Frequency-based decomposition of time series data is used in many visualization applications. Most of these decomposition methods (such as Fourier transform or singular spectrum analysis) only provide interaction via pre- and post-processing, but no means to influence the core algorithm. A method that also belongs to this class is Dynamic Mode Decomposition (DMD), a spectral decomposition method that extracts spatio-temporal patterns from data. In this paper, we incorporate frequency-based constraints into DMD for an adaptive decomposition that leads to user-controllable visualizations, allowing analysts to include their knowledge into the process. To accomplish this, we derive an equivalent reformulation of DMD that implicitly provides access to the eigenvalues (and therefore to the frequencies) identified by DMD. By utilizing a constrained minimization problem customized to DMD, we can guarantee the existence of desired frequencies by minimal changes to DMD. We complement this core approach by additional techniques for constrained DMD to facilitate explorative visualization and investigation of time series data. With several examples, we demonstrate the usefulness of constrained DMD and compare it to conventional frequency-based decomposition methods.

Dynamic Mode Decomposition (DMD) is a data-driven and model-free decomposition technique. It is suitable for revealing spatio-temporal features of both numerically and experimentally acquired data. Conceptually, DMD performs a low-dimensional spectral decomposition of the data into the following components: the modes, called DMD modes, encode the spatial contribution of the decomposition, whereas the DMD amplitudes specify their impact. Each associated eigenvalue, referred to as DMD eigenvalue, characterizes the frequency and growth rate of the DMD mode. In this paper, we demonstrate how the components of DMD can be utilized to obtain temporal and spatial information from time-dependent flow fields. We begin with the theoretical background of DMD and its application to unsteady flow. Next, we examine the conventional process with DMD mathematically and put it in relationship to the discrete Fourier transform. Our analysis shows that the current use of DMD components has several drawbacks. To resolve these problems we adjust the components and provide new and meaningful insights into the decomposition: we show that our improved components capture the spatio-temporal patterns of the flow better. Moreover, we remove redundancies in the decomposition and clarify the interplay between components, allowing users to understand the impact of components. These new representations, which respect the spatio-temporal character of DMD, enable two clustering methods that segment the flow into physically relevant sections and can therefore be used for the selection of DMD components. With a number of typical examples, we demonstrate that the combination of these techniques allows new insights with DMD for unsteady flow.

Abstract We present a method to simulate fluid flow on evolving surfaces, e.g., an oil film on a water surface. Given an animated surface (e.g., extracted from a particle‐based fluid simulation) in three‐dimensional space, we add a second simulation on this base animation. In general, we solve a partial differential equation (PDE) on a level set surface obtained from the animated input surface. The properties of the input surface are transferred to a sparse volume data structure that is then used for the simulation. We introduce one‐way coupling strategies from input properties to our simulation and we add conservation of mass and momentum to existing methods that solve a PDE in a narrow‐band using the Closest Point Method. In this way, we efficiently compute high‐resolution 2D simulations on coarse input surfaces. Our approach helps visual effects creators easily integrate a workflow to simulate material flow on evolving surfaces into their existing production pipeline.

We present a novel technique to correct errors introduced by the discretization of a fluid body when animating it with smoothed particle hydrodynamics (SPH). Our approach is based on the Shepard correction, which reduces the interpolation errors from irregularly spaced data. With Shepard correction, the smoothing kernel function is normalized using the weighted sum of the kernel function values in the neighborhood. To compute the correction factor, densities of neighboring particles are needed, which themselves are computed with the uncorrected kernel. This results in an inconsistent formulation and an error-prone correction of the kernel. As a consequence, the density computation may be inaccurate, thus the pressure forces are erroneous and may cause instabilities in the simulation process. We present a consistent formulation by using the corrected densities to compute the exact kernel correction factor and, thereby, increase the accuracy of the simulation. Employing our method, a smooth density distribution is achieved, i.e., the noise in the density field is reduced by orders of magnitude. To show that our method is independent of the SPH variant, we evaluate our technique on weakly compressible SPH and on divergence-free SPH. Incorporating the corrected density into the correction process, the problem cannot be stated explicitly anymore. We propose an efficient and easy-to-implement algorithm to solve the implicit problem by applying the power method. Additionally, we demonstrate how our model can be applied to improve the density distribution on rigid bodies when using a well-known rigid-fluid coupling approach.

Smoothed particle hydrodynamics (SPH) is a popular mesh-free, particle-based fluid simulation approach for a wide range of applications. There are several numerical variants of SPH along with a variety of models for aspects such as boundary conditions, compressibility or incompressibility, and surface tension. Different combinations of these models lead to varying effects that occur during simulation, and their analysis is a critical challenge for fluid mechanics. In this paper, we address this challenge by presenting a visual debugging application for simulations, which allows users to evaluate the properties of the models and to detect possible computational errors. Our multi-view application uses a combination of interactive 3D visualization of the particles and non-spatial visualizations from the field of information visualization, namely scatter plots and parallel coordinates plots. Our visual debugging environment thus enables a quantitative analysis of the multidimensional simulation attributes, including internal and physical properties contributing to the simulation process. All views support brushing and linking, that is, selections of interesting value ranges in the plots are directly visible in the 3D view and, conversely, the selection of particles in the 3D view highlights the corresponding data points in the plots. Since typical SPH simulations come with large numbers of data points, we employ stochastic subsampling to reduce visual clutter in the non-spatial views and accelerate the rendering speed. We discuss four real-world use cases for visual debugging of fluid simulations that showcase how our visual debugging environment is instrumental for identify code errors and increases the understanding of the simulation models. We also show how the combination of coupled views can reveal internal details, thus serving to improve simulation results.

High dynamic range imaging is currently being introduced to television, cinema and computer games. While it has been found that a fixed encoding for high dynamic range imagery needs at least 11 to 12 bits of tonal resolution, current mainstream image transmission interfaces, codecs and file formats are limited to 10 bits. To be able to use current generation imaging pipelines, this paper presents a baseband quantization scheme that exploits content characteristics to reduce the needed tonal resolution per image. The method is of low computational complexity and provides robust performance on a wide range of content types in different viewing environments and applications.

Abstract We present a robust and efficient method for the two‐way coupling between particle‐based fluid simulations and infinitesimally thin solids represented by triangular meshes. Our approach is based on a hybrid method that combines a repulsion force approach with a continuous intersection handling to guarantee that no penetration occurs. Moreover, boundary conditions for the tangential component of the fluid's velocity are implemented to model the different slip conditions. The proposed method is particularly useful for dynamic surfaces, like cloth and thin shells. In addition, we demonstrate how standard fluid surface reconstruction algorithms can be modified to prevent the calculated surface from intersecting close objects. For both the two‐way coupling and the surface reconstruction, we take into account that the fluid can wet the cloth. We have implemented our approach for the bidirectional interaction between liquid simulations based on Smoothed Particle Hydrodynamics (SPH) and standard mesh‐based cloth simulation systems.

Abstract For the rendering of multiple scattering effects in participating media, methods based on the diffusion approximation are an extremely efficient alternative to Monte Carlo path tracing. However, in sufficiently transparent regions, classical diffusion approximation suffers from non‐physical radiative fluxes which leads to a poor match to correct light transport. In particular, this prevents the application of classical diffusion approximation to heterogeneous media, where opaque material is embedded within transparent regions. To address this limitation, we introduce flux‐limited diffusion, a technique from the astrophysics domain. This method provides a better approximation to light transport than classical diffusion approximation, particularly when applied to heterogeneous media, and hence broadens the applicability of diffusion‐based techniques. We provide an algorithm for flux‐limited diffusion, which is validated using the transport theory for a point light source in an infinite homogeneous medium. We further demonstrate that our implementation of flux‐limited diffusion produces more accurate renderings of multiple scattering in various heterogeneous datasets than classical diffusion approximation, by comparing both methods to ground truth renderings obtained via volumetric path tracing.

The development of methods and tools for the generation of visually appealing motion sequences using prerecorded motion capture data has become an important research area in computer animation. In particular, data-driven approaches have been used for reconstructing high-dimensional motion sequences from low-dimensional control signals. In this article, we contribute to this strand of research by introducing a novel framework for generating full-body animations controlled by only four 3D accelerometers that are attached to the extremities of a human actor. Our approach relies on a knowledge base that consists of a large number of motion clips obtained from marker-based motion capturing. Based on the sparse accelerometer input a cross-domain retrieval procedure is applied to build up a lazy neighborhood graph in an online fashion. This graph structure points to suitable motion fragments in the knowledge base, which are then used in the reconstruction step. Supported by a kd-tree index structure, our procedure scales to even large datasets consisting of millions of frames. Our combined approach allows for reconstructing visually plausible continuous motion streams, even in the presence of moderate tempo variations which may not be directly reflected by the given knowledge base.

Particle systems have been widely employed to simulate deformable objects, most prominently virtual textiles. We introduce particle systems that adaptively generate new particles wherever this is required to model collisions correctly. When coarse meshes collide with an object they are automatically refined. Therefore this model allows physically accurate simulations that require a much smaller number of particles than regular particle systems. This allows for fast simulations with coarse meshes.

In this paper we will present a system to use three dimensional computer graphics in garment design. This system is capable to visualize the "real", i.e. the physically correct, appearance of a knitted fabric. A fast visualization of a physically correct micro‐structure garment is of crucial importance in textile industry, since it enables fast and less expensive product development. This system may be either used in the design of new products or teaching the art of knitted fabrics. We use in our system directly the produced machine‐code of the design system for knitting machines. A physical model, a particle system, is used to calculate the dynamics of the micro‐structure of the knitted garment.