Liverpool Centre for Mathematics in Healthcare

EPSRC Seminar - Solution Properties and Inverse Modeling in Variational Imaging - 16th, 17th, 18th October 2017

‌Speaker: Professor Mila Nikolova, Director of Research for the National Center for Scientific Research, CMLA Research Center for Applied Maths France

Venue:  MATH-209, 2nd Floor, Department of Mathematical Science Building

Times & Dates: 

2pm – 5pm 16th October 2017

2pm – 5pm 17th October 2017

2:30pm – 5:30pm 18th October 2017

Abstract- Numerous image processing tasks are defined as the solution of an optimization problem. The optimization problem (often called an objective) accounts for the model of the recording process and for the expected or desired features of the sought-after image. Usual approaches to construct an objective are Bayesian statistics, PDE’s, calculus of variations, and regularisation. In spite of their philosophical differences, they lead to quite similar objective functionals. Essentially, they amount to a weighted combination of a data-fidelity term and a (possibly adaptive) regularization term. Challenging theories has established bridges between disparate methods based on optimization, diffusion and frame representations. These results gave rise various practical ramifications of the objective functionals used in imaging sciences. However, these usual approaches to formulate an objective yield solutions whose features are hard to control. 

This short course presents a systematic approach to the problem of the choice of pertinent objective functionals for image processing. To this end, the focus is on the properties of the optimal solutions of an objective as an implicit function of both the data and the shape of the objective itself. This point of view leads to an intrinsic relationship between modelling and conception of relevant optimization problems. It thus provides a framework to unify the theory on optimization-based methods and to address rigorously the problem of the choice of objective functionals for image processing.

The goals of this course are the following:

  • to understand the practical issues governing the proper choice of an objective for image processing;
  • to show how to conceive objective functionals in such a way that their minimizers exhibit some desired or expected properties;
  • to provide a systematic way to compare existing objectives for image processing.

This course is based on a series of analytical results which characterize some salient properties exhibited by the optimal solutions of different families of objectives. Points of interest are, e.g., the recovery of edges and of regions that are homogeneous or sparse over some manifolds, the processing of textures, under different observations systems and noises. These are shown to be determined by some attributes of the objective functional relevant to its shape – (non)smoothness or (non)convexity, or more specific attributes. Numerical examples are used to illustrate the theory and stability results are provided. Applications where specific objectives are conceived using the mathematical results on optimal solutions are presented.

By way of conclusion, open questions ranging from concepts to practical imaging problems are discussed.

Course Notes

LCMH Seminar Notes - Prof Mila Nikolova