The problem of computing topological distance between two scalar fields based on Reeb graphs or contour trees has been studied and applied successfully to various problems in topological shape matching, data analysis, and visualization. However, generalizing such results for computing distance measures between two multi-fields based on their Reeb spaces is still in its infancy. Towards this, in the current article we propose a technique to compute an effective distance measure between two multi-fields by computing a novel multi-dimensional persistence diagram (MDPD) corresponding to each of the (quantized) Reeb spaces. First, we construct a multi-dimensional Reeb graph (MDRG), which is a hierarchical decomposition of the Reeb space into a collection of Reeb graphs. The MDPD corresponding to each MDRG is then computed based on the persistence diagrams of the component Reeb graphs of the MDRG. Our distance measure extends the Wasserstein distance between two persistence diagrams of Reeb graphs to MDPDs of MDRGs. We prove that the proposed measure is a pseudo-metric and satisfies a stability property. Effectiveness of the proposed distance measure has been demonstrated in (i) shape retrieval contest data - SHREC 2010 and (ii) Pt-CO bond detection data from computational chemistry. Experimental results show that the proposed distance measure based on the Reeb spaces has more discriminating power in clustering the shapes and detecting the formation of a stable Pt-CO bond as compared to the similar measures between Reeb graphs.

Searching similarity between a pair of shapes or data is an important problem in data analysis and visualization. The problem of computing similarity measures using scalar topology has been studied extensively and proven useful in the shape and data matching. Even though multi-field or multivariate (consists of multiple scalar fields) topology reveals richer topological features, research on building tools for computing similarity measures using multi-field topology is still in its infancy. In the current article, we propose a novel similarity measure between two piecewise-linear multi-fields based on their multi-resolution Reeb spaces - a newly developed data-structure that captures the topology of a multi-field. Overall, our method consists of two steps: (i) building a multi-resolution Reeb space corresponding to each of the multi-fields and (ii) proposing a similarity measure between two multi-resolution Reeb spaces by computing a list of topologically consistent matching pairs (of nodes) and the similarity between them. We demonstrate the effectiveness of the proposed similarity measure in detecting topological features from real time-varying multi-field data in two application domains - one from computational physics and one from computational chemistry.

Even though the Hartley-entropy-based contrast function guarantees an unmixing local minimum, the reported nonsmooth optimization techniques that minimize this nondifferentiable function encounter computational bottlenecks. Toward this, Powell's derivative-free optimization method has been extended to a Riemannian manifold, namely, oblique manifold, for the recovery of quasi-correlated sources by minimizing this contrast function. The proposed scheme has been demonstrated to converge faster than the related algorithms in the literature, besides the impressive source separation results in simulations involving synthetic sources having finite-support distributions and correlated images.

Abstract Scientific visualization has many effective methods for examining and exploring scalar and vector fields, but rather fewer for bivariate fields. We report the first general purpose approach for the interactive extraction of geometric separating surfaces in bivariate fields. This method is based on fiber surfaces: surfaces constructed from sets of fibers, the multivariate analogues of isolines. We show simple methods for fiber surface definition and extraction. In particular, we show a simple and efficient fiber surface extraction algorithm based on Marching Cubes. We also show how to construct fiber surfaces interactively with geometric primitives in the range of the function. We then extend this to build user interfaces that generate parameterized families of fiber surfaces with respect to arbitrary polygons. In the special case of isovalue‐gradient plots, fiber surfaces capture features geometrically for quantitative analysis that have previously only been analysed visually and qualitatively using multi‐dimensional transfer functions in volume rendering. We also demonstrate fiber surface extraction on a variety of bivariate data.

It is seemingly paradoxical to the classical definition of the independent component analysis (ICA), that in reality, the true sources are often not strictly uncorrelated. With this in mind, this letter concerns a framework to extract quasi-uncorrelated sources with finite supports by optimizing a range-based contrast function under unit-norm constraints (to handle the inherent scaling indeterminacy of ICA) but without orthogonality constraints. Albeit the appealing contrast properties of the range-based function (e.g., the absence of mixing local optima), the function is not differentiable everywhere. Unfortunately, there is a dearth of literature on derivative-free optimizers that effectively handle such a nonsmooth yet promising contrast function. This is the compelling reason for the design of a nonsmooth optimization algorithm on a manifold of matrices having unit-norm columns with the following objectives: to ascertain convergence to a Clarke stationary point of the contrast function and adhere to the necessary unit-norm constraints more naturally. The proposed nonsmooth optimization algorithm crucially relies on the design and analysis of an extension of the mesh adaptive direct search (MADS) method to handle locally Lipschitz objective functions defined on the sphere. The applicability of the algorithm in the ICA domain is demonstrated with simulations involving natural, face, aerial, and texture images.

The Morse-Smale complex is an important tool for global topological analysis in various problems of computational geometry and topology. Algorithms for Morse-Smale complexes have been presented in case of piecewise linear manifolds. However, previous research in this field does not provide certified methods in the case of smooth functions. In the current paper we use interval arithmetic to compute a topologically correct approximation of Morse-Smale complex of smooth functions of two variables. The algorithm can also compute geometrically close Morse-Smale complex.