Blind source separation for imaging

This work is concerned with the problem of blind source separation and its applications to imaging. We first establish a theoretical result that we stated in our previous article on imaging in diffusive environments. This result is a generalization of separability criteria found in the literature to arbitrary correlated complex-valued sources with additive noise. In a second step, we verify these separability conditions in two propagation regimes frequently encountered in imaging: the speckle regime and the random geometrical optics regime. Finally, we propose a new imaging method based on the blind source separation problem that improves on images obtained with the classical decomposition of the time reversal operator method.

💡 Research Summary

This paper addresses blind source separation (BSS) for imaging applications, extending the theoretical foundations of Independent Component Analysis (ICA) to settings that involve complex‑valued, correlated, non‑circular sources together with additive noise. The authors first derive explicit separability conditions that guarantee the critical points of the kurtosis functional—used by ICA algorithms such as FastICA and RobustICA—are close to the true columns of the inverse mixing matrix, even when the classical assumptions of independence and circularity are violated.

The key theoretical contribution is Theorem 2.1. By defining two quantities, M(s) (measuring the departure from independence and circularity of the sources) and M(n) (measuring the noise level), the theorem shows that if M(s)+M(n) is sufficiently small, the angular distance on the complex unit sphere between the global optimizer ŵ of the kurtosis (subject to a real‑part constraint) and the true direction w₀ (the normalized first column of (A⁻¹)*) scales as O(