Product of Gaussian Mixture Diffusion Models
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866914636382601216 |
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| author | Zach, Martin Kobler, Erich Chambolle, Antonin Pock, Thomas |
| author_facet | Zach, Martin Kobler, Erich Chambolle, Antonin Pock, Thomas |
| contents | 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 the diffusion partial differential equation $ (\partial_t - Δ_1)f_Y(\,\cdot\,, t) = 0 $ with initial condition $ f_Y(\,\cdot\,, 0) = f_X $. We propose a product-of-experts-type model utilizing Gaussian mixture experts and study configurations that admit an analytic expression for $ f_Y (\,\cdot\,, t) $. In particular, with a focus on image processing, we derive conditions for models acting on filter-, wavelet-, and shearlet responses. Our construction naturally allows the model to be trained simultaneously over the entire diffusion horizon using empirical Bayes. We show numerical results for image denoising where our models are competitive while being tractable, interpretable, and having only a small number of learnable parameters. As a byproduct, our models can be used for reliable noise level estimation, allowing blind denoising of images corrupted by heteroscedastic noise. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2310_12653 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Product of Gaussian Mixture Diffusion Models Zach, Martin Kobler, Erich Chambolle, Antonin Pock, Thomas Image and Video Processing 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 the diffusion partial differential equation $ (\partial_t - Δ_1)f_Y(\,\cdot\,, t) = 0 $ with initial condition $ f_Y(\,\cdot\,, 0) = f_X $. We propose a product-of-experts-type model utilizing Gaussian mixture experts and study configurations that admit an analytic expression for $ f_Y (\,\cdot\,, t) $. In particular, with a focus on image processing, we derive conditions for models acting on filter-, wavelet-, and shearlet responses. Our construction naturally allows the model to be trained simultaneously over the entire diffusion horizon using empirical Bayes. We show numerical results for image denoising where our models are competitive while being tractable, interpretable, and having only a small number of learnable parameters. As a byproduct, our models can be used for reliable noise level estimation, allowing blind denoising of images corrupted by heteroscedastic noise. |
| title | Product of Gaussian Mixture Diffusion Models |
| topic | Image and Video Processing |
| url | https://arxiv.org/abs/2310.12653 |