Gradient Decomposition Research Articles

The shape of dark matter haloes – V. Analysis of observations of edge-on galaxies

In the previous papers in this series, we have measured the stellar and \hi content in a sample of edge-on galaxies. In the present paper, we perform a simultaneous rotation curve and vertical force field gradient decomposition for five of these edge-on galaxies. The rotation curve decomposition provides a measure of the radial dark matter potential, while the vertical force field gradient provide a measure of the vertical dark matter potential. We fit dark matter halo models to these potentials. Using our \hi self-absorption results, we find that a typical dark matter halo has a less dense core ($0.094\pm0.230$\,M$_\odot$/pc$^3$) compared to an optically thin \hi model ($0.150\pm0.124$\,M$_\odot$/pc$^3$). The HI self-absorption dark matter halo has a longer scale length $R_c$ of $1.42\pm 3.48$\,kpc, versus $1.10\pm 1.81$\,kpc for the optically thin HI model. The median halo shape is spherical, at $q=1.0\pm0.6$ (self-absorbing \hi), while it is prolate at $q=1.5\pm0.6$ for the optically thin. Our best results were obtained for ESO\,274-G001 and UGC\,7321, for which we were able to measure the velocity dispersion in Paper III. These two galaxies have drastically different halo shapes, with one oblate and one strongly prolate. Overall, we find that the many assumptions required make this type of analysis susceptible to errors.

From dislocation motion to an additive velocity gradient decomposition, and some simple models of dislocation dynamics

A mathematical theory of time-dependent dislocation mechanics of unrestricted geometric and material nonlinearity is reviewed. Within a “small deformation” setting, a suite of simplified and interesting models consisting of a nonlocal Ginzburg Landau equation, a nonlocal level set equation, and a nonlocal generalized Burgers equation is derived. In the finite deformation setting, it is shown that an additive decomposition of the total velocity gradient into elastic and plastic parts emerges naturally from a micromechanical starting point that involves no notion of plastic deformation but only the elastic distortion, material velocity, dislocation density and the dislocation velocity. Moreover, a plastic spin tensor emerges naturally as well.

Hyperelastic Internal Balance by Multiplicative Decomposition of the Deformation Gradient

The multiplicative decomposition of the deformation gradient \({{\bf F} = {{\hat{\bf F}}}{\bf F}^*}\) is often used in finite deformation continuum mechanics as a basis for treating mechanical effects including plasticity, biological growth, material swelling, and notions of material morphogenesis. Evolution rules for the particular effect from this list are then posed for F*. The tensor \({{{\hat{\bf F}}}}\) is then invoked to describe a subsequent elastic accommodation, and a hyperelastic framework is put in place for its determination using an elastic energy density function, say \({W({\hat{\bf F}})}\), as a constitutive specification. Here we explore the theory that emerges if both F* and \({{\hat{\bf F}}}\) are governed by hyperelastic criteria; thus we consider energy densities \({W({{\hat{\bf F}}}, {\bf F}^*)}\). The decomposition of F is itself determined by energy minimization, and the variation associated with the multiplicative decomposition gives a tensor relation that is interpreted as an internal balance requirement. Our initial development purposefully proceeds with minimal presumptions on the kinematic interpretation of the factors in the deformation gradient decomposition. Connections are then made to treatments that ascribe particular kinematic properties to the decomposition factors—the theory of structured deformations is especially significant in this regard. Such theories have broad utility in describing certain substructural reconfigurations in solids. To demonstrate in the context of the present variational treatment we consider a boundary value problem that involves an imposed twist. If the twist is small then the minimizer is classically smooth. At larger values of twist the energy minimizer exhibits a non-smooth deformation that localizes slip at a singular surface.

Combinatorial Optimization Based on Conjugate Gradient and Orthogonal Triangular Decomposition for Sparse Recovery

Sparse recovery is a challenging theme in signal processing and image processing. The basic goal is to reconstruction of sparse images or signals from very few samples by means of solving a tractable optimization problem. An important aspect of sparse recovery is to develop the recovery performance in the presence of noise. In this article, we propose the matching pursuit algorithm of combinatorial optimization based Conjugate Gradient Lest Squares (CGLS) and Lest Squares QR (LSQR). We use non-negative matrix factorization for measuring discrepancy of solution sequence between CGLS and LSQR, and represent combinatorial optimization based CGLS and LSQR to choose optimal solution sequences. The experiments indicate our method is extended to the case where target signal has been corrupted by noise, it demonstrates perfectly recovery ability of signal with noise.

Image Fusion Algorithm Based on Gradient Pyramid and its Performance Evaluation

Image fusion algorithm based on gradient pyramid is one of the multi-scale, multi-resolution decomposition algorithms. Original image was decomposed into Gauss pyramid, after that, gradient decomposition was completed on each layer in four directions, and fusion effect was evaluated by taking using of entropy, average gradient, mean and standard deviation. Simulation results show that gradient pyramid algorithm is effective to multi-focus image and color image.

Genetic Algorithm for Opto-thermal Skin Hydration Depth Profiling Measurements

Stratum corneum is the outermost skin layer, and the water content in stratum corneum plays a key role in skin cosmetic properties as well as skin barrier functions. However, to measure the water content, especially the water concentration depth profile, within stratum corneum is very difficult. Opto-thermal emission radiometry, or OTTER, is a promising technique that can be used for such measurements. In this paper, a study on stratum corneum hydration depth profiling by using a genetic algorithm (GA) is presented. The pros and cons of a GA compared against other inverse algorithms such as neural networks, maximum entropy, conjugate gradient, and singular value decomposition will be discussed first. Then, it will be shown how to use existing knowledge to optimize a GA for analyzing the opto-thermal signals. Finally, these latest GA results on hydration depth profiling of stratum corneum under different conditions, as well as on the penetration profiles of externally applied solvents, will be shown.

Decomposing intensity gradients into information about shape and material

Recent work has shown that the perception of 3D shapes, material properties and illumination are inter-dependent, although for practical reasons, each set of experiments has probed these three causal factors independently. Most of these studies share a common observation though: that variations in image intensity (both their magnitude and direction) play a central role in estimating the physical properties of objects and illumination. Our aim is to separate retinal image intensity gradients into contributions of different shape and material properties, through a theoretical analysis of image formation. We find that gradients can be understood as the sum of three terms: variations of surface depth conveyed through surface-varying reflectance and near-field illumination effects (shadows and inter-reflections); variations of surface orientation conveyed through reflections and far-field lighting effects; and variations of surface micro-structures conveyed through anisotropic reflections. We believe our image gradient decomposition constitutes a solid and novel basis for perceptual inquiry. We first illustrate each of these terms with synthetic 3D scenes rendered with global illumination. We then show that it is possible to mimic the visual appearance of shading and reflections directly in the image, by distorting patterns in 2D. Finally, we discuss the consistency of our mathematical relations with observations drawn by recent perceptual experiments, including the perception of shape from specular reflections and texture. In particular, we show that the analysis can correctly predict certain specific illusions of both shape and material.

Modeling Growth in Biological Materials

The biomechanical modeling of growing tissues has recently become an area of intense interest. In particular, the interplay between growth patterns and mechanical stress is of great importance, with possible applications to arterial mechanics, embryo morphogenesis, tumor development, and bone remodeling. This review aims to give an overview of the theories that have been used to model these phenomena, categorized according to whether the tissue is considered as a continuum object or a collection of cells. Among the continuum models discussed is the deformation gradient decomposition method, which allows a residual stress field to develop from an incompatible growth field. The cell-based models are further subdivided into cellular automata, center-dynamics, and vertex-dynamics models. Of these the second two are considered in more detail, especially with regard to their treatment of cell-cell interactions and cell division. The review concludes by assessing the prospects for reconciliation between these two fundamentally different approaches to tissue growth, and by identifying possible avenues for further research.

Coupling ALE Technique and Harmonic Parametrization to Describe Concurrent and Successive Elementary Cell Deformations

We present here an extension of our finite element model of the Drosophila embryo to consider the interdependence of successive morphogenetic movements. A novel approach is used, which couples the Arbitrary Lagrangian Eulerian formulation with the harmonicparametrization. The combination of the two techniques allows to constantly update the deforming embryo geometry and simultaneously build an associated system of curvilinear coordinates. Thus, we are able to exactly describe the elementary cell deformations responsible for each biological event that are then defined with respect to the dynamic middle surface of the embryonic tissue and to their relative reference configuration. Both the active and the passive deformations occurring to the cells are considered through the deformation gradient decomposition. We develop a concurrent simulation of three morphogenetic movements: the ventral furrow invagination, the cephalic furrow formation and the germ band extension. The results show a consistent similarity with respect to the physical phenomena. More generally, the numerical approach that we propose could constitute a powerful tool to rigorously describe and simulate complex shape changes in biological systems.

Robust CoHOG Feature Extraction in Human-Centered Image/Video Management System

Many human-centered image and video management systems depend on robust human detection. To extract robust features for human detection, this paper investigates the following shortcomings of co-occurrence histograms of oriented gradients (CoHOGs) which significantly limit its advantages: 1) The magnitudes of the gradients are discarded, and only the orientations are used; 2) the gradients are not smoothed, and thus, aliasing effect exists; and 3) the dimensionality of the CoHOG feature vector is very large (e.g., 200,000). To deal with these problems, in this paper, we propose a framework that performs the following: 1) utilizes a novel gradient decomposition and combination strategy to make full use of the information of gradients; (2) adopts a two-stage gradient smoothing scheme to perform efficient gradient interpolation; and (3) employs incremental principal component analysis to reduce the large dimensionality of the CoHOG features. Experimental results on the two different human databases demonstrate the effectiveness of the proposed method.

An extensive numerical simulation of the cephalic furrow formation in Drosophila embryo

Here, we propose a new finite element model of the Drosophila embryo to simulate the cephalic furrow (CF) formation. Two different 3D geometries are introduced to reproduce the embryo. The two of them are parametrised through a novel system of curvilinear coordinates, well adapted for biological structures, which is obtained by three fictive electrostatic potentials. A gradient decomposition method is used to take into account both the active and the passive deformations occurring to the cells. Four simulations of the CF are proposed, which in turn considers singular and coupled aspects of the problem.

Simulation of multiple morphogenetic movements in the Drosophila embryo by a single 3D finite element model

The present work describes a 3D finite element model of the Drosophila embryo designed to simulate three morphogenetic movements during early gastrulation: ventral furrow invagination, cephalic furrow formation and germ band extension. The embryo is represented by a regular ellipsoid and only the mesoderm is modeled. Additionally, the parametric description of the biological structure in a special curvilinear system provides mesh-independent endogenous strains. A deformation gradient decomposition is used to couple an active deformation, specific for each morphogenetic movement, together with a passive deformation, which is due to the response of the continuous mesoderm. Boundary conditions such as the rigid contact with the external vitelline membrane and the yolk pressure are also taken into account. The results suggest that the number of active strains responsible for the morphogenetic events can be less than that deduced from direct experimental observations. Finally, the estimation of the non-local pressures induced during morphogenetic movements is in good agreement with the experimental data.

Decomposition of displacement gradient and strain definition

For continuum, the displacement gradient can be divided into symmetry part and asymmetry part. Based on Chen Zhi-da’s research, the asymmetry part is related with an orthogonal rotation. But the definition of rotation angle in Chen’s theory is not correct for large deformation or rotation. The definition of rotation angle was improved to be suitable for large deformation or rotation. The results show that the displacement gradient can be divided into symmetry part and orthogonal rotation part. This result improves Chen Zhi-da’s S-R decomposition theoretic research and Finger-Truesdell’s polar decomposition theorem. Based on the new formulation of displacement gradient decomposition, it shows that S-R decomposition can be extended to large deformation and large rotation cases. The symmetry part contains a coupling tensor directly related with rotation tensor, this coupling tensor causes that the strain defined significantly differ from actual value calculated based on measured stress field. The additional stress related with the coupling tensor is the function of square of elements of asymmetry part of displacement gradient. The strain defined on Finger-Truesdell’s polar decomposition theorem loses a factor related with expanding of rotation. As the rotation contains asymmetry stress, the traditional equilibrium equation only meets moment conservation, and the angular-moment conservation is derived to accept the asymmetry stress related with rotation. As the velocity gradient can be gotten by taking time differentiation of displacement, hence the results are also true for fluids.

A deformation gradient decomposition method for the analysis of the mechanics of morphogenesis

A new finite element model is proposed for the analysis of the mechanical aspects of morphogenesis and tested on the biologically well studied gastrulation phenomenon, in particular ventral furrow invagination of the Drosophila melanogaster embryo. A set of mechanisms are introduced in the numerical model, which lead to the observed deformed shapes. We split the total deformation into two parts: an imposed active deformation, and an elastic deformation superimposed onto the latter. The active deformation simulates the effects of apical constriction and apico-basal elongation. These mechanisms are associated with known gene expressions and so in this way we attempt to bridge the well explored signalling pathways, and their associated phenotypes in a mechanical model. While the former have been studied in depth, much less can be said about the forces they produce and the mechanisms involved. From the numerical results, we are able to test different plausible mechanical hypotheses that generate the necessary folding observed in the invagination process. In particular, we conclude that only certain ratios between both modes (apical constriction and apico-basal elongation) can successfully reproduce the invagination process. The model also supports the idea that this invagination requires the contribution of several mechanisms, and that their redundancy provides the necessary robustness.

Gradient decomposition method for the mechanical analysis of morphogenesis

Natural network-structured hydrogels (e.g. bacterial cellulose (BC)) can be synthesised with specific artificial hydrogels (e.g. poly(2-hydroxyethyl methacrylate) (PHEMA)) to form a tougher and stronger nanofibre-reinforced composite hydrogel, which possesses micro- and nano-porous structure. These synthetic hydrogels exhibit a number of advantages for biomedical applications, such as good biocompatibility and better permeability for molecules to pass through. In this paper, the mechanical properties of this nanofibre-reinforced hydrogel containing BC and PHEMA have been characterised in terms of their tangent modulus and fracture stress/strain by uniaxial compressive testing. Numerical simulations based on Mooney-Rivlin hyperelastic theory are also conducted to understand the internal stress distribution and possible failure of the nanofibre-reinforced hydrogel under compression. By comparing the mechanical characteristics of BC, PHEMA, and PHEMA-based nanofibre reinforced hydrogel (BC-PHEMA) under the compression, it is possible to develop a suitable scaffold for tissue engineering on the basis of fundamental understanding of mechanical and fracture behaviours of nanofibre-reinforced hydrogels.

Boundary Element Regularisation Methods for Solving the Cauchy Problem in Linear Elasticity

In this article several boundary element regularisation methods, such as iterative, conjugate gradient, Tikhonov regularisation and singular value decomposition methods, for solving the Cauchy problem in isotropic linear elasticity are developed and compared. Regularising stopping criteria are developed and the convergence, as well as the stability, of the numerical methods proposed are analysed. The Cauchy problem in isotropic linear elasticity can be regularised by various methods, such as the general regularisation methods presented in this article, but more accurate results are obtained by particular methods which take into account the particular structure of the problem, such as the alternating iterative algorithm originally proposed by [V.A. Kozlov, V.G. Maz'ya and A.F. Fomin, (1991). An iterative method for solving the Cauchy problem for elliptic equations. Comput. Maths. Math. Phys ., 31 , 45-52].

Use of the angle information in the wavelet transform maxima for image de-noising

In this article, a new method of de-noising is proposed, based on the wavelet maxima. The originality of this method is in the use of the gradient angle in a multi-scale framework as the discriminatory parameter. In order to use to the best advantage the angle information, the multi-scale gradient decomposition schema proposed by Mallat is modified thus enabling a computation of uncorrelated partial derivatives. From this computation, a selection method of multi-scale contours is put forward, having a lesser algorithmic complexity than processings based on the gradient norm. The performance of this new algorithm is illustrated using simulated data and angiography images.

Reduced gradient decomposition in multistage linear programming

It is shown that multistage linear programming problems without final constraints allow an ultimate decomposition based on reduced gradient computation. The decomposition results in one-stage linear programming problems of small dimensionality without the necessity of coordination involving master problems and subproblems.