OVERVIEW

The aim of this module is to provide students with basic and advanced mathematical methods for a theoretical analysis and practical solutions of problems in image reconstruction and image processing with specific applications in astronomical imaging. To this aim data from the most recent NASA and ESA missions will be at disposal. The first part of the module is dedicated to general notions and main operators for image processing. We will therefore address an image reconstruction problem within the Solar Orbiter (ESA) mission, an image processing problem within the SDO/AIA mission (NASA) and a prediction problem within the SDO/HMI mission (NASA). A computer exercise in MATLAB environment is foreseen for each of the problems addressed.

AIMS AND CONTENT

LEARNING OUTCOMES

The course aims to provide specialised mathematical skills on image reconstruction and processing techniques with particular focus on astronomical imaging. To this end, along with theoretical lectures, computer exercises are planned during which real data recorded by currently operational NASA and ESA satellites will be processed.

AIMS AND LEARNING OUTCOMES

The topics addressed in this module, together with the computer exercises, are aimed at the acquisition of knowledge and advanced skills in the field of:

• Reconstruction of X-ray images through Fourier Transform inversion from limited data
• Desaturation of EUV images by solving inverse diffraction problems
• Forecasting of solar flares by using machine learning techniques

PREREQUISITES

Each topic will be addressed in a self-consistent way

TEACHING METHODS

The module includes lectures and three computer exercises. Lectures will be held in person.

SYLLABUS/CONTENT

Basics of image processing: 1) Digital images: sampling and quantization. 2) Basic operators for image processing. 3) Image formation and image recording: blurring and noise. 4) Point Spread Function. 5) Imaging systems in the frequency domain. Transfer function. 6) Filtering in the frequency domain.

Astronomical image processing:

1. Image reconstruction from visibilities: definition of visibility; inverse Fourier Transform from limited data; deconvolution techniques; iterative methods for image reconstruction. The ESA STIX instrument in Solar Orbiter.
2. Image desaturazione: data formation process in the NASA SDO/AIA instrument; primary saturation, blooming, diffraction fringes; inverse diffraction; mosaicing.
3. Forecasting: feature extraction from SDO/HMI images; LASSO regression.

RECOMMENDED READING/BIBLIOGRAPHY

1. Bertero M and Boccacci P 1998 An Introduction to Linear Inverse Problems in Imaging (IOP, Bristol)
2. R.C. Gonzalez and R.E. Woods. Digital Image Processing 2nd edition. Prentice-Hall. 200
3. S. Giordano, N. Pinamonti, M. Piana, and A.M. Massone. The process of data formation for the Spectrometer/Telescope for Imaging X-rays (STIX) in Solar Orbiter. SIAM Journal on Imaging Sciences Vol. 8, No. 2, pp. 1315–1331, 2015
4. G. Torre, R.A. Schwartz, F. Benvenuto, A. M. Massone, and M. Piana. Inverse diffraction for the Atmospheric Imaging Assembly in the Solar Dynamics Observatory
5. S. Guastavino and F. Benvenuto. A mathematical model for image saturation with an application to the restoration of solar images via adaptive sparse deconvolution. Inverse problems vol. 37 issue 1 p 15010. 2020

TEACHERS AND EXAM BOARD

LESSONS

LESSONS START

In accordance with the academic calendar approved  by the Consiglio di Corso di Studi.

Class schedule

APPLICATIONS OF MATHEMATICS TO ASTROPHYSICS

EXAMS

EXAM DESCRIPTION

Three computer exercises will take place for which deadlines will be set. The admission to the final oral exam is subject to the approval of the practical exercises during the semester.

ASSESSMENT METHODS

Computer exercises are aimed at testing the practical skills acquired for the solution of the posed problems. They will be evaluated on the basis of the following criteria:

• accuracy and optimization of the code
• accuracy and presentation of the results (images, graphs, tables ...)
• comments on the procedures followed and on the results obtained

The oral exam is finally aimed at assessing the ability to communicate the knowledge acquired in a clear and competent manner

FURTHER INFORMATION

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.