The principle of computed tomography (CT) is to acquire a large number of x-rays all around the object, such so as
to measure for each orientation the x-ray image in that direction. From this set of x-ray projections, the reconstruction allows us to get a 3D image of this linear attenuation coefficient.
The reconstruction algorithm requires as input for each pixel of each projection an integral of the 3D scalar value. For the x-ray acquisition, this 3D scalar value to be reconstructed for each voxel \(v\) is simply the attenuation coefficient \(\mu\). Recalling the Beer-Lambert law (see section Attenuation law), we have one equation per pixel \(p\) as follows:
Suppose we have a flat-panel detector of \(N_X \times N_Y\) pixels and suppose we acquire \(N_P\) x-ray projections around the object, the number of integral data we get is simply \(N_X \times N_Y \times N_P\) which can be huge for standard imaging devices. To solve this inverse problem there are analytical methods (which solve the integral in the continuous domain before spatial discretization) and algebraic methods (which discretize the problem in the matrix form \(Ax=b\) before solving it). See for more details the book of Kak & Slaney available
This set of \(N_X \times N_Y \times N_P\) line integrals is known as the sinogram or Radon tranform. The name sinogram can easily be understood if we look at the same detecteor line in terms of the rotation angle. Fig. 146 shows such an image and the sine shape is clearly visible.
The most basic reconstruction is the simple back projection of the measured values along the
rays. The resulting 3D image is blurry. An analytical model image formation shows that it is
necessary to make a ramp filter in the Fourier domain before backproject to have an exact
reconstructed 3D image.
The circular trajectory does not allow reconstruction in a way exact as in the plan of
acquisition. The acquisition in helical mode allows for a more accurate reconstruction.
Some examples of sinograms and reconstructed volumes are given in the following.
An example in NDT.
Tomographic setup combining rotation and translation: helical mode. This mode is particularly suitable for long objects.
Tomographic setup with source motion parallel to the detector plane: Tomosynthesis or Laminography mode. This mode is interesting for flat cumbersome objects.
The presence of secondary radiation (scattered) is particularly important in conical beams because
the object is fully irradiated. On the other hand the fan beam geometry (with a detector linear)
very strongly limits the presence of secondary radiation on the detector. These scattered radiation
can greatly degrade the image 3D reconstructed: it is not quantitatively exploitable. Technological
solutions (eg with an anti-scatter grid) and methodologies (eg with an air-gap or a beam-stop array) exist to
reduce the impact of this scattered radiation.
An example of use of a beam-stop array (BSA) to compute and reduce the x-ray scatter contribution.
Some NDT applications of tomography taken from GrabCAD:
Dual-energy tomography can separate several phases according to their difference in attenuation.
Comparison of attenuation and phase imagery of a aluminum-silicon alloy. The sensitivity of the
imaging phase to density differences is much more important.