Abstract
In this work we tackle the problem of estimating the density \( f_X \) of a random variable \( X \) by successive smoothing, such that the smoothed random variable \( Y \) fulfills \( (\partial _t - \varDelta _1)f_Y(\,\cdot \,, t) = 0 \), \( f_Y(\,\cdot \,, 0) = f_X \). With a focus on image processing, we propose a product/fields-of-experts model with Gaussian mixture experts that admits an analytic expression for \(f_Y (\,\cdot \,, t)\) under an orthogonality constraint on the filters. This construction naturally allows the model to be trained simultaneously over the entire diffusion horizon using empirical Bayes. We show preliminary results on image denoising where our model leads to competitive results while being tractable, interpretable, and having only a small number of learnable parameters. As a byproduct, our model can be used for reliable noise estimation, allowing blind denoising of images corrupted by heteroscedastic noise.
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Notes
- 1.
For notational convenience, throughout this article we do not make a distinction between the distribution and density of a random variable.
- 2.
Without any reference to samples \( x_i \sim f_X \), an equivalent statement may be that \( f_X \) is (close to) zero almost everywhere (in the layman—not measure-theoretic—sense).
- 3.
For simplicity, we discard the normalization constant \( Z \), which is independent of \( t \).
- 4.
For visualization purposes, we normalized the negative-log density to have a minimum of zero over \( t \): \( l_\theta (x, t) = -\log \tilde{f}_\theta (x, t) - (\max _t \log \tilde{f}_\theta (x, t)) \).
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Zach, M., Pock, T., Kobler, E., Chambolle, A. (2023). Explicit Diffusion of Gaussian Mixture Model Based Image Priors. In: Calatroni, L., Donatelli, M., Morigi, S., Prato, M., Santacesaria, M. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2023. Lecture Notes in Computer Science, vol 14009. Springer, Cham. https://doi.org/10.1007/978-3-031-31975-4_1
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