Français
Bayesian inference and mathematical imaging - Part 4: mixture, random fields and hierarchical models
Multi angle
Authors : Pereyra, Marcelo (Author of the conference)
CIRM (Publisher )
49N45 - Inverse problems in calculus of variations
62C10 - Bayesian problems; characterization of Bayes procedures
62F15 - Bayesian inference
65C40 - Computational Markov chains (numerical analysis)
65C60 - Computational problems in statistics
65J22 - Inverse problems (numerical methods in abstract spaces)
68U10 - Image processing (computing aspects)
94A08 - Image processing (compression, reconstruction, etc.)
Additional resources :
https://www.cirm-math.fr/ProgWeebly/2019/Renc1993/PresPereyra4.pdf
CIRM (Publisher )
Loading the player...
|
Abstract :
This course presents an overview of modern Bayesian strategies for solving imaging inverse problems. We will start by introducing the Bayesian statistical decision theory framework underpinning Bayesian analysis, and then explore efficient numerical methods for performing Bayesian computation in large-scale settings. We will pay special attention to high-dimensional imaging models that are log-concave w.r.t. the unknown image, related to so-called “convex imaging problems”. This will provide an opportunity to establish connections with the convex optimisation and machine learning approaches to imaging, and to discuss some of their relative strengths and drawbacks. Examples of topics covered in the course include: efficient stochastic simulation and optimisation numerical methods that tightly combine proximal convex optimisation with Markov chain Monte Carlo techniques; strategies for estimating unknown model parameters and performing model selection, methods for calculating Bayesian confidence intervals for images and performing uncertainty quantification analyses; and new theory regarding the role of convexity in maximum-a-posteriori and minimum-mean-square-error estimation. The theory, methods, and algorithms are illustrated with a range of mathematical imaging experiments.
MSC Codes : 49N45 - Inverse problems in calculus of variations
62C10 - Bayesian problems; characterization of Bayes procedures
62F15 - Bayesian inference
65C40 - Computational Markov chains (numerical analysis)
65C60 - Computational problems in statistics
65J22 - Inverse problems (numerical methods in abstract spaces)
68U10 - Image processing (computing aspects)
94A08 - Image processing (compression, reconstruction, etc.)
Additional resources :
https://www.cirm-math.fr/ProgWeebly/2019/Renc1993/PresPereyra4.pdf
|
Information on the Video
Film maker : Hennenfent, GuillaumeLanguage : English Available date : 21/01/2019 Conference Date : 10/01/2019 Subseries : Research School arXiv category : Statistics Theory ; Optimization and Control ; Numerical Analysis Mathematical Area(s) : Probability & Statistics ; Control Theory & Optimization ; Numerical Analysis & Scientific Computing Format : MP4 (.mp4) - HD Video Time : 00:49:29 Targeted Audience : Researchers ; Graduate Students Download : https://videos.cirm-math.fr/2019-01-10_Peyrera_4.mp4 
|
Information on the Event
Event Title : IHP winter school: The mathematics of imaging / Ecole d'hiver IHP : Les mathématiques de l'imageEvent Organizers : Fadili, Jalal ; Melor, Clothilde ; Schönlieb, Carola-Bibiane ; Wirth, Benedikt Dates : 07/01/2019 - 11/01/2019 Event Year : 2019 Event URL : https://conferences.cirm-math.fr/1993.html
Citation Data
DOI : 10.24350/CIRM.V.19486903Cite this video as: Pereyra, Marcelo (2019). Bayesian inference and mathematical imaging - Part 4: mixture, random fields and hierarchical models. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19486903 URI : http://dx.doi.org/10.24350/CIRM.V.19486903 |
See Also
- [Multi angle] Bayesian inference and mathematical imaging - Part 2: Markov chain Monte Carlo / Author of the conference Pereyra, Marcelo.
- [Multi angle] Bayesian inference and mathematical imaging - Part 1: Bayesian analysis and decision theory / Author of the conference Pereyra, Marcelo.
- [Post-edited] Bayesian inference and mathematical imaging - Part 3: probability and convex optimisation / Author of the conference Pereyra, Marcelo.
Bibliography
- Ay, N., & Amari, S.-I. (2015). A novel approach to canonical divergences within information geometry. Entropy, 17(12), 8111-8129 - https://dx.doi.org/10.3390/e17127866
- Cai, X., Pereyra, M., & McEwen, J.D. (2018). Uncertainty quantification for radio interferometric imaging: II. MAP estimation. Monthly Notices of the Royal Astronomical Society, 480(3), 4170-4182 - https://doi.org/10.1093/mnras/sty2015
- Chambolle, A., & Pock, T. (2016). An introduction to continuous optimization for imaging. Acta Numerica, 25, 161-319 - https://doi.org/10.1017/S096249291600009X
- Deledalle, C.-A., Vaiter, S., Fadili, J., & Peyré, G. (2014). Stein Unbiased GrAdient estimator of the Risk (SUGAR) for multiple parameter selection. SIAM Journal on Imaging Sciences, 7(4), 2448-2487 - https://doi.org/10.1137/140968045
- Fernandez-Vidal, A. & Pereyra, M. (2018). Maximum likelihood estimation of regularisation parameters. In 2018 25th IEEE International Conference on Image Processing: Proceedings (pp. 1742-1746) - https://doi.org/10.1109/ICIP.2018.8451795
- Green, P.J., Latuszynski, K., Pereyra, M., & Robert, C.P. (2015). Bayesian computation: a summary of the current state, and samples backwards and forwards. Statistics and Computing, 25(4), 835-862 - http://dx.doi.org/10.1007/s11222-015-9574-5
- Pereyra, M. (2016). Maximum-a-posteriori estimation with bayesian confidence regions. SIAM Journal on Imaging Sciences, 10(1), 285–302 - https://doi.org/10.1137/16M1071249
- Pereyra, M. (2016). Revisiting maximum-a-posteriori estimation in log-concave models: from differential geometry to decision theory. 〈arXiv:1612.06149〉 - https://arxiv.org/abs/1612.06149
- Pereyra, M., Bioucas-Dias, J., & Figueiredo, M. (2015). Maximum-a-posteriori estimation with unknown regularisation parameters. In 2015 23rd European Signal Processing Conference (EUSIPCO): Proceedings (pp. 230-234) - https://doi.org/10.1109/EUSIPCO.2015.7362379
- Repetti, A., Pereyra, M., & Wiaux, Y. (2018). Scalable Bayesian uncertainty quantification in imaging inverse problems via convex optimisation.〈arXiv:1803.00889〉 - https://arxiv.org/abs/1803.00889
- Robert, C.P. (2001). The Bayesian Choice. From decision-theoretic foundations to computational implementation. 2nd ed., 1st paperback ed. New York, NY: Springer - http://dx.doi.org/10.1007/0-387-71599-1
- Zhu, L., Zhang, W., Elnatan, D., & Huang, B. (2012). Faster STORM using compressed sensing. Nature Methods, 9(7), 721-723 - https://doi.org/10.1038/nmeth.1978
