The humble loop shrinking property played a central role in the inception of modern topology but it has been eclipsed by more abstract algebraic formalism. This is particularly true in the context of detecting relevant non-contractible loops on surfa
ces where elaborate homological and/or graph theoretical constructs are favored in algorithmic solutions. In this work, we devise a variational analogy to the loop shrinking property and show that it yields a simple, intuitive, yet powerful solution allowing a streamlined treatment of the problem of handle and tunnel loop detection. Our formalization tracks the evolution of a diffusion front randomly initiated on a single location on the surface. Capitalizing on a diffuse interface representation combined with a set of rules for concurrent front interactions, we develop a dynamic data structure for tracking the evolution on the surface encoded as a sparse matrix which serves for performing both diffusion numerics and loop detection and acts as the workhorse of our fully parallel implementation. The substantiated results suggest our approach outperforms state of the art and robustly copes with highly detailed geometric models. As a byproduct, our approach can be used to construct Reeb graphs by diffusion thus avoiding commonly encountered issues when using Morse functions.
This paper introduces the first differentiable simulator of event streams, i.e., streams of asynchronous brightness change signals recorded by event cameras. Our differentiable simulator enables non-rigid 3D tracking of deformable objects (such as hu
man hands, isometric surfaces and general watertight meshes) from event streams by leveraging an analysis-by-synthesis principle. So far, event-based tracking and reconstruction of non-rigid objects in 3D, like hands and body, has been either tackled using explicit event trajectories or large-scale datasets. In contrast, our method does not require any such processing or data, and can be readily applied to incoming event streams. We show the effectiveness of our approach for various types of non-rigid objects and compare to existing methods for non-rigid 3D tracking. In our experiments, the proposed energy-based formulations outperform competing RGB-based methods in terms of 3D errors. The source code and the new data are publicly available.
Most 3D face reconstruction methods rely on 3D morphable models, which disentangle the space of facial deformations into identity geometry, expressions and skin reflectance. These models are typically learned from a limited number of 3D scans and thu
s do not generalize well across different identities and expressions. We present the first approach to learn complete 3D models of face identity geometry, albedo and expression just from images and videos. The virtually endless collection of such data, in combination with our self-supervised learning-based approach allows for learning face models that generalize beyond the span of existing approaches. Our network design and loss functions ensure a disentangled parameterization of not only identity and albedo, but also, for the first time, an expression basis. Our method also allows for in-the-wild monocular reconstruction at test time. We show that our learned models better generalize and lead to higher quality image-based reconstructions than existing approaches.
We suggest to represent an X-Field -a set of 2D images taken across different view, time or illumination conditions, i.e., video, light field, reflectance fields or combinations thereof-by learning a neural network (NN) to map their view, time or lig
ht coordinates to 2D images. Executing this NN at new coordinates results in joint view, time or light interpolation. The key idea to make this workable is a NN that already knows the basic tricks of graphics (lighting, 3D projection, occlusion) in a hard-coded and differentiable form. The NN represents the input to that rendering as an implicit map, that for any view, time, or light coordinate and for any pixel can quantify how it will move if view, time or light coordinates change (Jacobian of pixel position with respect to view, time, illumination, etc.). Our X-Field representation is trained for one scene within minutes, leading to a compact set of trainable parameters and hence real-time navigation in view, time and illumination.
We suggest representing light field (LF) videos as one-off neural networks (NN), i.e., a learned mapping from view-plus-time coordinates to high-resolution color values, trained on sparse views. Initially, this sounds like a bad idea for three main r
easons: First, a NN LF will likely have less quality than a same-sized pixel basis representation. Second, only few training data, e.g., 9 exemplars per frame are available for sparse LF videos. Third, there is no generalization across LFs, but across view and time instead. Consequently, a network needs to be trained for each LF video. Surprisingly, these problems can turn into substantial advantages: Other than the linear pixel basis, a NN has to come up with a compact, non-linear i.e., more intelligent, explanation of color, conditioned on the sparse view and time coordinates. As observed for many NN however, this representation now is interpolatable: if the image output for sparse view coordinates is plausible, it is for all intermediate, continuous coordinates as well. Our specific network architecture involves a differentiable occlusion-aware warping step, which leads to a compact set of trainable parameters and consequently fast learning and fast execution.
Trusses are load-carrying light-weight structures consisting of bars connected at joints ubiquitously applied in a variety of engineering scenarios. Designing optimal trusses that satisfy functional specifications with a minimal amount of material ha
s interested both theoreticians and practitioners for more than a century. In this paper, we introduce two main ideas to improve upon the state of the art. First, we formulate an alternating linear programming problem for geometry optimization. Second, we introduce two sets of complementary topological operations, including a novel subdivision scheme for global topology refinement inspired by Michells famed theoretical study. Based on these two ideas, we build an efficient computational framework for the design of lightweight trusses. AD{We illustrate our framework with a variety of functional specifications and extensions. We show that our method achieves trusses with smaller volumes and is over two orders of magnitude faster compared with recent state-of-the-art approaches.
In recent years, mesh subdivision---the process of forging smooth free-form surfaces from coarse polygonal meshes---has become an indispensable production instrument. Although subdivision performance is crucial during simulation, animation and render
ing, state-of-the-art approaches still rely on serial implementations for complex parts of the subdivision process. Therefore, they often fail to harness the power of modern parallel devices, like the graphics processing unit (GPU), for large parts of the algorithm and must resort to time-consuming serial preprocessing. In this paper, we show that a complete parallelization of the subdivision process for modern architectures is possible. Building on sparse matrix linear algebra, we show how to structure the complete subdivision process into a sequence of algebra operations. By restructuring and grouping these operations, we adapt the process for different use cases, such as regular subdivision of dynamic meshes, uniform subdivision for immutable topology, and feature-adaptive subdivision for efficient rendering of animated models. As the same machinery is used for all use cases, identical subdivision results are achieved in all parts of the production pipeline. As a second contribution, we show how these linear algebra formulations can effectively be translated into efficient GPU kernels. Applying our strategies to $sqrt{3}$, Loop and Catmull-Clark subdivision shows significant speedups of our approach compared to state-of-the-art solutions, while we completely avoid serial preprocessing.
Real-time marker-less hand tracking is of increasing importance in human-computer interaction. Robust and accurate tracking of arbitrary hand motion is a challenging problem due to the many degrees of freedom, frequent self-occlusions, fast motions,
and uniform skin color. In this paper, we propose a new approach that tracks the full skeleton motion of the hand from multiple RGB cameras in real-time. The main contributions include a new generative tracking method which employs an implicit hand shape representation based on Sum of Anisotropic Gaussians (SAG), and a pose fitting energy that is smooth and analytically differentiable making fast gradient based pose optimization possible. This shape representation, together with a full perspective projection model, enables more accurate hand modeling than a related baseline method from literature. Our method achieves better accuracy than previous methods and runs at 25 fps. We show these improvements both qualitatively and quantitatively on publicly available datasets.
