This course introduces students to the theory of wave propagation (i.e., direct scattering), both acoustic and electromagnetic, and to the resolution of nonlinear inverse scattering problems in biomedical applications. In the direct scattering problem, by means of information about the source of the wave and the so-called scatterer (i.e. the medium), the scattered wave has to be determined. In the inverse problem, from measurements of the scattered wave in an external domain, the sources and characteristics (imaging) of the scatterer are determined.
The course aims to describe the modeling of sound waves in perfect fluids and of direct and inverse scattering problems.
The course allows students to understand the basic mathematical tools for the solution of direct and inverse scattering problems, both acoustic and electromagnetic. Nonlinear imaging includes nonlinear problems of domain restoration by measurement of the transmitted wave away from the scatterer domain.
At the end of the course the student will have acquired sufficient theoretical knowledge:
• to identify and understand the main mathematical models of acoustic and electromagnetic wave propagation, on the basis of the involved physical models;
• to analyse and mathematically solve direct scattering problems, by computing analytical explicit solutions in the simplest cases, or approximated solution in more general contexts;
• to manage specific mathematical tools for solving wave propagation problems;
• to correctly treat the resolution of inverse problems associated with propagation models;
• to apply numerical-computational tools to direct and inverse scattering problems, useful for a subsequent resolution in real applications, with attention to ecography and tomography.
The teaching activity consists of traditional lectures, in which the subjects are introduced and explained in their classical theoretical setting. Although attendance is optional, it is strongly recommended.
Students with DSA certification ("specific learning disabilities"), disability or other special educational needs are advised to contact the teacher at the beginning of the course to agree on teaching and examination methods that, in compliance with the teaching objectives, take account of individual learning arrangements and provide appropriate compensatory tools.
(Direct) scattering of acoustic waves:
Euler equations, D’Alembert (or wave) equation. Harmonic waves: Helmholtz equation. Sommerfeld condition. Foundamental solution. Helmholtz representation Theorems. Far field pattern.
Resolution of the Helmholtz equation by single components analysis:
Spherical harmonic functions. Bessel, Neumann e Hankel spherical functions and their properties (outline). Rellich Lemma. Lippmann-Schwinger equation.
Inverse scattering problem of acoustic waves:
Lippmann-Schwinger as nonlinear fixed point equation. Born approximations. Integro-differential formulation.
Ultrasound imaging:
Mathematical model for the ecography and its implementation; diagnostic imaging for biological tissues by reflection of ultrasoud acoustic waves.
Inverse scattering problem of electromagnetic waves:
Lippmann-Schwinger equation for per electrodynamics scattering. Far field and its properties. Eskin Theorem. Ill-posedness and linearization by the Fréchet derivative. Gauss-Newton approach.
Qualitative methods for inverse scattering:
Linear sampling method and its implementation. Fundamental Theorem. Ill-posedness of the inverse problem. Tikhonov regularization. Tubes of energy flow.
Electrical impedance tomography (EIT):
Derivation from Maxwell’s equations. Continuum model and boundary value problem. Nonlinearity of the problem and Gauss Newton approach. Existence, uniqueness and Ill-posedness of the inverse problem (infinite/finite dimensional data).
Towards applications of EIT:
Different electrode models and choices of current patterns. Variational formulation and discretization. Reconstruction methods: direct, iterative, inclusion methods, machine learning methods. Application to stroke detection and monitoring.
In general, the notes taken during class lessons and some downloadable materials from the course web page are sufficient. Course handouts will be also given to the students. Any additional texts will be indicated during the lessons.
Ricevimento: Office hours by appointment via email (piana@dima.unige.it)
Ricevimento: Students may contact the professor by e-mail
CLAUDIO ESTATICO (President)
FEDERICO BENVENUTO
MICHELE PIANA (President Substitute)
The class will start according to the academic calendar.
The exam is oral.
The oral exam focuses on the theory and applications, as well as on the discussion of applications and their examples.
