Affine Deformations Research Articles

Affine and non-affine deformations quantified in cytoskeletal networks through three-dimensional form-finding model

Actin filaments and cross-linkers are main components of cytoskeletal networks in eukaryotic cells, and they support bending moments and axial forces respectively. A three-dimensional form-finding model is proposed in this work to investigate affine and non-affine deformations in cytoskeletal networks. In recent studies, modeling of cytoskeletal networks turns out to be a key piece in the cell mechanics puzzle. We used form-finding analysis to compute and analyze cytoskeletal models. A three-dimensional model is much more flexible and contains more elements than a two-dimensional model, and non-linear finite element analysis is difficult to converge. Thus, vector form intrinsic finite element analysis is employed here for valid results. The three-dimensional model reveals new behaviors beyond earlier two-dimensional models and better aligns with available data. Relative density of actin filaments and height of the form-finding model both play important roles in determining cytoskeletal stiffness, positively and negatively, respectively. Real cytoskeletal networks are quite mixed in terms of affine and non-affine deformations, which are quantified by internal strain energy in actin filaments and cross-linkers. Results are also influenced by actin filament relative density and height of the model. The three-dimensional form-finding model does provide much more room for intensive studies on cytoskeletal networks. In our future study, microtubules, fluidics, viscoelastic-plastic cross-linkers and even the whole cell model may be taken into account gradually to improve the cytoskeletal form-finding model.

Classical Dynamics of Fullerenes

The classical mechanics of large molecules and fullerenes is studied. The approach is based on the model of collective motion of these objects. The mixed Lagrangian (material) and Eulerian (space) description of motion is used. In particular, the Green and Cauchy deformation tensors are geometrically defined. The important issue is the group-theoretical approach to describing the affine deformations of the body. The Hamiltonian description of motion based on the Poisson brackets methodology is used. The Lagrange and Hamilton approaches allow us to formulate the mechanics in the canonical form. The method of discretization in analytical continuum theory and in classical dynamics of large molecules and fullerenes enable us to formulate their dynamics in terms of the polynomial expansions of configurations. Another approach is based on the theory of analytical functions and on their approximations by finite-order polynomials. We concentrate on the extremely simplified model of affine deformations or on their higher-order polynomial perturbations.

LRA: Local Rigid Averaging of Stretchable Non-rigid Shapes

We present a novel algorithm for generating the mean structure of non-rigid stretchable shapes. Following an alignment process, which supports local affine deformations, we translate the search of the mean shape into a diagonalization problem where the structure is hidden within the kernel of a matrix. This is the first step required in many practical applications, where one needs to model bendable and stretchable shapes from multiple observations.

Motion Estimation and Compensation Strategies in Dynamic Computerized Tomography

A main challenge in computerized tomography consists in imaging moving objects. Temporal changes during the measuring process lead to inconsistent data sets, and applying standard reconstruction techniques causes motion artefacts which can severely impose a reliable diagnostics. Therefore, novel reconstruction techniques are required which compensate for the dynamic behavior. This article builds on recent results from a microlocal analysis of the dynamic setting, which enable us to formulate efficient analytic motion compensation algorithms for contour extraction. Since these methods require information about the dynamic behavior, we further introduce a motion estimation approach which determines parameters of affine and certain non-affine deformations directly from measured motion-corrupted Radon-data. Our methods are illustrated with numerical examples for both types of motion.

Data-Driven Affine Deformation Estimation and Correction in Cone Beam Computed Tomography.

In computed tomography (CT), motion and deformation during the acquisition lead to streak artefacts and blurring in the reconstructed images. To remedy these artefacts, we introduce an efficient algorithm to estimate and correct for global affine deformations directly on the cone beam projections. The proposed technique is data driven and thus removes the need for markers and/or a tracking system. A relationship between affine transformations and the cone beam transform is proved and used to correct the projections. The deformation parameters that describe deformation perpendicular to the projection direction are estimated for each projection by minimizing a plane-based inconsistency criterion. The criterion compares each projection of the main scan with all projections of a fast reference scan, which is acquired prior or posterior to the main scan. Experiments with simulated and experimental data show that the proposed affine deformation estimation method is able to substantially reduce motion artefacts in cone beam CT images.

Motion Detection and Compensation of Affine Deformations in Infrared Retinal Videos

Diabetic macular edema (DME) affects the retina and reduces the visual acuity of patients with severe diabetic retinopathy. Its conventional treatment involves laser photocoagulation combined with infrared (IR) imaging. However, the laser beam may hit healthy retinal areas and cause unintentional retinal damages if retinal motion occurs. We propose a method for retinal motion detection and compensation that relies on phase correlations in the image and in the log-polar domains, and is robust to affine retinal deformations (e.g., rotations, scales, and translations) in IR videos. The proposed method is also robust to the background noise and illumination changes commonly occurring in this retinal imaging modality. The proposed method can be used to estimate the retinal affine motion parameters and compensate for small retinal motions (nearly 50 $\mu \text{m}$ ). The critical method parameters are selected and adjusted optimally, which improves the robustness of the method. The experimental results suggest that the proposed method potentially can be more robust for detecting retinal motion and for estimating the parameters of affine retinal deformations than comparable methods that currently represent the state of the art, which helps to improve the reliability of laser treatments for DME.

McShane-type identities for affine deformations

We derive an identity for Margulis invariants of affine deformations of a complete orientable one-ended hyperbolic sur- face following the identities of McShane, Mirzakhani and Tan- Wong-Zhang. As a corollary, a deformation of the surface which infinitesimally lengthens all interior simple closed curves must in- finitesimally lengthen the boundary.

Contrasting the dynamics of elastic and non-elastic deformations across an experimental colloidal Martensitic transition.

We present a framework to segregate the roles of elastic and non-elastic deformations in the examination of real-space experiments of solid-solid Martensitic transitions. The Martensitic transformation of a body-centred-tetragonal (BCT) to a body-centred-orthorhombic (BCO) crystal structure has been studied in a model system of micron-scale ionic microgel colloids (P. S. Mohanty, P. Bagheri, S. Nöjd, A. Yethiraj and P. Schurtenberger, Phys. Rev. X, 2015, 5, 011030). Non-affine fluctuations, i.e., displacement fluctuations that do not arise from purely elastic (affine) deformations, are detected in particle configurations acquired from the experiment. Tracking these fluctuations serves as a highly sensitive tool in signaling the onset of the Martensitic transition and precisely locating particle rearrangements occurring at length scales of a few particle diameters. Particle rearrangements associated with non-affine displacement modes become increasingly favorable during the transformation process. The nature of the displacement fluctuation modes that govern the transformation are shown to be different from those predominant in an equilibrium crystal. We show that BCO crystallites formed through shear may, remarkably, co-exist with those resulting from local rearrangements within the same sample.

Infinitesimally affine deformations of a hypersurface

Affine deformations serve as basic examples in the continuum mechanics of deformable three-dimensional bodies (usually referred to as homogeneous deformations). They preserve parallelism of straight lines, and are often used as an approximation to general deformations. However, when the deformable body is a membrane, a shell or an interface modeled by a surface, parallelism is defined by the affine connection of this surface. In this work we study the infinitesimally affine time-dependent deformations (motions) of such a continuum, but in a more general context, by considering that it is modeled by a Riemannian hypersurface. First we prove certain equivalent formulas for the variation of the connection of the hypersurface. Some of these formulas are expressed in terms of geometrical quantities, and others in terms of kinematical quantities of the deforming continuum. The latter is achieved by using an adapted version of the polar decomposition theorem, frequently used in continuum mechanics to analyze motion. We also apply our results to special motions like tangential and normal motions. Further, we find necessary and sufficient conditions for this variation to be zero (infinitesimal affine motions), providing insight on the form of these motions and the kind of hypersurfaces that allow such motions. Finally, we give some specific examples of mechanical interest which demonstrate motions that are infinitesimally affine but not infinitesimally isometric.

An affine invariant approach for dense wide baseline image matching

Visual sensor networks have emerged as an important class of sensor-based distributed intelligent systems, where image matching is one of the key technologies. This article presents an affine invariant method to produce dense correspondences between uncalibrated wide baseline images. Under affine transformations, both point location and its neighborhood texture are changed between views, so dense matching becomes a tough task. The proposed approach tends to solve this problem within a sparse-to-dense framework. The contribution of this article is in threefolds. First, a strategy of reliable sparse matching is proposed, which starts from affine invariant features extraction and matching and then these initial matches are utilized as spatial prior to produce more sparse matches. Second, match propagation from sparse feature points to its neighboring pixels is conducted in the way of region growing in an affine invariant framework. Third, the unmatched points are handled by low-rank matrix recovery technique. Comparison experiments of the proposed method versus existing ones show a significant improvement in the presence of large affine deformations.

ICA-ASIFT-Based Multi-Temporal Matching of High-Resolution Remote Sensing Urban Images

Abstract While SIFT (Scale Invariant Feature Transform) features are used to match High-Resolution (HR) remote sensing urban images captured at different phases with large scale and view variations, feature points are few and the matching accuracy is low. Although replacing SIFT with fully affine invariant features ASIFT (Affine-SIFT) can increase the number of feature points, it results in matching inefficiency and a non-uniform distribution of matched feature point pairs. To address these problems, this paper proposes the novel matching method ICA-ASIFT, which matches HR remote sensing urban images captured at different phases by using an Independent Component Analysis algorithm (ICA) and ASIFT features jointly. First, all possible affine deformations are modeled for the image transform, extracting ASIFT features of remote sensing images captured at different times. The ICA algorithm reduces the dimensionality of ASIFT features and improves matching efficiency of subsequent ASIFT feature point pairs. Next, coarse matching is performed on ASIFT feature point pairs through the algorithms of Nearest Vector Angle Ratio (NVAR), Direction Difference Analysis (DDA) and RANdom SAmple Consensus (RANSAC), eliminating apparent mismatches. Then, fine matching is performed on rough matched point pairs using a Neighborhoodbased Feature Graph Matching algorithm (NFGM) to obtain final ASIFT matching point pairs of remote sensing images. Finally, final matching point pairs are used to compute the affine transform matrix. Matching HR remote sensing images captured at different phases is achieved through affine transform. Experiments are used to compare the performance of ICA-ASFIT and three other algorithms (i.e., Harris- SIFT, PCA-SIFT, TD-ASIFT) on HR remote sensing images captured at different times in different regions. Experimental results show that the proposed ICA-ASFIT algorithm effectively matches HR remote sensing urban images and outperforms other algorithms in terms of matching accuracy and efficiency.

The Material Point Monte Carlo model: A discrete, off-lattice method for microstructural evolution simulations

A new kinetic Monte Carlo model, termed the Material Point Monte Carlo (MPMC) method, is developed to simulate microstructural evolution under a variety of driving forces. The MPMC operates on a spatially random set of discrete material points; at each time step, points are relocated and their material data are updated according to a statistical mechanics scheme. Because material points are randomly distributed, and remain so under affine deformations, the MPMC eliminates artifacts of the simulation grid that occur in lattice-based methods such as the Monte Carlo Potts model. For model systems the MPMC is verified to correctly reproduce grain boundary motion caused by bulk and interfacial driving forces. It is then applied to larger and more complex problems, including grain growth during plastic deformation.

Creep of sound paths in consolidated granular material detected through coda wave interferometry.

The time evolution of the contact force structure of a consolidated granular material subjected to a constant stress is monitored using the coda wave interferometry method. In addition, the nature of the aging and rejuvenation processes are investigated. These processes are interpreted in terms of affine and nonaffine structural path deformations. During the later stages of creep, the rearrangements of subgrains are so small that they only produce affine deformations in the contact paths, without any significant changes in the structural configuration. As a result, the strain path distribution follows the macroscopic strain. Conversely, in the presence of ultrasonic perturbations, the nonaffine grain buckling mechanism dominates, producing relatively drastic changes in the structural configuration accompanied by path deformations of the order of the grain size. This plastic mechanism induces material rejuvenation that is observed macroscopically as an ultrasonically accelerated creep.

Universal Manifold Embedding for Geometrically Deformed Functions

Assume we have a set of observations (for example, images) of different objects, each undergoing a different geometric deformation, yet all the deformations belong to the same family. As a result of the action of these deformations, the set of different observations on each object is generally a manifold in the ambient space of observations. In this paper we show that in those cases where the set of deformations admits a finite-dimensional representation, there is a mapping from the space of observations to a low-dimensional linear space. The manifold corresponding to each object is mapped to a distinct linear subspace of Euclidean space. The dimension of the subspace is the same as that of the manifold. This mapping, which we call universal manifold embedding, enables the estimation of geometric deformations using the classical linear theory. The universal manifold embedding further enables the representation of the object classification and detection problems in a linear subspace matching framework. The embedding of the space of observations depends on the deformation model, and is independent of the specific observed object; hence, it is universal. We study two cases of this embedding: that of elastic deformations of 1-D signals, and the case of affine deformations of $n$ -dimensional signals.

Null space and resolution in dynamic computerized tomography

One major challenge in computerized tomography is to image objects which change during the data acquisition and hence lead to inconsistent data sets. Motion artefacts in the reconstructions can be reduced by applying specially adapted algorithms which take the dynamic behaviour into account. Within this article, we analyse the achievable resolution in the dynamic setting in case of two-dimensional affine deformations. To this end, we characterize the null space of the operator describing the dynamic case, using its singular value decomposition and a necessary dynamic consistency condition. This shows that independent of any reconstruction method, the specimen’s dynamics results in a loss of resolution compared to the stationary setting. Our theoretical results are illustrated at a numerical example.

Importance of non-affine viscoelastic response in disordered fibre networks.

Disordered fibre networks are ubiquitous in nature and have a wide range of industrial applications as novel biomaterials. Predicting their viscoelastic response is straightforward for affine deformations that are uniform over all length scales, but when affinity fails, as has been observed experimentally, modelling becomes challenging. Here we present a numerical methodology, related to an existing framework for amorphous packings, to predict the steady-state viscoelastic spectra and degree of affinity for disordered fibre networks driven at arbitrary frequencies. Applying this method to a peptide gel model reveals a monotonic increase of the shear modulus as the soft, non-affine normal modes are successively suppressed as the driving frequency increases. In addition to being dominated by fibril bending, these low frequency network modes are also shown to be delocalised. The presented methodology provides insights into the importance of non-affinity in the viscoelastic response of peptide gels, and is easily extendible to all types of fibre networks.

Effects of Various Shape Functions and Subset Size in Local Deformation Measurements Using DIC

The digital image correlation (DIC) method obtains comparable results with strain gauges and its reliability and accuracy are commonly accepted in the measurement of affine deformations. However, in engineering measurements, there are always substantial local deformations with high strain gradients, such as the Portevin-Le Chatelier (PLC) shear bands, deformations near gaps, and crack tips. In these situations, strain gauges are restricted because the results within the contact areas are smoothed. Although the DIC method can be employed to measure these local deformations, the calculation parameters (e.g., the order of the shape functions, and template size) seriously impact the results. By analyzing PLC shear bands with different gradients in tensile tests and simulated bands, the deep mechanism on how shape functions and templates impact on the accuracy of DIC results is established. This study also demonstrates that second-order shape functions are more suitable than first-order shape functions to describe local deformations. The theory that the results of second-order shape functions are reliable and accurate when the relative error between first- and second-order shape functions is less than 10 %, is proposed. In addition, improving the spatial resolution and the acquisition frequency is proposed, and proved to achieve reliable results.

Developing a Novel Stereovision Scheme for Improving Accuracy of Depth Estimation in Robotic Vision Systems

Acquiring three-dimensional data from a pair of stereo images is called stereovision that has been studied by researchers for decades. However, most of the previous studies on this topic focused on establishment of stereovision parameter matching and made conclusions on the premise of fixed focus. With the rapid development of multimedia technology, varifocal digital cameras have been widely used in many robotic applications recently. In general, error in the depth estimate becomes bigger when the focus and aperture is unknown or not fixed. For that reason, a three-stage framework is proposed in this paper to modify the conventional stereovision model for improving accuracy of depth estimation. The first stage is to modify the computational model of conventional stereovision for varifocal cameras. Then, the spacing of depth interval in non-uniform spacing of discrete depth levels can be altered, in particular, it is unaffected by changes in focal length. Finally, with considering the affine transformation, we add the deformation coefficient into the modified stereovision model for correcting three-dimensional affine deformations. Experimental results demonstrated that the depth estimation from stereo images using the proposed scheme was more accurate than conventional method.

Summation rules for a fully nonlocal energy-based quasicontinuum method

The quasicontinuum (QC) method coarse-grains crystalline atomic ensembles in order to bridge the scales from individual atoms to the micro- and mesoscales. A crucial cornerstone of all QC techniques, summation or quadrature rules efficiently approximate the thermodynamic quantities of interest. Here, we investigate summation rules for a fully nonlocal, energy-based QC method to approximate the total Hamiltonian of a crystalline atomic ensemble by a weighted sum over a small subset of all atoms in the crystal lattice. Our formulation does not conceptually differentiate between atomistic and coarse-grained regions and thus allows for seamless bridging without domain-coupling interfaces. We review traditional summation rules and discuss their strengths and weaknesses with a focus on energy approximation errors and spurious force artifacts. Moreover, we introduce summation rules which produce no residual or spurious force artifacts in centrosymmetric crystals in the large-element limit under arbitrary affine deformations in two dimensions (and marginal force artifacts in three dimensions), while allowing us to seamlessly bridge to full atomistics. Through a comprehensive suite of examples with spatially non-uniform QC discretizations in two and three dimensions, we compare the accuracy of the new scheme to various previous ones. Our results confirm that the new summation rules exhibit significantly smaller force artifacts and energy approximation errors. Our numerical benchmark examples include the calculation of elastic constants from completely random QC meshes and the inhomogeneous deformation of aggressively coarse-grained crystals containing nano-voids. In the elastic regime, we directly compare QC results to those of full atomistics to assess global and local errors in complex QC simulations. Going beyond elasticity, we illustrate the performance of the energy-based QC method with the new second-order summation rule by the help of nanoindentation examples with automatic mesh adaptation. Overall, our findings provide guidelines for the selection of summation rules for the fully nonlocal energy-based QC method.

Structural control of elastic moduli in ferrogels and the importance of non-affine deformations.

One of the central appealing properties of magnetic gels and elastomers is that their elastic moduli can reversibly be adjusted from outside by applying magnetic fields. The impact of the internal magnetic particle distribution on this effect has been outlined and analyzed theoretically. In most cases, however, affine sample deformations are studied and often regular particle arrangements are considered. Here we challenge these two major simplifications by a systematic approach using a minimal dipole-spring model. Starting from different regular lattices, we take into account increasingly randomized structures, until we finally investigate an irregular texture taken from a real experimental sample. On the one hand, we find that the elastic tunability qualitatively depends on the structural properties, here in two spatial dimensions. On the other hand, we demonstrate that the assumption of affine deformations leads to increasingly erroneous results the more realistic the particle distribution becomes. Understanding the consequences of the assumptions made in the modeling process is important on our way to support an improved design of these fascinating materials.