The inverse problems research group at the University of Helsinki is a part of the Finnish Centre of Excellence of Inverse Modelling and Imaging and has an active role in the Finnish Inverse Problems Society. Our research group is also further divided into two subgroups: *Geometric Inverse Problems and Applications* led by Matti Lassas and *Computational Inverse Problems* led by Samuli Siltanen.

Inverse problems are best understood by comparing them to the corresponding direct (or forward) problems. In direct problems we have an input which we run through some kind of process to get an output. We know what the input is and we also know what the process is, so we can always figure out what the output will be.

For inverse problems we reverse the order. Now we have a given output and some idea of what kind of process acted on it, and possibly some information about the input. Our task is to figure out what the input was which resulted in this output.

One might ask if the inverse problem is equivalent to a direct problem if we know the process exactly, so that we could then simply run the process backwards. Unfortunately this is rarely possible since most processes cannot be reversed. As a simple example one could consider the process which takes two numbers X and Y, and gives the sum X+Y. But if we are given some number Z and are asked to determine which X and Y sum to this Z, we would quickly find that there are many possible solutions without any other restrictions on the allowed inputs.

The 3×3 grid below contains an example of a direct problem. Given the values in the squares, what is the sum of the values along the rows and columns?

Then we have the corresponding inverse problem below, where instead of being given the numbers in the squares we are given the sums of the rows and columns and are tasked with determining the values in the squares.

- Does this problem have a solution? Maybe it's impossible to have 9 numbers in the squares such that we get the given sums.
- If there is a solution, is it unique? There could be more than one set/arrangement of 9 numbers that give the same sums.
- Do we have some prior information about the numbers that could help us? Maybe we know that the numbers cannot be negative, or that they have to be whole numbers. More information can help us narrow down the possibilities.

This example may seem to be somewhat arbitrary but it is actually analogous to X-ray tomography, where internal features of people or objects are determined by firing an X-ray through the person/object from different directions and a detector on the other side measures how much the X-ray beam has been blocked by dense features. Essentially, the beam calculates the total mass along its path, similar to the sums in the 3×3 grids earlier.

When taking photographs, there are several ways in which the photo can come out blurry. The most common causes are motion blur which is caused by relative motion between the camera and the target while a picture is being taken, and defocus blur which is caused by the focal point of the lenses being in front of or behind the object being photographed. Regardless of the cause the outcome is a blurry picture, and the scene that was being captured is now lost. Can this blurring be undone? Let's consider the defocus blur in this example and see what can be done.

First we have to formulate the forward problem. How do we go from a sharp picture to a defocused picture? The idea in defocus blur is that the light hitting the camera's sensor is not being focused to a single point, but rather to some larger region. Photographing a point source of light would produce an image of this defocus region, but it can generally be approximated to be a circle. The size of the circle then describes how out-of-focus the picture is; larger circles smear the image more. Mathematically, this smearing can be modelled by convolution.

Below is an example of the forward problem of blurring a sharp picture using a defocus kernel through the application of the convolution operator (indicated by the asterisk '*'):

The corresponding inverse problem of deblurring an out-of-focus photograph can be expressed as follows:

Where we have the blurry photograph and we have a model for how the perfect picture was blurred, and need to determine what the sharp picture looks like.

Using the theory of inverse problems, we can achieve fairly good reconstructions of the sharp image, such as the one below:

The reconstruction is not perfect but it does make the text readable again, unlike in the blurry picture.