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The variability and interdependence of local porosity and local mean geodesic tortuosity, which is a measure for the sinuosity of shortest transportation paths, is investigated at the example of the microstructure in sack paper. By means of statistical image analysis, these two morphological characteristics are computed for several cutouts of 3D image data obtained by X-ray microcomputed tomography. Considering cutouts of different sizes allows us to study the influence of the sample size on the local variability of the considered characteristics. Moreover, the interdependence between local porosity and local mean geodesic tortuosity is quantified by modeling their joint distribution parametrically using Archimedean copulas. It turns out that the family of Gumbel copulas is an appropriate model type, which is formally validated by a goodness of fit test. Besides mean geodesic tortuosity, we consider further related morphological characteristics, describing the sinuosity of those shortest transportation paths, whose minimum diameter exceeds a predefined threshold. Moreover, we show that the copula approach investigated in this paper can also be used to quantify the negative correlation between local porosity and these modified versions of local mean geodesic tortuosity. Our results elucidate the impact of local porosity on various kinds of morphological characteristics, which are not experimentally accessible and which are important for local air permeance – a key property of sack paper.

This paper is intended as a road map for the literature on Synthetic Aperture Radar (SAR) image processing and analysis with Information-Theoretic and Information-Geometric Statistical tools. We do this by commenting upon seemingly disjoint problems, and showing that they can be tackled successfully within the same framework: tools that arise from using Information Theory (entropies and divergences) and Information Geometry (geodesic distances) enhanced with Statistics.

In this second paper we define a post-minkowskian (PM) weak field approximation leading to a linearization of the Hamilton equations of Arnowitt–Deser–Misner (ADM) tetrad gravity in the York canonical basis in a family of nonharmonic 3-orthogonal Schwinger time gauges. The York time 3K (the relativistic inertial gauge variable, not existing in newtonian gravity, parametrizing the family, and connected to the freedom in clock synchronization, i.e., to the definition of the the shape of the instantaneous 3-spaces) is set equal to an arbitrary numerical function. The matter are considered point particles, with a Grassmann regularization of self-energies, and the electromagnetic field in the radiation gauge: an ultraviolet cutoff allows a consistent linearization, which is shown to be the lowest order of a hamiltonian PM expansion. We solve the constraints and the Hamilton equations for the tidal variables and we find PM gravitational waves with asymptotic background (and the correct quadrupole emission formula) propagating on dynamically determined non-euclidean 3-spaces. The conserved ADM energy and the Grassmann regularization of self-energies imply the correct energy balance. A generalized transverse–traceless gauge can be identified and the main tools for the detection of gravitational waves are reproduced in these nonharmonic gauges. In conclusion, we get a PM solution for the gravitational field and we identify a class of PM Einstein space–times, which will be studied in more detail in a third paper together with the PM equations of motion for the particles and their post-newtonian expansion (but in the absence of the electromagnetic field). Finally we make a discussion on the gauge problem in general relativity to understand which type of experimental observations may lead to a preferred choice for the inertial gauge variable 3K in PM space–times. In the third paper we will show that this choice is connected with the problem of dark matter.

We present a novel and effective method for modeling a developable surface to simulate paper bending in interactive and animation applications. The method exploits the representation of a developable surface as the envelope of rectifying planes of a curve in 3D, which is therefore necessarily a geodesic on the surface. We manipulate the geodesic to provide intuitive shape control for modeling paper bending. Our method ensures a natural continuous isometric deformation from a piece of bent paper to its flat state without any stretching. Test examples show that the new scheme is fast, accurate, and easy to use, thus providing an effective approach to interactive paper bending. We also show how to handle non-convex piecewise smooth developable surfaces.

Question: 1—Echocardiography images Images were acquired from end of diastole to end of systole at different phases (7 phases) within a cardiac cycle by a Vivid 3 GE Health echocardiography machine. To obtain worthy images at denoted phases, it would be dependent to operator (for the best clinical examination). Of course, having assumed images not only were acquired from A4C and A2C but also from short-axis views. Short axis views have not been used at your papers, so some radial, circumferential and sample rotatory data would be missed. Response: It is simply possible for us to record and analyze images of short-axis views. However, according to American Society of Echocardiography (ASE) [1, 2], in the modified Simpson’s rule which is a method for LV volume computation from 2-D echocardiography images, it suffices to acquire apical two and four chamber views (A4C, A2C) for left ventricular (LV) volume computation. Question: 2—Observed data yi′ s Echocardiography images paly role of observed data and they are symbolized by yi′ s. there is a sequence of observed data y1, y2,…,yN. N is the number of obtained images. In fact we have made a chain of displacement, rotation and pure strain or nonrigid transformation (deformation) by these observed data that is started from y1 and is ended up to yN. This chain has some conceptual interpretations of the elasticity of the global left ventricular motion/function and the regional LV fiber arrangement/movement. Response: Yes, it is correct. Question: 3—Embed observed data to a high dimensional space Response: Yes, it is correct. Question: 4—Image processing on yi′ s and geodesic distance between yi′ s Observed data y1, y2,…,yN are embed to a meshed surface sized number of pixels of an image. Translation, rotation and pure deformation of y1, y2,…,yN are occurred/and studied at this surface utilizing the mathematical elasticity theory. A distance metric is defined that computes distances between observed points. Medical interpretations of observed points over the time should be checked/and stated clearly at your manuscripts. Images are not usual images but also are images from the left ventricle. Motion (displacement and velocity), deformation (strain and strain rate) and torsion of each myocardial sample have to be extracted during a cardiac cycle in the mentioned surface. These are used on the creation of graph G referred to their papers. Response: In this stage, we have just computed the distance between the two images in a cardiac cycle using non-rigid image registration regardless of relationship between images over the time. Therefore by this method, the motion and longitudinal deformation (short-axis views are needed to get for radial and circumferential deformations) of LV myocardium between all of two images in a cardiac cycle are extracted (for example, between end systole image and end diastole image, etc.). Then, in other stage of our method (calculation of the matrix of pairwise geodesic distance and apply multidimensional scaling on the resulting geodesic distance matrix to construct the low dimensional data points xi), motion and longitudinal deformation of LV myocardium during a cardiac cycle are extracted. Question: 5—An Isomap f The main tool at this method is the function “f” which is an isometric map [2, 3]. I will be interested to know the structure/formula or relationship of this isometric function which has most probably a lot of practical information of the left ventricular function and structure. I mean, how we can get a good representation of such a this function “f” for heart science study? Is it “f” known as an algebraic function? What information has been coded to fibers (f−1) of this isometric function “f”? Response: Isomap algorithm is one of the most popular sets of nonlinear dimensionality reduction (NLDR) algorithms. This algorithm attempts to extract low-dimensional data points from input points in a high-dimensional space in such a way that pairwise geodesic distances (the distance between two points measured over the manifold) are preserved. So that nearby and far points in high-dimensional space map to nearby and far points in low-dimensional space. This method is frequently used for visualization of medial image set. Detail of this method is described in Tenenbaum et al. and Borg et al. [3, 4]. It should be noted that we do not change the structure or formula of isomap algorithm. We only define a new image distance function based on image registration in calculation of geodesic distance between yi′ s in isomap algorithm. With this modification, Isomap algorithm is specified for assessment of the left ventricular function and structure. However, precise mathematical descriptions of f and (f−1) applied to echocardiography images and using the other NLDR algorithms need further researches in the following of this article. Question: 6—A reconstructed curve on a 2D manifold space & hidden data xi′ s; f(yi) = xi Isometric function results in a reconstructive curve crossing from hidden data (xi′ s) in a surface. It’s natural that some information (maybe clinical information) would betransferred to these hidden points like the curve of LV volume changes and so on and so on. A main question is: what are these new points xi′ s exactly (medically point of view)? What the other data has been come out from these hidden points? I think all of these questions back to gain a good understanding of isometric function “f”. Response: These points xi′ s will probably enable physicians to diagnose and follow up many cardiac structures and functions that also open doors to much more research at imaging cardiology in the future. References 1. AlizadehSani Z, Shalbaf A, Behnam H, Shalbaf R: Automatic computation of left ventricularvolume changes over a cardiac cycle from echocardiography images by nonlinear dimensionality reduction. J Digit Imaging : July, 2014 2. Tenenbaum JB, de Silva V, Langford JC: global geometric framework for nonlinear dimensionalityreduction. Science 290:2319–2323, 2000. Reprint available online: http://web.mit.edu/cocosci/Papers/sci_reprint.pdf 3. Ledesma-Carbayo MJ, Kybic J, Desco M, et al.: Spatio-temporal nonrigid registration for ultrasoundcardiac motion estimation. IEEE Trans Med Imaging 24:1113–1126, 2005.

We present a new approach and improvements to the recent results of Gabai, Meyerhoff and Milley concerning tubes and short geodesics in hyperbolic 3 3 -manifolds. We establish the following two facts: if a hyperbolic 3 3 -manifold admits an embedded tubular neighbourhood of radius r 0 > 1.32 r_0>1.32 about any closed geodesic, then its volume exceeds that of the Weeks manifold. If the shortest geodesic of M M has length less than ℓ 0 > 0.1 \ell _0>0.1 , then its volume also exceeds that of the Weeks manifold.

One cannot assume that resemblances between the mechanical devices of human technology and those produced by the evolutionary process reflect either specific copying of nature by people or some particular point of functional superiority. A third alternative is that the two mechanical contexts derive quite different advantages from a given arrangement. While this latter might appear unlikely, one can argue that it underlies such things as the use of conical shapes, helical tensile structures, spheres and cylinders, beams and columns of relatively low torsional stiffness, and geodesic shells.

We would like to start by congratulating Byrne and Girolami for writing such a thoughtful and extremely interesting paper. This is in fact a worthy addition to other high impact papers recently published by Professor Girolami’s lab in this field. The common theme of these papers is to use geometrically motivated methods to improve efficiency of sampling algorithms. In their seminal paper, Girolami & Calderhead (2011) proposed a novel Hamiltonian Monte Carlo (HMC) method, called Riemannian manifold HMC, that exploits the Riemannian geometry of the target distribution to improve standard HMC’s efficiency by automatically adapting to the local structure. Although this is a natural and beautiful idea, there are significant computational difficulties, which arise in effectively implementing this algorithm. In contrast, in this current contribution, Byrne and Girolami focused on special probability distributions, which give rise to particularly nice Riemannian geometries. In particular, the examples under consideration described in Section 4 allow for closed-form solutions to the geodesic equation, which can be used to reduce computational cost of geometrically motivated Monte Carlo methods. Although the proposed splitting algorithm is quiet interesting, we initially doubted its impact because Riemannian metrics with closed-form geodesics are extremely rare. However, we are now convinced that this approach will likely see application beyond what is outlined herein. For example, we believe that this approach can be used to improve computational efficiency of sampling algorithms when the parameter space is constrained. The standard HMC algorithm needs to evaluate each proposal to ensure that it is within the boundaries imposed by the constraints. Alternatively, as discussed by Neal (2011), one could modify standard HMC so the sampler bounces back after hitting the boundaries. In Appendix A, Byrne and Girolami discussed this approach for geodesic updates on the simplex. In many cases, a constrained parameter space can be bijectively mapped to a unit ball

Methods to rectify distortion of digital camera document images of curved papers have become important for camera-based image recognition. In this paper we propose a novel distortion rectification method based on "shape from parallel geodesies." This method considers the following features: parallel lines corresponding to character strings or ruled lines of tables on extended surface become parallel geodesies on a curved paper surface and a smoothly curved paper can be modeled by a ruled surface, that is, a sweep surface of rulings. The projected geodesies and the projected rulings exist in the input image derived from perspective transformation. The presented method extracts the projected geodesies, estimates the projected rulings in the input image, estimates the ruled surface that models the curved paper, and generates the corrected image, in this order. It can estimate the ruled surface model directly by numerical operations of differentiation, integration and matrix inversion without any iterative calculation. We also report on experiments that show the effectiveness of the proposed method.

We correct two clerical errors made in the paper ”A Lower Bound for the Length of Closed Geodesics on a Finsler Manifold”.

The metric $D_{\alpha}(q,q')$ on the set $Q$ of particle locations of a homogeneous Poisson process on $\mathbb{R}^d$ , defined as the infimum of $(\sum_i |q_i - q_{i+1}|^{\alpha})^{1/\alpha}$ over sequences in $Q$ starting with $q$ and ending with $q'$ (where $|·|$ denotes Euclidean distance) has nontrivial geodesics when $\alpha>1$. The cases $1< \alpha

The intrinsic geometry of a space of distributions is represented by means of an embedding in Hilbert space, which has been previously studied.

Motivation: Digital reconstruction, or tracing, of 3D neuron structures is critical toward reverse engineering the wiring and functions of a brain. However, despite a number of existing studies, this task is still challenging, especially when a 3D microscopic image has low signal-to-noise ratio (SNR) and fragmented neuron segments. Published work can handle these hard situations only by introducing global prior information, such as where a neurite segment starts and terminates. However, manual incorporation of such global information can be very time consuming. Thus, a completely automatic approach for these hard situations is highly desirable. Results: We have developed an automatic graph algorithm, called the all-path pruning (APP), to trace the 3D structure of a neuron. To avoid potential mis-tracing of some parts of a neuron, an APP first produces an initial over-reconstruction, by tracing the optimal geodesic shortest path from the seed location to every possible destination voxel/pixel location in the image. Since the initial reconstruction contains all the possible paths and thus could contain redundant structural components (SC), we simplify the entire reconstruction without compromising its connectedness by pruning the redundant structural elements, using a new maximal-covering minimal-redundant (MCMR) subgraph algorithm. We show that MCMR has a linear computational complexity and will converge. We examined the performance of our method using challenging 3D neuronal image datasets of model organisms (e.g. fruit fly). Availability: The software is available upon request. We plan to eventually release the software as a plugin of the V3D-Neuron package at http://penglab.janelia.org/proj/v3d. Contact: pengh@janelia.hhmi.org

Summary Three-dimensional geometry of leaf margins is an important shape characteristic to distinguish different leaf phenotypes. Novel geometrical methods were defined, measured, and used to quantify waviness and lobiness of leaves., The geometry of leaf margins is an important shape characteristic that distinguishes among different leaf phenotypes. Current definitions of leaf shape are qualitative and do not allow quantification of differences in shape between phenotypes. This is especially true for leaves with some non-trivial three-dimensional (3D) configurations. Here we present a novel geometrical method novel geometrical methods to define, measure, and quantify waviness and lobiness of leaves. The method is based on obtaining the curve of the leaf rim from a 3D surface measurement and decomposing its local curvature vector into the normal and geodesic components. We suggest that leaf waviness is associated with oscillating normal curvature along the margins, while lobiness is associated with oscillating geodesic curvature. We provide a way to integrate these local measures into global waviness and lobiness quantities. Using these novel definitions, we analysed the changes in leaf shape of two Arabidopsis genotypes, either as a function of gene mis-expression induction level or as a function of time. These definitions and experimental methods open the way for a more quantitative study of the shape of leaves and other growing slender organs.

In this short paper, we will prove the following Theorem: Let \(\mathcal {F}\) be a codimension-one foliation of a complete and connected Riemannian manifold \(M^{n+1}\) with constant sectional curvature \(c {\leq} 0\). Suppose that \(\mathcal {F} \) is transversely orientable, i.e., there exists an unit vector field \(N \epsilon {\mathfrak{X}}(M)\) such that N is normal to the leaves of \(\mathcal {F} \). Suppose that N is geodesic. Then, if the foliation \(\mathcal {F} \) has an umbilical leaf, every leaf of foliation \(\mathcal {F} \) must be umbilical.

Motivation: Digital reconstruction of 3D neuron structures is an important step toward reverse engineering the wiring and functions of a brain. However, despite a number of existing studies, this task is still challenging, especially when a 3D microscopic image has low single-to-noise ratio and discontinued segments of neurite patterns. Results: We developed a graph-augmented deformable model (GD) to reconstruct (trace) the 3D structure of a neuron when it has a broken structure and/or fuzzy boundary. We formulated a variational problem using the geodesic shortest path, which is defined as a combination of Euclidean distance, exponent of inverse intensity of pixels along the path and closeness to local centers of image intensity distribution. We solved it in two steps. We first used a shortest path graph algorithm to guarantee that we find the global optimal solution of this step. Then we optimized a discrete deformable curve model to achieve visually more satisfactory reconstructions. Within our framework, it is also easy to define an optional prior curve that reflects the domain knowledge of a user. We investigated the performance of our method using a number of challenging 3D neuronal image datasets of different model organisms including fruit fly, Caenorhabditis elegans, and mouse. In our experiments, the GD method outperformed several comparison methods in reconstruction accuracy, consistency, robustness and speed. We further used GD in two real applications, namely cataloging neurite morphology of fruit fly to build a 3D ‘standard’ digital neurite atlas, and estimating the synaptic bouton density along the axons for a mouse brain. Availability: The software is provided as part of the V3D-Neuron 1.0 package freely available at http://penglab.janelia.org/proj/v3d Contact: pengh@janelia.hhmi.org

Humans interact with their environment through sensory information and motor actions. These interactions may be understood via the underlying geometry of both perception and action. While the motor space is typically considered by default to be Euclidean, persistent behavioral observations point to a different underlying geometric structure. These observed regularities include the “two-thirds power law”, which connects path curvature with velocity, and “local isochrony”, which prescribes the relation between movement time and its extent. Starting with these empirical observations, we have developed a mathematical framework based on differential geometry, Lie group theory and Cartan’s moving frame method for the analysis of human hand trajectories. We also use this method to identify possible motion primitives, i.e., elementary building blocks from which more complicated movements are constructed. We show that a natural geometric description of continuous repetitive hand trajectories is not Euclidean but equi-affine. Specifically, equi-affine velocity is piecewise constant along movement segments, and movement execution time for a given segment is proportional to its equi-affine arc-length. Using this mathematical framework, we then analyze experimentally recorded drawing movements. To examine movement segmentation and classification, the two fundamental equi-affine differential invariants—equi-affine arc-length and curvature are calculated for the recorded movements. We also discuss the possible role of conic sections, i.e., curves with constant equi-affine curvature, as motor primitives and focus in more detail on parabolas, the equi-affine geodesics. Finally, we explore possible schemes for the internal neural coding of motor commands by showing that the equi-affine framework is compatible with the common model of population coding of the hand velocity vector when combined with a simple assumption on its dynamics. We then discuss several alternative explanations for the role that the equi-affine metric may play in internal representations of motion perception and production.