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

TL;DR

This paper introduces the Signals as Parametric Curves (SPC) model for blind source separation, proposing geometrical objective functions and FT-PCA that leverage signal derivatives, offering alternatives to traditional mutual information-based ICA methods.

Contribution

The paper extends the ISPS model to 1D signals, developing new geometrical objective functions and FT-PCA for BSS that do not rely on independence assumptions or stochastic process models.

Findings

Geometrical objective functions outperform traditional MI-based methods in synthetic tests.

FT-PCA effectively separates sources without independence assumptions.

Proposed methods are simple, powerful, and applicable across various signal types.

Abstract

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.