4. Virtual monochromatic DT imaging using DE

#### 4.1. Theory of material decomposition processing

Given that the photoelectric effect and Compton scattering of X-ray photons in the diagnostic range (E < 140 keV) are the predominant mechanisms responsible for X-ray attenuation (monochromatic X-ray), the mass attenuation coefficient of any material can be expressed with sufficient accuracy as a linear combination of the photoelectric and Compton attenuation coefficients. Consequently, the mass attenuation coefficient can also be expressed as a linear combination of the attenuation coefficients of two basis materials [10]:

$$
\mu(r,E) = \left(\frac{\mu}{\rho}\right)\_1(E) \cdot \rho\_1(r) + \left(\frac{\mu}{\rho}\right)\_2(E) \cdot \rho\_2(r) \tag{11}
$$

where the basis materials exhibit different photoelectric and Compton characteristics, (μ/r)i(E) and i = 1, 2 denote the mass attenuation coefficients of the two basis materials, and ri(r) and i = 1, 2 denotes the local densities (g/cm<sup>3</sup> ) of the two basis materials at location r.

In principle, DE can only accurately decompose a mixture into two materials. Therefore, for DE measurement–based mixture decomposition into three constitutive materials, a third constituent must be provided to solve for three unknowns by using only two spectral measurements. In one solution, the sum of the volumes of the three constituent materials is assumed to be equivalent to the volume of the mixture (volume or mass conservation) [11]. In this work, we used a simple projection space (prereconstruction) decomposition method to estimate the material fractions ( fn) of the CaCO3 ( fcaco3, local density; 2.711 g/cm<sup>3</sup> ), PMMA ( fPMMA, local density; 1.17 g/cm3 ), and epoxy resin ( fepoxy, local density; 1.11 g/cm3 ) in the phantom.

Three basis materials can also be expressed as a linear combination of the attenuation coefficients:

$$
\mu(r,E) = \left(\frac{\mu}{\rho}\right)\_1(E) \cdot \rho\_1(r) + \left(\frac{\mu}{\rho}\right)\_2(E) \cdot \rho\_2(r) + \left(\frac{\mu}{\rho}\right)\_3(E) \cdot \rho\_3(r) \tag{12}
$$

In DE acquisition, the detected image intensity can be expressed as follows:

$$I\_L = \int P\_L(E) \exp\left\{-\left(\frac{\mu}{\rho}\right)\_1 (E) \cdot L\_1 - \left(\frac{\mu}{\rho}\right)\_2 (E) \cdot L\_2 - \left(\frac{\mu}{\rho}\right)\_3 (E) \cdot L\_3\right\} dE \tag{13}$$

$$I\_H = \int P\_H(E) \exp\left\{-\left(\frac{\mu}{\rho}\right)\_1 (E) \cdot L\_1 - \left(\frac{\mu}{\rho}\right)\_2 (E) \cdot L\_2 - \left(\frac{\mu}{\rho}\right)\_3 (E) \cdot L\_3\right\} dE \tag{14}$$

$$L\_1 \, ^\ast L\_2 \, ^\ast L\_3 = 1.0\tag{15}$$

Eqs. (13) and (14) were used to calculate the values for IL\_caco3, IL\_PMMAr, IL\_epoxy, IH\_caco3, IH\_PMMA, and IH\_epoxy as simulated attenuation intensities of these materials at the two energy levels. These values were then used to construct a sensitivity matrix, and the material fractions

Figure 2. The linear attenuation coefficients of CaCO3, epoxy, and PMMA with respect to the photons.

IL\_CaCO<sup>3</sup> IL\_PMMA IL\_epoxy IH\_CaCO<sup>3</sup> IH\_PMMA IH\_epoxy 1:0 1:0 1:0

f CaCO3IL\_CaCO<sup>3</sup> þ f PMMAIL\_PMMA þ f epoxyIL\_epoxy ¼ DTEL

f CaCO3IH\_CaCO<sup>3</sup> þ f PMMAIH\_PMMA þ f epoxyIH\_epoxy ¼ DTEH

f CaCO<sup>3</sup> þ f PMMA þ f epoxy ¼ 1:0

After decomposition by matrix inversion, the "inv" function available in MATLAB was used (Mathworks; Natick, MA, USA); this function constrains the possible fraction to [0,1] while imposing a sum of one. Accordingly, the processing pipeline yields three material fraction

�<sup>1</sup> DTEL

DTEH 1:0

State-Of-The-Art X-Ray Digital Tomosynthesis Imaging http://dx.doi.org/10.5772/intechopen.81667

(16)

75

were obtained from the inverse of this matrix (Eq. 16):

f CaCO<sup>3</sup> f PMMA f epoxy

4.2. Virtual monochromatic image processing

outputs corresponding to CaCO3, epoxy, and PMMA.

Virtual monochromatic processing is performed according to Eq. 15:

$$L\_1 = \int \rho\_1(r) dl$$

$$L\_2 = \int \rho\_2(r) dl$$

$$L\_3 = \int \rho\_3(r) dl$$

where PL(E) represents the low-energy primary intensities, PH(E) represents the high-energy primary intensities, IL represents the low-energy attenuated intensities, and IH denotes the high-energy attenuated intensities. The equivalent densities (g/cm<sup>2</sup> ; L1, L2, and L3) of the three basis materials must be determined for each ray path. Eqs. (13, 14, 15) can be solved for the equivalent area density, where L1, L2, and L<sup>3</sup> are the unknown materials. Therefore, the basis material decomposition can be accomplished by solving simultaneous equations to calculate the values of L1, L2, and L<sup>3</sup> from the measured projection pixel values [12]. By using the density corresponding to the area with the three basis materials, the linear attenuation coefficient μ(r, E) can be calculated for any photon.

We used the local and area densities for each material to calculate the theoretical linear attenuation coefficient curves shown in Figure 2; these values were generated by inputting the chemical compositions of the CaCO3, epoxy resin, and PMMA into the XCOM program developed by Berger and Hubbell [27]. The curves show that the linear attenuation coefficient of CaCO3 decreases more rapidly than those of the foam epoxy and PMMA in the energy band <100 keV. Finally, we used the projection space decomposition approach to generate material decomposition images for the CaCO3, epoxy resin image, and PMMA by using the following process.

Figure 2. The linear attenuation coefficients of CaCO3, epoxy, and PMMA with respect to the photons.

Eqs. (13) and (14) were used to calculate the values for IL\_caco3, IL\_PMMAr, IL\_epoxy, IH\_caco3, IH\_PMMA, and IH\_epoxy as simulated attenuation intensities of these materials at the two energy levels. These values were then used to construct a sensitivity matrix, and the material fractions were obtained from the inverse of this matrix (Eq. 16):

$$
\begin{bmatrix} f\_{\text{CaCO3}} \\ f\_{\text{PMMA}} \\ f\_{\text{epoxy}} \end{bmatrix} = \begin{bmatrix} I\_{L\\_\text{CaCO3}} & I\_{L\\_\text{PMMA}} & I\_{L\\_\text{epoxy}} \\ I\_{H\\_\text{CaCO3}} & I\_{H\\_\text{PMMA}} & I\_{H\\_\text{epoxy}} \\ 1.0 & 1.0 & 1.0 \end{bmatrix}^{-1} \begin{bmatrix} DT\_{EL} \\ DT\_{EH} \\ 1.0 \end{bmatrix} \tag{16}
$$

$$
f\_{\text{CaCO3}}I\_{L\\_\text{CaCO3}} + f\_{\text{PMMA}}I\_{L\\_\text{PMMA}} + f\_{\text{epoxy}}I\_{L\\_\text{epoxy}} = DT\_{EL}
$$

$$
f\_{\text{CaCO3}}I\_{H\\_\text{CaCO3}} + f\_{\text{PMMA}}I\_{H\\_\text{PMMA}} + f\_{\text{epoxy}}I\_{H\\_\text{epury}} = DT\_{EH}
$$

$$
f\_{\text{CaCO3}} + f\_{\text{PMMA}} + f\_{\text{epury}} = 1.0
$$

#### 4.2. Virtual monochromatic image processing

equivalent to the volume of the mixture (volume or mass conservation) [11]. In this work, we used a simple projection space (prereconstruction) decomposition method to estimate the

Three basis materials can also be expressed as a linear combination of the attenuation coeffi-

r � �

ð Þ� <sup>E</sup> <sup>L</sup><sup>1</sup> � <sup>μ</sup>

ð Þ� <sup>E</sup> <sup>L</sup><sup>1</sup> � <sup>μ</sup>

2

r � �

r � �

r1ð Þr dl

r2ð Þr dl

r3ð Þr dl

where PL(E) represents the low-energy primary intensities, PH(E) represents the high-energy primary intensities, IL represents the low-energy attenuated intensities, and IH denotes the

basis materials must be determined for each ray path. Eqs. (13, 14, 15) can be solved for the equivalent area density, where L1, L2, and L<sup>3</sup> are the unknown materials. Therefore, the basis material decomposition can be accomplished by solving simultaneous equations to calculate the values of L1, L2, and L<sup>3</sup> from the measured projection pixel values [12]. By using the density corresponding to the area with the three basis materials, the linear attenuation coefficient

We used the local and area densities for each material to calculate the theoretical linear attenuation coefficient curves shown in Figure 2; these values were generated by inputting the chemical compositions of the CaCO3, epoxy resin, and PMMA into the XCOM program developed by Berger and Hubbell [27]. The curves show that the linear attenuation coefficient of CaCO3 decreases more rapidly than those of the foam epoxy and PMMA in the energy band <100 keV. Finally, we used the projection space decomposition approach to generate material decomposition images for the CaCO3, epoxy resin image, and PMMA by using the following process.

2

2

� �

� �

ð Þ� <sup>E</sup> <sup>r</sup>2ð Þþ <sup>r</sup> <sup>μ</sup>

ð Þ� <sup>E</sup> <sup>L</sup><sup>2</sup> � <sup>μ</sup>

ð Þ� <sup>E</sup> <sup>L</sup><sup>2</sup> � <sup>μ</sup>

r � �

3

r � �

r � �

3 ð Þ� E L<sup>3</sup>

3 ð Þ� E L<sup>3</sup>

<sup>∗</sup> <sup>L</sup><sup>3</sup> <sup>¼</sup> <sup>1</sup>:<sup>0</sup> (15)

), and epoxy resin ( fepoxy, local density; 1.11 g/cm3

ð Þ� <sup>E</sup> <sup>r</sup>1ð Þþ <sup>r</sup> <sup>μ</sup>

In DE acquisition, the detected image intensity can be expressed as follows:

1

1

L1 <sup>∗</sup> L<sup>2</sup>

> L<sup>1</sup> ¼ ð

> L<sup>2</sup> ¼ ð

> L<sup>3</sup> ¼ ð

r � �

r � �

high-energy attenuated intensities. The equivalent densities (g/cm<sup>2</sup>

), PMMA ( fPMMA, local

ð Þ� E r3ð Þr (12)

; L1, L2, and L3) of the three

dE (13)

dE (14)

) in the phantom.

material fractions ( fn) of the CaCO3 ( fcaco3, local density; 2.711 g/cm<sup>3</sup>

<sup>μ</sup>ð Þ¼ <sup>r</sup>; <sup>E</sup> <sup>μ</sup>

IL ¼ ð

74 Medical Imaging and Image-Guided Interventions

IH ¼ ð

μ(r, E) can be calculated for any photon.

r � �

PLð Þ <sup>E</sup> exp � <sup>μ</sup>

PHð Þ <sup>E</sup> exp � <sup>μ</sup>

1

density; 1.17 g/cm3

cients:

After decomposition by matrix inversion, the "inv" function available in MATLAB was used (Mathworks; Natick, MA, USA); this function constrains the possible fraction to [0,1] while imposing a sum of one. Accordingly, the processing pipeline yields three material fraction outputs corresponding to CaCO3, epoxy, and PMMA.

Virtual monochromatic processing is performed according to Eq. 15:

$$\text{Mono\\_p\\_img} = f\_{\text{CaCO3}} \, ^\ast \left( \frac{\mu}{\rho} \right)\_{\text{CaCO3}} (E) + f\_{\text{PMMA}} \, ^\ast \left( \frac{\mu}{\rho} \right)\_{\text{PMMA}} (E) + f\_{\text{epoxy}} \, ^\ast \left( \frac{\mu}{\rho} \right)\_{\text{epury}} (E) \tag{17}$$

6. Optimization parameter

compounds in the analyzed set.

extracting structure-based information [34].

The RMSE was defined in this study as follows:

The experiments were performed according to the scheme shown in Figure 3. A range of optional parameters have been identified for IR algorithms. Among these parameters, some are important for determining algorithmic behavior. In this study, we compared the root-meansquare error (RMSE) and mean structural similarity (MSSIM; reconstructed volume image from the previous iterations between the current iteration) to optimize the iteration numbers (i).

> P<sup>n</sup> <sup>i</sup>¼<sup>1</sup> <sup>y</sup> ∧ <sup>k</sup> � yk � �<sup>2</sup>

referenced image [previous reconstructed image (in-focus plane)], and n is the number of

The MSSIM of local patterns of luminance- and contrast-normalized pixel intensity were compared to determine the structural similarity (SSIM) index of contrast preservation. This image quality metric is based on the assumed suitability of the human visual system for

where yk is the observed image [current reconstructed image (in-focus plane)], y

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

vuut (18)

State-Of-The-Art X-Ray Digital Tomosynthesis Imaging http://dx.doi.org/10.5772/intechopen.81667

∧

<sup>k</sup> is the

77

n

RMSE ¼

Figure 3. For DT acquisition, the phantom was arranged parallel to the x–y detector plane.

where Mono\_p\_img is the virtual monochromatic projection image, and [μ/r]caco3(E), [μ/ r]PMMA(E), and [μ/r]epoxy(E) are the mass attenuation coefficients of each material. The generated virtual monochromatic X-ray projection image was reconstructed by using each algorithm for energies of 60, 80, 100, 120, and 140 keV. The real projection data acquired on a DT system were used for reconstruction. All image reconstruction calculations, including DE material decomposition processing and reconstruction, as well as FBP, SART, SART-TV, virtual monochromatic processing, and MLEM, were implemented in MATLAB.
