Abstract The humble loop shrinking property played a central role in the inception of modern topology but it has been eclipsed by more abstract algebraic formalisms. This is particularly true in the context of detecting relevant non‐contractible loops on surfaces 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.

High-fidelity reconstruction of dynamic fluids from sparse multiview RGB videos remains a formidable challenge, due to the complexity of the underlying physics as well as the severe occlusion and complex lighting in the captured data. Existing solutions either assume knowledge of obstacles and lighting, or only focus on simple fluid scenes without obstacles or complex lighting, and thus are unsuitable for real-world scenes with unknown lighting conditions or arbitrary obstacles. We present the first method to reconstruct dynamic fluid phenomena by leveraging the governing physics (ie, Navier -Stokes equations) in an end-to-end optimization from a mere set of sparse video frames without taking lighting conditions, geometry information, or boundary conditions as input. Our method provides a continuous spatio-temporal scene representation using neural networks as the ansatz of density and velocity solution functions for fluids as well as the radiance field for static objects. With a hybrid architecture that separates static and dynamic contents apart, fluid interactions with static obstacles are reconstructed for the first time without additional geometry input or human labeling. By augmenting time-varying neural radiance fields with physics-informed deep learning, our method benefits from the supervision of images and physical priors. Our progressively growing model with regularization further disentangles the density-color ambiguity in the radiance field, which allows for a more robust optimization from the given input of sparse views. A pretrained density-to-velocity fluid model is leveraged in addition as the data prior to avoid suboptimal velocity solutions which underestimate vorticity but trivially fulfill physical equations. Our method exhibits high-quality results with relaxed constraints and strong flexibility on a representative set of synthetic and real flow captures. Code and sample tests are at https://people.mpi-inf.mpg.de/~mchu/projects/PI-NeRF/.

While modern fluid simulation methods achieve high-quality simulation results, it is still a big challenge to interpret and control motion from visual quantities, such as the advected marker density. These visual quantities play an important role in user interactions: Being familiar and meaningful to humans, these quantities have a strong correlation with the underlying motion. We propose a novel data-driven conditional adversarial model that solves the challenging and theoretically ill-posed problem of deriving plausible velocity fields from a single frame of a density field. Besides density modifications, our generative model is the first to enable the control of the results using all of the following control modalities: obstacles, physical parameters, kinetic energy, and vorticity. Our method is based on a new conditional generative adversarial neural network that explicitly embeds physical quantities into the learned latent space, and a new cyclic adversarial network design for control disentanglement. We show the high quality and versatile controllability of our results for density-based inference, realistic obstacle interaction, and sensitive responses to modifications of physical parameters, kinetic energy, and vorticity. Code, models, and results can be found at https://github.com/RachelCmy/den2vel.

Abstract Subdivision surfaces have become an invaluable asset in production environments. While progress over the last years has allowed the use of graphics hardware to meet performance demands during animation and rendering, high‐performance is limited to immutable mesh connectivity scenarios. Motivated by recent progress in mesh data structures, we show how the complete Catmull‐Clark subdivision scheme can be abstracted in the language of linear algebra. While this high‐level formulation allows for a fully parallel implementation with significant performance gains, the underlying algebraic operations require further specialization for modern parallel hardware. Integrating domain knowledge about the mesh matrix data structure, we replace costly general linear algebra operations like matrix‐matrix multiplication by specialized kernels. By further considering innate properties of Catmull‐Clark subdivision, like the quad‐only structure after refinement, we achieve an additional order of magnitude in performance and significantly reduce memory footprints. Our approach can be adapted seamlessly for different use cases, such as regular subdivision of dynamic meshes, fast evaluation for immutable topology and feature‐adaptive subdivision for efficient rendering of animated models. In this way, patchwork solutions are avoided in favor of a streamlined solution with consistent performance gains throughout the production pipeline. The versatility of the sparse matrix linear algebra abstraction underlying our work is further demonstrated by extension to other schemes such as and Loop subdivision.

Abstract The rich and evocative patterns of natural tessellations endow them with an unmistakable artistic appeal and structural properties which are echoed across design, production, and manufacturing. Unfortunately, interactive control of such patterns‐as modeled by Voronoi diagrams, is limited to the simple two dimensional case and does not extend well tofreeform surfaces. We present an approach for direct modeling and editing of such cellular structures on surface meshes. The overall modeling experience is driven by a set of editing primitives which are efficiently implemented on graphics hardware. We feature a novel application for 3D printing on modern support‐free additive manufacturing platforms. Our method decomposes the input surface into a cellular skeletal structure which hosts a set of overlay shells. In this way, material saving can be channeled to the shells while structural stability is channeled to the skeleton. To accommodate the available printer build volume, the cellular structure can be further split into moderately sized parts. Together with shells, they can be conveniently packed to save on production time. The assembly of the printed parts is streamlined by a part numbering scheme which respects the geometric layout of the input model.

In the ongoing efforts targeting the vectorization of linear algebra primitives, sparse matrix-matrix multiplication (SpGEMM) has received considerably less attention than sparse Matrix-Vector multiplication (SpMV). While both are equally important, this disparity can be attributed mainly to the additional formidable challenges raised by SpGEMM.

We introduce a hierarchical sparse matrix representation (HiSparse) tailored for the graphics processing unit (GPU). The representation adapts to the local nonzero pattern at all levels of the hierarchy and uses reduced bit length for addressing the entries. This allows a smaller memory footprint than standard formats. Executing algorithms on a hierarchical structure on the GPU usually entails significant synchronization and management overhead or slowdowns due to diverging execution paths and memory access patterns. We address these issues by means of a dynamic scheduling strategy specifically designed for executing algorithms on top of a hierarchical matrix on the GPU. The evaluation of our implementation of basic linear algebra routines, suggests that our hierarchical format is competitive to highly optimized standard libraries and significantly outperforms them in the case of transpose matrix operations. The results point towards the viability of hierarchical matrix formats on massively parallel devices such as the GPU.

The rising popularity of the graphics processing unit (GPU) across various numerical computing applications triggered a breakneck race to optimize key numerical kernels and in particular, the sparse matrix-vector product (SpMV). Despite great strides, most existing GPU-SpMV approaches trade off one aspect of performance against another. They either require preprocessing, exhibit inconsistent behavior, lead to execution divergence, suffer load imbalance or induce detrimental memory access patterns. In this paper, we present an uncompromising approach for SpMV on the GPU. Our approach requires no separate preprocessing or knowledge of the matrix structure and works directly on the standard compressed sparse rows (CSR) data format. From a global perspective, it exhibits a homogeneous behavior reflected in efficient memory access patterns and steady per-thread workload. From a local perspective, it avoids heterogeneous execution paths by adapting its behavior to the work load at hand, it uses an efficient encoding to keep temporary data requirements for on-chip memory low, and leads to divergence-free execution. We evaluate our approach on more than 2500 matrices comparing to vendor provided, and state-of-the-art SpMV implementations. Our approach not only significantly outperforms approaches directly operating on the CSR format ( 20% average performance increase), but also outperforms approaches that preprocess the matrix even when preprocessing time is discarded. Additionally, the same strategies lead to significant performance increase when adapted for transpose SpMV.

A key advantage of working with structured grids (e.g., images) is the ability to directly tap into the powerful machinery of linear algebra. This is not much so for unstructured grids where intermediate bookkeeping data structures stand in the way. On modern high performance computing hardware, the conventional wisdom behind these intermediate structures is further challenged by costly memory access, and more importantly by prohibitive memory resources on environments such as graphics hardware. In this paper, we bypass this problem by introducing a sparse matrix representation for unstructured grids which not only reduces the memory storage requirements but also cuts down on the bulk of data movement from global storage to the compute units. In order to take full advantage of the proposed representation, we augment ordinary matrix multiplication by means of action maps, local maps which encode the desired interaction between grid vertices. In this way, geometric computations and topological modifications translate into concise linear algebra operations. In our algorithmic formulation, we capitalize on the nature of sparse matrix-vector multiplication which allows avoiding explicit transpose computation and storage. Furthermore, we develop an efficient vectorization to the demanding assembly process of standard graph and finite element matrices.

This paper revisits variational multi-view stereo and identifies two issues pertaining to matching and view merging: i) regions with low visibility and relatively high depth variation are only resolved by the sole regularizer contribution. This often induces wrong matches which tend to bleed into neigh boring regions, and more importantly distort nearby features. ii) small matching errors can lead to overlapping surface layers which cannot be easily addressed by standard outlier removal techniques. In both scenarios, we rely on the analysis of the distortion of spatial and planar maps in order to improve the quality of the reconstruction. At the matching level, an anisotropic diffusion driven by spatial grid distortion is proposed to steer grid lines away from those problematic regions. At the merging level, advantage is taken of Lambert's cosine law to favor contributions from image areas where the cosine angle between the surface normal and the line of sight is maximal. Tests on standard benchmarks suggest a good blend between computational efficiency, ease of implementation, and reconstruction quality.

Among all forms of sculpture, bas-relief is arguably the closest to painting. Although inherently a two dimensional sculpture, a bas-relief suggests a visual spatial extension of the scene in depth through the combination of composition, perspective, and shading. There have been significant results on digital bas-relief generation but many of the existing techniques may wash out high level surface detail during the compression process. The primary goal of this work is to address the problem of fine features by tailoring a filtering technique that achieves good compression without compromising the quality of surface details. As a secondary application we explore the generation of artistic relief which mimic cubism in painting and we show how it could be used for generating Picasso like portraits.

Quadrangular remeshing of triangulated surfaces has received an increasing attention in recent years. A particularly elegant approach is the extraction of quads from the streamlines of a harmonic field. While the construction of such fields is by now a standard technique in geometry processing, enforcing design constraints is still not fully investigated. This work presents a technique for handling directional constraints by directly controlling the gradient of the field. In this way, line constraints sketched by the user or automatically obtained as feature lines can be fulfilled efficiently. Furthermore, we show the potential of quasi-harmonic fields as a flexible tool for controlling the behavior of the field over the surface. Treating the surface as an inhomogeneous domain we can endow specific surface regions with field attraction/repulsion properties.

We present an efficient approach for solving the spherical parameterization problem. The essence of the approach is to look for a solution in the curvilinear coordinate system without requiring the additional spherical constraints usually needed in Cartesian formulations. This setup allows us to take full advantage of some existing techniques originally developed for planar parameterization. Our results substantiate the efficiency of the method and confirm its robustness. Meshes of non-trivial geometry with tens of thousands of triangles are processed in a few seconds, always yielding bijective maps. This computational achievement bridges a so far wide gap in performance between spherical and planar parameterization

In the last decade, surface mesh parameterization has emerged as a standard technique in computer graphics. The ever increasing need for processing large and highly detailed data sets fosters the development of efficient parameterization techniques that can capture the geometry of the input meshes and produce low distortion planar maps. We present a set of novel techniques allowing for low distortion parameterization. In particular, we address one of the major shortcomings of linear methods by allowing the parametric representation to evolve freely on the plane without any fixed boundary vertices. Our method consists of several simple steps, each solving a linear problem. Our results exhibit a fair balance between high-quality and computational efficiency.

We introduce new linear operators for surface parameterization. Given an initial mapping from the parametric plane onto a surface mesh, we establish a secondary map from the plane onto itself that mimics the initial one. The resulting low-distortion parameterization is smooth as it stems from solving a quasi-harmonic equation. Our parameterization method is robust and independent of (the quality of) the initial map.

We present an interactive method for applying deformations to a surface mesh while preserving its global shape and local properties. Two surface editing scenarios are discussed, which conceptually differ in the specification of deformations: Either interpolation constraints are imposed explicitly, e.g., by dragging a subset of vertices, or, deformation of a reference surface is mimicked. The contribution of this paper is a novel approach for interpolation of local deformations over the manifold and for efficiently establishing correspondence to a reference surface from only few pairs of markers. As a general tool for both scenarios, a harmonic field is constructed to guide the interpolation of constraints and to find correspondence required for deformation transfer. We show that our approach fits nicely in a unified mathematical framework, where the same type of linear operator is applied in all phases, and how this approach can be used to create an intuitive and interactive editing tool. Figure 1: A simple edit: The visualized harmonic field is used as guidance for bending the cactus (left). Here, the field is defined by one source (red) at the tip of the left arm and one sink (blue) below the middle of the trunk. The result is shown in the center image. Notice the different propagation of the rotation compared to the edit on the right, where three sources on all arms were chosen (without picture). 1.

We introduce a new technique for estimating the curvature tensor of a triangular mesh. The input of the algorithm is only a single triangle equipped with its (exact or estimated) vertex normals. This way we get a smooth junction of the curvature tensor inside each triangle of the mesh. We show that the error of the new method is comparable with the error of a cubic fitting approach if the incorporated normals are estimated. If the exact normals of the underlying surface are available at the vertices, the error drops significantly. We demonstrate the applicability of the new estimation at a rather complex data set.

We propose a polygonal decomposition of the 1-ring neighborhood of a quadrilateral mesh, which is suitable for the study of the Catmull-Clark subdivision scheme. The initial configuration consists of 2n planar 2n-gons and under the Catmull-Clark subdivision they transform into 4n planar n-gons coming in pairs of coplanar polygons and quadruples of parallel polygons. We calculate the eigenvalues and eigenvectors of the transformations of these configurations showing their relation with the tangent plane and the curvature properties of the subdivision surface. Using direct computations on circulant-block matrices, we show how the same eigenvalues can be analytically deduced from the subdivision matrix.