Explicit Diffusion of Gaussian Mixture Model Based Image Priors

Martin Zach; Thomas Pock; Erich Kobler; Antonin Chambolle

doi:10.1007/978-3-031-31975-4_1

Explicit Diffusion of Gaussian Mixture Model Based Image Priors

Martin Zach^*, Thomas Pock, Erich Kobler, Antonin Chambolle

^*Corresponding author for this work

Institute of Visual Computing (7100)

Research output: Chapter in Book/Report/Conference proceeding › Conference paper › peer-review

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 (∂t-Δ1)fY(·,t)=0, fY(·,0)=fX. With a focus on image processing, we propose a product/fields-of-experts model with Gaussian mixture experts that admits an analytic expression for fY(·,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.

Original language	English
Title of host publication	Scale Space and Variational Methods in Computer Vision - 9th International Conference, SSVM 2023, Proceedings
Editors	Luca Calatroni, Marco Donatelli, Serena Morigi, Marco Prato, Matteo Santacesaria
Publisher	Springer Science and Business Media Deutschland GmbH
Pages	3-15
Number of pages	13
ISBN (Print)	9783031319747
DOIs	https://doi.org/10.1007/978-3-031-31975-4_1
Publication status	Published - 2023
Event	9th International Conference on Scale Space and Variational Methods in Computer Vision: SSVM 2023 - Santa Margherita di Pula, Italy Duration: 21 May 2023 → 25 May 2023

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	14009 LNCS
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

Conference

Conference	9th International Conference on Scale Space and Variational Methods in Computer Vision
Abbreviated title	SSVM 2023
Country/Territory	Italy
City	Santa Margherita di Pula
Period	21/05/23 → 25/05/23

Keywords

Blind Denoising
Diffusion Models
Empirical Bayes
Gaussian Mixture

ASJC Scopus subject areas

Theoretical Computer Science
General Computer Science

Access to Document

10.1007/978-3-031-31975-4_1

Cite this

Zach, M., Pock, T., Kobler, E., & Chambolle, A. (2023). Explicit Diffusion of Gaussian Mixture Model Based Image Priors. In L. Calatroni, M. Donatelli, S. Morigi, M. Prato, & M. Santacesaria (Eds.), Scale Space and Variational Methods in Computer Vision - 9th International Conference, SSVM 2023, Proceedings (pp. 3-15). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 14009 LNCS). Springer Science and Business Media Deutschland GmbH. https://doi.org/10.1007/978-3-031-31975-4_1

Explicit Diffusion of Gaussian Mixture Model Based Image Priors. / Zach, Martin; Pock, Thomas; Kobler, Erich et al.
Scale Space and Variational Methods in Computer Vision - 9th International Conference, SSVM 2023, Proceedings. ed. / Luca Calatroni; Marco Donatelli; Serena Morigi; Marco Prato; Matteo Santacesaria. Springer Science and Business Media Deutschland GmbH, 2023. p. 3-15 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 14009 LNCS).

Research output: Chapter in Book/Report/Conference proceeding › Conference paper › peer-review

Zach, M, Pock, T, Kobler, E & Chambolle, A 2023, Explicit Diffusion of Gaussian Mixture Model Based Image Priors. in L Calatroni, M Donatelli, S Morigi, M Prato & M Santacesaria (eds), Scale Space and Variational Methods in Computer Vision - 9th International Conference, SSVM 2023, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 14009 LNCS, Springer Science and Business Media Deutschland GmbH, pp. 3-15, 9th International Conference on Scale Space and Variational Methods in Computer Vision, Santa Margherita di Pula, Italy, 21/05/23. https://doi.org/10.1007/978-3-031-31975-4_1

Zach M, Pock T, Kobler E, Chambolle A. Explicit Diffusion of Gaussian Mixture Model Based Image Priors. In Calatroni L, Donatelli M, Morigi S, Prato M, Santacesaria M, editors, Scale Space and Variational Methods in Computer Vision - 9th International Conference, SSVM 2023, Proceedings. Springer Science and Business Media Deutschland GmbH. 2023. p. 3-15. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). doi: 10.1007/978-3-031-31975-4_1

Zach, Martin ; Pock, Thomas ; Kobler, Erich et al. / Explicit Diffusion of Gaussian Mixture Model Based Image Priors. Scale Space and Variational Methods in Computer Vision - 9th International Conference, SSVM 2023, Proceedings. editor / Luca Calatroni ; Marco Donatelli ; Serena Morigi ; Marco Prato ; Matteo Santacesaria. Springer Science and Business Media Deutschland GmbH, 2023. pp. 3-15 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

@inproceedings{66867e50a01c4944b1f6b0f09caf06a4,

title = "Explicit Diffusion of Gaussian Mixture Model Based Image Priors",

abstract = "In this work we tackle the problem of estimating the density fX of a random variable X by successive smoothing, such that the smoothed random variable Y fulfills (∂t-Δ1)fY(·,t)=0, fY(·,0)=fX. With a focus on image processing, we propose a product/fields-of-experts model with Gaussian mixture experts that admits an analytic expression for fY(·,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.",

keywords = "Blind Denoising, Diffusion Models, Empirical Bayes, Gaussian Mixture",

author = "Martin Zach and Thomas Pock and Erich Kobler and Antonin Chambolle",

note = "Publisher Copyright: {\textcopyright} 2023, The Author(s), under exclusive license to Springer Nature Switzerland AG.; 9th International Conference on Scale Space and Variational Methods in Computer Vision : SSVM 2023, SSVM 2023 ; Conference date: 21-05-2023 Through 25-05-2023",

year = "2023",

doi = "10.1007/978-3-031-31975-4_1",

language = "English",

isbn = "9783031319747",

series = "Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)",

publisher = "Springer Science and Business Media Deutschland GmbH",

pages = "3--15",

editor = "Luca Calatroni and Marco Donatelli and Serena Morigi and Marco Prato and Matteo Santacesaria",

booktitle = "Scale Space and Variational Methods in Computer Vision - 9th International Conference, SSVM 2023, Proceedings",

address = "Germany",

}

TY - GEN

T1 - Explicit Diffusion of Gaussian Mixture Model Based Image Priors

AU - Zach, Martin

AU - Pock, Thomas

AU - Kobler, Erich

AU - Chambolle, Antonin

PY - 2023

Y1 - 2023

N2 - In this work we tackle the problem of estimating the density fX of a random variable X by successive smoothing, such that the smoothed random variable Y fulfills (∂t-Δ1)fY(·,t)=0, fY(·,0)=fX. With a focus on image processing, we propose a product/fields-of-experts model with Gaussian mixture experts that admits an analytic expression for fY(·,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.

AB - In this work we tackle the problem of estimating the density fX of a random variable X by successive smoothing, such that the smoothed random variable Y fulfills (∂t-Δ1)fY(·,t)=0, fY(·,0)=fX. With a focus on image processing, we propose a product/fields-of-experts model with Gaussian mixture experts that admits an analytic expression for fY(·,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.

KW - Blind Denoising

KW - Diffusion Models

KW - Empirical Bayes

KW - Gaussian Mixture

UR - http://www.scopus.com/inward/record.url?scp=85161183670&partnerID=8YFLogxK

U2 - 10.1007/978-3-031-31975-4_1

DO - 10.1007/978-3-031-31975-4_1

M3 - Conference paper

AN - SCOPUS:85161183670

SN - 9783031319747

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 3

EP - 15

BT - Scale Space and Variational Methods in Computer Vision - 9th International Conference, SSVM 2023, Proceedings

A2 - Calatroni, Luca

A2 - Donatelli, Marco

A2 - Morigi, Serena

A2 - Prato, Marco

A2 - Santacesaria, Matteo

PB - Springer Science and Business Media Deutschland GmbH

T2 - 9th International Conference on Scale Space and Variational Methods in Computer Vision

Y2 - 21 May 2023 through 25 May 2023

ER -