Learning Object-Independent Modes of Variation with Feature Flow Fields

We present a unifying framework in which "object-independent" modes of variation are learned from continuous-time data such as video sequences. These modes of variation can be used as "generators" to produce a manifold of images of a new object from a single example of that object. We develop the framework in the context of a well-known example: analyzing the modes of spatial deformations of a scene under camera movement. Our method learns a close approximation to the standard affine deformations that are expected from the geometry of the situation, and does so in a completely unsupervised (i.e. ignorant of the geometry of the situation) fashion. We stress that it is learning a "parameterization", not just the parameter values, of the data. We then demonstrate how we have used the same framework to derive a novel data-driven model of joint color change in images due to common lighting variations. The model is superior to previous models of color change in describing non-linear color changes due to lighting

Critical Point Identification In 3D Velocity Fields

Classification of flow fields involving strong vortices such as those from bluff body wakes and animal locomotion can provide important insight to their hydrodynamic behavior. Previous work has successfully classified 2D flow fields based on critical points of the velocity field and the structure of an associated weighted graph using the critical points as vertices. The present work focuses on extension of this approach to 3D flows. To this end, we have used the Gauss-Bonnet theorem to find critical points and their indices in the 3D velocity vector field, which functions similarly to the Poincare-Bendixon theorem in 2D flow fields. The approach utilizes an initial search for critical points based on local flow field direction, and the Gauss-Bonnet theorem is used to refine the location of critical points by dividing the volume integral form of the Gauss-Bonnet theorem into smaller regions. The developed method is cable of locating critical points at sub-grid level precision, which is a key factor for locating critical points and determining their associated eigenvalues on coarse grids. To verify this approach, we have applied this method on sample flow fields generated from potential flow theory and numerical methods

TROPHY: A Topologically Robust Physics-Informed Tracking Framework for Tropical Cyclones

Tropical cyclones (TCs) are among the most destructive weather systems. Realistically and efficiently detecting and tracking TCs are critical for assessing their impacts and risks. Recently, a multilevel robustness framework has been introduced to study the critical points of time-varying vector fields. The framework quantifies the robustness of critical points across varying neighborhoods. By relating the multilevel robustness with critical point tracking, the framework has demonstrated its potential in cyclone tracking. An advantage is that it identifies cyclonic features using only 2D wind vector fields, which is encouraging as most tracking algorithms require multiple dynamic and thermodynamic variables at different altitudes. A disadvantage is that the framework does not scale well computationally for datasets containing a large number of cyclones. This paper introduces a topologically robust physics-informed tracking framework (TROPHY) for TC tracking. The main idea is to integrate physical knowledge of TC to drastically improve the computational efficiency of multilevel robustness framework for large-scale climate datasets. First, during preprocessing, we propose a physics-informed feature selection strategy to filter 90% of critical points that are short-lived and have low stability, thus preserving good candidates for TC tracking. Second, during in-processing, we impose constraints during the multilevel robustness computation to focus only on physics-informed neighborhoods of TCs. We apply TROPHY to 30 years of 2D wind fields from reanalysis data in ERA5 and generate a number of TC tracks. In comparison with the observed tracks, we demonstrate that TROPHY can capture TC characteristics that are comparable to and sometimes even better than a well-validated TC tracking algorithm that requires multiple dynamic and thermodynamic scalar fields

Lifted Wasserstein Matcher for Fast and Robust Topology Tracking

This paper presents a robust and efficient method for tracking topological features in time-varying scalar data. Structures are tracked based on the optimal matching between persistence diagrams with respect to the Wasserstein metric. This fundamentally relies on solving the assignment problem, a special case of optimal transport, for all consecutive timesteps. Our approach relies on two main contributions. First, we revisit the seminal assignment algorithm by Kuhn and Munkres which we specifically adapt to the problem of matching persistence diagrams in an efficient way. Second, we propose an extension of the Wasserstein metric that significantly improves the geometrical stability of the matching of domain-embedded persistence pairs. We show that this geometrical lifting has the additional positive side-effect of improving the assignment matrix sparsity and therefore computing time. The global framework implements a coarse-grained parallelism by computing persistence diagrams and finding optimal matchings in parallel for every couple of consecutive timesteps. Critical trajectories are constructed by associating successively matched persistence pairs over time. Merging and splitting events are detected with a geometrical threshold in a post-processing stage. Extensive experiments on real-life datasets show that our matching approach is an order of magnitude faster than the seminal Munkres algorithm. Moreover, compared to a modern approximation method, our method provides competitive runtimes while yielding exact results. We demonstrate the utility of our global framework by extracting critical point trajectories from various simulated time-varying datasets and compare it to the existing methods based on associated overlaps of volumes. Robustness to noise and temporal resolution downsampling is empirically demonstrated

Meshing Deforming Spacetime for Visualization and Analysis

We introduce a novel paradigm that simplifies the visualization and analysis of data that have a spatially/temporally varying frame of reference. The primary application driver is tokamak fusion plasma, where science variables (e.g., density and temperature) are interpolated in a complex magnetic field-line-following coordinate system. We also see a similar challenge in rotational fluid mechanics, cosmology, and Lagrangian ocean analysis; such physics implies a deforming spacetime and induces high complexity in volume rendering, isosurfacing, and feature tracking, among various visualization tasks. Without loss of generality, this paper proposes an algorithm to build a simplicial complex -- a tetrahedral mesh, for the deforming 3D spacetime derived from two 2D triangular meshes representing consecutive timesteps. Without introducing new nodes, the resulting mesh fills the gap between 2D meshes with tetrahedral cells while satisfying given constraints on how nodes connect between the two input meshes. In the algorithm we first divide the spacetime into smaller partitions to reduce complexity based on the input geometries and constraints. We then independently search for a feasible tessellation of each partition taking nonconvexity into consideration. We demonstrate multiple use cases for a simplified visualization analysis scheme with a synthetic case and fusion plasma applications