Geodesic Transport Barriers in Jupiter's Atmosphere: A Video-Based Analysis

Jupiter's zonal jets and Great Red Spot are well known from still images. Yet the planet's atmosphere is highly unsteady, which suggests that the actual material transport barriers delineating its main features should be time-dependent. Rare video footages of Jupiter's clouds provide an opportunity to verify this expectation from optically reconstructed velocity fields. Available videos, however, provide short-time and temporally aperiodic velocity fields that defy classical dynamical systems analyses focused on asymptotic features. To this end, we use here the recent theory of geodesic transport barriers to uncover finite-time mixing barriers in the wind field extracted from a video captured by NASA's Cassini space mission. More broadly, the approach described here provides a systematic and frame-invariant way to extract dynamic coherent structures from time-resolved remote observations of unsteady continua

Objective Eulerian Coherent Structures

We define objective Eulerian Coherent Structures (OECSs) in two-dimensional, non-autonomous dynamical systems as instantaneously most influential material curves. Specifically, OECSs are stationary curves of the averaged instantaneous material stretching-rate or material shearing-rate functionals. From these objective (frame-invariant) variational principles, we obtain explicit differential equations for hyperbolic, elliptic and parabolic OECSs. As illustration, we compute OECSs in an unsteady ocean velocity data set. In comparison to structures suggested by other common Eulerian diagnostic tools, we find OECSs to be the correct short-term cores of observed trajectory deformation patterns

A Novel Space-Time Representation on the Positive Semidefinite Con for Facial Expression Recognition

In this paper, we study the problem of facial expression recognition using a novel space-time geometric representation. We describe the temporal evolution of facial landmarks as parametrized trajectories on the Riemannian manifold of positive semidefinite matrices of fixed-rank. Our representation has the advantage to bring naturally a second desirable quantity when comparing shapes -- the spatial covariance -- in addition to the conventional affine-shape representation. We derive then geometric and computational tools for rate-invariant analysis and adaptive re-sampling of trajectories, grounding on the Riemannian geometry of the manifold. Specifically, our approach involves three steps: 1) facial landmarks are first mapped into the Riemannian manifold of positive semidefinite matrices of rank 2, to build time-parameterized trajectories; 2) a temporal alignment is performed on the trajectories, providing a geometry-aware (dis-)similarity measure between them; 3) finally, pairwise proximity function SVM (ppfSVM) is used to classify them, incorporating the latter (dis-)similarity measure into the kernel function. We show the effectiveness of the proposed approach on four publicly available benchmarks (CK+, MMI, Oulu-CASIA, and AFEW). The results of the proposed approach are comparable to or better than the state-of-the-art methods when involving only facial landmarks.Comment: To be appeared at ICCV 201

A Nonlinear Dimensionality Reduction Framework Using Smooth Geodesics

Existing dimensionality reduction methods are adept at revealing hidden underlying manifolds arising from high-dimensional data and thereby producing a low-dimensional representation. However, the smoothness of the manifolds produced by classic techniques over sparse and noisy data is not guaranteed. In fact, the embedding generated using such data may distort the geometry of the manifold and thereby produce an unfaithful embedding. Herein, we propose a framework for nonlinear dimensionality reduction that generates a manifold in terms of smooth geodesics that is designed to treat problems in which manifold measurements are either sparse or corrupted by noise. Our method generates a network structure for given high-dimensional data using a nearest neighbors search and then produces piecewise linear shortest paths that are defined as geodesics. Then, we fit points in each geodesic by a smoothing spline to emphasize the smoothness. The robustness of this approach for sparse and noisy datasets is demonstrated by the implementation of the method on synthetic and real-world datasets.Comment: 13 pages, 7 figures, submitted to Pattern Recognitio

A Critical Comparison of Lagrangian Methods for Coherent Structure Detection

We review and test twelve different approaches to the detection of finite-time coherent material structures in two-dimensional, temporally aperiodic flows. We consider both mathematical methods and diagnostic scalar fields, comparing their performance on three benchmark examples: the quasiperiodically forced Bickley jet, a two-dimensional turbulence simulation, and an observational wind velocity field from Jupiter's atmosphere. A close inspection of the results reveals that the various methods often produce very different predictions for coherent structures, once they are evaluated beyond heuristic visual assessment. As we find by passive advection of the coherent set candidates, false positives and negatives can be produced even by some of the mathematically justified methods due to the ineffectiveness of their underlying coherence principles in certain flow configurations. We summarize the inferred strengths and weaknesses of each method, and make general recommendations for minimal self-consistency requirements that any Lagrangian coherence detection technique should satisfy

Visualizing Time-Varying Particle Flows with Diffusion Geometry

The tasks of identifying separation structures and clusters in flow data are fundamental to flow visualization. Significant work has been devoted to these tasks in flow represented by vector fields, but there are unique challenges in addressing these tasks for time-varying particle data. The unstructured nature of particle data, nonuniform and sparse sampling, and the inability to access arbitrary particles in space-time make it difficult to define separation and clustering for particle data. We observe that weaker notions of separation and clustering through continuous measures of these structures are meaningful when coupled with user exploration. We achieve this goal by defining a measure of particle similarity between pairs of particles. More specifically, separation occurs when spatially-localized particles are dissimilar, while clustering is characterized by sets of particles that are similar to one another. To be robust to imperfections in sampling we use diffusion geometry to compute particle similarity. Diffusion geometry is parameterized by a scale that allows a user to explore separation and clustering in a continuous manner. We illustrate the benefits of our technique on a variety of 2D and 3D flow datasets, from particles integrated in fluid simulations based on time-varying vector fields, to particle-based simulations in astrophysics.Comment: 14 pages, 16 figures, under revie

A landmark-based algorithm for automatic pattern recognition and abnormality detection

We study a class of mathematical and statistical algorithms with the aim of establishing a computer-based framework for fast and reliable automatic abnormality detection on landmark represented image templates. Under this framework, we apply a landmark-based algorithm for finding a group average as an estimator that is said to best represent the common features of the group in study. This algorithm extracts information of momentum at each landmark through the process of template matching. If ever converges, the proposed algorithm produces a local coordinate system for each member of the observing group, in terms of the residual momentum. We use a Bayesian approach on the collected residual momentum representations for making inference. For illustration, we apply this framework to a small database of brain images for detecting structure abnormality. The brain structure changes identified by our framework are highly consistent with studies in the literature

Facial Behavior Analysis using 4D Curvature Statistics for Presentation Attack Detection

The uniqueness, complexity, and diversity of facial shapes and expressions led to success of facial biometric systems. Regardless of the accuracy of current facial recognition methods, most of them are vulnerable against the presentation of sophisticated masks. In the highly monitored application scenario at airports and banks, fraudsters probably do not wear masks. However, a deception will become more probable due to the increase of unsupervised authentication using kiosks, eGates and mobile phones in self-service. To robustly detect elastic 3D masks, one of the ultimate goals is to automatically analyze the plausibility of the facial behavior based on a sequence of 3D face scans. Most importantly, such a method would also detect all less advanced presentation attacks using static 3D masks, bent photographs with eyeholes, and replay attacks using monitors. Our proposed method achieves this goal by comparing the temporal curvature change between presentation attacks and genuine faces. For evaluation purposes, we recorded a challenging database containing replay attacks, static and elastic 3D masks using a high-quality 3D sensor. Based on the proposed representation, we found a clear separation between the low facial expressiveness of presentation attacks and the plausible behavior of genuine faces.Comment: Manuscript submitted for publication in IEEE International Conference on Automatic Face & Gesture Recognition (FG

Fitting tree model with CNN and geodesics to track vesselsand application to Ultrasound Localization Microscopy data

Segmentation of tubular structures in vascular imaging is a well studied task, although it is rare that we try to infuse knowledge of the tree-like structure of the regions to be detected. Our work focuses on detecting the important landmarks in the vascular network (via CNN performing both localization and classification of the points of interest) and representing vessels as the edges in some minimal distance tree graph. We leverage geodesic methods relevant to the detection of vessels and their geometry, making use of the space of positions and orientations so that 2D vessels can be accurately represented as trees. We build our model to carry tracking on Ultrasound Localization Microscopy (ULM) data, proposing to build a good cost function for tracking on this type of data. We also test our framework on synthetic and eye fundus data. Results show that scarcity of well annotated ULM data is an obstacle to localization of vascular landmarks but the Orientation Score built from ULM data yields good geodesics for tracking blood vessels.Comment: This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessibl

Elastic Functional Coding of Riemannian Trajectories

Visual observations of dynamic phenomena, such as human actions, are often represented as sequences of smoothly-varying features . In cases where the feature spaces can be structured as Riemannian manifolds, the corresponding representations become trajectories on manifolds. Analysis of these trajectories is challenging due to non-linearity of underlying spaces and high-dimensionality of trajectories. In vision problems, given the nature of physical systems involved, these phenomena are better characterized on a low-dimensional manifold compared to the space of Riemannian trajectories. For instance, if one does not impose physical constraints of the human body, in data involving human action analysis, the resulting representation space will have highly redundant features. Learning an effective, low-dimensional embedding for action representations will have a huge impact in the areas of search and retrieval, visualization, learning, and recognition. The difficulty lies in inherent non-linearity of the domain and temporal variability of actions that can distort any traditional metric between trajectories. To overcome these issues, we use the framework based on transported square-root velocity fields (TSRVF); this framework has several desirable properties, including a rate-invariant metric and vector space representations. We propose to learn an embedding such that each action trajectory is mapped to a single point in a low-dimensional Euclidean space, and the trajectories that differ only in temporal rates map to the same point. We utilize the TSRVF representation, and accompanying statistical summaries of Riemannian trajectories, to extend existing coding methods such as PCA, KSVD and Label Consistent KSVD to Riemannian trajectories or more generally to Riemannian functions.Comment: Under major revision at IEEE T-PAMI, 201