Jun 19 2015

Beam Hardening 2-the no-linearize theorem

Tag: Implementation,Math,Physicsadmin @ 1:45 pm
The last post showed that beam hardening causes a nonlinearity between the log of the measurements and the A-vector. It is natural to think that we can eliminate the beam hardening artifacts by measuring the nonlinearity and then “linearizing” it with an inverse transformation. In this post, I will show that this is not possible in general. Although there are some special cases when we can linearize and a linearizing transformation may reduce the artifacts, we cannot do this for every object. I will show that this is due to the fact that we need at least a two dimension basis set to represent the attenuation coefficient.

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Jun 12 2015

Beam hardening 1

Tag: Clinical hardware,Physicsadmin @ 2:50 pm
Beam hardening artifacts were seen soon after the introduction of CT. Radiologists noticed a ring of increased Hounsfield numbers against the inside of the skull. At first they thought the increase was due to the difference between the white matter in the interior and the gray matter in the cortex of the brain but images of skulls filled with only water also showed the ring so it was obvious the increased values were an artifact.
The EMI corporation, which produced the first CT scanners, must have known about the artifact but they were notoriously close mouthed about the scanner design. In their first scanner the patient stuck his head into a plastic bladder filled with water and the x-ray system measured through the head surrounded by the water. This reduced the dynamic range requirements for the electronics but it also reduced the beam hardening nonlinearity as well as other artifacts as I will show.
In Al Macovski’s group at Stanford, we quickly figured out that the change in average energy of the transmitted photons as the object thickness increases, spectral shift as we called it, would produce a nonlinear relationship between the logarithm of the measurements and the line integral of the attenuation coefficient. We also showed that this nonlinearity could produce the artifact. We were quite interested in it because it was an effect of x-ray energy on the image and we wanted to extract energy dependent information.
Fig. 1↓ shows that the change in average energy and the effective attenuation coefficient as object thickness increases are both quite large. In this post, I will show how this change leads to a nonlinearity between the log of the measurements and the line integral of the object. I will derive expressions for the magnitude of the nonlinearity. These will lead to ways to reduce the nonlinearity and therefore the artifacts. In later posts I will show that the nonlinearity cannot in general be corrected using a lookup table, the no-linearize theorem. I will then describe a general way to understand the effect of the nonlinearity on the reconstructed CT image. Finally, I will examine whether iterative reconstruction methods can be used to correct the artifacts by making the projections and the image consistent.

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