X-ray Computed Tomography (CT), first introduced by Cormack and Hounseld (1963), is an X-ray imaging modality that enables the generation of cross-sectional slices of an object, often the human body. Today, X-ray CT continues to be one of the leading clinical imaging modalities. For the computed tomography a CT scanner records object projections at many different angles. From these data a 3D image of the object structure is computed.

The task of computed tomography:

<= From measured set of 2D projections (left)

Compute 3D reconstruction of  the object structure (right) =>

(Mouse kidney, microfocus tungsten X-ray tube, 40kV)

There is a variety of methods to reconstruct clinically useful 3D images from CT measurements. The most dominant CT reconstruction technique, is Filtered Back Projection (FBP) method. FBP is analytic and its practical implementations take advantage of the Fast Fourier Transform (FFT). FBP is fast and deterministic and its properties are well understood. However, FBP is fully correct only when the noise influence can be neglected and when the number of projections is infinite which in real experiments is hardly achievable . Moreover, it is difficult to incorporate more elaborate physical models into the FBP method (complex geometry, influence of scattering, fluorescence, beam hardening, phase effects, etc.). Therefore, FBP can lead to artifacts in reconstructed images.

Another approach uses an iterative scheme of tomographic reconstruction. A typical algorithm which uses iterative approach is based on Expectation Maximization (EM). In the first iteration the uniform “trial” object is taken into account and its projections are computed (using even a very sophisticated physical model). The projections obtained are compared with those acquired by measurement. Using this comparison the trial object is modified to produce projections which are closer to the measured data. Then the algorithm iteratively repeats. The trial object is modified in each iteration and its projections converge to measured data. The main disadvantage of the iterative approach are the high computational requirements. A single iteration demands usually the same power as two full reconstructions in FBP.

To accelerate convergence speed of iterative algorithms a technique of Ordered Subsets (OS) can be used. When combined with the EM method it is called OSEM. OS technique splits each iteration into several subiterations. In each subiteration just a selected subset of all projections is used for trial object modification. The following subiteration uses a different subset of projections and so on. After all projections are used, the single full iteration is finished. The number of projections in each subset can be as low as 3 or 4 (OSEM method is then labeled OSEM3 or OSEM4). The speed of an iterative process is multiplied by number of subsets used (with some remarks).

Iterative methods are advantageous when the projection data are noisy, when the number of projections is low or incomplete and when the physical model is too complex.

Cylindrical bone cut (cow): Photograph (a), single projection with slice position indicated by line (b), FBP reconstruction of a single slice with streak artifacts caused by low number of projections (c) and OSEM3 reconstruction of the same slice (d).

Reconstructions done from 90 projections measured with tungsten X-ray tube at 40kV.

For most CT reconstruction algorithms it is necessary to preprocess the measured transmission data to be made linearly dependent on the object absorption. For monochromatic radiation the dependence is exponential and can be linearized taking the logarithm of the data. In the polychromatic case (X-ray tube as source) the dependence is generally unknown.

Computed tomography with Medipix2 pixel detector

Medipix2 pixel detectors equipped with a appropriate sensor chip (Si for small or light objects e.g. soft tissue, GaAs or CdTe for larger or heavier objects) are well suited for CT. The measured projections can be linearized using technique of per pixel signal-to-equivalent-thickness calibration. It has to be emphasized that this linearization method is fully precize just for ideal material composition of the object (soft tissue is fine from this point of view).

Histogram of raw image of overlapping 50um thick aluminium foils (top). Histogram of image corrected by signal-to-equivalent-thickness calibration (bottom). Peaks in the second histogram are narrow and equidistant (linearization feature).

Several examples of CT reconstructions by modified OSEM algorithm from data measured by X-ray radiographic setup consisting the Medipix2 device with 300um thick silicon sensor and a microfocus X-ray tube are shown in following figures, more examples can be found in applications (biomedical and material science).

Several Examples

Mouse paw (180 projections, Tungsten X-ray tube at 40kV), click to images to enlarge them:

LEMO connector, golden contacts are shown in red
(96 projections, thermal neutrons, adapted Medipix2 device):

Cartridge, exposive filling is shown in red
(100 projections, thermal neutrons, adapted Medipix2 device):

Fresh tooth,
(96 projections, thermal neutrons, adapted Medipix2 device):