Multidimensional Iterative Filtering method for the decomposition of high-dimensional non-stationary signals

Iterative Filtering (IF) is an alternative technique to the Empirical Mode Decomposition (EMD) algorithm for the decomposition of non-stationary and non-linear signals. Recently in [1] IF has been proved to be convergent for any $L^2$ signal and its stability has been also showed through examples. Furthermore in [1] the so called Fokker-Planck (FP) filters have been introduced. They are smooth at every point and have compact supports. Based on those results, in this paper we introduce the Multidimensional Iterative Filtering (MIF) technique for the decomposition and time-frequency analysis of non-stationary high-dimensional signals. And we present the extension of FP filters to higher dimensions. We illustrate the promising performance of MIF algorithm, equipped with high-dimensional FP filters, when applied to the decomposition of 2D signals. [1] A. Cicone, J. Liu, and H. Zhou, Adaptive local iterative filtering for signal decomposition and instantaneous frequency analysis, arXiv:1411.6051, 2014

A Weak Galerkin Finite Element Method for A Type of Fourth Order Problem Arising From Fluorescence Tomography

In this paper, a new and efficient numerical algorithm by using weak Galerkin (WG) finite element methods is proposed for a type of fourth order problem arising from fluorescence tomography(FT). Fluorescence tomography is an emerging, in vivo non-invasive 3-D imaging technique which reconstructs images that characterize the distribution of molecules that are tagged by fluorophores. Weak second order elliptic operator and its discrete version are introduced for a class of discontinuous functions defined on a finite element partition of the domain consisting of general polygons or polyhedra. An error estimate of optimal order is derived in an $H^2$-equivalent norm for the WG finite element solutions. Error estimates in the usual $L^2$ norm are established, yielding optimal order of convergence for all the WG finite element algorithms except the one corresponding to the lowest order (i.e., piecewise quadratic elements). Some numerical experiments are presented to illustrate the efficiency and accuracy of the numerical scheme.Comment: 27 pages,6 figures, 2 tables. arXiv admin note: substantial text overlap with arXiv:1309.5560; substantial text overlap with arXiv:1303.0927 by other author

Numerical Analysis for Iterative Filtering with New Efficient Implementations Based on FFT

Real life signals are in general non--stationary and non--linear. The development of methods able to extract their hidden features in a fast and reliable way is of high importance in many research fields. In this work we tackle the problem of further analyzing the convergence of the Iterative Filtering method both in a continuous and a discrete setting in order to provide a comprehensive analysis of its behavior. Based on these results we provide new ideas for efficient implementations of Iterative Filtering algorithm which are based on Fast Fourier Transform (FFT), and the reduction of the original iterative algorithm to a direct method

Double Descent and Intermittent Color Diffusion for Global Optimization and landscape exploration

In this work, we present a method to explore the landscape of a smooth potential in the search of global minimizers,combining a double-descent technique and a basin-escaping technique based on intermittent colored diffusion. Numerical results illustrate the performance of the method

Multi-robot motion planning via optimal transport theory

In this work we establish a simple yet effective strategy, based on optimal transport theory, for enabling a group of robots to accomplish complex tasks, such as shape formation and assembly. We demonstrate the feasibility of this approach and rigorously prove collision avoidance and convergence properties of the proposed algorithms

Hyperspectral Chemical Plume Detection Algorithms Based On Multidimensional Iterative Filtering Decomposition

Chemicals released in the air can be extremely dangerous for human beings and the environment. Hyperspectral images can be used to identify chemical plumes, however the task can be extremely challenging. Assuming we know a priori that some chemical plume, with a known frequency spectrum, has been photographed using a hyperspectral sensor, we can use standard techniques like the so called matched filter or adaptive cosine estimator, plus a properly chosen threshold value, to identify the position of the chemical plume. However, due to noise and sensors fault, the accurate identification of chemical pixels is not easy even in this apparently simple situation. In this paper we present a post-processing tool that, in a completely adaptive and data driven fashion, allows to improve the performance of any classification methods in identifying the boundaries of a plume. This is done using the Multidimensional Iterative Filtering (MIF) algorithm (arXiv:1411.6051, arXiv:1507.07173), which is a non-stationary signal decomposition method like the pioneering Empirical Mode Decomposition (EMD) method. Moreover, based on the MIF technique, we propose also a pre-processing method that allows to decorrelate and mean-center a hyperspectral dataset. The Cosine Similarity measure, which often fails in practice, appears to become a successful and outperforming classifier when equipped with such pre-processing method. We show some examples of the proposed methods when applied to real life problems

A discrete Schrodinger equation via optimal transport on graphs

In 1966, Edward Nelson presented an interesting derivation of the Schrodinger equation using Brownian motion. Recently, this derivation is linked to the theory of optimal transport, which shows that the Schrodinger equation is a Hamiltonian system on the probability density manifold equipped with the Wasserstein metric. In this paper, we consider similar matters on a finite graph. By using discrete optimal transport and its corresponding Nelson's approach, we derive a discrete Schrodinger equation on a finite graph. The proposed system is quite different from the commonly referred discretized Schrodinger equations. It is a system of nonlinear ordinary differential equations (ODEs) with many desirable properties. Several numerical examples are presented to illustrate the properties

Wasserstein Hamiltonian flows

We establish kinetic Hamiltonian flows in density space embedded with the $L^2$-Wasserstein metric tensor. We derive the Euler-Lagrange equation in density space, which introduces the associated Hamiltonian flows. We demonstrate that many classical equations, such as Vlasov equation, Schr{\"o}dinger equation and Schr{\"o}dinger bridge problem, can be rewritten as the formalism of Hamiltonian flows in density space

Optimal Sensor Positioning (OSP); A Probability Perspective Study

We propose a method to optimally position a sensor system, which consists of multiple sensors, each has limited range and viewing angle, and they may fail with a certain failure rate. The goal is to find the optimal locations as well as the viewing directions of all the sensors and achieve the maximal surveillance of the known environment. We setup the problem using the level set framework. Both the environment and the viewing range of the sensors are represented by level set functions. Then we solve a system of ordinary differential equations (ODEs) to find the optimal viewing directions and locations of all sensors together. Furthermore, we use the intermittent diffusion, which converts the ODEs into stochastic differential equations (SDEs), to find the global maximum of the total surveillance area. The numerical examples include various failure rates of sensors, different rate of importance of surveillance region, and 3-D setups. They show the effectiveness of the proposed method

Equilibrium selection via Optimal transport

We propose a new dynamics for equilibrium selection of finite player discrete strategy games. The dynamics is motivated by optimal transportation, and models individual players' myopicity, greedy and uncertainty when making decisions. The stationary measure of the dynamics provides each pure Nash equilibrium a probability by which it is ranked. For potential games, its dynamical properties are characterized by entropy and Fisher information.Comment: Game theory; Optimal transport; Gradient flow; Gibbs measure; Entropy; Fisher informatio