Scalable Bayesian Uncertainty Quantification in Imaging Inverse Problems via Convex Optimization

Research output: Contribution to journalArticle

Abstract

We propose a Bayesian uncertainty quantification method for large-scale imaging inverse problems. Our method applies to all Bayesian models that are log-concave, where maximum a posteriori (MAP) estimation is a convex optimization problem. The method is a framework to analyze the confidence in specific structures observed in MAP estimates (e.g., lesions in medical imaging, celestial sources in astronomical imaging), to enable using them as evidence to inform decisions and conclusions. Precisely, following Bayesian decision theory, we seek to assert the structures under scrutiny by performing a Bayesian hypothesis test that proceeds as follows: first, it postulates that the structures are not present in the true image, and then seeks to use the data and prior knowledge to reject this null hypothesis with high probability. Computing such tests for imaging problems is generally very difficult because of the high dimensionality involved. A main feature of this work is to leverage probability concentration phenomena and the underlying convex geometry to formulate the Bayesian hypothesis test as a convex problem, which we then efficiently solve by using scalable optimization algorithms. This allows scaling to high-resolution and high-sensitivity imaging problems that are computationally unaffordable for other Bayesian computation approaches. We illustrate our methodology, dubbed BUQO (Bayesian Uncertainty Quantification by Optimization), on a range of challenging Fourier imaging problems arising in astronomy and medicine. MATLAB code for the proposed uncertainty quantification method is available on GitHub.
Original languageEnglish
Pages (from-to)87-118
Number of pages32
JournalSIAM Journal on Imaging Sciences
Volume12
Issue number1
Early online date22 Jan 2019
DOIs
Publication statusPublished - 2019

Fingerprint

Convex optimization
Inverse problems
Imaging techniques
Decision theory
Astronomy
Medical imaging
MATLAB
Medicine
Uncertainty
Geometry

Keywords

  • Bayesian inference
  • Convex optimization
  • Hypothesis testing
  • Image processing
  • Inverse problems
  • Uncertainty quantification

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

@article{94bbe1702c5b483b86f56a9646e434b3,
title = "Scalable Bayesian Uncertainty Quantification in Imaging Inverse Problems via Convex Optimization",
abstract = "We propose a Bayesian uncertainty quantification method for large-scale imaging inverse problems. Our method applies to all Bayesian models that are log-concave, where maximum a posteriori (MAP) estimation is a convex optimization problem. The method is a framework to analyze the confidence in specific structures observed in MAP estimates (e.g., lesions in medical imaging, celestial sources in astronomical imaging), to enable using them as evidence to inform decisions and conclusions. Precisely, following Bayesian decision theory, we seek to assert the structures under scrutiny by performing a Bayesian hypothesis test that proceeds as follows: first, it postulates that the structures are not present in the true image, and then seeks to use the data and prior knowledge to reject this null hypothesis with high probability. Computing such tests for imaging problems is generally very difficult because of the high dimensionality involved. A main feature of this work is to leverage probability concentration phenomena and the underlying convex geometry to formulate the Bayesian hypothesis test as a convex problem, which we then efficiently solve by using scalable optimization algorithms. This allows scaling to high-resolution and high-sensitivity imaging problems that are computationally unaffordable for other Bayesian computation approaches. We illustrate our methodology, dubbed BUQO (Bayesian Uncertainty Quantification by Optimization), on a range of challenging Fourier imaging problems arising in astronomy and medicine. MATLAB code for the proposed uncertainty quantification method is available on GitHub.",
keywords = "Bayesian inference, Convex optimization, Hypothesis testing, Image processing, Inverse problems, Uncertainty quantification",
author = "Audrey Repetti and Marcelo Pereyra and Yves Wiaux",
year = "2019",
doi = "10.1137/18M1173629",
language = "English",
volume = "12",
pages = "87--118",
journal = "SIAM Journal on Imaging Sciences",
issn = "1936-4954",
publisher = "Society for Industrial and Applied Mathematics Publications",
number = "1",

}

TY - JOUR

T1 - Scalable Bayesian Uncertainty Quantification in Imaging Inverse Problems via Convex Optimization

AU - Repetti, Audrey

AU - Pereyra, Marcelo

AU - Wiaux, Yves

PY - 2019

Y1 - 2019

N2 - We propose a Bayesian uncertainty quantification method for large-scale imaging inverse problems. Our method applies to all Bayesian models that are log-concave, where maximum a posteriori (MAP) estimation is a convex optimization problem. The method is a framework to analyze the confidence in specific structures observed in MAP estimates (e.g., lesions in medical imaging, celestial sources in astronomical imaging), to enable using them as evidence to inform decisions and conclusions. Precisely, following Bayesian decision theory, we seek to assert the structures under scrutiny by performing a Bayesian hypothesis test that proceeds as follows: first, it postulates that the structures are not present in the true image, and then seeks to use the data and prior knowledge to reject this null hypothesis with high probability. Computing such tests for imaging problems is generally very difficult because of the high dimensionality involved. A main feature of this work is to leverage probability concentration phenomena and the underlying convex geometry to formulate the Bayesian hypothesis test as a convex problem, which we then efficiently solve by using scalable optimization algorithms. This allows scaling to high-resolution and high-sensitivity imaging problems that are computationally unaffordable for other Bayesian computation approaches. We illustrate our methodology, dubbed BUQO (Bayesian Uncertainty Quantification by Optimization), on a range of challenging Fourier imaging problems arising in astronomy and medicine. MATLAB code for the proposed uncertainty quantification method is available on GitHub.

AB - We propose a Bayesian uncertainty quantification method for large-scale imaging inverse problems. Our method applies to all Bayesian models that are log-concave, where maximum a posteriori (MAP) estimation is a convex optimization problem. The method is a framework to analyze the confidence in specific structures observed in MAP estimates (e.g., lesions in medical imaging, celestial sources in astronomical imaging), to enable using them as evidence to inform decisions and conclusions. Precisely, following Bayesian decision theory, we seek to assert the structures under scrutiny by performing a Bayesian hypothesis test that proceeds as follows: first, it postulates that the structures are not present in the true image, and then seeks to use the data and prior knowledge to reject this null hypothesis with high probability. Computing such tests for imaging problems is generally very difficult because of the high dimensionality involved. A main feature of this work is to leverage probability concentration phenomena and the underlying convex geometry to formulate the Bayesian hypothesis test as a convex problem, which we then efficiently solve by using scalable optimization algorithms. This allows scaling to high-resolution and high-sensitivity imaging problems that are computationally unaffordable for other Bayesian computation approaches. We illustrate our methodology, dubbed BUQO (Bayesian Uncertainty Quantification by Optimization), on a range of challenging Fourier imaging problems arising in astronomy and medicine. MATLAB code for the proposed uncertainty quantification method is available on GitHub.

KW - Bayesian inference

KW - Convex optimization

KW - Hypothesis testing

KW - Image processing

KW - Inverse problems

KW - Uncertainty quantification

UR - http://www.scopus.com/inward/record.url?scp=85064217473&partnerID=8YFLogxK

U2 - 10.1137/18M1173629

DO - 10.1137/18M1173629

M3 - Article

VL - 12

SP - 87

EP - 118

JO - SIAM Journal on Imaging Sciences

JF - SIAM Journal on Imaging Sciences

SN - 1936-4954

IS - 1

ER -