The course aims at providing a solid mathematical presentation of the main analytical tools used in the theories of signal processing and imaging. At the end of the course, the students will be able to calculate multidimensional integrals and will have a familiarity with spaces of integrable functions. They will have understood how these spaces are realizations of more general mathematical structures and learned several techniques to analyze them. They will have become acquainted with integral transforms and developed methods to compute them. Furthermore, they will know how these theoretical concepts are applied in concrete problems arising in the theories of signal processing and imaging.
Course Prerequisites
Mathematical Analysis: differential calculus in one and several variables, integration in one variable, numerical sequences and series. Linear algebra: vector spaces, matrices, linear applications.
Teaching Methods
Lectures delivered at the blackboard and/or with a tablet. Attending the lectures is strongly recommended.
Assessment Methods
The exam consists of a written test aimed at evaluating both the students' knowledge of the theoretical aspects of the course and their ability to solve exercises. At the discretion of the teacher, this test might be complemented by an oral examination.
Texts
The course is based on topics mostly covered by the following textbooks: - G.B. Folland, "Real Analysis. Modern techniques and their applications", J. Wiley, 1999. - S. Mallat, "A wavelet tour of signal processing", Academic Press, 1998. - K Bredies, D. Lorenz, "Mathematical Image Processing", Springer, 2018. Transcripts of the lectures will be made available on Kiro.
Contents
Selected topics in Mathematical Analysis, including: measure theory and Lebesgue integration, Banach and Hilbert spaces, Lebesgue spaces, functions of one complex variable and the residue theorem. Applications to signal processing and imaging, including: Fourier transform and series, wavelets, sampling, denoising, inverse problems.
