign ambiguities of low order eigenfunctions can be inferred by minimizing the difference of their third order moments. The sign ambiguities of antisymmetric eigenfunctions can be resolved by exploiting isometric invariant relations between the gradients of the eigenfunctions and the surface normal. We present experiments demonstrating the success of the proposed method applied to feature point correspondence. Full article

(This article belongs to the Special Issue Discrete Differential Geometry and Its Applications to Imaging and Graphics)

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4100 KiB

Open AccessArticle

A Comparative Study of Several Classical, Discrete Differential and Isogeometric Methods for Solving Poisson’s Equation on the Disk

by Thien Nguyen, Keçstutis Karčiauskas and Jörg Peters

Axioms 2014, 3(2), 280-299; https://doi.org/10.3390/axioms3020280 - 11 Jun 2014

Cited by 68 | Viewed by 9494

Abstract

This paper outlines and qualitatively compares the implementations of seven different methods for solving Poisson’s equation on the disk. The methods include two classical finite elements, a cotan formula-based discrete differential geometry approach and four isogeometric constructions. The comparison reveals numerical convergence rates [...] Read more.

This paper outlines and qualitatively compares the implementations of seven different methods for solving Poisson’s equation on the disk. The methods include two classical finite elements, a cotan formula-based discrete differential geometry approach and four isogeometric constructions. The comparison reveals numerical convergence rates and, particularly for isogeometric constructions based on Catmull–Clark elements, the need to carefully choose quadrature formulas. The seven methods include two that are new to isogeometric analysis. Both new methods yield O(h³) convergence in the L² norm, also when points are included where n 6≠ 4 pieces meet. One construction is based on a polar, singular parameterization; the other is a G¹ tensor-product construction. Full article

(This article belongs to the Special Issue Discrete Differential Geometry and Its Applications to Imaging and Graphics)

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19270 KiB

Open AccessArticle

Conformal-Based Surface Morphing and Multi-Scale Representation

by Ka Chun Lam, Chengfeng Wen and Lok Ming Lui

Axioms 2014, 3(2), 222-243; https://doi.org/10.3390/axioms3020222 - 20 May 2014

Cited by 4 | Viewed by 6360

Abstract

This paper presents two algorithms, based on conformal geometry, for the multi-scale representations of geometric shapes and surface morphing. A multi-scale surface representation aims to describe a 3D shape at different levels of geometric detail, which allows analyzing or editing surfaces at the [...] Read more.

This paper presents two algorithms, based on conformal geometry, for the multi-scale representations of geometric shapes and surface morphing. A multi-scale surface representation aims to describe a 3D shape at different levels of geometric detail, which allows analyzing or editing surfaces at the global or local scales effectively. Surface morphing refers to the process of interpolating between two geometric shapes, which has been widely applied to estimate or analyze deformations in computer graphics, computer vision and medical imaging. In this work, we propose two geometric models for surface morphing and multi-scale representation for 3D surfaces. The basic idea is to represent a 3D surface by its mean curvature function, H, and conformal factor function λ, which uniquely determine the geometry of the surface according to Riemann surface theory. Once we have the (λ, H) parameterization of the surface, post-processing of the surface can be done directly on the conformal parameter domain. In particular, the problem of multi-scale representations of shapes can be reduced to the signal filtering on the λ and H parameters. On the other hand, the surface morphing problem can be transformed to an interpolation process of two sets of (λ, H) parameters. We test the proposed algorithms on 3D human face data and MRI-derived brain surfaces. Experimental results show that our proposed methods can effectively obtain multi-scale surface representations and give natural surface morphing results. Full article

(This article belongs to the Special Issue Discrete Differential Geometry and Its Applications to Imaging and Graphics)

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1880 KiB

Open AccessArticle

Characteristic Number: Theory and Its Application to Shape Analysis

by Xin Fan, Zhongxuan Luo, Jielin Zhang, Xinchen Zhou, Qi Jia and Daiyun Luo

Axioms 2014, 3(2), 202-221; https://doi.org/10.3390/axioms3020202 - 15 May 2014

Cited by 6 | Viewed by 6232

Abstract

Geometric invariants are important for shape recognition and matching. Existing invariants in projective geometry are typically defined on the limited number (e.g., five for the classical cross-ratio) of collinear planar points and also lack the ability to characterize the curve or surface underlying [...] Read more.

Geometric invariants are important for shape recognition and matching. Existing invariants in projective geometry are typically defined on the limited number (e.g., five for the classical cross-ratio) of collinear planar points and also lack the ability to characterize the curve or surface underlying the given points. In this paper, we present a projective invariant named after the characteristic number of planar algebraic curves. The characteristic number in this work reveals an intrinsic property of an algebraic hypersurface or curve, which relies no more on the existence of the surface or curve as its planar version. The new definition also generalizes the cross-ratio by relaxing the collinearity and number of points for the cross-ratio. We employ the characteristic number to construct more informative shape descriptors that improve the performance of shape recognition, especially when severe affine and perspective deformations occur. In addition to the application to shape recognition, we incorporate the geometric constraints on facial feature points derived from the characteristic number into facial feature matching. The experiments show the improvements on accuracy and robustness to pose and view changes over the method with the collinearity and cross-ratio constraints. Full article

(This article belongs to the Special Issue Discrete Differential Geometry and Its Applications to Imaging and Graphics)

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7202 KiB

Open AccessArticle

Deterministic Greedy Routing with Guaranteed Delivery in 3D Wireless Sensor Networks

by Su Xia, Xiaotian Yin, Hongyi Wu, Miao Jin and Xianfeng David Gu

Axioms 2014, 3(2), 177-201; https://doi.org/10.3390/axioms3020177 - 15 May 2014

Cited by 3 | Viewed by 5787

Abstract

With both computational complexity and storage space bounded by a small constant, greedy routing is recognized as an appealing approach to support scalable routing in wireless sensor networks. However, significant challenges have been encountered in extending greedy routing from 2D to 3D space. [...] Read more.

With both computational complexity and storage space bounded by a small constant, greedy routing is recognized as an appealing approach to support scalable routing in wireless sensor networks. However, significant challenges have been encountered in extending greedy routing from 2D to 3D space. In this research, we develop decentralized solutions to achieve greedy routing in 3D sensor networks. Our proposed approach is based on a unit tetrahedron cell (UTC) mesh structure. We propose a distributed algorithm to realize volumetric harmonic mapping (VHM) of the UTC mesh under spherical boundary condition. It is a one-to-one map that yields virtual coordinates for each node in the network without or with one internal hole. Since a boundary has been mapped to a sphere, node-based greedy routing is always successful thereon. At the same time, we exploit the UTC mesh to develop a face-based greedy routing algorithm and prove its success at internal nodes. To deliver a data packet to its destination, face-based and node-based greedy routing algorithms are employed alternately at internal and boundary UTCs, respectively. For networks with multiple internal holes, a segmentation and tunnel-based routing strategy is proposed on top of VHM to support global end-to-end routing. As far as we know, this is the first work that realizes truly deterministic routing with constant-bounded storage and computation in general 3D wireless sensor networks. Full article

(This article belongs to the Special Issue Discrete Differential Geometry and Its Applications to Imaging and Graphics)

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Graphical abstract

291 KiB

Open AccessArticle

Ricci Curvature on Polyhedral Surfaces via Optimal Transportation

by Benoît Loisel and Pascal Romon

Axioms 2014, 3(1), 119-139; https://doi.org/10.3390/axioms3010119 - 6 Mar 2014

Cited by 27 | Viewed by 5440

Abstract

The problem of correctly defining geometric objects, such as the curvature, is a hard one in discrete geometry. In 2009, Ollivier defined a notion of curvature applicable to a wide category of measured metric spaces, in particular to graphs. He named it coarse [...] Read more.

The problem of correctly defining geometric objects, such as the curvature, is a hard one in discrete geometry. In 2009, Ollivier defined a notion of curvature applicable to a wide category of measured metric spaces, in particular to graphs. He named it coarse Ricci curvature because it coincides, up to some given factor, with the classical Ricci curvature, when the space is a smooth manifold. Lin, Lu and Yau and Jost and Liu have used and extended this notion for graphs, giving estimates for the curvature and, hence, the diameter, in terms of the combinatorics. In this paper, we describe a method for computing the coarse Ricci curvature and give sharper results, in the specific, but crucial case of polyhedral surfaces. Full article

(This article belongs to the Special Issue Discrete Differential Geometry and Its Applications to Imaging and Graphics)

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533 KiB

Open AccessArticle

Canonical Coordinates for Retino-Cortical Magnification

by Luc Florack

Axioms 2014, 3(1), 70-81; https://doi.org/10.3390/axioms3010070 - 24 Feb 2014

Viewed by 5962

Abstract

A geometric model for a biologically-inspired visual front-end is proposed, based on an isotropic, scale-invariant two-form field. The model incorporates a foveal property typical of biological visual systems, with an approximately linear decrease of resolution as a function of eccentricity, and by a [...] Read more.

A geometric model for a biologically-inspired visual front-end is proposed, based on an isotropic, scale-invariant two-form field. The model incorporates a foveal property typical of biological visual systems, with an approximately linear decrease of resolution as a function of eccentricity, and by a physical size constant that measures the radius of the geometric foveola, the central region characterized by maximal resolving power. It admits a description in singularity-free canonical coordinates generalizing the familiar log-polar coordinates and reducing to these in the asymptotic case of negligibly-sized geometric foveola or, equivalently, at peripheral locations in the visual field. It has predictive power to the extent that quantitative geometric relationships pertaining to retino-cortical magnification along the primary visual pathway, such as receptive field size distribution and spatial arrangement in retina and striate cortex, can be deduced in a principled manner. The biological plausibility of the model is demonstrated by comparison with known facts of human vision. Full article

(This article belongs to the Special Issue Discrete Differential Geometry and Its Applications to Imaging and Graphics)

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