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arXiv2018

Signals as Parametric Curves: Application to Independent Component Analysis and Blind Source Separation

Images Stacks as Parametric Surfaces (ISPS) is a powerful model that was originally proposed for image registration. Being closely related to mutual information (MI) - the most classic similarity measure for image registration, ISPS works well across different categories of registration problems. The Signals as Parametric Curves (SPC) model is derived from ISPS extended to 1-dimensional signals. Blind Source Separation (BSS) is a classic problem in signal processing, where Independent Component Analysis (ICA) based approaches are popular and effective. Since MI plays an important role in ICA, based on the close relationship with MI, we apply SPC model to BSS in this paper, and propose a group of geometrical objective functions that are simple yet powerful, and serve as replacements of original MI-based objective functions. Motivated by the geometrical objective functions, we also propose a second-order-statistics approach, FT-PCA. Both geometrical objective functions and FT-PCA consider signals as functions instead of stochastic processes, make use of derivative information of signals, and do not rely on the independence assumption. In this paper, we discuss the reasonability of the assumptions of geometrical objective functions and FT-PCA, and show their effectiveness by synthetic experiments, comparing with other previous classic approaches.

eess.SP↗

arXiv2015

A fast eikonal equation solver using the Schrodinger wave equation

We use a Schrödinger wave equation formalism to solve the eikonal equation. In our framework, a solution to the eikonal equation is obtained in the limit as Planck's constant $\hbar$ (treated as a free parameter) tends to zero of the solution to the corresponding linear Schrödinger equation. The Schrödinger equation corresponding to the eikonal turns out to be a \emph{generalized, screened Poisson equation}. Despite being linear, it does not have a closed-form solution for arbitrary forcing functions. We present two different techniques to solve the screened Poisson equation. In the first approach we use a standard perturbation analysis approach to derive a new algorithm which is guaranteed to converge provided the forcing function is bounded and positive. The perturbation technique requires a sequence of discrete convolutions which can be performed in $O(N\log N)$ using the Fast Fourier Transform (FFT) where $N$ is the number of grid points. In the second method we discretize the linear Laplacian operator by the finite difference method leading to a sparse linear system of equations which can be solved using the plethora of sparse solvers. The eikonal solution is recovered from the exponent of the resultant scalar field. Our approach eliminates the need to explicitly construct viscosity solutions as customary with direct solutions to the eikonal. Since the linear equation is computed for a small but non-zero $\hbar$, the obtained solution is an approximation. Though our solution framework is applicable to the general class of eikonal problems, we detail specifics for the popular vision applications of shape-from-shading, vessel segmentation, and path planning.

math.NA↗