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In this work we present a novel bulk-surface virtual element method (BSVEM) for the numerical approximation of elliptic bulk-surface partial differential equations (BSPDEs) in three space dimensions. The BSVEM is based on the discretisation of the bulk domain into polyhedral elements with arbitrarily many faces. The polyhedral approximation of the bulk induces a polygonal approximation of the surface. Firstly, we present a geometric error analysis of bulk-surface polyhedral meshes independent of the numerical method. Then, we show that BSVEM has optimal second-order convergence in space, provided the exact solution is $H^{2+3/4}$ in the bulk and $H^2$ on the surface, where the additional $\frac{3}{4}$ is due to the combined effect of surface curvature and polyhedral elements close to the boundary. We show that general polyhedra can be exploited to reduce the computational time of the matrix assembly. To demonstrate optimal convergence results, a numerical example is presented on the unit sphere.

Differentiable physics is a powerful tool in computer vision and robotics for scene understanding and reasoning about interactions. Existing approaches have frequently been limited to objects with simple shape or shapes that are known in advance. In this paper, we propose a novel approach to differentiable physics with frictional contacts which represents object shapes implicitly using signed distance fields (SDFs). Our simulation supports contact point calculation even when the involved shapes are nonconvex. Moreover, we propose ways for differentiating the dynamics for the object shape to facilitate shape optimization using gradient-based methods. In our experiments, we demonstrate that our approach allows for model-based inference of physical parameters such as friction coefficients, mass, forces or shape parameters from trajectory and depth image observations in several challenging synthetic scenarios and a real image sequence.

An essential need for many model-based robot control algorithms is the ability to quickly and accurately compute partial derivatives of the equations of motion. State of the art approaches to this problem often use analytical methods based on the chain rule applied to existing dynamics algorithms. Although these methods are an improvement over finite differences in terms of accuracy, they are not always the most efficient. In this paper, we contribute new closed-form expressions for the first-order partial derivatives of inverse dynamics, leading to a recursive algorithm. The algorithm is benchmarked against chain-rule approaches in Fortran and against an existing algorithm from the Pinocchio library in C++. Tests consider computing the partial derivatives of inverse and forward dynamics for robots ranging from kinematic chains to humanoids and quadrupeds. Compared to the previous open-source Pinocchio implementation, our new analytical results uncover a key computational restructuring that enables efficiency gains. Speedups of up to 1.4x are reported for calculating the partial derivatives of inverse dynamics for the 50-dof Talos humanoid.

The Mat\'ern covariance function is ubiquitous in the application of Gaussian processes to spatial statistics and beyond. Perhaps the most important reason for this is that the smoothness parameter $\nu$ gives complete control over the mean-square differentiability of the process, which has significant implications for the behavior of estimated quantities such as interpolants and forecasts. Unfortunately, derivatives of the Mat\'ern covariance function with respect to $\nu$ require derivatives of the modified second-kind Bessel function $\mathcal{K}_\nu$ with respect to $\nu$. While closed form expressions of these derivatives do exist, they are prohibitively difficult and expensive to compute. For this reason, many software packages require fixing $\nu$ as opposed to estimating it, and all existing software packages that attempt to offer the functionality of estimating $\nu$ use finite difference estimates for $\partial_\nu \mathcal{K}_\nu$. In this work, we introduce a new implementation of $\mathcal{K}_\nu$ that has been designed to provide derivatives via automatic differentiation (AD), and whose resulting derivatives are significantly faster and more accurate than those computed using finite differences. We provide comprehensive testing for both speed and accuracy and show that our AD solution can be used to build accurate Hessian matrices for second-order maximum likelihood estimation in settings where Hessians built with finite difference approximations completely fail.

The total variation (TV) regularization has phenomenally boosted various variational models for image processing tasks. We propose combining the backward diffusion process in the earlier literature of image enhancement with the TV regularization and show that the resulting enhanced TV minimization model is particularly effective for reducing the loss of contrast, which is often encountered by models using the TV regularization. We establish stable reconstruction guarantees for the enhanced TV model from noisy subsampled measurements; non-adaptive linear measurements and variable-density sampled Fourier measurements are considered. In particular, under some weaker restricted isometry property conditions, the enhanced TV minimization model is shown to have tighter reconstruction error bounds than various TV-based models for the scenario where the level of noise is significant and the amount of measurements is limited. The advantages of the enhanced TV model are also numerically validated by preliminary experiments on the reconstruction of some synthetic, natural, and medical images.

Fracture produces new mesh fragments that introduce additional degrees of freedom in the system dynamics. Existing finite element method (FEM) based solutions suffer from an explosion in computational cost as the system matrix size increases. We solve this problem by presenting a graph-based FEM model for fracture simulation that is remeshing-free and easily scales to high-resolution meshes. Our algorithm models fracture on the graph induced in a volumetric mesh with tetrahedral elements. We relabel the edges of the graph using a computed damage variable to initialize and propagate fracture. We prove that non-linear, hyper-elastic strain energy is expressible entirely in terms of the edge lengths of the induced graph. This allows us to reformulate the system dynamics for the relabeled graph without changing the size of system dynamics matrix and thus prevents the computational cost from blowing up. The fractured surface has to be reconstructed explicitly only for visualization purposes. We simulate standard laboratory experiments from structural mechanics and compare the results with corresponding real-world experiments. We fracture objects made of a variety of brittle and ductile materials, and show that our technique offers stability and speed that is unmatched in current literature.

We propose a new method for reconstructing controllable implicit 3D human models from sparse multi-view RGB videos. Our method defines the neural scene representation on the mesh surface points and signed distances from the surface of a human body mesh. We identify an indistinguishability issue that arises when a point in 3D space is mapped to its nearest surface point on a mesh for learning surface-aligned neural scene representation. To address this issue, we propose projecting a point onto a mesh surface using a barycentric interpolation with modified vertex normals. Experiments with the ZJU-MoCap and Human3.6M datasets show that our approach achieves a higher quality in a novel-view and novel-pose synthesis than existing methods. We also demonstrate that our method easily supports the control of body shape and clothes.

This work focuses on mitigating two limitations in the joint learning of local feature detectors and descriptors. First, the ability to estimate the local shape (scale, orientation, etc.) of feature points is often neglected during dense feature extraction, while the shape-awareness is crucial to acquire stronger geometric invariance. Second, the localization accuracy of detected keypoints is not sufficient to reliably recover camera geometry, which has become the bottleneck in tasks such as 3D reconstruction. In this paper, we present ASLFeat, with three light-weight yet effective modifications to mitigate above issues. First, we resort to deformable convolutional networks to densely estimate and apply local transformation. Second, we take advantage of the inherent feature hierarchy to restore spatial resolution and low-level details for accurate keypoint localization. Finally, we use a peakiness measurement to relate feature responses and derive more indicative detection scores. The effect of each modification is thoroughly studied, and the evaluation is extensively conducted across a variety of practical scenarios. State-of-the-art results are reported that demonstrate the superiority of our methods.

With the advent of deep neural networks, learning-based approaches for 3D reconstruction have gained popularity. However, unlike for images, in 3D there is no canonical representation which is both computationally and memory efficient yet allows for representing high-resolution geometry of arbitrary topology. Many of the state-of-the-art learning-based 3D reconstruction approaches can hence only represent very coarse 3D geometry or are limited to a restricted domain. In this paper, we propose occupancy networks, a new representation for learning-based 3D reconstruction methods. Occupancy networks implicitly represent the 3D surface as the continuous decision boundary of a deep neural network classifier. In contrast to existing approaches, our representation encodes a description of the 3D output at infinite resolution without excessive memory footprint. We validate that our representation can efficiently encode 3D structure and can be inferred from various kinds of input. Our experiments demonstrate competitive results, both qualitatively and quantitatively, for the challenging tasks of 3D reconstruction from single images, noisy point clouds and coarse discrete voxel grids. We believe that occupancy networks will become a useful tool in a wide variety of learning-based 3D tasks.

Limited capture range, and the requirement to provide high quality initialization for optimization-based 2D/3D image registration methods, can significantly degrade the performance of 3D image reconstruction and motion compensation pipelines. Challenging clinical imaging scenarios, which contain significant subject motion such as fetal in-utero imaging, complicate the 3D image and volume reconstruction process. In this paper we present a learning based image registration method capable of predicting 3D rigid transformations of arbitrarily oriented 2D image slices, with respect to a learned canonical atlas co-ordinate system. Only image slice intensity information is used to perform registration and canonical alignment, no spatial transform initialization is required. To find image transformations we utilize a Convolutional Neural Network (CNN) architecture to learn the regression function capable of mapping 2D image slices to a 3D canonical atlas space. We extensively evaluate the effectiveness of our approach quantitatively on simulated Magnetic Resonance Imaging (MRI), fetal brain imagery with synthetic motion and further demonstrate qualitative results on real fetal MRI data where our method is integrated into a full reconstruction and motion compensation pipeline. Our learning based registration achieves an average spatial prediction error of 7 mm on simulated data and produces qualitatively improved reconstructions for heavily moving fetuses with gestational ages of approximately 20 weeks. Our model provides a general and computationally efficient solution to the 2D/3D registration initialization problem and is suitable for real-time scenarios.