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H 2 Bayesian inference and mathematical imaging - Part 3: probability and convex optimisation

Auteurs : Pereyra, Marcelo (Auteur de la Conférence)
CIRM (Editeur )

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imaging inverse problems Bayesian statistical inference uncertainty quantification hypothesis testing maximum-a-posteriori estimation regularisation parameters questions from the audience

Résumé : 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.

Codes MSC :
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.)

Ressources complémentaires :
https://www.cirm-math.fr/ProgWeebly/2019/Renc1993/PresPereyra3.pdf

    Informations sur la Vidéo

    Langue : Anglais
    Date de publication : 21/01/2019
    Date de captation : 10/01/2019
    Collection : Research schools
    Format : MP4 (.mp4) - HD
    Durée : 01:47:34
    Domaine : Probability & Statistics ; Control Theory & Optimization ; Numerical Analysis & Scientific Computing
    Audience : Chercheurs ; Etudiants Science Cycle 2 ; Doctorants , Post - Doctorants
    Download : https://videos.cirm-math.fr/2019-01-10_Pereyra_3.mp4

Informations sur la rencontre

Nom de la rencontre : IHP winter school: The mathematics of imaging / Ecole d'hiver IHP : Les mathématiques de l'image
Organisateurs de la rencontre : Fadili, Jalal ; Melor, Clothilde ; Schönlieb, Carola-Bibiane ; Wirth, Benedikt
Dates : 07/01/2019 - 11/01/2019
Année de la rencontre : 2019
URL Congrès : https://conferences.cirm-math.fr/1993.html

Citation Data

DOI : 10.24350/CIRM.V.19486103
Cite this video as: Pereyra, Marcelo (2019). Bayesian inference and mathematical imaging - Part 3: probability and convex optimisation. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19486103
URI : http://dx.doi.org/10.24350/CIRM.V.19486103

Voir aussi

Bibliographie

  • 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



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