Lagrangian theory provides a rich set of tools for analyzing non-local, long-term motion information in computer vision applications. Based on this theory, we present a specialized Lagrangian technique for the automated detection of violent scenes in video footage. We present a novel feature using Lagrangian direction fields that is based on a spatio-temporal model and uses appearance, background motion compensation, and long-term motion information. To ensure appropriate spatial and temporal feature scales, we apply an extended bag-of-words procedure in a late-fusion manner as a classification scheme on a per-video basis. We demonstrate that the temporal scale, captured by the Lagrangian integration time parameter, is crucial for violence detection and show how it correlates to the spatial scale of characteristic events in the scene. The proposed system is validated on multiple public benchmarks and non-public, real-world data from the London Metropolitan Police. Our experiments confirm that the inclusion of Lagrangian measures is a valuable cue for automated violence detection and increases the classification performance considerably compared with the state-of-the-art methods.

Abstract Traditionally, Lagrangian fields such as finite‐time Lyapunov exponents (FTLE) are precomputed on a discrete grid and are ray casted afterwards. This, however, introduces both grid discretization errors and sampling errors during ray marching. In this work, we apply a progressive, view‐dependent Monte Carlo‐based approach for the visualization of such Lagrangian fields in time‐dependent flows. Our approach avoids grid discretization and ray marching errors completely, is consistent, and has a low memory consumption. The system provides noisy previews that converge over time to an accurate high‐quality visualization. Compared to traditional approaches, the proposed system avoids explicitly predefined fieldline seeding structures, and uses a Monte Carlo sampling strategy named Woodcock tracking to distribute samples along the view ray. An acceleration of this sampling strategy requires local upper bounds for the FTLE values, which we progressively acquire during the rendering. Our approach is tailored for high‐quality visualizations of complex FTLE fields and is guaranteed to faithfully represent detailed ridge surface structures as indicators for Lagrangian coherent structures (LCS). We demonstrate the effectiveness of our approach by using a set of analytic test cases and real‐world numerical simulations.

Abstract Lagrangian coherent structures (LCSs) have become a widespread and powerful method to describe dynamic motion patterns in time‐dependent flow fields. The standard way to extract LCS is to compute height ridges in the finite‐time Lyapunov exponent field. In this work, we present an alternative method to approximate Lagrangian features for 2D unsteady flow fields that achieve subgrid accuracy without additional particle sampling. We obtain this by a geometric reconstruction of the flow map using additional material constraints for the available samples. In comparison to the standard method, this allows for a more accurate global approximation of LCS on sparse grids and for long integration intervals. The proposed algorithm works directly on a set of given particle trajectories and without additional flow map derivatives. We demonstrate its application for a set of computational fluid dynamic examples, as well as trajectories acquired by Lagrangian methods, and discuss its benefits and limitations.

Understanding the hemodynamics of blood flow in vascular pathologies such as intracranial aneurysms is essential for both their diagnosis and treatment. Computational fluid dynamics (CFD) simulations of blood flow based on patient-individual data are performed to better understand aneurysm initiation and progression and more recently, for predicting treatment success. In virtual stenting, a flow-diverting mesh tube (stent) is modeled inside the reconstructed vasculature and integrated in the simulation. We focus on steady-state simulation and the resulting complex multiparameter data. The blood flow pattern captured therein is assumed to be related to the success of stenting. It is often visualized by a dense and cluttered set of streamlines.We present a fully automatic approach for reducing visual clutter and exposing characteristic flow structures by clustering streamlines and computing cluster representatives. While individual clustering techniques have been applied before to streamlines in 3D flow fields, we contribute a general quantitative and a domain-specific qualitative evaluation of three state-of-the-art techniques. We show that clustering based on streamline geometry as well as on domain-specific streamline attributes contributes to comparing and evaluating different virtual stenting strategies. With our work, we aim at supporting CFD engineers and interventional neuroradiologists.

Abstract Recent research in flow visualization is focusing on the analysis of time‐dependent, but mass‐less particles. However, in many application scenarios, the mass of particles – and their resulting inertia – is essential in understanding fluid mechanics. This includes critical processes, such as dust particles interacting with aircraft (e.g., brown‐or white‐out effects) and particle separation based on density variation. In this paper, we contribute a generalized description of mass‐dependent particle trajectories and apply existing unsteady flow visualization methods to the mass‐dependent case. This comprises the extension of common concepts, i.e., path lines, streak lines, and time lines. Furthermore, we introduce a new class of integral curves, called mass lines that effectively visualizes mass separation and captures mass‐related features in unsteady flow fields that are inaccessible using traditional methods. We demonstrate the applicability of our method, using a number of real‐world and artificial data sets, in which mass is a crucial parameter. In particular, we focus on the analysis of brown‐out conditions, introduced by a helicopter in forward flight close to the ground.

The Finite Time Lyapunov Exponent (FTLE) has become a widespread tool for analyzing unsteady flow behavior. For its computation, several numerical methods have been introduced, which provide trade-offs between performance and accuracy. In order to decide which methods and parameter settings are suitable for a particular application, an evaluation of the different FTLE methods is necessary. We propose a general benchmark for FTLE computation, which consists of a number of 2D time-dependent flow fields and error measures. Evaluating the accuracy of a numerically computed FTLE field requires a ground truth, which is not available for realistic flow data sets, since such fields can generally not be described in a closed form. To overcome this, we introduce approaches to create non-trivial vector fields with a closed-form formulation of the FTLE field. Using this, we introduce a set of benchmark flow data sets that resemble relevant geometric aspects of Lagrangian structures, but have an analytic solution for FTLE. Based on this ground truth, we perform a comparative evaluation of three standard FTLE concepts. We suggest error measures based on the variance of both, the fields and the extracted ridge structures.

The availability of dense motion information in computer vision domain allows for the effective application of Lagrangian techniques that have their origin in fluid flow analysis and dynamical systems theory. A well established technique that has been proven to be useful in image-based crowd analysis are Finite Time Lyapunov Exponents (FTLE). Based on this, we present a method to detect people carrying object and describe a methodology how to apply established flow field methods onto the problem of describing individuals. Further, we reinterpret Lagrangian features in relation to the underlying motion process and show their applicability towards the appearance modeling of pedestrians. This definition allows to increase performance of state-of-the-art methods and is shown to be robust under varying parameter settings and different optical flow extraction approaches.

Human action recognition requires the description of complex motion patterns in image sequences. In general, these patterns span varying temporal scales. In this context, Lagrangian methods have proven to be valuable for crowd analysis tasks such as crowd segmentation. In this paper, we show that, besides their potential in describing large scale motion patterns, Lagrangian methods are also well suited to model complex individual human activities over variable time intervals. We use Finite Time Lyapunov Exponents and time-normalized arc length measures in a linear SVM classification scheme. We evaluated our method on the Weizmann and KTH datasets. The results demonstrate that our approach is promising and that human action recognition performance is improved by fusing Lagrangian measures.

The extraction of motion patterns from image sequences based on the optical flow methodology is an important and timely topic among visual multi media applications. In this work we will present a novel framework that combines the optical flow methodology from image processing with methods developed for the Lagrangian analysis of time-dependent vector fields. The Lagrangian approach has been proven to be a valuable and powerful tool to capture the complex dynamic motion behavior within unsteady vector fields. To come up with a compact and applicable framework, this paper will provide concepts on how to compute trajectory-based Lagrangian measures in series of optical flow fields, a set of basic measures to capture the essence of the motion behavior within the image, and a compact hierarchical, feature-based description of the resulting motion features. The resulting framework will bee shown to be suitable for an automated image analysis as well as compact visual analysis of image sequences in its spatio-temporal context. We show its applicability for the task of motion feature description and extraction on different temporal scales, crowd motion analysis, and automated detection of abnormal events within video sequences.

Abstract Vector fields are a common concept for the representation of many different kinds of flow phenomena in science and engineering. Methods based on vector field topology are known for their convenience for visualizing and analysing steady flows, but a counterpart for unsteady flows is still missing. However, a lot of good and relevant work aiming at such a solution is available. We give an overview of previous research leading towards topology‐based and topology‐inspired visualization of unsteady flow, pointing out the different approaches and methodologies involved as well as their relation to each other, taking classical (i.e. steady) vector field topology as our starting point. Particularly, we focus on Lagrangian methods, space–time domain approaches, local methods and stochastic and multifield approaches. Furthermore, we illustrate our review with practical examples for the different approaches.