2.2 Beam hardening (BH) artefacts and correction techniques in CT
2.2.2 Linearisation BH software correction methods
The more commonly applied correction method is based on applying a linearisation procedure on the projection data [Van de Casteele et al., 2002; Herman, 1979; Ham- mersberg and Måns, 1998], henceforth called BH linearisation methods in this thesis. The Beer-Lambert law for a homogeneous sample and a monochromatic source is shown in Eqn. 1.2. One can plot the measured attenuation coefficient versus the sample thickness, that is “beam hardening curve" (BH curve), and is expected to reflect a linear function, while the experimental attenuation data has a significant de- viation from the linear trend. I measured the BH curves of several material imaged at the ANUµCT facility that is shown in Fig. 3.4. To correct the BH curves, the mea- sured nonlinear relation can be fitted with a model. The inverse of the model should be applied on the BH curve to linearise the data. This method is most applicable for the samples that are composed of only one material.
Several linearisation methods have been developed for BH correction [Herman, 1979; Van de Casteele et al., 2002; Kingston et al., 2012]. Herman [Herman, 1979] mod- elled the BH curve with a polynomial. I implemented the polynomial BH correction method using the projection data obtained at the ANUµCT facility and presented the results in section 3.3.1. Van de Casteele et al., [Van de Casteele et al., 2002] modelled the BH curve based on the physical model. I implemented BH correction using this model in section 3.3.2. These two methods offer acceptable results, however the first requires several parameters to be estimated depending to the order of the polyno- mial and the second needs some initial information about the attenuation coefficient values or these information should be estimated and doesn’t provide good results for all materials.
Kingston et al., [Kingston et al., 2012] developed a method to minimise the re- projection distance. This method, that is explained in section 2.2.2.1, provided a good BH artefact correction for the ANU µCT system but was not compatible for cylindrical samples. More than 50% of samples at the ANUµCT facility have a cylin- drical shape, thus, further investigation was required to find a proper BH linearisa- tion model for correction of BH artefacts of cylindrical sample, which is surveyed in chapter 3. I applied the chosen model of chapter 3 on nested-cylinder samples and presented the results in chapter 4. Ketcham and Hanna, presented an iterative optimisation algorithm on the regions-of-interest (ROI)s chosen by an expert user, to find a generalised spline-interpolated transform function for BH correction of het- erogenous specimens. This method is explained in section 2.2.2.2. In section 3.3.4,
I applied a non-iterative procedure using a cubic spline model, on the BH curve of the homogeneous cylindrical specimens imaged at ANUµCT facility, to linearise the data.
2.2.2.1 Linearisation BH correction by minimising re-projection distance
This method assumes an average material for the heterogeneous samples and uses a procedure that minimises the re-projection distance to find the best correction curve. Function B, that is an eight order polynomial, is used to model the BH curve, as follows: Ba(x) = 8 ∑ i=1 ai bi(x); b(x) = ∑8 j=1 mjxj (2.1) Where ai andmi are the constants of the eight order polynomials. This process in- volves eight reconstructions and re-projections of the experiment data. The best BH curve generates a projection set that minimises the re-projection distance, di, that is the difference between the uncorrected projection Pu and the projection, which is re- constructed and then re-projected,RR−1Pu, as follows:
minimise {ai} k ∑8 i=1 ai(RR−1−I)bi(Pu)k k ∑8 i=1 aibi(Pu)k (2.2)
A flowchart for this method is shown in Fig. 2.5. This technique doesn’t requirea pri- oriinformation of the X-ray spectrum or materials. It works well for single-material samples, and also multiple-material samples if the BH curve determined for the av- erage of all the materials of the sample.
The work presented in chapter 4 is complementary to the BH correction method developed by Kingston et al. [Kingston et al., 2012] explained in this section. That method enforces self-consistency in the data but does not work for cylindrical sam- ples. The model can not determine any objective where attenuation is function a function of radius because BH projections still exists in the re-projections [Kingston et al., 2012]. Chapter 4 considers the specimens composed of nested cylinders, e.g., a rock core within a container to correct the BH artefacts in their tomograms. This assumption covers a significant fraction (> 50%) of the specimens currently imaged at ANUµCT facility.
Data acquisition
Intensity images (projections);
P
uDown-sampling
Generate and store
8 projection sets
b
1(P
u),
b
2(P
u), ...
b
8(P
u)
from Eqn. 2.1
Generate and store
8 re-projection sets
RR
−1b
1(P
u),
RR
−1b
2(P
u),
...
RR
−1b
8(P
u)
from Eqn. 2.1;
Calculate re-projection differences;
d
i=
RR
−1b
i(P
u)−b
i(P
u)
i
∈
[1, 8]
Parameter extraction;
a
iof Eqn. 2.2
Remap the polyno-
mial of Eqn. 2.1 to
P
uCorrected intensity images
2.2.2.2 Linearisation iterative expert-guided BH correction for heterogeneous spec- imens
This method [Ketcham and Hanna, 2014] requires an expert user to identify and choose some regions-of-interest (ROI)s that most clearly indicate BH artefacts in a reconstructed image. These ROIs ideally should include a single material or void space. These regions are used to construct a transform function, f, that transforms each polychromatic projection data, P, into the value it should be if the X-ray beam were mono-energetic, M, i.e., M= f(P).
A set of uniform (evenly-spaced) nodes of splines are distributed over the range ofP, with the endpoints of the function fixed to be endpoints of the data values. A cubic spline interpolates between these nodes.
Next, an iterative optimisation procedure, using a simplex method [Press et al., 1988], is used to estimate the the coefficients of the spline. In every iteration, the estimated coefficients are used to generate the transformed projection data, M. This projection is then reconstructed using the filtered back-projection algorithm (see section 1.2.7.3) and used to test the correction. The iterative process runs until the BH is minimised. This process is depicted the flowchart of Fig. 2.6.