### Interaction

All the course materials will be posted on the course MOOC-page. Make sure you register there as a student as soon as possible!

### Timetable

### Conduct of the course

You gain credits from this course by completing the online course, Introduction to Computational Inverse Problems (https://mooc.helsinki.fi), given in 4.9. - 22.10. and passing a home exam given after the online course in 18. - 26.10. Both, the online course and the home exam, contribute equally to your final grade (0-5 scale). You are required to obtain at least 40 % of the total points from the online course in order to take the exam.

The online course consists of lecture materials (notes, videos etc.) and six weekly exercise sets of 10 questions. There are also weekly lectures and tutorial sessions which you are encouraged to take part. The tutorial sessions are guided teaching events where you can get help to solve exercises of the online course and other course matters.

### Description

Optional course.

Master's Programme in Mathematics and Statistics is responsible for the course.

The course belongs to the Mathematics and Applied mathematics module.

The course is available to students from other degree programmes.

Preliminaries: basic linear algebra and matrix calculus. Familiarity with Matlab programming and Fast Fourier transform FFT is useful but not mandatory. For example, the course “Applications of matrix computations” is a suitable prerequisite.

Master studies

The goals of the course:

(1) understand convolution as a matrix model,

(2) learn least-squares solution technique and see that it is not enough to solve deconvolution,

(3) show how to use SVD to detect ill-posedness in a matrix-based inverse problem,

(4) understand why deconvolution needs special regularised methods,

(5) write robust Matlab programs for signal deconvolution and image deblurring,

(6) learn how to extend the solution methods to large-scale deconvolution problems.

Recommended time/stage of studies for completion: 1. or 2. year

Term/teaching period when the course will be offered: varying

Inverse problems is the scientific art of going from effect to cause.

For example, the “cause” can be a clear digital recording of a spoken message. The “effect” is a mumbled, noisy, incomprehensible sound signal coming through a bad communications line. Recovering the clean signal from the noisy measurement is an inverse problem.

Convolution is a linear operation widely used in signal and image processing, where it is often called “filtering.” It is a kind of a moving average weighted by a “convolution kernel.” In a discrete setting, convolution acts on one-dimensional data (vectors) or images (matrices). It is a model for imperfect measurements; for example a poorly focused photograph is a convolved version of a sharp photo, and in spectroscopy, crisp spectral lines are blurred by a device function. Convolution is also very popular in machine learning in the form of convolutional neural networks (CNN). The so-called Deep Learning is based on CNN’s and fast convolution operations implemented using graphics processing units (GPU).

Convolution is a useful tool in pure mathematics as well, especially in harmonic analysis and the study of partial differential equations.

The inverse problem related to convolution is called deconvolution. The observed data is interpreted as clean signal convolved with a kernel and corrupted with random noise. The goal of deconvolution is to reconstruct the clean signal from the noisy data. This is an ill-posed inverse problem, meaning that the solution is highly sensitive to modelling errors and measurement noise. Robust solution of deconvolution problems is based on regularization.

The course starts by introducing convolution for discrete signals and images. Naive deconvolution is attempted computationally using matrix inversion, only to find out that the ill-posedness of the inverse problem spells trouble for this simple approach. The source of the difficulties is identified using Singular Value Decomposition (SVD), which will be introduced and discussed in detail in the course.

The difficulties can be overcome by a technique called “regularization.” Two regularized reconstruction methods are discussed both theoretically and computationally: truncated SVD and Tikhonov regularization.

The course has lectures, weekly exercises and an exam. It is possible to continue from this course to Inverse Problems Project Work course (5 credit units).

Samuli Siltanen