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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.
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