Descriptif

Digital images are ubiquitous : from professional and smartphone cameras to remote sensing and medical
imaging, technology steadily improves, allowing to obtain ever more accurate images under ever more
extreme acquisition conditions (shorter exposures, low light imaging, finer resolution, indirect
computational imaging methods, to name a few).
This course introduces inverse problems in imaging (aka image restoration), namely the mathematical
models and algorithms that allow to obtain high quality images from partial, indirect or noisy observa-
tions. After a short introduction of the physical modeling of image acquisition systems, we introduce the
mathematical and computational tools required to achieve that goal. The course is structured in two parts.
The first part deals with well-posed inverse problems where perfect reconstruction is possible under
certain hypotheses. We first introduce the theory of continuous and discrete (fast) Fourier transforms,
convolutions, and several versions of the Shannon sampling theorem, aliasing and the Gibbs effect. Then
we review how imaging technology ensures the necessary band-limited hypothesis, and a few applications
including: antialiasing and multi-image super-resolution, exact interpolation and registration for stereo
vision, synthesis of stationary textures.
In the second part we deal with ill-posed inverse problems and the variational and Bayesian formulations,
leading to regularized optimization problems (for posterior maximization) and to posterior sampling (not
covered in this course). This part starts with a review of optimization algorithms including gradient descent,
and the most simple splitting and proximal algorithms. Then we review increasingly powerful regularization
techniques in historical order: from Wiener filters and Tikhonov regularization, to total variation, and non-
local self-similarity. By the end of the course we briefly introduce an overture to recent approaches using
pretrained denoisers as implicit regularizers of inverse problems via RED and plug and play algorithms for
posterior maximization. The theory is illustrated by applications to image denoising, deblurring and
inpainting.