What Are Inverse Problems?
Inverse problems appear when we want to see or examine something that we cannot access directly. What we have is an indirect measurement that contains hidden information.
An inverse problem is always a counterpart of a direct problem, as shown in the schematic diagram below. The direct problem is going from object to data, and the inverse problem is about finding the object back from the data.
Let me give a few examples.
The first example is moving between a positive image and a negative image. (Those of you who have experience with black&white film photography know that images were negative on the film and turned to positive when printed.)
The second example is blurring and deblurring. Blurring occurs when a camera is misfocused.
The third example is X-ray tomography. On the left you see the standard Shepp-Logan phantom that models a cross-section of a human head. White is scull bone and the grey stuff models the brain. On the right you see a so-called sinogram, which is a neat way of organizing tomographic data, or line integrals over the X-ray attenuation function.
All of the above examples are linear inverse problems, meaning that the forward map taking the object to the data is linear.
Our fourth example is the non-linear imaging problem called electrical impedance tomography. There we probe an unknown physical body (such as a patient) with electrical currents fed to the body using electrodes. The resulting voltages at the electrodes are measured, and the measurement is repeated with several current patterns. The data is organised into a current-to-voltage matrix.
The fifth and final example is inverse obstacle scattering. There one sends plane waves (or waves emanating from a distant point source) towards objects impenetrable to the waves. Measuring the scattered wave far away from the objects allows one to approximately determine the so-called far field pattern.
The general mathematical framework of inverse problems can be expressed schematically as maps between model space and data space.
Hadamard's famous definition is used for dividing inverse problems into well-posed and ill-posed ones. Among the above examples, all but the positive-negative photograph example are ill-posed. In the research field of inverse problems, ill-posed problems are considered to be the interesting ones. That's because they are much harder to solve than well-posed inverse problems!
What does it mean to solve an inverse problem? In practical situations, one has the measured data and a computational model for the forward map. Then the task is to design a computer program that computes an approximation to the object. The computation needs to be stable, so that moderate errors in data will not change the reconstructed object too much.
The solution of an ill-posed inverse problem needs to be robust against modelling errors and noise in the data. Such stability is provided by regularisation.