### Introduction to Computational Inverse Problems

Convolution is a linear operation widely used in signal and image processing, where it is used as a model for imperfect measurements. 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.

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 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. 