Source-weighting method

Heismann et al., [Heismann et al., 2003] used the projection images recorded at two energy levels to quantitatively estimateZandρof the materials of a sample such that:

µ1(ρ,Z) µ2(ρ,Z) → ρ(µ1,µ2) Z(µ1,µ2) (2.5) Substituting Eqn. 1.4 in the above equation, one can obtain:

µ1(ρ,Z) µ2(ρ,Z) = K2 g1 K2 g2 . ρ ρZn (2.6) where gε = K1 R Sε(E)

Em dE. From thereρ and Z are calculated as the weighted differ-

ence ofµ1 andµ2, and Z as a non-linear function of the ratio of the µ_µ1₁ as follows:

       ρ= _K1 2 g2µ1−g1µ2 g2−g1 ; Z= K2 1−µ₁ µ₂ g2µµ1₂−g1 !1_n (2.7) Figure 2.12 (a) shows Z as a nonlinear function of the ratio of dual-energy attenu- ation coefficients and Fig. 2.12 (b) shows ρ as a weighted difference of attenuation coefficients at two energy spectra.

(a)

(b)

Figure 2.12: Plots of the numerical results ofZandρprojections: a) shows a function, F(Z) = µ1

µ2, and b) showsρ(µ1,µ2)for recorded images at 80 and 140 keV (µ1 andµ2

2.4

Conclusion

This chapter described the two main energy related issues inµCT: 1) BH and 2) ma- terial characterisation. Several energy selective techniques have been surveyed, that lies in this two main categories. These methods should consider the polychromatic nature of the X-ray source and energy dependency of attenuation coefficient.

Standard tomographic reconstruction algorithms assume monochromatic radiation. Ignoring the polychromatic nature of the radiation in reconstruction produces in- consistent attenuation coefficient values in the tomogram which is identifiable in the form of cupping and streaking artefacts. Section 2.2 reviewed the description of the BH artefact and many available techniques for its correction.

Section 2.3 included the energy selective methods to characterise the constituent ma- terials of a sample. X-ray interactions with matter and lower and higher energies occur in form of PE and CS (see section 1.2.6). As a consequence, imaging a sample at two energy spectra provides information relevant to PE and CS effects, which in turn, correspond to atomic number and density of sample materials, thus, making material characterisation possible. Both approaches improve images such that seg- mentation is possible.

In future chapters of this thesis, I implemented several mentioned methods of this chapter. The results of these implementation are compared with the results of the developed methods in this thesis, to assess the accuracy and applicability of the methods. For BH correction, chapter 3 includes implementation of several lineari- sation BH correction models including some models mentioned in this chapter. In chapter 4, I am using the most appropriate model of chapter 3 to correct the BH of nested-cylinders. For material characterisation methods, I have implemented AMP and SK models of this chapter, in chapter 6. As previously discussed, all the previous application of AM model were using a simplified form that because of its limitations and the considerable computation required. I have calibrated and used the full AM model in chapter 5. The results of this chapter is compared with the results of SK and AMP model in section 6.

Assessment of several linearisation

X-ray beam hardening correction

methods

3.1

Introduction

Micrometer-scale computed tomography (µCT) systems (see section 1.2) in the lab- oratory use a micro-focus X-ray source (see section 1.2.3.1) that emits polychromatic (Bremsstrahlung) radiation (see section 1.2.3). As mentioned in section 2.2, low- energy X-rays are attenuated more readily as the X-ray beam passes through a sam- ple, causing the resulting X-ray beam to have a higher proportion of high-energy X-rays, i.e., X-ray beam-hardening (BH). Therefore the total attenuation, given by Beer-Lamber law in Eqn. 1.2, is no longer a linear function of material thickness. This effect can produce severe BH artefacts, including cupping and streaking artefacts, in the reconstructed images. Apart from these visual aspects, quantitative problems may arise, thus, these artefacts make subsequent tomogram segmentation and anal- ysis difficult.

There are a number of techniques that can be used to correct this artefacts (see section 2.2). BH can be reduced by physically filtering the beam, effectively pre- hardening the beam (see section 2.2.1). However, sometimes sufficient filtering is not used or may not be feasible due to reduced X-ray flux. In these cases, the use of some form of software correction method is required. Several correction methods have been developed that apply a linearisation curve (see section 1.2.6.1), called BH curve. The BH curve is the nonlinear relationship between sample thickness and the measured intensity. It can be measured or estimated and fitted using a BH model, e.g., a polynomial [Herman, 1979]. The inverse of the BH curve is applied to remap the measured intensity. In order to directly measure the curve, one can image some kind of wedge phantom [Van de Casteele et al., 2002; Herman, 1979]. Methods to estimate the curve without imaging phantom are typically iterative and computa- tionally intensive (see section 2.2.3), therefore, I chose to work with BH curves to correct the data captured using ANUµCT facility [Sakellariou et al., 2004].

In this work, I considered using cylindrical specimens, e.g., brines that are imaged in a cylindrical container. This assumption covers a significant fraction (> 50%) of the specimens currently imaged at ANU µCT facility. Section 3.2.1 describes how to generate BH curves directly from the projection data of a homogeneous cylindrical specimen, similar to that obtained by imaging wedge phantoms. Section 3.2 plots the BH curves of several frequently used samples at the ANU µCT facility to view the nonlinear trend of beam hardened data versus the sample thickness. The BH curve is applied in linearisation methods to correct the BH artefacts. An overview of the existing linearisation BH correction methods in tomography is presented in sec- tion 1.2.6.1. Here I am implementing two of the mentioned non-iterative linearisation methods: the polynomial model [Herman, 1979] in section 3.3.1 and the bimodal en- ergy model [Van de Casteele et al., 2002] in section 3.3.2. This is followed by a study of three further BH models including the power law model in section 3.3.3 [Paziresh et al., 2013], the cubic spline model [Ketcham and Hanna, 2014] (see section 2.2.2.2) in a non-iterative form for the homogeneous samples of this chapter in section 3.3.4, and finally a linear spline model in section 3.3.5 that is an extension of the bimodal model [Van de Casteele et al., 2002]. These techniques will be investigated to deter- mine the most appropriate model to be used at the ANU µCT facility for correction of artefacts caused by beam hardening. Section 3.4 covers the performance assess- ment of the reviewed methods.

This research has been carried out within the “Computed Tomography" group of the Applied Mathematics Department at the Australian National University (ANU). I applied the linearisation models and performed the assessment to find out the most appropriate model for BH correction of cylindrical samples at ANU µCT facility. A survey of the applied models in this chapter is in preparation for publication with the title and author list as follows:

“Paziresh M, Kingston A. M, Myers G. M, Latham S. J, and Adrian Sheppard. 2016. Assessment of several linearisation models for X-ray beam hardening correction of cylindrical specimens. In preparation."

The linearisation BH correction only is applicable for homogeneous samples because different materials attenuate X-rays differently and thus have different BH curves. From the five models described in this chapter, I have chosen one with the lowest er- ror and least number of parameters to estimate to correct the BH of the heterogeneous (concentric) cylinders in chapter 4. That research was presented at the International Conference on Tomography of Materials and Structures 2013 (ICTM 2013). As explained in section 3.2.2, the brine containers used in the department are mostly made of poly- oxymethylene, which is a low density material. However, if a container is made of a high density material, because X-rays pass through more material at the edges than the centre of the container, in a fan beam radiation, an inverse BH effect is evident in the tomogram, that intensifies the difficulties in segmentation. For the correction of

this effect, we refer to:

“Holt J, Paziresh M, Kingston A. M, and Sheppard A. 2014. Correction of beam hard- ening artefacts in microtomography for samples imaged in containers. SPIE Optical Engineering+Applications. Vol. 9212, 92120A."