INVERSE PROBLEMS

Main lecturer

Mail address

Phone number

Christian Heinrich, Professor

christian.heinrich@unistra.fr, office C209,

+33 (0)3 68 85 44 88

Other instructor(s)

N/A

APOGEE code

Track - Year - Option - Semester

Coefficient = ECTS

Duration

EP013M63

Engineer - TIS DTMI - S9

Master - 2Y ID G + HCI + HealthTech-DTMI - S3

1 / 1.5 (ID G + HCI)

10.5h CM

EXAMS

Duration

Authorized documents

      If yes, which ones :

School calculator authorized

Session 1

1h30 - computer room

Yes

One handwritten A4 page

No

Session 2

1h30 - computer room

Yes

One handwritten A4 page

No

Prerequisites

Basic knowledge in signal and image processing ; algebra, calculus, probability, optimization ; matlab.

Lecture goals

The goals of this lecture are to highlight the difficulties arising in the inversion of a physical model, and to present the main frameworks and methods addressing the resolution of inverse problems.

Detailed outline

Context: case studies; difficulties arising in the inversion of an ill-posed problem (interpretation considering singular values and interpretation in the Fourier domain); information loss, necessity to regularize the problem, concept of prior information.

Deterministic approaches: truncated singular value decomposition; estimation by minimization of a cost function.

Overview of Bayesian statistics (Bayes’ rule, choice of an estimator, Monte Carlo methods).

Stochastic approaches: maximum likelihood estimation – maximum a posteriori estimation – Markovian models – estimation of hyperparameters.

Conclusion: general perspective, bibliography.

Applications

Laboratory session 1: deterministic methods – application to spike trains deconvolution and tomography (matlab).

Laboratory session 2: stochastic methods – application seismic exploration (matlab).

Acquired skills

After this lecture, the student will be able to evaluate an inverse problem resolution method. The student will also be able to propose approaches for the resolution of a given inverse problem.