Topological data analysis is becoming a popular way to study high dimensional feature spaces without any contextual clues or assumptions. This paper concerns itself with one popular topological feature, which is the number of @math dimensional holes in the dataset, also known as the Betti @math number. The persistence of the Betti numbers over various scales is encoded into a persistence diagram (PD), which indicates the birth and death times of these holes as scale varies. A common way to compare PDs is by a point-to-point matching, which is given by the @math -Wasserstein metric. However, a big drawback of this approach is the need to solve correspondence between points before computing the distance; for @math points, the complexity grows according to @math n @math . Instead, we propose to use an entirely new framework built on Riemannian geometry, that models PDs as 2D probability density functions that are represented in the square-root framework on a Hilbert Sphere. The resulting space is much more intuitive with closed form expressions for common operations. The distance metric is 1) correspondence-free and also 2) independent of the number of points in the dataset. The complexity of computing distance between PDs now grows according to @math , for a @math discretization of @math . This also enables the use of existing machinery in differential geometry towards statistical analysis of PDs such as computing the mean, geodesics, classification etc. We report competitive results with the Wasserstein metric, at a much lower computational load, indicating the favorable properties of the proposed approach.
The square-root representation for 1D probability density functions (pdfs) was proposed in @cite_4 and was used for shape classification. It has since been used in several applications, including activity recognition @cite_1 - where the square-root velocity representation is used to model the space of time warping functions. This representation extends quite easily to arbitrary high-dimensional pdfs as well.
@cite_1 @cite_4
2012048508 2098213935 2016316738 1979158214
In this paper, we cast the problem of point cloud matching as a shape matching problem by transforming each of the given point clouds into a shape representation called the Schrodinger distance transform (SDT) representation. This is achieved by solving a static Schrodinger equation instead of the corresponding static Hamilton-Jacobi equation in this setting. The SDT representation is an analytic expression and following the theoretical physics literature, can be normalized to have unit 2 norm -- making it a square-root density, which is identified with a point on a unit Hilbert sphere, whose intrinsic geometry is fully known. The Fisher-Rao metric, a natural metric for the space of densities leads to analytic expressions for the geodesic distance between points on this sphere. In this paper, we use the well known Riemannian framework never before used for point cloud matching, and present a novel matching algorithm. We pose point set matching under rigid and non-rigid transformations in this framework and solve for the transformations using standard nonlinear optimization techniques. Finally, to evaluate the performance of our algorithm -- dubbed SDTM -- we present several synthetic and real data examples along with extensive comparisons to state-of-the-art techniques. The experiments show that our algorithm outperforms state-of the-art point set registration algorithms on many quantitative metrics. In this letter, the joint probability density function (PDF) for the eigenvalues of a complex Wishart matrix and a perturbed version of it are derived. The latter version can be used to model channel estimation errors and variations over time or frequency. As an example, the joint PDF is used to calculate the transition probabilities between modulation states in an adaptive MIMO system. This leads to a Markov model for the system. We then use the model to investigate the modulation state entering rates (MSER), the average stay duration (ASD), and the effects of feedback delay on the accuracy of modulation state selection in mobile radio systems. Other applications of this PDF are also discussed. A large amount of 2D drawings have been produced in engineering fields. To reuse and share the available drawings efficiently, we propose two methods in this paper, namely 2.5D spherical harmonics transformation and 2D shape histogram, to retrieve 2D drawings by measuring their shape similarity. The first approach represents a drawing as a spherical function by transforming it from a 2D space into a 3D space. Then a fast spherical harmonics transformation is employed to get a rotation invariant descriptor. The second statistics-based approach represents the shape of a 2D drawing using a distance distribution between two randomly sampled points. To allow users to interactively emphasize certain local shapes that they are interested in, we have adopted a flexible sampling strategy by specifying a bias sampling density upon these local shapes. The two proposed methods have many valuable properties, including transform invariance, efficiency, and robustness. In addition, their insensitivity to noise allows for the user's causal input, thus supporting a freehand sketch-based retrieval user interface. Experiments show that a better performance can be achieved by combining them together using weights. The aim of this paper is to introduce a new descriptor for the spatio-temporal volume (STV). Human motion is completely represented by STV (action volume) which is constructed over successive frames by stacking human silhouettes in consecutive frames. Action volume comprehensively contains spatial and temporal information about an action. The main contribution of this paper is to propose a new affine invariant action volume descriptor based on a function of spherical harmonic coefficients. This means, it is invariant under rotation, non-uniform scaling and translation. In the 3D shape analysis literature, there have been a few attempts to use coefficients of spherical harmonics to describe a 3D shape. However, those descriptors are not affine invariant and they are only rotation invariant. In addition, the proposed approach employs a parametric form of spherical harmonics that handles genus zero surfaces regardless of whether they are stellar or not. Another contribution of this paper is the way that action volume is constructed. We applied the proposed descriptor to the KTH, Weizmann, IXMAS and Robust datasets and compared the performance of our algorithm to competing methods available in the literature. The results of our experiments show that our method has a comparable performance to the most successful and recent existing algorithms.