Statistical Component Separation for Targeted Signal Recovery in Noisy Mixtures
Bruno Régaldo-Saint Blancard, Michael Eickenberg
TL;DR
The paper tackles recovering statistical descriptors $\phi(x_0)$ from noisy mixtures $y = x_0 + \epsilon_0$ when samples of the noise are available, rather than full signal recovery, by optimizing $\mathcal{L}(x) = \mathbb{E}_{\epsilon}[ \lVert \phi(x+\epsilon) - \phi(y) \rVert^2_2 ]$. It first derives global minimizers for simple representations (linear, quadratic, and power-spectrum), showing that linear $\phi$ reduces to mean subtraction while certain nonlinear/$\phi$ choices can recover $\phi(x_0)$. The authors then implement a vanilla optimization for image denoising using two representations—Wavelet Phase Harmonics (WPH) and ConvNet-based descriptors—and introduce a diffusive, stepwise variant that decomposes Gaussian noise into smaller-variance components, with experiments on dust, large-scale structure, and ImageNet data. Results indicate WPH-based statistics improve descriptor recovery and achieve competitive PSNR in non-Gaussian settings, while ConvNet-based descriptors are less effective for regular denoising; the diffusive approach provides additional gains and illuminates optimization dynamics. These methods offer a robust, descriptor-focused alternative to full signal reconstruction in noisy scientific and imaging contexts, with code and data available for public use.
Abstract
Separating signals from an additive mixture may be an unnecessarily hard problem when one is only interested in specific properties of a given signal. In this work, we tackle simpler "statistical component separation" problems that focus on recovering a predefined set of statistical descriptors of a target signal from a noisy mixture. Assuming access to samples of the noise process, we investigate a method devised to match the statistics of the solution candidate corrupted by noise samples with those of the observed mixture. We first analyze the behavior of this method using simple examples with analytically tractable calculations. Then, we apply it in an image denoising context employing 1) wavelet-based descriptors, 2) ConvNet-based descriptors on astrophysics and ImageNet data. In the case of 1), we show that our method better recovers the descriptors of the target data than a standard denoising method in most situations. Additionally, despite not constructed for this purpose, it performs surprisingly well in terms of peak signal-to-noise ratio on full signal reconstruction. In comparison, representation 2) appears less suitable for image denoising. Finally, we extend this method by introducing a diffusive stepwise algorithm which gives a new perspective to the initial method and leads to promising results for image denoising under specific circumstances.