Inverse Problems Group

Statistical inversion for 3D dental X-ray imaging

Diagnostic and operational tasks based on dental radiology often require three-dimensional (3D) information that is not available in a single x-ray projection image. A computerized tomography (CT) scan can provide the dentist with comprehensive three-dimensional data. However, in many situations a complete CT-scan is not available or practical because of i) high radiation dose or ii) high cost and availability of the CT-scanner equipment. Because of these limitations, we have considered a novel type 3D imaging modality for dental radiology. The modality can be viewed as an intermediate between traditional x-ray imaging and CT-imaging; the main idea is to construct a system which could provide the needed 3D information about the tissues based on a small number of projection images and could handle incomplete imaging geometry. With this kind of construction, the projection images could be taken by any existing digital x-ray system, including dental panoramic and intraoral systems.

A complication in these experiments is that the set of a few projections that are collected from sparsely distributed directions does not contain sufficient information to completely describe the 3D x-ray attenuation function. In other words, the reconstruction problem becomes an ill-posed inverse problem. It is well-known that traditional reconstruction methods, such as filtered backprojection (FBP), do not perform well with such limited data.

Bayesian inversion is a natural framework to tackle the reconstruction problem with sparse projection data. In Bayesian inversion, a priori knowledge of the tissue is used in the image reconstruction problem to compensate for the incomplete information of the sparse projection data. For example, in dental x-ray imaging we know that the attenuation function is a nonnegative, piecewise regular function, and different tissue types are separated by well-defined boundaries. The unknown x-ray attenuation function and projection data are considered as random variables, and separate statistical models (probability distributions) are formulated for (1) the acquisition of the projection data and (2) the a priori information. Based on these models and the Bayes formula, the complete solution of the inverse problem is provided by the posterior probability distribution. Final images of the tissue are then obtained as point estimates from the posterior distribution.

The key problems with respect to the application of statistical inversion to 3D X-ray imaging are:

How to write the prior information in such a form that it can be coded into an algorithm. The prior information is often in qualitative form that cannot be directly applied to the algorithms. The problem is then how to transform and parameterize this information into quantitative form so that it can be implemented into the algorithms.
Another practical difficulty in applying Bayesian methods to 3D x-ray imaging is the heavy computational requirements. If realistic resolution in a 3D problem is used, the number of unknown voxel values is typically in the range 1 - 10 millions, and thus, the computation of the posterior statistics leads to large-scale optimization or integration problems. Thus, powerful computers and efficient numerical algorithms are required to compute the 3D reconstruction in clinically acceptable time.

In this project we seek to answer these questions up to the level of implementation that can be integrated to commercial imaging devices.

Figure 1 shows one horizontal and vertical slice from a reconstruction that was computed from projection data of a tooth specimen. Left column shows a Bayesian MAP estimate from full-angle projection data (23 projections spanning a view angle of 187 degrees). The prior model for the oral structures was a weighted L1 and total variation prior together with the positivity prior. The computation of the MAP estimate was implemented with parallel computing techniques on a decicately build Beowulf computer cluster. The middle column shows traditional backprojected reconstruction from limited-angle data (9 projections spanning a view angle of 68 degrees). The right column shows MAP estimate from the limited angle data.

Figure 1: Reconstructions from projection data of a tooth specimen. Top row shows one horizontal slice and bottom row shows one vertical slice from the 3D reconstructions, respectively. Left column: Bayesian MAP estimate from full-angle data (23 projections spanning a view angle of 187 degrees). Middle column: Backprojection from limited-angle data (9 projections spanning a view angle of 68 degrees). Right column: MAP estimate from the limited angle data.

People working on x-ray imaging:

Ville Kolehmainen, Ph.D.
Antti Vanne, M.Sc.