Introduction to X-Ray CT and Artefacts Sophia B. Coban, William R.B. Lionheart, Philip J. Withers

[1] A. Beer, Determination of the absorption of red light in coloured liquids (in German). Annalender Physik und Chemie, 86:78–88, 1852.

[2] G.R. Davis and J.C. Elliott, Artefacts in X-ray microtomography of materials. Materials Science and Technology, 2006.

[3] J.H. Lambert, On the measure and gradations of light, colors, and shade (published in German). Augsburg (“Augusta Vindelicorum”), p. 391,1760.

References

RingsThe images above are taken from [2].

• Caused by sensitivity differences between detector ele-ments.

• Usually present in experiments where materials are tested.

There is always noise in real data so it is important to beable to distinguish the different types of artefacts!

Centre of RotationMotion

• Common problem with 3D scans with2D detectors.

• Problem due to the centre of rotationnot correctly detected and aligned.

• The motion errors = the object (couldbe source or detector) moving dur-ing the scan.

Beam Hardening Streaking

• The photons with lower energy areabsorbed by the material quicker, i.e.beam hardens.

• This results in information not beingclear.

• Cupping and streaking effects occurin the reconstructed image.

Phantom Image

• Artefacts are the limitations of X-ray CT technique.

• They are due to the discretisation, scattering of the X-rays or motion during the scan.

• Practical ways available to reduce them, but we need toknow what they are.

• The most common artefact types are...

Artefacts = Error!

•MXIF is the combination of Henry Mose-ley X-ray Imaging Facilty and ResearchComplex at Harwell.

• Investment of £11m, with 15 academicsand over 100 users.

• 2D, 3D and 4D imaging in multiscaletomography!

• In collaboration with big names, suchas Nikon XTek, Rapiscan and Xradia.

Manchester X-Ray Imaging Facility

• Integrating this to find intensity over the line, L, and rearranging for I , we get∫L

dI

I0= −

∫L

µ dL =⇒ lnI

I0= −

∫L

µ dL =⇒ I = I0e−

∫L µ dL.

• For a polychromatic beam, this equation becomes

I(E) =

∫I0(E)e−

∫L µ dL dE.

•Mathematically, the goal of X-ray CT is to recover the attenuation coefficient,µ, from the information at the detectors, I .

• There are a number of exact methods used to solve this problem, such as theInverse Radon Transform, the Fourier Transform or the back projection methods.

•We also have iterative methods that solve Aµ = I , where A is a very large andsparse matrix.

• This problem is sensitive to the initial data, failing to satisfy the third of Hadamard’swell-posedness conditions.

• Therefore this is classified as an Inverse Problem.

•We begin with a small intensity dI atposition dL and derive the exponentialabsorption law.

• So, for a monochromatic beam, the in-tensity dI at position dL is

dI

I0= −µ dL.

Formulation

• The proportionality is called the linear attenuation coefficient, µ.

• Important: This model assumes the X-rays travel in straight lines!

Beer–Lambert Law

Light is attenuated exponentially as ittravels through an object. Mathemat-ically, this means that absorption =− ln (I/I0).

The mathematical model is also very simple. We know two quantities:

• The initial intensity (energy) of the X-ray beam, I0, when it leaves the source, and

• The final intensity of the beam, I , at the detector.

We take the logarithm of the ratio of these quantities and work out a map of what isinside. This is due to the exponential absorption law in materials, discovered by J.H.Lambert in 1760 [3] and extended to concentrations by A. Beer in 1852 [1].

Mathematics of X-Ray CT

• These rays travel through the object andreach the detector at the other end withless energy.

• In other words, the rays are attenuated.

• The detector records how intense thearriving rays are.

• Finally, the comparison of the intensi-ties gives us an idea of what is insidethe object.

X-Ray computed tomography (CT)

• is a very important and popular tool in imaging.

• allows us to see insides of an object... without destroying it!

• is commonly used in medical imaging and non-destructive material testing.

The physical model is straightforward:

•We have a source that shoots out X-rays at a given energy.

Introduction to X-Ray CT

Sophia B. Coban, William R.B. Lionheart, Philip J. Withers

Introduction to X-Ray Computed Tomography and Artefact Types