Affine Deformations Research Articles

UnWarpME: Unsupervised warping map estimation in real-world scenarios

This paper introduces a novel approach to warp an RGB image to a new target pose, where it learns to inpaint the invisible parts of the image by sampling pixels from the visible parts. This technique is particularly relevant in applications such as novel pose image generation, where the quality of the generated samples heavily relies on the warped images. Conventional methods typically utilize an affine warping map estimator and learn to estimate the warping map through a downstream image generation task. However, this approach results in an affine function being returned regardless of the complexity of the deformation between the source and target poses. In contrast, our proposed method involves estimating the warping map using a convolutional function, which learns through alternating between two downstream tasks. Initially, the estimation treats the warping as part of an image generation task, similar to existing methods but without the constraint of an affine transformation. This encourages the estimation of complex transformations beyond affine deformations. Subsequently, disregarding the image generation task, we supervise the convolutional warping map to learn any potential affine transformation between the guiding poses of the sample. Our experimental results demonstrate the high effectiveness of our method in preserving image textures while transforming it into highly complex poses.

Information geometry of Wasserstein statistics on shapes and affine deformations

AbstractInformation geometry and Wasserstein geometry are two main structures introduced in a manifold of probability distributions, and they capture its different characteristics. We study characteristics of Wasserstein geometry in the framework of [32] for the affine deformation statistical model, which is a multi-dimensional generalization of the location-scale model. We compare merits and demerits of estimators based on information geometry and Wasserstein geometry. The shape of a probability distribution and its affine deformation are separated in the Wasserstein geometry, showing its robustness against the waveform perturbation in exchange for the loss in Fisher efficiency. We show that the Wasserstein estimator is the moment estimator in the case of the elliptically symmetric affine deformation model. It coincides with the information-geometrical estimator (maximum-likelihood estimator) when the waveform is Gaussian. The role of the Wasserstein efficiency is elucidated in terms of robustness against waveform change.

Elastic–plastic intermittent re-arrangements of frictionless, soft granular matter under very slow isotropic deformations

How do soft granular materials (or dense amorphous systems) respond to externally applied deformations at different rates and for different system sizes? This long-standing question was intensively studied for shear deformations but only more recently for isotropic deformations, like compression–decompression cycles. For moderate strain rates, in the solid-like state, above jamming, the system appears to evolve more or less smoothly in time/strain, whereas for slow enough deformations, the material flips intermittently between the elastic, reversible base state and plastic, dynamic “events.” Only during the latter events, the microstructure re-arranges irreversibly. The reversible base state involves both affine and non-affine deformations, while the events are purely non-affine. The system size and rate dependence of the events are studied, providing reference data for comparison in future studies evaluating materials like hydrogel particles.

Neural Photometry-Guided Visual Attribute Transfer.

We present a deep learning-based method for propagating spatially-varying visual material attributes (e.g., texture maps or image stylizations) to larger samples of the same or similar materials. For training, we leverage images of the material taken under multiple illuminations and a dedicated data augmentation policy, making the transfer robust to novel illumination conditions and affine deformations. Our model relies on a supervised image-to-image translation framework and is agnostic to the transferred domain; we showcase a semantic segmentation, a normal map, and a stylization. Following an image analogies approach, the method only requires the training data to contain the same visual structures as the input guidance. Our approach works at interactive rates, making it suitable for material edit applications. We thoroughly evaluate our learning methodology in a controlled setup providing quantitative measures of performance. Last, we demonstrate that training the model on a single material is enough to generalize to materials of the same type without the need for massive datasets.

Strong Ellipticity and Infinitesimal Stability within Nth-Order Gradient Elasticity

We formulate a series of strong ellipticity inequalities for equilibrium equations of the gradient elasticity up to the Nth order. Within this model of a continuum, there exists a deformation energy introduced as an objective function of deformation gradients up to the Nth order. As a result, the equilibrium equations constitute a system of 2N-order nonlinear partial differential equations (PDEs). Using these inequalities for a boundary-value problem with the Dirichlet boundary conditions, we prove the positive definiteness of the second variation of the functional of the total energy. In other words, we establish sufficient conditions for infinitesimal instability. Here, we restrict ourselves to a particular class of deformations which includes affine deformations.

An energetically consistent surface correction method for bond-based peridynamics

A novel surface correction method is proposed for bond-based peridynamics which ensures energy consistency with a classical reference body for general affine deformations. This method is validated for simple geometries and then applied to a typical surface-dominated problem, namely the indentation of a surface in the shallow to moderate-depth regime.

DeformRS: Certifying Input Deformations with Randomized Smoothing

Deep neural networks are vulnerable to input deformations in the form of vector fields of pixel displacements and to other parameterized geometric deformations e.g. translations, rotations, etc. Current input deformation certification methods either (i) do not scale to deep networks on large input datasets, or (ii) can only certify a specific class of deformations, e.g. only rotations. We reformulate certification in randomized smoothing setting for both general vector field and parameterized deformations and propose DeformRS-VF and DeformRS-Par, respectively. Our new formulation scales to large networks on large input datasets. For instance, DeformRS-Par certifies rich deformations, covering translations, rotations, scaling, affine deformations, and other visually aligned deformations such as ones parameterized by Discrete-Cosine-Transform basis. Extensive experiments on MNIST, CIFAR10, and ImageNet show competitive performance of DeformRS-Par achieving a certified accuracy of 39\% against perturbed rotations in the set [-10 degree, 10 degree] on ImageNet.

Implementation of cellular bulk stresses in vertex models of biological tissues.

Vertex models describe biological tissues as tilings of polygons. In standard vertex models, the tissue dynamics result from a balance between isotropic stresses, which are associated with the bulk of the cells, and tensions associated with cell-cell interfaces. However, in this framework it is less obvious how to describe anisotropic stresses arising from the bulk of cells. In epithelia, such bulk anisotropic stresses could arise for instance through medial myosin fluctuations. Two recent publications-Tlili et al. (Proc Natl Acad Sci USA 116(51):25430-25439, 2019) and Comelles et al. (eLife 10:e57730, 2021)-have proposed different schemes to implement bulk anisotropic stresses in vertex models. Here we show that while both schemes transform in the same way under affine deformations, they lead to significantly different tissue dynamics. Our results are consistent with the interpretation that the Tilli et al. scheme describes bulk stresses that are uniform within each cell, while the Comelles et al. scheme corresponds to non-uniform bulk stresses. Finally, we wondered whether a standard vertex model can be fully expressed in terms of bulk cellular stresses alone. We find that, in general, neither scheme can mimic the vertex forces created by cell-cell interface tensions.

Multiscale Computational Model Predicts Mouse Skin Kinematics Under Tensile Loading.

Skin is a complex tissue whose biomechanical properties are generally understood in terms of an incompressible material whose microstructure undergoes affine deformations. A growing number of experiments, however, have demonstrated that skin has a high Poisson's ratio, substantially decreases in volume during uniaxial tensile loading, and demonstrates collagen fiber kinematics that are not affine with local deformation. In order to better understand the mechanical basis for these properties, we constructed multiscale mechanical models (MSM) of mouse skin based on microstructural multiphoton microscopy imaging of the dermal microstructure acquired during mechanical testing. Three models that spanned the cases of highly aligned, moderately aligned, and nearly random fiber networks were examined and compared to the data acquired from uniaxially stretched skin. Our results demonstrate that MSMs consisting of networks of matched fiber organization can predict the biomechanical behavior of mouse skin, including the large decrease in tissue volume and nonaffine fiber kinematics observed under uniaxial tension.

Pattern stabilization in swarms of programmable active matter: A probe for turbulence at large length scales.

We propose an algorithm for creating stable, ordered, swarms of active robotic agents arranged in any given pattern. The strategy involves suppressing a class of fluctuations known as "nonaffine" displacements, viz., those involving nonlinear deformations of a reference pattern, while all (or most) affine deformations are allowed. We show that this can be achieved using precisely calculated, fluctuating, thrust forces associated with a vanishing average power input. A surprising outcome of our study is that once the structure of the swarm is maintained at steady state, the statistics of the underlying flow field is determined solely from the statistics of the forces needed to stabilize the swarm.

Non-affinity in multi-material mechanical metamaterials

Non-affine deformations enable mechanical metamaterials to achieve their unusual properties while imposing implications for their structural integrity. The presence of multiple phases with different mechanical properties results in additional non-affinity of the deformations, a phenomenon that has never been studied before in the area of extremal mechanical metamaterials. Here, we studied the degree of non-affinity, Gamma , resulting from the random substitution of a fraction of the struts,{rho }_{h}, that make up a lattice structure and are printed using a soft material (elastic modulus = {E}_{s}) by those printed using a hard material ({E}_{h}). Depending on the unit cell angle (i.e., theta = 60°, 90°, or 120°), the lattice structures exhibited negative, near-zero, or positive values of the Poisson’s ratio, respectively. We found that the auxetic structures exhibit the highest levels of non-affinity, followed by the structures with positive and near-zero values of the Poisson’s ratio. We also observed an increase in Gamma with frac{{E}_{h}}{{E}_{s}} and {rho }_{h} until frac{{E}_{h}}{{E}_{s}} =104 and {rho }_{h}= 75%-90% after which Gamma saturated. The dependency of Gamma upon {rho }_{h} was therefore found to be highly asymmetric. The positive and negative values of the Poisson’s ratio were strongly correlated with Gamma . Interestingly, achieving extremely high or extremely low values of the Poisson’s ratio required highly affine deformations. In conclusion, our results clearly show the importance of considering non-affinity when trying to achieve a specific set of mechanical properties and underscore the structural integrity implications in multi-material mechanical metamaterials.

The designs and deformations of rigidly and flat-foldable quadrilateral mesh origami

Rigidly and flat-foldable quadrilateral mesh origami is the class of quadrilateral mesh crease patterns with one fundamental property: the patterns can be folded from flat to fully-folded flat by a continuous one-parameter family of piecewise affine deformations that do not stretch or bend the mesh-panels. In this work, we explicitly characterize the designs and deformations of all possible rigidly and flat-foldable quadrilateral mesh origami. Our key idea is a rigidity theorem (Theorem 3.1) that characterizes compatible crease patterns surrounding a single panel and enables us to march from panel to panel to compute the pattern and its corresponding deformations explicitly. The marching procedure is computationally efficient. So we use it to formulate the inverse problem: to design a crease pattern to achieve a targeted shape along the path of its rigidly and flat-foldable motion. The initial results on the inverse problem are promising and suggest a broadly useful engineering design strategy for shape-morphing with origami.

Conformal equivalence of measures and dynamics of orthogonal polynomials

ABSTRACTWe introduce a notion of asymptotically orthonormal polynomials for a Borel measure μ with compact non-polar support in . Such sequences of polynomials have similar convergence properties of the sequences of Julia sets and filled Julia sets to those for sequences of orthonormal polynomials, see also Christiansen et al. [Julia sets of orthogonal polynomials, Potential Anal. 50(3) (2019), pp. 401–413]. We give examples of measures for which the monic orthogonal polynomials are asymptotically orthonormal. Combining this with observations on conformal invariance of orthogonal polynomials we explore the measure dependency of the associated dynamics of orthogonal polynomials. Concretely, we study the dynamics of sequences of asymptotically orthonormal polynomials for the pullback measure under affine mappings φ. We prove that the sequences of Julia sets and filled Julia sets of affine deformations of sequences of asymptotically orthonormal polynomials for μ also have the same convergence properties as the Julia sets and filled Julia sets of the orthonormal polynomials. This leads to theorems on the convergence properties of affine deformations of the family of iterates of any fixed monic centred polynomial and, in the case the polynomial is hyperbolic, on the corresponding family of affine parameter spaces.

Smoothed particle hydrodynamics simulation of granular system under cyclic compressions

The response of two-dimensional frictionless granular material to isotropic cyclic compression is simulated using an improved version of the Smoothed Particle Hydrodynamics (SPH) method, which includes realistic constitutive model for deformation of individual grains. The study reveals the evolution of mean coordination number and global pressure over cycles. The probability distribution function (PDF) of contact forces for different compression cycles is also reported. The global pressure at maximum compression shows downward trend for packing fractions below a certain value. The structural rearrangement that can give rise to such stress relaxation is studied by mapping relative particle mobilities and quantifying dynamic heterogeneity using a four-point susceptibility measure. The four-point susceptibility measure reveals length and time scales that can characterize the dynamics of driven system. Meso-scale structural rearrangement is studied using Falk-Langer measure of affine and non-affine deformation. The affine and non-affine deformations drop to a stable value and oscillates around it, which suggest that the structure is driving towards a more stable configuration. A negative correlation is found between the local packing fraction and the non-affine squared displacement. Finally, a complex network analysis is employed to better understand the structural rearrangement at meso-scale. The average degree and average clustering coefficient obtained from the complex network analysis show peaks at maximum compressions, but the peak values increase with cycles. The degree per particle is found to be positively correlated with local packing fraction and negatively correlated with the non-affine squared deformation. An enrichment of three-cycle population is seen, suggesting it as the most preferred conformation for particles at the meso-scale.

On the Relation of Lie Algebroids to Constrained Systems and their BV/BFV Formulation

We observe that a system of irreducible, fiber-linear, first class constraints on T*M is equivalent to the definition of a foliation Lie algebroid over M. The BFV formulation of the constrained system is given by the Hamiltonian lift of the Vaintrob description (E[1],Q) of the Lie algebroid to its cotangent bundle T*E[1]. Affine deformations of the constraints are parametrized by the first Lie algebroid cohomology H^1_Q and lead to irreducible constraints also for much more general Lie algebroids such as Dirac structures; the modified BFV function follows by the addition of a representative of the deformation charge. Adding a Hamiltonian to the system corresponds to a metric g on M. Evolution invariance of the constraint surface introduces a connection nabla on E and one reobtains the compatibility of g with (E,rho,nabla) found previously in the literature. The covariantization of the Hamiltonian to a function on T*E[1] serves as a BFV-Hamiltonian, iff, in addition, this connection is compatible with the Lie algebroid structure, turning (E, rho, [ , ], nabla) into a Cartan-Lie algebroid. The BV formulation of the system is obtained from BFV by a (time-dependent) AKSZ procedure.

Computational investigation of the impact of assumption of affine deformation on constitutive models of soft tissues

All constitutive models proposed during the last decades for large strain composites with hyperelastic matrix use an intrinsic assumption of affine deformation between the matrix and reinforcing fibres. While for typical technical composites the affinity of deformation between fibres and matrix till high loads is ensured by targeted creation of their chemical bonds, this need not to be the case with soft biological tissues. For instance, this assumption might be disputable for arterial tissues with their matrix consisting of very compliant gel-like proteoglycans. On the other hand, no constitutive model proposed till now has been capable to give a reasonable fit of all mechanical tests of some pathological tissues with low initial stiffness, such as aortic aneurysm wall. Thus, a question occurs whether this discrepancy could not be caused by the intrinsic assumption of affine deformation used in all models. To test the impact of this assumption on the simulated response in some mechanical tests, two finite element models of specimens of arterial tissues were created, both including matrix and fibres separately. The former model mimicked the affine deformation of matrix and fibres by merging all the nodes of both components, while the deformation of fibres was independent of the matrix in the latter model, with exception of both fibre ends. Differences in reaction forces of specimens were evaluated in various strain states and directions with respect to the orientation of fibres. The evaluated differences between models with affine and non-affine deformations were significant but smaller than typical inaccuracies of constitutive models when fitting aortic aneurysm tissues.

ADAIS: Automatic Derivation of Anisotropic Ideal Strength via high-throughput first-principles computations

Anisotropic ideal strength is a fundamental and important plasticity parameter in scaling the intrinsic strength of strong crystalline materials, and is a potential descriptor in searching and designing novel hard/superhard materials. However, to the best of our knowledge, an automatic derivation of anisotropic ideal strength has not been implemented in any open-source code available so far. In this paper, we present our developed ADAIS code, an automatic derivation of anisotropic ideal strength via high-throughput first-principles computations for both three-dimensional and two-dimensional crystalline materials with any symmetry, as well as for an ideal interface model. Several fundamental mechanical quantities can be automatically derived, including ideal tensile and shear strengths through affine deformation, universal binding energy and generalized stacking fault energy, as well as the ideal cleavage and slide stresses through alias deformation. The implementation of this code has been comprehensively demonstrated and critically validated by a lot of evaluations and tests of various crystalline materials with different symmetry, indicating that our code could provide a high-efficiency solution to quantify the strength of strong solids. Program summaryProgram title: ADAISProgram Files doi:http://dx.doi.org/10.17632/336hry5g66.1Licensing provisions: BSD 3-ClauseProgramming language: Fortran90Nature of problem: A scheme adapted to high-throughput first-principles computations is much necessary to automatically determine the anisotropic ideal strength along any crystallographic orientation under affine and alias deformations for crystalline solids with any symmetry.Solution method: To derive the ideal strength of a crystalline solid along a specific crystallographic orientation, a projection and/or redefinition of a lattice are firstly performed. The affine (alias) deformation is then applied to the projected (redefined) lattice, and a relaxation is done on the distorted structure by the simply modified VASP code optimizer with certain cell constraints. Afterwards, the resultant total energies and stresses vs strain are generated accordingly. Finally, the anisotropic ideal strengths are derived from the calculated stress/energy–strain relationship, and meanwhile the variation of bond length as a function of strain is revealed to illustrate the failure mode. Whenever an unexpected error happens, a message will be printed for further evaluation.

Automatic Registration of Images With Inconsistent Content Through Line-Support Region Segmentation and Geometrical Outlier Removal.

The implementation of automatic image registration is still difficult in various applications. In this paper, an automatic image registration approach through line-support region segmentation and geometrical outlier removal is proposed. This new approach is designed to address the problems associated with the registration of images with affine deformations and inconsistent content, such as remote sensing images with different spectral content or noise interference, or map images with inconsistent annotations. To begin with, line-support regions, namely a straight region whose points share roughly the same image gradient angle, are extracted to address the issues of inconsistent content existing in images. To alleviate the incompleteness of line segments, an iterative strategy with multi-resolution is employed to preserve global structures that are masked at full resolution by image details or noise. Then, geometrical outlier removal is developed to provide reliable feature point matching, which is based on affine-invariant geometrical classifications for corresponding matches initialized by scale invariant feature transform. The candidate outliers are selected by comparing the disparity of accumulated classifications among all matches, instead of conventional methods which only rely on local geometrical relations. Various image sets have been considered in this paper for the evaluation of the proposed approach, including aerial images with simulated affine deformations, remote sensing optical and synthetic aperture radar images taken at different situations (multispectral, multisensor, and multitemporal), and map images with inconsistent annotations. Experimental results demonstrate the superior performance of the proposed method over the existing approaches for the whole data set.

Combining phase-field crystal methods with a Cahn-Hilliard model for binary alloys.

Diffusion-induced phase transitions typically change the lattice symmetry of the host material. In battery electrodes, for example, Li ions (diffusing species) are inserted between layers in a crystalline electrode material (host). This diffusion induces lattice distortions and defect formations in the electrode. The structural changes to the lattice symmetry affect the host material's properties. Here, we propose a 2D theoretical framework that couples a Cahn-Hilliard (CH) model, which describes the composition field of a diffusing species, with a phase-field crystal (PFC) model, which describes the host-material lattice symmetry. We couple the two continuum models via coordinate transformation coefficients. We introduce the transformation coefficients in the PFC method to describe affine lattice deformations. These transformation coefficients are modeled as functions of the composition field. Using this coupled approach, we explore the effects of coarse-grained lattice symmetry and distortions on a diffusion-induced phase transition process. In this paper, we demonstrate the working of the CH-PFC model through three representative examples: First, we describe base cases with hexagonal and square symmetries for two composition fields. Next, we illustrate how the CH-PFC method interpolates lattice symmetry across a diffuse phase boundary. Finally, we compute a Cahn-Hilliard type of diffusion and model the accompanying changes to lattice symmetry during a phase transition process.

Kinematics of collagen fibers in carotid arteries under tension-inflation loading.

Biomechanics of the extracellular matrix in arteries determines their macroscopic mechanical behavior. In particular, the distribution of collagen fibers and bundles plays a significant role. Experimental data showed that, in most arterial walls, there are preferred fiber directions. However, the realignment of collagen fibers during tissue deformation is still controversial: whilst authors claim that fibers should undergo affine deformations, others showed the contrary. In order to have an insight about this important question of affine deformations at the microscopic scale, we measured the realignment of collagen fibers in the adventitia layer of carotid arteries using multiphoton microscopy combined with an unprecedented Fourier based method. We compared the realignment for two types of macroscopic loading applied on arterial segments: axial tension under constant pressure (scenario 1) and inflation under constant axial length (scenario 2). Results showed that, although the tissue underwent macroscopic stretches beyond 1.5 in the circumferential direction, fiber directions remained unchanged during scenario 2 loading. Conversely, fibers strongly realigned along the axis direction for scenario 1 loading. In both cases, the motion of collagen fibers did not satisfy affine deformations, with a significant difference between both cases: affine predictions strongly under-estimated fiber reorientations in uniaxial tension and over-estimated fiber reorientations during inflation at constant length. Finally, we explained this specific kinematics of collagen fibers by the complex tension-compression interactions between very stiff collagen fibers and compliant surrounding proteins. A tensegrity representation of the extracellular matrix in the adventitia taking into account these interactions was proposed to model the motion of collagen fibers during tissue deformation.