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.
Our approach employs diffusion geometry, which relies on a notion of scale in constructing similarities between particle trajectories. This is related to scale-space approaches @cite_61 which analyze field-based data in multiple scales, typically to denoise or find optimal spatial scales for filtering. Flow analysis has employed scale-space techniques in various contexts, such as vortex tracking @cite_2 and detection of FTLE ridges @cite_59 . However, the construction of multi-scale distances requires different mathematical tools compared to traditional scale-space methods on fields. Furthermore, it is nontrivial to extend these techniques to particle data, as they utilize a vector field for their respective scale-space approaches.
@cite_61 @cite_59 @cite_2
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We investigate techniques for analysis and retrieval of object trajectories in two or three dimensional space. Such data usually contain a large amount of noise, that has made previously used metrics fail. Therefore, we formalize non-metric similarity functions based on the longest common subsequence (LCSS), which are very robust to noise and furthermore provide an intuitive notion of similarity between trajectories by giving more weight to similar portions of the sequences. Stretching of sequences in time is allowed, as well as global translation of the sequences in space. Efficient approximate algorithms that compute these similarity measures are also provided. We compare these new methods to the widely used Euclidean and time warping distance functions (for real and synthetic data) and show the superiority of our approach, especially in the strong presence of noise. We prove a weaker version of the triangle inequality and employ it in an indexing structure to answer nearest neighbor queries. Finally, we present experimental results that validate the accuracy and efficiency of our approach. We propose a multiscale approach to anomaly detection in images, combining spectral dimensionality reduction and a nearest-neighbor-based anomaly score. We use diffusion maps to embed the data in a low dimensional representation, which separates the anomaly from the background. The diffusion distance between points is then used to estimate the local density of each pixel in the new embedding. The diffusion map is constructed based on a subset of samples from the image and then extended to all other pixels. Due to the interpolative nature of extension methods, this may limit the ability of the diffusion map to reveal the presence of the anomaly in the data. To overcome this limitation, we propose a multiscale approach based on Gaussian pyramid representation, which drives the sampling process to ensure separability of the anomaly from the background clutter. The algorithm is successfully tested on side-scan sonar images of sea-mines. This paper describes a generalized axiomatic scale-space theory that makes it possible to derive the notions of linear scale-space, affine Gaussian scale-space and linear spatio-temporal scale-space using a similar set of assumptions (scale-space axioms). The notion of non-enhancement of local extrema is generalized from previous application over discrete and rotationally symmetric kernels to continuous and more general non-isotropic kernels over both spatial and spatio-temporal image domains. It is shown how a complete classification can be given of the linear (Gaussian) scale-space concepts that satisfy these conditions on isotropic spatial, non-isotropic spatial and spatio-temporal domains, which results in a general taxonomy of Gaussian scale-spaces for continuous image data. The resulting theory allows filter shapes to be tuned from specific context information and provides a theoretical foundation for the recently exploited mechanisms of shape adaptation and velocity adaptation, with highly useful applications in computer vision. It is also shown how time-causal spatio-temporal scale-spaces can be derived from similar assumptions. The mathematical structure of these scale-spaces is analyzed in detail concerning transformation properties over space and time, the temporal cascade structure they satisfy over time as well as properties of the resulting multi-scale spatio-temporal derivative operators. It is also shown how temporal derivatives with respect to transformed time can be defined, leading to the formulation of a novel analogue of scale normalized derivatives for time-causal scale-spaces. The kernels generated from these two types of theories have interesting relations to biological vision. We show how filter kernels generated from the Gaussian spatio-temporal scale-space as well as the time-causal spatio-temporal scale-space relate to spatio-temporal receptive field profiles registered from mammalian vision. Specifically, we show that there are close analogies to space-time separable cells in the LGN as well as to both space-time separable and non-separable cells in the striate cortex. We do also present a set of plausible models for complex cells using extended quasi-quadrature measures expressed in terms of scale normalized spatio-temporal derivatives. The theories presented as well as their relations to biological vision show that it is possible to describe a general set of Gaussian and or time-causal scale-spaces using a unified framework, which generalizes and complements previously presented scale-space formulations in this area. We introduce a volumetric space-time technique for the reconstruction of moving and deforming objects from point data. The output of our method is a four-dimensional space-time solid, made up of spatial slices, each of which is a three-dimensional solid bounded by a watertight manifold. The motion of the object is described as an incompressible flow of material through time. We optimize the flow so that the distance material moves from one time frame to the next is bounded, the density of material remains constant, and the object remains compact. This formulation overcomes deficiencies in the acquired data, such as persistent occlusions, errors, and missing frames. We demonstrate the performance of our flow-based technique by reconstructing coherent sequences of watertight models from incomplete scanner data.