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The main goal of this article is to describe a relation between the asymptotic properties of filtrations on section rings and the geometry at infinity of the space of Kähler potentials. More precisely, for a polarized projective manifold and an ample test configuration, Phong and Sturm associated a geodesic ray of plurisubharmonic metrics on the polarizing line bundle. On the other hand, for the same data, Witt Nyström associated a filtration on the section ring of the polarized manifold. In this article, we establish a folklore conjecture that the pluripotential chordal distance between the geodesic rays associated with two ample test configurations coincides with the spectral distance between the associated filtrations on the section ring. This gives an algebraic description of the boundary at infinity of the space of positive metrics, viewed – as it is usually done for spaces of negative curvature – through geodesic rays. [ABSTRACT FROM AUTHOR]

Multiphysics simulations frequently require transferring solution fields between subproblems with nonmatching spatial discretizations, typically using interpolation techniques. Standard methods are usually based on measuring the closeness between points by means of the Euclidean distance, which does not account for curvature, cuts, cavities, or other nontrivial geometrical or topological features of the domain. This may lead to spurious oscillations in the interpolant in proximity to these features. To overcome this issue, we propose a modification to rescaled localized radial basis function (RL-RBF) interpolation to account for the geometry of the interpolation domain, by yielding conformity and fidelity to geometrical and topological features. The proposed method, referred to as RL-RBF-G, relies on measuring the geodesic distance between data points. RL-RBF-G removes spurious oscillations appearing in the RL-RBF interpolant, resulting in increased accuracy in domains with complex geometries. We demonstrate the effectiveness of RL-RBF-G interpolation through a convergence study in an idealized setting. Furthermore, we discuss the algorithmic aspects and the implementation of RL-RBF-G interpolation in a distributed-memory parallel framework, and present the results of a strong scalability test yielding nearly ideal results. Finally, we show the effectiveness of RL-RBF-G interpolation in multiphysics simulations by considering an application to a whole-heart cardiac electromechanics model. [ABSTRACT FROM AUTHOR]

This paper describes a method for fast simplification of surface meshes. Whereas past methods focus on visual appearance, our goal is to solve equations on the surface. Hence, rather than approximate the extrinsic geometry, we construct a coarse intrinsic triangulation of the input domain. In the spirit of the quadric error metric (QEM), we perform greedy decimation while agglomerating global information about approximation error. In lieu of extrinsic quadrics, however, we store intrinsic tangent vectors that track how far curvature "drifts" during simplification. This process also yields a bijective map between the fine and coarse mesh, and prolongation operators for both scalar- and vector-valued data. Moreover, we obtain hard guarantees on element quality via intrinsic retriangulation---a feature unique to the intrinsic setting. The overall payoff is a "black box" approach to geometry processing, which decouples mesh resolution from the size of matrices used to solve equations. We show how our method benefits several fundamental tasks, including geometric multigrid, all-pairs geodesic distance, mean curvature flow, geodesic Voronoi diagrams, and the discrete exponential map. [ABSTRACT FROM AUTHOR]

Random fields are widely used to characterize the inherent spatial uncertainty in engineering applications. However, for widely existing manifolds, few studies focus on the simulation of Gaussian/non‐Gaussian fields over manifolds due to the geometry complexity. Recently, we proposed a two‐stage method to simulate Gaussian fields over manifolds. In this study, we extend it to a three‐stage method for non‐Gaussian fields. In the first stage, the original manifold is dimensional reduced to a Euclidean domain by the isometric feature mapping. By this step, the geodesic distance is preserved to guarantee the correlation structure unchanged. In the second stage, conventional representations can be used to simulate the underlying Gaussian field over the Euclidean domain. In the third stage, an enhanced translation procedure is proposed to translate the underlying Gaussian field to the target non‐Gaussian field over the manifold. To explain the applicability and accuracy of the three‐stage method, two kinds of non‐Gaussian fields over three different manifolds are simulated. Then, a global error is proposed to evaluate the performance of the method, and the error analysis demonstrates the method is accurate enough. Finally, two engineering structures, that is, a compressed half‐cylinder and a Canopy shell with complex geometry, are investigated to illustrate the potential applications of the method. [ABSTRACT FROM AUTHOR]

In this work, we study the set of strictly accretive matrices, that is, the set of matrices with positive definite Hermitian part, and show that the set can be interpreted as a smooth manifold. Using the recently proposed symmetric polar decomposition for sectorial matrices, we show that this manifold is diffeomorphic to a direct product of the manifold of (Hermitian) positive definite matrices and the manifold of strictly accretive unitary matrices. Utilizing this decomposition, we introduce a family of Finsler metrics on the manifold and characterize their geodesics and geodesic distances. Finally, we apply the geodesic distance to a matrix approximation problem and also give some comments on the relation between the introduced geometry and the geometric mean of strictly accretive matrices as defined by Drury [1]. [ABSTRACT FROM AUTHOR]

Prevalent neuroscientific theories are combined with acoustic observations from various studies to create a consistent geometric model for music perception in order to rationalize, explain and predict psycho-acoustic phenomena. The space of all chords is shown to be a Whitney stratified space. Each stratum is a Riemannian manifold which naturally yields a geodesic distance across strata. The resulting metric is compatible with voice-leading satisfying the triangle inequality. The geometric model allows for rigorous studies of psychoacoustic quantities such as roughness and harmonicity as height functions. In order to show how to use the geometric framework in psychoacoustic studies, concepts for the perception of chord resolutions are introduced and analyzed. [ABSTRACT FROM AUTHOR]

• Using the global information of structure tensor, we define a robust intrinsic feature of images. • We convert the structure tensor to the symmetric positive definite matrix and using the tensor manifold to define our feature. • Tensors are studied in a Riemannian manifold for measuring the natural attributes of tensors. • The defined feature can accurately measure image similarity. [Display omitted] As an important feature of images, the structure tensor (ST) provides properties of the local image intensities. However, it is a challenging task to analyse images directly using STs since STs are 2nd-order symmetric positive semi-definite matrices. In this paper, we define a robust intrinsic feature of images using ST. In addition, using our feature, we propose an improved image similarity measure. By converting the ST into a symmetric positive definite (called tensor) matrix, we define our feature using the extended geodesic distance of tensors calculated in a Riemannian manifold. Our feature shows the essential natural properties of ST and images as tensors are analysed on the tensor manifold. Moreover, defined by the global embedded geometry of the structure tensor, our feature provides a stable intrinsic property of images. The experiments show that our feature performs well in representing the essential attributes of images, especially the edges and important structures. It also shows that our image similarity measure can accurately detect similar images or patches. [ABSTRACT FROM AUTHOR]

By using the theory of maximal Lq-regularity and methods of singular analysis, we show a Taylor's type expansion–with respect to the geodesic distance around an arbitrary point–for solutions of quasilinear parabolic equations on closed manifolds. The powers of the expansion are determined explicitly by the local geometry, whose reflection to the solutions is established through the local space asymptotics. [ABSTRACT FROM AUTHOR]

Geometric analysis of three-dimensional (3D) surfaces with local deformations is a challenging task, required by mobile devices. In this paper, we propose a new local feature-based method derived from diffusion geometry, including a keypoint detector named persistence-based Heat Kernel Signature (pHKS), and a feature descriptor named Heat Propagation Strips (HeaPS). The pHKS detector first constructs a scalar field using the heat kernel signature function. The scalar field is generated at a small scale to capture fine geometric information of the local surface. Persistent homology is then computed to extract all the local maxima from the scalar field, and to provide a measure of persistence. Points with a high persistence are selected as pHKS keypoints. In order to describe a keypoint, an intrinsic support region is generated by the diffusion area. This support region is more robust than its geodesic distance counterpart, and provides a local surface with adaptive scale for subsequent feature description. The HeaPS descriptor is then developed by encoding the information contained in both the spatial and temporal domains of the heat kernel. We conducted several experiments to evaluate the effectiveness of the proposed method. On the TOSCA Dataset, the HeaPS descriptor achieved a high performance in terms of descriptiveness. The feature detector and descriptor were then tested on the SHREC 2010 Feature Detection and Description Dataset, and produced results that were better than the state-of-the-art methods. Finally, their application to shape retrieval was evaluated. The proposed pHKS detector and HeaPS descriptor achieved a notable improvement on the SHREC 2014 Human Dataset. [ABSTRACT FROM AUTHOR]

Classical geometry can be described either in terms of a metric tensor g a b (x) or in terms of the geodesic distance σ 2 (x , x ′). Recent work, however, has shown that the geodesic distance is better suited to describe the quantum structure of spacetime. This is because one can incorporate some of the key quantum effects by replacing σ 2 by another function S [ σ 2 ] such that S [ 0 ] = L 0 2 is nonzero. This allows one to introduce a zero-point-length in the spacetime. I show that the geodesic distance can be an emergent construct, arising in the form of a correlator S [ σ 2 (x , y) ] = 〈 J (x) J (y) 〉 , of a pregeometric variable J (x) , which can be interpreted as the quantum density of spacetime events. This approach also shows why null surfaces play a special role in the interface of quantum theory and gravity. I describe several technical and conceptual aspects of this construction and discuss some of its implications. [ABSTRACT FROM AUTHOR]

Calling mammals, ships, and many other objects have been commonly located during the last century with two-dimensional (2D) models from measurements of a signal's Time Differences of Arrivals (TDOA) when the objects are not on the 2D surface. The overwhelmingly common method for locating signals with 2D models takes signal speed as constant and location is derived by intersecting hyperbolas. However, when correct locations are required for 2D models, the speed used to derive location must depend on the geodesic distance along the 2D model surface between the object and the instrument. For example, when this distance is zero, the speed needed for correct location must also be zero. The dimension reduction from three to two introduces large errors in 2D models both near and far from the instruments unless the variable speeds induced by the dimensional reduction are explicitly accounted for. In light of these findings, methods are derived for generating extremely reliable confidence intervals for estimated locations in 2D models and identifying regions of the 2D model where a 3D model is needed. Because speeds needed for correct location are spatially inhomogeneous in the extreme, isodiachrons emerge as a natural geometry for interpreting location instead of hyperbolas. These issues are caused by choice of coordinates, and the same phenomena occur when coordinate transformations are applied in other fields of physics. [ABSTRACT FROM AUTHOR]

By exploiting the relationship between probability density and the differential geometry structure of received data and geodesic distance, the recently proposed information geometry (IG) method can provide higher accuracy and resolution ability for direction of arrival (DOA) estimation than many existing methods. However, its performance is not robust even for high signal-to-noise ratio. To have a deep understanding of its unstable performance, a theoretical analysis of the IG method is presented by deriving the relationship between the cost function and the number of array elements, powers and DOAs of source signals, and noise power. Then, to make better use of the nonlinear and super resolution property of the cost function, a Scaling TRansform based INformation Geometry (STRING) method is proposed, which simply scales the array received data or its covariance matrix by a real number. However, the expression for the optimum value of the scalar is complicated and related to the unknown signal DOAs and powers. Hence, a decision criterion and a simple search based procedure are developed, guaranteeing a robust performance. As demonstrated by computer simulations, the proposed STRING method has the best and robust angle resolution performance compared with many existing high resolution methods and even outperforms the classic Cramer–Rao bound, although at the cost of a bias in the estimation results. [ABSTRACT FROM AUTHOR]

• Proposal of a minimalist parallel method for computing geodesic distances on meshes, and easy to implement. • No heavy pre-processing steps are required. • Yields quite good accuracy results even without preprocessing obtuse triangles. • It is able to solve the problem for very large meshes, regular and irregular. • Can be used with both single and double precision configurations. • Produces speedup results comparable to the state-of-art methods, or better, in some cases. The computation of geodesic distances is an important research topic in Geometry Processing and 3D Shape Analysis as it is a basic component of many methods used in these areas. In this work, we present a minimalistic parallel algorithm based on front propagation to compute approximate geodesic distances on meshes. Our method is practical and simple to implement, and does not require any heavy pre-processing. The convergence of our algorithm depends on the number of discrete level sets around the source points from which distance information propagates. To appropriately implement our method on GPUs taking into account memory coalescence problems, we take advantage of a graph representation based on a breadth-first search traversal that works harmoniously with our parallel front propagation approach. We report experiments that show how our method scales with the size of the problem. We compare the mean error and processing time obtained by our method with such measures computed using other methods. Our method produces results in competitive times with almost the same accuracy, especially for large meshes. We also demonstrate its use for solving two classical geometry processing problems: the regular sampling problem and the Voronoi tessellation on meshes. [ABSTRACT FROM AUTHOR]

In this work we demonstrate how Finsler geometry---and specifically the related geodesic tractography--- can be levied to analyze structural connections between different brain regions. We present new theoretical developments which support the definition of a novel Finsler metric and associated connectivity measures, based on closely related works on the Riemannian framework for diffusion MRI. Using data from the Human Connectome Project, as well as population data from an autism spectrum disorder study, we demonstrate that this new Finsler metric, together with the new connectivity measures, results in connectivity maps that are much closer to known tract anatomy compared to previous geodesic connectivity methods. Our implementation can be used to compute geodesic distance and connectivity maps for segmented areas and is publicly available. [ABSTRACT FROM AUTHOR]

In this paper, we study the limit of compactness which is a graph index originally introduced for measuring structural characteristics of hypermedia. Applying compactness to large scale small-world graphs (Mehler, 2008) observed its limit behaviour to be equal 1. The striking question concerning this finding was whether this limit behaviour resulted from the specifics of small-world graphs or was simply an artefact. In this paper, we determine the necessary and sufficient conditions for any sequence of connected graphs resulting in a limit value of CB = 1 which can be generalized with some consideration for the case of disconnected graph classes (Theorem 3). This result can be applied to many well-known classes of connected graphs. Here, we illustrate it by considering four examples. In fact, our proof-theoretical approach allows for quickly obtaining the limit value of compactness for many graph classes sparing computational costs. [ABSTRACT FROM AUTHOR]

We develop a full theory for the new class of Optimal Entropy-Transport problems between nonnegative and finite Radon measures in general topological spaces. These problems arise quite naturally by relaxing the marginal constraints typical of Optimal Transport problems: given a pair of finite measures (with possibly different total mass), one looks for minimizers of the sum of a linear transport functional and two convex entropy functionals, which quantify in some way the deviation of the marginals of the transport plan from the assigned measures. As a powerful application of this theory, we study the particular case of Logarithmic Entropy-Transport problems and introduce the new Hellinger–Kantorovich distance between measures in metric spaces. The striking connection between these two seemingly far topics allows for a deep analysis of the geometric properties of the new geodesic distance, which lies somehow between the well-known Hellinger–Kakutani and Kantorovich–Wasserstein distances. [ABSTRACT FROM AUTHOR]

Given a thin-sheet surface S in 3D, as well as a collection of sites located on S , it is a fundamental operation in digital geometry processing to partition S into a set of constituent surface patches in terms of proximity. Geodesic Voronoi diagram (GVD) provides a theoretically sound solution to this problem, but has limited use due to its high computational cost. Restricted Voronoi Diagram (RVD) considers the partitioning of S based on the straight-line distance. Under general conditions, RVD can yield a practicable solution with much less running time. However, when the input model contains thin sheets but the sites are sparse, RVD may present some issues due to the significant difference between the straight-line distance and the geodesic distance. The resulting flawed RVD cells limit the use of RVDs on further geometry processing occasions. In this paper, we address two kinds of commonly encountered issues: (1) a site dominates multiple disconnected regions, and (2) a site dominates one single region but the region is kidney-shaped. Both of the issues occur frequently, on a thin-sheet model with a relatively small number of sites. The first kind of flawed RVD cells occurs when the dominance of a site penetrates the other side of the surface. For such flawed RVD cells, it is easy to separate the remote parts from the principal region, based on connectivity. The second kind of flawed RVD cells occurs when a site is nearby a geometry edge and its dominating area crosses the geometry edge, leading to a kidney-shaped region. We observe that for a roughly flat RVD cell, the sum of its turning angles is close to 2 π. For a kidney-shaped region, however, the total angle becomes much larger than 2 π , which inspires us to develop an angle-based checking rule. We further propose a concept of "virtual site" to cut the region at the narrowest part and re-partition the ownerless region by the surrounding sites (may be virtual). All of the above-mentioned operations can be parallelized and generally only require lightweight computational relative to the RVD. After post-processing, the improved RVD has inconspicuous difference from the GVD, and is of much higher quality than LRVD (Yan et al., 2014) and EDBVD (Xin et al., 2022). We present its usage in intrinsic Delaunay triangulation (IDT) and centroidal Voronoi tessellation (CVT) based meshing. • Propose a parallel post-processing algorithm to fix the issues of the RVD. • We classify flawed RVD cells into two types T1 and T2. • For T1, isolate one or more ownerless regions from the principal region. • For T2, cut the kidney-shaped region into two parts and introduce a virtual site. • We further give a set of strategies for re-partitioning ownerless regions locally. [ABSTRACT FROM AUTHOR]

Introduction: Focal cortical dysplasia (FCD) is a congenital malformation of cortical development that often leads to medically refractory epilepsy. Focal resection can be an effective treatment, but is challenging as the surgically relevant abnormality may exceed the MR-visible lesion. The aim of the current study is to develop methodology to characterize the profile of functional connectivity around FCDs using resting-state functional MRI and in the individual patient. The detection of aberrant connectivity may provide a means to more completely delineate the clinically relevant lesion. Materials and Methods: Fifteen FCD patients (age, mean±SD: 31±11 years; 11 males) and 16 matched healthy controls (35±9 years; 7 males) underwent structural and functional imaging at 3 Tesla. The cortical surface was reconstructed from the T1-weighted scan and the registered functional MRI data was spatially normalized to a common anatomical standard space employing the gyral pattern. Seed-based functional connectivity was determined in all subjects for all dysplasia locations. A single patient was excluded based on an aberrant FCD seed time series. Functional connectivity as a function of geodesic distance (along the cortical surface) was compared between the individual patients and the homotopic normative connectivity profiles derived from the controls. Results: In 12/14 patients, aberrant profiles of functional connectivity were found, which demonstrated both hyper- and hypoconnectivity as well as combinations. Abnormal functional connectivity was typically found (also) beyond the lesion visible on structural MRI, while functional connectivity profiles not related to a lesion appeared normal in patients. Conclusion: This novel functional MRI technique has potential for delineating functionally aberrant from normal cortex beyond the structural lesion in FCD, which remains to be confirmed in future research. [ABSTRACT FROM AUTHOR]

In the last decades, several three-dimensional face recognition algorithms have been thought, designed, and assessed. What they have in common can be hardly said, as they differ in theoretical background, tools, and method. Here we propose a new 3D face recognition algorithm, entirely developed in Matlab ® , whose framework totally comes from differential geometry. First, 17 soft-tissue landmarks are automatically extracted relying on geometrical properties of facial shape. We made use of derivatives, coefficients of the fundamental forms, principal, mean, and Gaussian curvatures, and shape and curvedness indexes. Then, a set of geodesic and Euclidean distances, together with nose volume and ratios between geodesic and Euclidean distances, has been computed and summed in a final score, used to compare faces. The highest contribution of this work, we believe, is that its theoretical substratum is differential geometry with its various descriptors, which is something totally new in the field. [ABSTRACT FROM AUTHOR]

Many computer vision and pattern recognition problems may be posed as the analysis of a set of dissimilarities between objects. For many types of data, these dissimilarities are not euclidean (i.e., they do not represent the distances between points in a euclidean space), and therefore cannot be isometrically embedded in a euclidean space. Examples include shape-dissimilarities, graph distances and mesh geodesic distances. In this paper, we provide a means of embedding such non-euclidean data onto surfaces of constant curvature. We aim to embed the data on a space whose radius of curvature is determined by the dissimilarity data. The space can be either of positive curvature (spherical) or of negative curvature (hyperbolic). We give an efficient method for solving the spherical and hyperbolic embedding problems on symmetric dissimilarity data. Our approach gives the radius of curvature and a method for approximating the objects as points on a hyperspherical manifold without optimisation. For objects which do not reside exactly on the manifold, we develop a optimisation-based procedure for approximate embedding on a hyperspherical manifold. We use the exponential map between the manifold and its local tangent space to solve the optimisation problem locally in the euclidean tangent space. This process is efficient enough to allow us to embed data sets of several thousand objects. We apply our method to a variety of data including time warping functions, shape similarities, graph similarity and gesture similarity data. In each case the embedding maintains the local structure of the data while placing the points in a metric space. [ABSTRACT FROM AUTHOR]

We introduce a novel method for computing the earth mover's distance (EMD) between probability distributions on a discrete surface. Rather than using a large linear program with a quadratic number of variables, we apply the theory of optimal transportation and pass to a dual differential formulation with linear scaling. After discretization using finite elements (FEM) and development of an accompanying optimization method, we apply our new EMD to problems in graphics and geometry processing. In particular, we uncover a class of smooth distances on a surface transitioning from a purely spectral distance to the geodesic distance between points; these distances also can be extended to the volume inside and outside the surface. A number of additional applications of our machinery to geometry problems in graphics are presented. [ABSTRACT FROM AUTHOR]

A long-standing open problem asks whether a Riemannian metric on the real projective space with the same volume as the canonical metric carries a periodic geodesic whose length is at most π. A contact-geometric reformulation of systolic geometry and the use of canonical perturbation theory allow us to solve a parametric version of this problem: if g is a smooth, constant-volume deformation of the canonical metric that is not formally trivial, the length of the shortest periodic geodesic of the metric g attains π as a strict local maximum at s = 0. This result still holds for complex and quaternionic projective spaces as well as for the Cayley plane. Moreover, the same techniques can be applied to show that Zoll Finsler manifolds are the unique smooth critical points of the systolic volume. [ABSTRACT FROM AUTHOR]

This article presents a new, unified technique to perform general edge-sensitive editing operations on n-dimensional images and videos efficiently. The first contribution of the article is the introduction of a Generalized Geodesic Distance Transform (GGDT), based on soft masks. This provides a unified framework to address several edge-aware editing operations. Diverse tasks such as denoising and nonphotorealistic rendering are all dealt with fundamentally the same, fast algorithm. Second, a new Geodesic Symmetric Filter (GSF) is presented which imposes contrast-sensitive spatial smoothness into segmentation and segmentation-based editing tasks (cutout, object highlighting, colorization, panorama stitching). The effect of the filter is controlled by two intuitive, geometric parameters. In contrast to existing techniques, the GSF filter is applied to real-valued pixel likelihoods (soft masks), thanks to GGDTs and it can be used for both interactive and automatic editing. Complex object topologies are dealt with effortlessly. Finally, the parallelism of GGDTs enables us to exploit modem multicore CPU architectures as well as powerful new GPUs, thus providing great flexibility of implementation and deployment. Our technique operates on both images and videos, and generalizes naturally to n-dimensional data. The proposed algorithm is validated via quantitative and qualitative comparisons with existing, state-of-the-ann approaches. Numerous results on a variety of image and video editing tasks further demonstrate the effectiveness of our method. [ABSTRACT FROM AUTHOR]

Abstract: Tensors are nowadays an increasing research domain in different areas, especially in image processing, motivated for example by diffusion tensor magnetic resonance imaging (DT-MRI). Up to now, algorithms and tools developed to deal with tensors were founded on the assumption of a matrix vector space with the constraint of remaining symmetric positive definite matrices. On the contrary, our approach is grounded on the theoretically well-founded differential geometrical properties of the space of multivariate normal distributions, where it is possible to define an affine-invariant Riemannian metric and express statistics on the manifold of symmetric positive definite matrices. In this paper, we focus on the contribution of these tools to the anisotropic filtering and regularization of tensor fields. To validate our approach we present promising results on both synthetic and real DT-MRI data. [Copyright &y& Elsevier]

Bayesian methods have been rapidly developed due to the important role of explicable causality in practical problems. We develope geometric approaches to Bayesian inference of Pareto models, and give an application to the analysis of sea clutter. For Pareto two-parameter model, we show the non-existence of α-parallel prior in general, hence we adopt Jeffreys prior to deal with the Bayesian inference. Considering geodesic distance as the loss function, an estimation in the sense of minimal mean geodesic distance is obtained. Meanwhile, by involving Al-Bayyati's loss function we gain a new class of Bayesian estimations. In the simulation, for sea clutter, we adopt Pareto model to acquire various types of parameter estimations and the posterior prediction results. Simulation results show the advantages of the Bayesian estimations proposed and the posterior prediction. [ABSTRACT FROM AUTHOR]

A new Riemannian geometry for the zero-mean Compound Gaussian distribution with deterministic textures is proposed. In particular, the Fisher information metric (up to a factor) is obtained, along with corresponding geodesics and distance function. This new geometry is applied on a change detection problem on Multivariate Image Times Series: a recursive approach based on Riemannian optimization is developed. As shown on simulated data, it allows to reach optimal performance while being computationally more efficient. [ABSTRACT FROM AUTHOR]

Due to the scarcity of radio spectrum resources and the growing demand, the use of spectrum sensing technology to improve the utilization of spectrum resources has become a hot research topic. In order to improve the utilization of spectrum resources, this paper proposes a spectrum sensing method that combines information geometry and deep learning. Firstly, the covariance matrix of the sensing signal is projected onto the statistical manifold. Each sensing signal can be regarded as a point on the manifold. Then, the geodesic distance between the signals is perceived as its statistical characteristics. Finally, deep neural network is used to classify the dataset composed of the geodesic distance. Simulation experiments show that the proposed spectrum sensing method based on deep neural network and information geometry has better performance in terms of sensing precision. [ABSTRACT FROM AUTHOR]

This paper presents a novel approach to the formulation and solution of discrete geometry processing problems including mesh denoising. The main quantity of interest is the piecewise constant unit normal vector field, which we consider on piecewise flat, triangulated surfaces. In a similar fashion as one does for images defined on 'flat' domains, our goal is to remove noise while preserving shape features such as sharp edges. To this end, we model a denoising problem via a quadratic vertex tracking term and a regularizer based on the total variation of the normal vector field. Since the latter has values on the unit sphere, its total variation reduces to a finite sum of the geodesic distances of neighboring normals. We solve the model numerically by applying split‐Bregmann iterations and provide results for the 'fandisk' benchmark. [ABSTRACT FROM AUTHOR]