There are a few technical approaches to computed tomography (CT). Nevertheless, you should be familiar with some basic mathematical properties of the problem if you want to understand the science and numerics in this field.
What is an inverse problem? They appear in several fields, including medical imaging, image processing, mathematical finance, astronomy, geophysics, nondestructive material testing and sub-surface prospecting. Inverse problems can be thought of as questions asked “backwards”, for example, “If we had a set of X-ray images from a patient, what is the three-dimensional structure of her inner organs?” This problem is essential in computed tomography, CT. Inverse problems can be linear or non-linear depending on the underlying physics that governs the problem. A typical way of describing an inverse problem in mathematical ways is to find the best solution for the unknown x from the equation:
m = Ax + e
Where m is the recorded measurement, A is an operator that describes the relationship between the measurement and model parameters, e is an error term, i.e. noise. When solving these kinds of problems numerically, there are always trade-offs. One important trade-off is regularization, which usually has in-built forehand information about the phenomenon, but also has some unwanted effects on the results.
Inverse problems are a fairly young branch of mathematics, discovered and introduced during the 20th century. Since the nature of inverse problems links them with real world phenomena, they're a very active area of modern applied mathematics and a real crossroad of several fields of sciences. The development of computers in the late 20th century and the beginning of 21st has opened these problems to numerical calculations and thus bound them to industrial processes. Even today, the new features of computers and GPU open new possibilities for advanced methods that have been impossible to implement before, since they've been numerically too demanding for industrial purposes.
Statistical inversion theory
Statistical inversion theory is a sub branch of inverse problems, where these “backward” questions are answered using statistical tools, including sampling, distributions and probabilities. Bayesian methods and formula govern this area of mathematics and offer vast possibilities to bring a priori knowledge to ease the solving process and enhance the solutions. The statistical inversion approach is based on the following principles:
- All variables included in the model are modeled as random variables.
- The randomness describes our degree of information concerning their realizations.
- The degree of information concerning these values is coded in the probability distributions.
- The solution of the inverse problem is the posterior probability distribution.
The last item, in particular, makes the statistical approach quite different from the traditional approach. Regularization methods produce single estimates of the unknowns while the statistical method produces a distribution that can be used to obtain estimates that, loosely speaking, have different probabilities.
Statistical methods require vastly more numerical calculation power than traditional approaches and solving this isn't straightforward, since the nature of statistical inverse problems makes them really hard to parallelize and thus utilize calculation capabilities of GPUs. Trade-offs are also part of statistical methods. Tuning the parameters for a given case can take a while, since the relationship between parameters isn't obvious and thus pinpointing the right values isn't easy. You have to had deep understanding about the nature of the problem and your assumptions to fully benefit from SI methods.
In computed tomography, SI methods offer vast improvements and enable the use of 3D X-ray in situations where it has been considered impossible due to a limited and/or sparse measurement angle.
Traditional methods produce poor quality reconstructions in limited and/or sparse angle situations.
Statistical inversion for X-ray tomography with few radiographs
Below are two of the most known articles of SI methods in X-ray tomography. Among the authors are employes of Eigenors predecessor, Invers Oy. We have continued R&D from these basic tenets.
Download part 1: General theory (.pdf)
Download part 2: Application to dental radiology (.pdf)