Learning single-image 3D reconstruction by generative modelling of shape, pose and shading

We present a unified framework tackling two problems: class-specific 3D reconstruction from a single image, and generation of new 3D shape samples. These tasks have received considerable attention recently; however, most existing approaches rely on 3D supervision, annotation of 2D images with keypoints or poses, and/or training with multiple views of each object instance. Our framework is very general: it can be trained in similar settings to existing approaches, while also supporting weaker supervision. Importantly, it can be trained purely from 2D images, without pose annotations, and with only a single view per instance. We employ meshes as an output representation, instead of voxels used in most prior work. This allows us to reason over lighting parameters and exploit shading information during training, which previous 2D-supervised methods cannot. Thus, our method can learn to generate and reconstruct concave object classes. We evaluate our approach in various settings, showing that: (i) it learns to disentangle shape from pose and lighting; (ii) using shading in the loss improves performance compared to just silhouettes; (iii) when using a standard single white light, our model outperforms state-of-the-art 2D-supervised methods, both with and without pose supervision, thanks to exploiting shading cues; (iv) performance improves further when using multiple coloured lights, even approaching that of state-of-the-art 3D-supervised methods; (v) shapes produced by our model capture smooth surfaces and fine details better than voxel-based approaches; and (vi) our approach supports concave classes such as bathtubs and sofas, which methods based on silhouettes cannot learn.Comment: Extension of arXiv:1807.09259, accepted to IJCV. Differentiable renderer available at https://github.com/pmh47/dir

A variational model of fracture for tearing brittle thin sheets

Tearing of brittle thin elastic sheets, possibly adhered to a substrate, involves a rich interplay between nonlinear elasticity, geometry, adhesion, and fracture mechanics. In addition to its intrinsic and practical interest, tearing of thin sheets has helped elucidate fundamental aspects of fracture mechanics including the mechanism of crack path selection. A wealth of experimental observations in different experimental setups is available, which has been often rationalized with insightful yet simplified theoretical models based on energetic considerations. In contrast, no computational method has addressed tearing in brittle thin elastic sheets. Here, motivated by the variational nature of simplified models that successfully explain crack paths in tearing sheets, we present a variational phase-field model of fracture coupled to a nonlinear Koiter thin shell model including stretching and bending. We show that this general yet straightforward approach is able to reproduce the observed phenomenology, including spiral or power-law crack paths in free standing films, or converging/diverging cracks in thin films adhered to negatively/positively curved surfaces, a scenario not amenable to simple models. Turning to more quantitative experiments on thin sheets adhered to planar surfaces, our simulations allow us to examine the boundaries of existing theories and suggest that homogeneous damage induced by moving folds is responsible for a systematic discrepancy between theory and experiments. Thus, our computational approach to tearing provides a new tool to understand these complex processes involving fracture, geometric nonlinearity and delamination, complementing experiments and simplified theories.Fil: Li, Bin. Universidad Politécnica de Catalunya; España. Sorbonne Université; Francia. Centre National de la Recherche Scientifique; FranciaFil: Millán, Raúl Daniel. Universidad Nacional de Cuyo. Facultad de Ciencias Aplicadas a la Industria; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Mendoza; Argentina. Universidad Politécnica de Catalunya; EspañaFil: Torres Sánchez, Alejandro. Universidad Politécnica de Catalunya; EspañaFil: Roman, Benoît. Centre National de la Recherche Scientifique; Francia. Sorbonne Université; FranciaFil: Arroyo Balaguer, Marino. Universidad Politécnica de Catalunya; Españ

Symmetric confidence regions and confidence intervals for normal map formulations of stochastic variational inequalities

Stochastic variational inequalities (SVI) model a large class of equilibrium problems subject to data uncertainty, and are closely related to stochastic optimization problems. The SVI solution is usually estimated by a solution to a sample average approximation (SAA) problem. This paper considers the normal map formulation of an SVI, and proposes a method to build asymptotically exact confidence regions and confidence intervals for the solution of the normal map formulation, based on the asymptotic distribution of SAA solutions. The confidence regions are single ellipsoids with high probability. We also discuss the computation of simultaneous and individual confidence intervals

Phase-field boundary conditions for the voxel finite cell method: surface-free stress analysis of CT-based bone structures

The voxel finite cell method employs unfitted finite element meshes and voxel quadrature rules to seamlessly transfer CT data into patient-specific bone discretizations. The method, however, still requires the explicit parametrization of boundary surfaces to impose traction and displacement boundary conditions, which constitutes a potential roadblock to automation. We explore a phase-field based formulation for imposing traction and displacement constraints in a diffuse sense. Its essential component is a diffuse geometry model generated from metastable phase-field solutions of the Allen-Cahn problem that assumes the imaging data as initial condition. Phase-field approximations of the boundary and its gradient are then employed to transfer all boundary terms in the variational formulation into volumetric terms. We show that in the context of the voxel finite cell method, diffuse boundary conditions achieve the same accuracy as boundary conditions defined over explicit sharp surfaces, if the inherent length scales, i.e., the interface width of the phase-field, the voxel spacing and the mesh size, are properly related. We demonstrate the flexibility of the new method by analyzing stresses in a human femur and a vertebral body

Subdivision Shell Elements with Anisotropic Growth

A thin shell finite element approach based on Loop's subdivision surfaces is proposed, capable of dealing with large deformations and anisotropic growth. To this end, the Kirchhoff-Love theory of thin shells is derived and extended to allow for arbitrary in-plane growth. The simplicity and computational efficiency of the subdivision thin shell elements is outstanding, which is demonstrated on a few standard loading benchmarks. With this powerful tool at hand, we demonstrate the broad range of possible applications by numerical solution of several growth scenarios, ranging from the uniform growth of a sphere, to boundary instabilities induced by large anisotropic growth. Finally, it is shown that the problem of a slowly and uniformly growing sheet confined in a fixed hollow sphere is equivalent to the inverse process where a sheet of fixed size is slowly crumpled in a shrinking hollow sphere in the frictionless, quasi-static, elastic limit.Comment: 20 pages, 12 figures, 1 tabl

The diffuse Nitsche method: Dirichlet constraints on phase-field boundaries

We explore diffuse formulations of Nitsche's method for consistently imposing Dirichlet boundary conditions on phase-field approximations of sharp domains. Leveraging the properties of the phase-field gradient, we derive the variational formulation of the diffuse Nitsche method by transferring all integrals associated with the Dirichlet boundary from a geometrically sharp surface format in the standard Nitsche method to a geometrically diffuse volumetric format. We also derive conditions for the stability of the discrete system and formulate a diffuse local eigenvalue problem, from which the stabilization parameter can be estimated automatically in each element. We advertise metastable phase-field solutions of the Allen-Cahn problem for transferring complex imaging data into diffuse geometric models. In particular, we discuss the use of mixed meshes, that is, an adaptively refined mesh for the phase-field in the diffuse boundary region and a uniform mesh for the representation of the physics-based solution fields. We illustrate accuracy and convergence properties of the diffuse Nitsche method and demonstrate its advantages over diffuse penalty-type methods. In the context of imaging based analysis, we show that the diffuse Nitsche method achieves the same accuracy as the standard Nitsche method with sharp surfaces, if the inherent length scales, i.e., the interface width of the phase-field, the voxel spacing and the mesh size, are properly related. We demonstrate the flexibility of the new method by analyzing stresses in a human vertebral body

A variational nonlinear Hausdorff-Young inequality in the discrete setting

Following the works of Lyons and Oberlin, Seeger, Tao, Thiele and Wright, we relate the variation of certain discrete curves on the Lie group $\text{SU}(1,1)$ to the corresponding variation of their linearized versions on the Lie algebra. Combining this with a discrete variational Menshov-Paley-Zygmund theorem, we establish a variational Hausdorff-Young inequality for a discrete version of the nonlinear Fourier transform on $\text{SU}(1,1)$.Comment: 16 page