correction models
After investigating five different models for BH correction I found that the spline model, has the lowest error, followed by the eight order polynomial model, then power law model with very slight differences. The overall models mentioned have compensated the BH effect well, which is expected since this is a single material. Figure 3.17 shows the RMS error chart of the applied models.
Figure 3.17: The average RMS error of corrected images compared with uniform disc. Using the BH linearisation method produced a good correction method but the ma- jority of them i.e., polynomial, power law and spline models, had no physical back- ground. Furthermore, higher degree polynomials produce better fit, but also caused an increase in the number of parameters to be estimated. Bimodal considered PE and CS energy region and fitted the BH curve differently for in the BH curve but the
results were found to be noise sensitive. Even though spline and power law models don’t have a defined physical background, the correction result was superior. The polynomial of degree eight requires estimation of nine parameters and cubic spline require four parameters for each spline, while power law has the least number of parameters to be estimated, only two. Considering the small difference in the ap- plied BH correction, power law is performing as a good, simple model for ANUµCT system.
3.5
Conclusion
As mentioned in chapter 2.2, there are three main methods for BH correction. Firstly, hardware filtering that only offers a reduction of the problem. Secondly, iterative software correction methods that are computationally unfeasible, therefore, the main procedure under investigation in this chapter is linearisation technique from which I chose five different BH correction models including: 1) bimodal, and 2) polynomial, 3) power law behaviour, 4) cubic spline, and 4) linear spline models. Every value on the models is corrected towards the linear trend line, which is expected in the monochromatic case.
To prepare the data, several solid materials and several fluids (oil/water) with dif- ferent salinity concentrations as well as other materials has been imaged at the ANU µCT facility. Materials which have the most usage in our imaging system have been used as samples. Higher salinity, and thus higher effective atomic number, leads to more severe hardening of a beam as it passes though the sample material. Different materials and salinity levels have been chosen for imaging to visualise their BH ef- fect. CsI , NaI and CsCl with 0.1, 0.19, 0.25, 0.357, 0.5, 0.75, 1, 1.5, 2 M, BrC8 in Oil
with 2.5, 5, 10, 20, and 100M of BrC8, 100M of BrC10, 100M of BrC11, 100M of BrC14,
and IC6, IC8, IC10have been imaged at the ANUµCT facility separately. This chapter included measurement the BH curves of cylinder samples directly from their projec- tions versus thickness of the cylinder and fit the BH curves with the five mentioned models. Next, the inverse of the model remapped to linearise the data. Finally, this section determined the model which has average lowest error for all the materials and salinities that used in this chapter.
I acknowledge the existence of ring artefacts that was introduced in section 3.3 in the RMS error estimations of table 3.17, however these errors are estimated using the same images for all applied BH models. Also the measured errors are already small. Furthermore, the main focus of this study about assessing the accuracy of the applied models could be presented in section 3.4. Therefore in this case, the ring artefacts can be ignored.
the only model that has a physical background by considering the PE and CS en- ergy regions, and so the bimodal model fit the BH curve with different models in PE and CS energy regions. The linear spline model was introduced as the approxi- mation and extension of the bimodal model. These models were found to be noise sensitive. Even though the spline, power law, and polynomial models don’t have a defined physical origin, they were found to have the lowest RMS errors, respectively, 6.36×10−1%, 6.24×10−1%, and 6.41×10−1% on average for all the materials and densities that I investigated (see Fig. 3.17). Considering that only two parameters must be estimated in the power law model, and the small difference in error when using the power law model when compared to the spline and polynomial models, supports the statement that the power law model is a good and simple model for the ANUµCT system. However, spline is the most accurate model for complicated and heterogeneous specimens.
Beam hardening correction of
concentric cylindrical specimens
using power law model
4.1
Introduction
As described in section 2.2, X-ray beam-hardening (BH) effect produces artefacts in form of cupping or streaking artefacts in the reconstructed images. As a result, the quantitative analysis and specifically segmentation will be very difficult. The amount of BH varies depending on the material composition of the specimen and the incident X-ray spectrum. In section 3.2.2, I measured the BH curve of several single-material cylindrical specimens, using ANUµCT facility [Sakellariou et al., 2004], and plotted their BH curves. Next, in section 3.3 I used five BH linearisation models to linearise the data and assessed the accuracy an applicability of the models for ANU µCT in section 3.4.
This chapter considers the specimens composed of nested-cylinders, e.g., a rock core within a container and uses the chosen model of chapter 3 to correct the BH artefacts of these specimens. As mentioned in section 3.1, this assumption covers a signifi- cant fraction (> 50%) of the specimens currently imaged at ANU µCT facility. By assuming a uniform material for each cylinder, one can generate BH curves directly from the projection data in a manner similar to that obtained by imaging wedge phantoms described in section 3.2.1. Here, I demonstrate the principle for concentric multi-component cylinders using the power law model (see section 3.3.3). The work presented here is complementary to the BH correction method previously developed by Kingston et al., [Kingston et al., 2012] in our group at ANU, that minimises re- projection distance (see section 1.2.6.1). That method enforces self-consistency in the data but does not work for objects where attenuation is a function of radius.
This chapter shows 1) how to determine the centre and radius of the cylinders of nested-cylinders in section 4.2, 2) generate BH curves using these cylinder parame- ter values in section 4.3, and 3) how to linearise the projection data of the nested-
cylinders by fitting a power law model to the BH curves in section 4.4. Section 3.5 demonstrate that the BH artefacts are significantly reduced in the tomographic re- constructions resulting from these corrected projections.
This research has been carried out within the “Computed Tomography" group of the Applied Mathematics Department at the Australian National University (ANU). I am the first author of a conference proceeding publication in the “Proceeding of the 1st International Conference on Tomography of Materials and Structures (ICTMS 2013)". I have applied the beam hardening corrections on the samples and prepared the manuscript. In relation to estimation of the radii of cylinders that are used in beam hardening correction of multiple nested-cylinder specimens, I sough the advice of Dr Shane Latham. This research is published with the title and author list as follows: “Paziresh M, Kingston A. M, Myers G. M, and Latham S. J. 2013. Software X-ray Beam Hardening Correction of Cylindrical Specimens. Proceeding of the 1st Interna- tional Conference on Tomography of Materials and Structures (ICTMS 2013). PP. 187-9190" [Paziresh et al., 2016]