*4.2.1. Internal scanning*

The second modality has originated from an proposal in [41]. At the time only numerical simulation and reconstruction using the Singular Value Decomposition method were performed on a turbine blade with encouraging results. One may conceive this modality as an evolved Prettyman's scatterometer, which can rotate around an axis perpendicular to its plane, realizing what Prettyman had foreseen long ago to generate more useful data for his numerical reconstruction method. In fact, as will be shown the rotational motion is necessary for the reconstruction the electron density in a trans-axial slice.

**Figure 9.** Second CST modality: internal scanning

The apparatus is is sketched in Fig. 9. An emitting radiation point source **S** is placed at a distance 2*p* from a point detector **D**. The segment *SD* joining them rotates around its middle point **O**. At site **D** is collected the single-scattered radiation flux density from the scanned object for a given angular position of the line *SD* and at a given scattering energy *E*(*ω*), (equivalently at scattering angle *ω*). Then at fixed *φ*, a multichannel analyzer records the photon counts in each energy channel. Data acquisition is performed for every angular position *φ* of *SD*.

Thus, thanks to the physics of the Compton effect, the detected radiation flux density is proportional to the integral of the electron density *n*(**M**) on a class of circular arcs sharing a chord of fixed length 2*p*, which rotates about its fixed middle point **O**.

With a polar coordinate system centered at **O**, the equation of a circular arc lying inside a circle of center *O* and radius *p* reads (see [42])

$$r = r(\cos(\theta - \phi)) = p\left(\sqrt{1 + \tau^2 \cos^2(\theta - \phi)} - \tau \cos(\theta - \phi)\right),\tag{13}$$

which is defined by two parameters *τ* and *φ*: a) *φ* is the angle made by its symmetry axis with the reference direction *Ox* and b) *τ* is related to the scattering angle *ω* by *τ* = cot *ω*. Note that *τ* is positive for 0 *< ω < π*/2 and the range of *θ* is (*φ* − *π*/2) *< θ <* (*φ* + *π*/2).

Again to simplify the notation by assuming that attenuation and beam spreading are neglected as well as by absorbing the Compton kinematic factor into one single function, we can say that the detected photon flux density *<sup>n</sup>*�(*τ*, *<sup>φ</sup>*) is the Radon transform of the electron density *n*(*r*, *θ*) along arcs of circle given by equation 13. Thus using the auxiliary angle *<sup>γ</sup>* = (*<sup>θ</sup>* <sup>−</sup> *<sup>φ</sup>*) as in the previous subsection, we may express *<sup>n</sup>*�(*τ*, *<sup>φ</sup>*) with *ds*, the integration arc element given by

$$ds = r \, d\gamma \, \sqrt{\frac{1 + \tau^2}{1 + \tau^2 \cos^2 \gamma}} = dr \, \frac{\sqrt{1 + \tau^2}}{\tau \sin \gamma}. \tag{14}$$

Now introducing the angular Fourier components of *<sup>n</sup>*�(*τ*, *<sup>φ</sup>*) and *<sup>n</sup>*(*r*, *<sup>θ</sup>*) as given by equation 9, we see that they are related by

$$\frac{\pi \widehat{n}\_l(\tau)}{\sqrt{1+\tau^2}} = 2 \int\_{p(\sqrt{1+\tau^2}-\tau)}^p \frac{dr}{\sqrt{1-\frac{1}{4\tau^2}\left(\frac{p}{r}-\frac{r}{p}\right)^2}} \cos\left[l\cos^{-1}\left(\frac{1}{2\tau}\left(\frac{p}{r}-\frac{r}{p}\right)\right)\right] \, n\_l(r). \tag{15}$$

At first this Chebyshev transform looks a bit hopeless. However if one introduces a new variable *g* defined by

$$g = \frac{1}{2} \left( \frac{p}{r} - \frac{r}{p} \right) . \tag{16}$$

one can put it under the form

$$\frac{\tau \widehat{n}\_l(\tau)}{\sqrt{1+\tau^2}} = 2 \int\_0^\tau d\mathbf{g} \, \frac{\cos l \cos^{-1}\left(\frac{\mathbf{g}}{\tau}\right)}{\sqrt{1-\frac{\mathbf{g}^2}{\tau^2}}} \, \frac{p(\sqrt{1+\mathbf{g}^2}-\mathbf{g})}{\sqrt{1+\mathbf{g}^2}} \, n\_l(p(\sqrt{1+\mathbf{g}^2}-\mathbf{g})) . \tag{17}$$

Then defining new functions by

$$\hat{N}\_l(\tau) = \frac{\tau \hat{n}\_l(\tau)}{\sqrt{1 + \tau^2}} \quad \text{and} \quad N\_l(\mathbf{g}) = \frac{p(\sqrt{1 + \mathbf{g}^2} - \mathbf{g})}{\sqrt{1 + \mathbf{g}^2}} n\_l(p(\sqrt{1 + \mathbf{g}^2} - \mathbf{g})), \tag{18}$$

we obtain

12 Will-be-set-by-IN-TECH

There are two scanning modes in this modality. If the object is "small", or can be put inside a circle of radius *p* (a mechanical adjustable parameter), one can make an "internal" scanning inside this circle. If the object is "large" or situated far way from the observer, one can use the

The second modality has originated from an proposal in [41]. At the time only numerical simulation and reconstruction using the Singular Value Decomposition method were performed on a turbine blade with encouraging results. One may conceive this modality as an evolved Prettyman's scatterometer, which can rotate around an axis perpendicular to its plane, realizing what Prettyman had foreseen long ago to generate more useful data for his numerical reconstruction method. In fact, as will be shown the rotational motion is necessary

The apparatus is is sketched in Fig. 9. An emitting radiation point source **S** is placed at a distance 2*p* from a point detector **D**. The segment *SD* joining them rotates around its middle point **O**. At site **D** is collected the single-scattered radiation flux density from the scanned object for a given angular position of the line *SD* and at a given scattering energy *E*(*ω*), (equivalently at scattering angle *ω*). Then at fixed *φ*, a multichannel analyzer records the photon counts in each energy channel. Data acquisition is performed for every angular

Thus, thanks to the physics of the Compton effect, the detected radiation flux density is proportional to the integral of the electron density *n*(**M**) on a class of circular arcs sharing

With a polar coordinate system centered at **O**, the equation of a circular arc lying inside a circle

<sup>1</sup> + *<sup>τ</sup>*<sup>2</sup> cos2 (*<sup>θ</sup>* − *<sup>φ</sup>*) − *<sup>τ</sup>* cos (*<sup>θ</sup>* − *<sup>φ</sup>*)

, (13)

a chord of fixed length 2*p*, which rotates about its fixed middle point **O**.

**4.2. Second CST modality (Nguyen-Truong 2010)**

*4.2.1. Internal scanning*

"external" scanning mode. The two modes shall be discussed separately.

for the reconstruction the electron density in a trans-axial slice.

**Figure 9.** Second CST modality: internal scanning

of center *O* and radius *p* reads (see [42])

*r* = *r*(cos(*θ* − *φ*)) = *p*

position *φ* of *SD*.

$$\hat{N}\_l(\tau) = 2 \int\_0^\tau d\mathbf{g} \, \frac{\cos l \cos^{-1} \left(\frac{\mathcal{S}}{\tau}\right)}{\sqrt{1 - \frac{\mathcal{S}^2}{\tau^2}}} \, N\_l(\mathbf{g})\_\prime \tag{19}$$

which is precisely of the form of equation 10, obtained in the Radon problem on circles passing through a fixed point. Hence an inversion formula exists in the (*τ*, *g*) variables, *i.e.*

$$N\_l(\mathbf{g}) = \frac{1}{\pi \mathcal{g}} \left( \int\_0^\mathcal{g} dp \, \frac{d\widehat{\mathbf{N}}\_l(\tau)}{d\tau} \, \frac{\left( (\mathbf{g}/\tau) - \sqrt{(\mathbf{g}/\tau)^2 - 1} \right)^l}{\sqrt{(\mathbf{g}/\tau)^2 - 1}} - \int\_\mathcal{g}^\infty d\tau \, \frac{d\widehat{\mathbf{N}}\_l(\tau)}{d\tau} \, \mathcal{U}\_{l-1}(\mathbf{g}/\tau) \right). \tag{20}$$

#### 14 Will-be-set-by-IN-TECH 114 Numerical Simulation – From Theory to Industry Recent Developments on Compton Scatter Tomography: Theory and Numerical Simulations <sup>15</sup>

A closed form of the inversion formula can be deduced from equation 8

$$N(\mathbf{g}, \theta) = \frac{1}{2\pi^2 \mathcal{g}} \int\_0^{2\pi} d\phi \int\_0^\infty d\tau \,\frac{\partial \dot{N}(\tau, \phi)}{\partial \tau} \frac{1}{\mathbf{g}/\tau - \cos(\theta - \phi)}.\tag{21}$$

on this arc of circle is now expressed by the following Chebyshev integral transform for its

This equation will take the form of equation 10, as equation 19, when the intermediate variable

) = *<sup>p</sup>*(

are used. Finally, following the same steps as in the previous scanning mode, the

In this subsection we discuss two scanning modes of a third CST modality proposed in [54]. This modality has originated from a search for curves in the plane such that the Radon transform on these curves may be reduced to a Chebyshev integral transform of the form of equation 10, for circles passing through a fixed point. It turns out that Radon transforms on arcs of circles orthogonal to a fixed circle *do have this property*. The pair source-detector still moves on a circle of radius *p* (still an adjustable parameter) and centered at **O**, the origin of a polar coordinate system. But their separation distance is no longer constant as in the second CST modality: it depends on the scattering angle *ω*. The positions of **S** and **D** are given by an

1 2*τ r <sup>p</sup>* <sup>−</sup> *<sup>p</sup> r* 

*nl*(*r*) cos

Recent Developments on Compton Scatter Tomography: Theory and Numerical Simulations 115

1 + *g*�<sup>2</sup> + *g*�

1 + *g*�<sup>2</sup>

)

1

*nl*(*p*( 

− cos(*θ* − *φ*)

*<sup>τ</sup>*<sup>2</sup> cos2(*<sup>θ</sup>* − *<sup>φ</sup>*) − <sup>1</sup>), (28)

*l* cos−<sup>1</sup>

 1 2*τ*  *r <sup>p</sup>* <sup>−</sup> *<sup>p</sup> r*

, (25)

1 + *g*�<sup>2</sup> + *g*�

*∂*

*∂τ <sup>τ</sup> <sup>n</sup>*(*τ*, *<sup>φ</sup>*) √ 1 + *τ*<sup>2</sup>

. (24)

)), (26)

 .

(27)

*dr*

*<sup>g</sup>*� <sup>=</sup> <sup>1</sup> 2 *r <sup>p</sup>* <sup>−</sup> *<sup>p</sup> r* 

angular components

*p*(

*p*

*<sup>N</sup><sup>l</sup>*(*τ*) = *<sup>τ</sup> <sup>n</sup><sup>l</sup>*(*τ*) √ 1 + *τ*<sup>2</sup>

> 1 + <sup>1</sup> 4 *r <sup>p</sup>* <sup>−</sup> *<sup>p</sup> r* 2

reconstruction formula for *n*(*r*, *θ*) reads

*r* 2 *r <sup>p</sup>* <sup>−</sup> *<sup>p</sup> r* 

**4.3. Third CST modality (Truong-Nguyen 2011)**

<sup>√</sup>1+*τ*<sup>2</sup>+*τ*)

<sup>1</sup> <sup>−</sup> <sup>1</sup> 4*τ*<sup>2</sup> *r <sup>p</sup>* <sup>−</sup> *<sup>p</sup> r* 2

and *Nl*(*g*�

 2*π* 0

opening angle *γ*0, measured from the symmetry axis of the circular arc.

*r* = *p*(*τ* cos(*θ* − *φ*) −

In this case, the Radon transform is defined on the following circular arc of equation

where *τ* = 1/ cos *γ*<sup>0</sup> and −*γ*<sup>0</sup> *<* (*θ* − *φ*) *< γ*0. Inspection of Fig. 11 shows that *γ*<sup>0</sup> = (*π*/2 − *ω*) for 0 *< ω < π*/2. Hence *τ* = 1/ sin *ω >* 1. So data acquisition works as follows. For fixed *φ*, the pair source - detector must be *simultaneously* displaced on the circle of radius *<sup>p</sup>* so that the opening angle *SOD* <sup>=</sup> <sup>2</sup>*γ*<sup>0</sup> = (*<sup>π</sup>* <sup>−</sup> <sup>2</sup>*ω*), before registering at **<sup>D</sup>** a photon count. The number of isogonic lines is thus equal to the number of positions of the pair (**S**, **D**).

*dφ* ∞ 0 *dτ*

<sup>1</sup> <sup>+</sup> *<sup>τ</sup>*<sup>2</sup> <sup>=</sup> <sup>2</sup>

and the functions

*<sup>n</sup>*(*r*, *<sup>θ</sup>*) = <sup>1</sup>

2*π*<sup>2</sup>

*4.3.1. Internal scanning*

*<sup>τ</sup> <sup>n</sup><sup>l</sup>*(*τ*) √

Finally going back to the original functions *<sup>n</sup>*(*r*, *<sup>θ</sup>*) and *<sup>n</sup>*(*τ*, *<sup>φ</sup>*), via equations 18, the reconstructed electron density is

$$n(r,\theta) = \frac{1}{2\pi^2} \frac{\sqrt{1 + \frac{1}{4}\left(\frac{p}{r} - \frac{r}{p}\right)^2}}{\frac{r}{2}\left(\frac{p}{r} - \frac{r}{p}\right)} \int\_0^{2\pi} d\phi \int\_0^\infty d\tau \frac{1}{\frac{1}{2\pi}\left(\frac{p}{r} - \frac{r}{p}\right) - \cos(\theta - \phi)} \frac{\partial}{\partial\tau} \left(\frac{\tau\hat{n}(\tau,\phi)}{\sqrt{1 + \tau^2}}\right). \tag{22}$$

#### *4.2.2. External scanning*

This scanning mode is illustrated by Fig. 10. Data acquisition is the same as for internal scanning, except for an object of compact support, the rotational motion may be replaced by a back and forth radar type sweeping motion.

**Figure 10.** Second CST modality: external scanning

The working is similar to the internal scanning mode, except that one uses the scattering angle range *π*/2 *< ω < π* and the "external" arc of circle given by the equation

$$r = r(\cos(\theta - \phi)) = p\left(\sqrt{1 + \tau^2 \cos^2(\theta - \phi)} + \tau \cos(\theta - \phi)\right),\tag{23}$$

where *τ* = − cot *ω >* 0 and (*φ* − *π*/2) *< θ <* (*φ* + *π*/2). This arc of circle Radon transform has not yet been considered in [42]. Similarly the Radon transform of the electron density on this arc of circle is now expressed by the following Chebyshev integral transform for its angular components

$$\frac{2\pi\hat{n}\_l(\tau)}{\sqrt{1+\tau^2}} = 2\int\_p^{p(\sqrt{1+\tau^2}+\tau)} \frac{dr}{\sqrt{1-\frac{1}{4\tau^2}\left(\frac{r}{p}-\frac{p}{r}\right)^2}} \,\eta\_l(r)\,\cos\left[l\cos^{-1}\left(\frac{1}{2\tau}\left(\frac{r}{p}-\frac{p}{r}\right)\right)\right]. \tag{24}$$

This equation will take the form of equation 10, as equation 19, when the intermediate variable

$$g' = \frac{1}{2} \left( \frac{r}{p} - \frac{p}{r} \right),\tag{25}$$

and the functions

14 Will-be-set-by-IN-TECH

Finally going back to the original functions *<sup>n</sup>*(*r*, *<sup>θ</sup>*) and *<sup>n</sup>*(*τ*, *<sup>φ</sup>*), via equations 18, the

This scanning mode is illustrated by Fig. 10. Data acquisition is the same as for internal scanning, except for an object of compact support, the rotational motion may be replaced by

The working is similar to the internal scanning mode, except that one uses the scattering angle

where *τ* = − cot *ω >* 0 and (*φ* − *π*/2) *< θ <* (*φ* + *π*/2). This arc of circle Radon transform has not yet been considered in [42]. Similarly the Radon transform of the electron density

<sup>1</sup> + *<sup>τ</sup>*<sup>2</sup> cos2 (*<sup>θ</sup>* − *<sup>φ</sup>*) + *<sup>τ</sup>* cos (*<sup>θ</sup>* − *<sup>φ</sup>*)

, (23)

range *π*/2 *< ω < π* and the "external" arc of circle given by the equation

1 2*τ p <sup>r</sup>* <sup>−</sup> *<sup>r</sup> p* 

*∂N*(*τ*, *φ*) *∂τ*

1 *g*/*τ* − cos(*θ* − *φ*)

− cos(*θ* − *φ*)

*∂ ∂τ*

1

. (21)

 *<sup>τ</sup> <sup>n</sup>*(*τ*, *<sup>φ</sup>*) √ 1 + *τ*<sup>2</sup>

 .

(22)

A closed form of the inversion formula can be deduced from equation 8

 2*π* 0

> 2*π* 0

*dφ* ∞ 0 *dτ*

*dφ* ∞ 0 *dτ*

*<sup>N</sup>*(*g*, *<sup>θ</sup>*) = <sup>1</sup>

reconstructed electron density is

 1 + <sup>1</sup> 4 *p <sup>r</sup>* <sup>−</sup> *<sup>r</sup> p* 2

> *r* 2 *p <sup>r</sup>* <sup>−</sup> *<sup>r</sup> p*

a back and forth radar type sweeping motion.

**Figure 10.** Second CST modality: external scanning

*r* = *r*(cos(*θ* − *φ*)) = *p*

2*π*<sup>2</sup>

*4.2.2. External scanning*

*<sup>n</sup>*(*r*, *<sup>θ</sup>*) = <sup>1</sup>

2*π*2*g*

$$\hat{N}\_l(\tau) = \frac{\tau \,\hat{n}\_l(\tau)}{\sqrt{1+\tau^2}} \quad \text{and} \quad N\_l(\mathbf{g'}) = \frac{p(\sqrt{1+\mathbf{g'}^2} + \mathbf{g'})}{\sqrt{1+\mathbf{g'}^2}} \, n\_l(p(\sqrt{1+\mathbf{g'}^2} + \mathbf{g'})) , \tag{26}$$

are used. Finally, following the same steps as in the previous scanning mode, the reconstruction formula for *n*(*r*, *θ*) reads

$$m(r,\theta) = \frac{1}{2\pi^2} \frac{\sqrt{1 + \frac{1}{4}\left(\frac{r}{p} - \frac{p}{r}\right)^2}}{\frac{r}{2}\left(\frac{r}{p} - \frac{p}{r}\right)} \int\_0^{2\pi} d\phi \int\_0^\infty d\tau \frac{1}{\frac{1}{2\tau}\left(\frac{r}{p} - \frac{p}{r}\right) - \cos(\theta - \phi)} \frac{\partial}{\partial\tau} \left(\frac{\tau\hat{n}(\tau,\phi)}{\sqrt{1 + \tau^2}}\right). \tag{27}$$

#### **4.3. Third CST modality (Truong-Nguyen 2011)**

In this subsection we discuss two scanning modes of a third CST modality proposed in [54]. This modality has originated from a search for curves in the plane such that the Radon transform on these curves may be reduced to a Chebyshev integral transform of the form of equation 10, for circles passing through a fixed point. It turns out that Radon transforms on arcs of circles orthogonal to a fixed circle *do have this property*. The pair source-detector still moves on a circle of radius *p* (still an adjustable parameter) and centered at **O**, the origin of a polar coordinate system. But their separation distance is no longer constant as in the second CST modality: it depends on the scattering angle *ω*. The positions of **S** and **D** are given by an opening angle *γ*0, measured from the symmetry axis of the circular arc.

#### *4.3.1. Internal scanning*

In this case, the Radon transform is defined on the following circular arc of equation

$$r = p(\tau \cos(\theta - \phi) - \sqrt{\tau^2 \cos^2(\theta - \phi) - 1}),\tag{28}$$

where *τ* = 1/ cos *γ*<sup>0</sup> and −*γ*<sup>0</sup> *<* (*θ* − *φ*) *< γ*0. Inspection of Fig. 11 shows that *γ*<sup>0</sup> = (*π*/2 − *ω*) for 0 *< ω < π*/2. Hence *τ* = 1/ sin *ω >* 1. So data acquisition works as follows. For fixed *φ*, the pair source - detector must be *simultaneously* displaced on the circle of radius *<sup>p</sup>* so that the opening angle *SOD* <sup>=</sup> <sup>2</sup>*γ*<sup>0</sup> = (*<sup>π</sup>* <sup>−</sup> <sup>2</sup>*ω*), before registering at **<sup>D</sup>** a photon count. The number of isogonic lines is thus equal to the number of positions of the pair (**S**, **D**).

**Figure 11.** Third CST modality: internal scanning

The integration arc element being now

$$ds = r d\gamma \sqrt{\frac{\tau^2 - 1}{\tau^2 \cos^2 \gamma} - 1} = dr \, \frac{\sqrt{\tau^2 - 1}}{\tau \sin \gamma} \, \tag{29}$$

in [54]. Consequently we recover the electron density in closed form as

 2*π* 0

*dφ* ∞ 0 *dτ*

1 2*τ p <sup>r</sup>* <sup>+</sup> *<sup>r</sup> p* 

This scanning mode is shown in Fig. 12. It has the same data acquisition procedure as for internal scanning. It is appropriate for large objects or objects situated far away from the scanner, *e.g.* buried objects underground or undersea, with a back and forth radar type of

The Radon transform is now defined on the external circular arc with respect to the reference

Here *<sup>π</sup>*/2 *<sup>&</sup>lt; <sup>ω</sup> <sup>&</sup>lt; <sup>π</sup>* and *<sup>γ</sup>*<sup>0</sup> = (*<sup>ω</sup>* <sup>−</sup> *<sup>π</sup>*/2) 2. As for the case of internal scanning, we have also

The integration arc element is the same as in the previous scanning mode, see equation 29. Then the Radon transform of the electron density *<sup>n</sup>*(*τ*, *<sup>φ</sup>*) becomes, in terms of function

*nl*(*r*) cos

*l* cos−<sup>1</sup>

*<sup>r</sup>* <sup>=</sup> *<sup>p</sup>*(*<sup>τ</sup>* cos(*<sup>θ</sup>* <sup>−</sup> *<sup>φ</sup>*) +

angular components, the following Chebyshev transform

<sup>2</sup> Note that the two arcs of equations 28 and 35 belong to the same circle.

<sup>1</sup> <sup>−</sup> <sup>1</sup> 4*τ*<sup>2</sup> *p <sup>r</sup>* <sup>+</sup> *<sup>r</sup> p* 2

<sup>√</sup>*τ*<sup>2</sup>−1)

*τ* = 1/ cos *γ*<sup>0</sup> and −*γ*<sup>0</sup> *<* (*θ* − *φ*) *< γ*<sup>0</sup> with the same data acquisition procedure.

*dr*

1

Recent Developments on Compton Scatter Tomography: Theory and Numerical Simulations 117

− cos(*θ* − *φ*)

*<sup>τ</sup>*<sup>2</sup> cos2(*<sup>θ</sup>* − *<sup>φ</sup>*) − <sup>1</sup>). (35)

 1 2*τ*  *p r* + *r p* . (36)

*∂*

*∂τ <sup>τ</sup> <sup>n</sup>*(*τ*, *<sup>φ</sup>*) <sup>√</sup>*τ*<sup>2</sup> <sup>−</sup> <sup>1</sup>  .

(34)

*<sup>n</sup>*(*r*, *<sup>θ</sup>*) = <sup>1</sup>

2*π*<sup>2</sup>

*4.3.2. External scanning*

scanning motion.

 1 4 *p <sup>r</sup>* <sup>+</sup> *<sup>r</sup> p* 2 − 1

> *r* 2 *p <sup>r</sup>* <sup>+</sup> *<sup>r</sup> p*

**Figure 12.** Third CST modality: external scanning

circle of radius *p* of equation

*p*(*τ*−

*p*

*<sup>τ</sup> <sup>n</sup>l*(*τ*) <sup>√</sup>*τ*<sup>2</sup> <sup>−</sup> <sup>1</sup> <sup>=</sup> <sup>2</sup>

the Radon transform of the electron density *<sup>n</sup>*(*τ*, *<sup>φ</sup>*) becomes, in terms of the angular components *nl*(*r*) and *<sup>n</sup><sup>l</sup>*(*τ*), a Chebyshev transform similar to equation 17, *i.e.*

$$\frac{\pi \,\hat{n}\_l(\tau)}{\sqrt{\tau^2 - 1}} = 2 \int\_{p(\tau - \sqrt{\tau^2 - 1})}^p \frac{dr}{\sqrt{1 - \frac{1}{4\tau^2} \left(\frac{p}{r} + \frac{r}{p}\right)^2}} \, n\_l(r) \, \cos\left[l \cos^{-1}\left(\frac{1}{2\tau} \left(\frac{p}{r} + \frac{r}{p}\right)\right)\right]. \tag{30}$$

The structural similarity with equations 15 and 24 suggests an intermediate variable *g*" of the form

$$g'' = \frac{1}{2} \left( \frac{r}{p} + \frac{p}{r} \right) \, \tag{31}$$

which yields the following Chebyshev transform

$$\frac{\tau \widehat{n}\_l(\tau)}{\sqrt{\tau^2 - 1}} = 2 \int\_1^\tau d\mathbf{g''} \, \frac{\cos l \cos^{-1} \left(\frac{\mathbf{g''}}{\tau}\right)}{\sqrt{1 - \frac{\mathbf{g''}^2}{\tau^2}}} \, \frac{p(\mathbf{g''} - \sqrt{\mathbf{g''}^2 - 1})}{\sqrt{\mathbf{g''}^2 - 1}} \, n\_l(p(\mathbf{g''} - \sqrt{\mathbf{g''}^2 - 1})). \tag{32}$$

Now redefining the functions by

$$
\hat{N}\_l(\tau) = \frac{\tau \hat{n}\_l(\tau)}{\sqrt{\tau^2 - 1}} \quad \text{and} \quad N\_l(\mathbf{g''}) = \frac{p(\sqrt{\mathbf{g''} - \mathbf{g''}^2 - 1})}{\sqrt{\mathbf{g''}^2 - 1}} \, n\_l(p(\mathbf{g''} - \sqrt{\mathbf{g''}^2 - 1})), \tag{33}
$$

we obtain the form of equation 10, nevertheless with a lower integration bound equal to *τ* = 1 and not zero. This does not spoil the inversion procedure based on the identity 11, as shown in [54]. Consequently we recover the electron density in closed form as

$$n(r,\theta) = \frac{1}{2\pi^2} \frac{\sqrt{\frac{1}{4}\left(\frac{p}{r} + \frac{r}{p}\right)^2 - 1}}{\frac{r}{2}\left(\frac{p}{r} + \frac{r}{p}\right)} \int\_0^{2\pi} d\phi \int\_0^\infty d\tau \frac{1}{\frac{1}{2\tau}\left(\frac{p}{r} + \frac{r}{p}\right) - \cos(\theta - \phi)} \frac{\partial}{\partial\tau}\left(\frac{\tau\hat{n}(\tau,\phi)}{\sqrt{\tau^2 - 1}}\right). \tag{34}$$

#### *4.3.2. External scanning*

16 Will-be-set-by-IN-TECH

**Figure 11.** Third CST modality: internal scanning

*ds* = *r dγ*

cos *l* cos−<sup>1</sup>

 <sup>1</sup> <sup>−</sup> *<sup>g</sup>*"2 *τ*2

<sup>1</sup> <sup>−</sup> <sup>1</sup> 4*τ*<sup>2</sup> *p <sup>r</sup>* <sup>+</sup> *<sup>r</sup> p* 2

components *nl*(*r*) and *<sup>n</sup><sup>l</sup>*(*τ*), a Chebyshev transform similar to equation 17, *i.e.*

*dr*

*<sup>g</sup>*" <sup>=</sup> <sup>1</sup> 2 *r p* <sup>+</sup> *<sup>p</sup> r* 

 *g*" *τ* 

and *Nl*(*g*") = *<sup>p</sup>*(

*<sup>τ</sup>*<sup>2</sup> − <sup>1</sup>

*<sup>τ</sup>*<sup>2</sup> cos2 *<sup>γ</sup>* <sup>−</sup> <sup>1</sup> <sup>=</sup> *dr*

the Radon transform of the electron density *<sup>n</sup>*(*τ*, *<sup>φ</sup>*) becomes, in terms of the angular

The structural similarity with equations 15 and 24 suggests an intermediate variable *g*" of the

*nl*(*r*) cos

*<sup>p</sup>*(*g*" <sup>−</sup> *g*"<sup>2</sup> <sup>−</sup> <sup>1</sup>) *g*"<sup>2</sup> <sup>−</sup> <sup>1</sup>

*g*" <sup>−</sup> *<sup>g</sup>*"<sup>2</sup> <sup>−</sup> <sup>1</sup>) *g*"<sup>2</sup> <sup>−</sup> <sup>1</sup>

we obtain the form of equation 10, nevertheless with a lower integration bound equal to *τ* = 1 and not zero. This does not spoil the inversion procedure based on the identity 11, as shown

<sup>√</sup>*τ*<sup>2</sup> <sup>−</sup> <sup>1</sup>

 *l* cos−<sup>1</sup>

*<sup>τ</sup>* sin *<sup>γ</sup>* , (29)

 *p r* + *r p*

, (31)

*<sup>g</sup>*"<sup>2</sup> − <sup>1</sup>)). (32)

*<sup>g</sup>*"<sup>2</sup> − <sup>1</sup>)), (33)

. (30)

 1 2*τ*

*nl*(*p*(*g*" −

*nl*(*p*(*g*" −

The integration arc element being now

 *p p*(*τ*−

<sup>√</sup>*τ*<sup>2</sup>−1)

which yields the following Chebyshev transform

 *τ* 1 *dg*"

Now redefining the functions by

<sup>√</sup>*τ*<sup>2</sup> <sup>−</sup> <sup>1</sup>

*<sup>N</sup><sup>l</sup>*(*τ*) = *<sup>τ</sup> <sup>n</sup><sup>l</sup>*(*τ*)

*<sup>τ</sup> <sup>n</sup><sup>l</sup>*(*τ*) <sup>√</sup>*τ*<sup>2</sup> <sup>−</sup> <sup>1</sup> <sup>=</sup> <sup>2</sup>

> *<sup>τ</sup> <sup>n</sup><sup>l</sup>*(*τ*) <sup>√</sup>*τ*<sup>2</sup> <sup>−</sup> <sup>1</sup> <sup>=</sup> <sup>2</sup>

form

This scanning mode is shown in Fig. 12. It has the same data acquisition procedure as for internal scanning. It is appropriate for large objects or objects situated far away from the scanner, *e.g.* buried objects underground or undersea, with a back and forth radar type of scanning motion.

**Figure 12.** Third CST modality: external scanning

The Radon transform is now defined on the external circular arc with respect to the reference circle of radius *p* of equation

$$r = p(\tau \cos(\theta - \phi) + \sqrt{\tau^2 \cos^2(\theta - \phi) - 1}). \tag{35}$$

Here *<sup>π</sup>*/2 *<sup>&</sup>lt; <sup>ω</sup> <sup>&</sup>lt; <sup>π</sup>* and *<sup>γ</sup>*<sup>0</sup> = (*<sup>ω</sup>* <sup>−</sup> *<sup>π</sup>*/2) 2. As for the case of internal scanning, we have also *τ* = 1/ cos *γ*<sup>0</sup> and −*γ*<sup>0</sup> *<* (*θ* − *φ*) *< γ*<sup>0</sup> with the same data acquisition procedure.

The integration arc element is the same as in the previous scanning mode, see equation 29. Then the Radon transform of the electron density *<sup>n</sup>*(*τ*, *<sup>φ</sup>*) becomes, in terms of function angular components, the following Chebyshev transform

$$\frac{2\pi\widehat{n}\_l(\tau)}{\sqrt{\tau^2 - 1}} = 2\int\_p^{p(\tau-\sqrt{\tau^2-1})} \frac{dr}{\sqrt{1 - \frac{1}{4\tau^2}\left(\frac{p}{r} + \frac{r}{p}\right)^2}} \, n\_l(r) \, \cos\left[l \cos^{-1}\left(\frac{1}{2\tau}\left(\frac{p}{r} + \frac{r}{p}\right)\right)\right]. \tag{36}$$

<sup>2</sup> Note that the two arcs of equations 28 and 35 belong to the same circle.

#### 18 Will-be-set-by-IN-TECH 118 Numerical Simulation – From Theory to Industry Recent Developments on Compton Scatter Tomography: Theory and Numerical Simulations <sup>19</sup>

We see that the intermediate variable *g*" of equation (31) can be used again so that

$$\frac{\tau \widehat{m}\_l(\tau)}{\sqrt{\tau^2 - 1}} = 2 \int\_1^\tau d\mathbf{g''} \, \frac{\cos l \cos^{-1} \left(\frac{\mathbf{g''}}{\tau}\right)}{\sqrt{1 - \frac{\mathbf{g''}^2}{\tau^2}}} \, \frac{p(\mathbf{g''} + \sqrt{\mathbf{g''}^2 - 1})}{\sqrt{\mathbf{g''}^2 - 1}} \, n\_l(p(\mathbf{g''} + \sqrt{\mathbf{g'}^2 - 1})). \tag{37}$$

where *K* is the maximal value taken by *k*. Now introducing the indefinite integrals

and *Jl*(*x*) = � *<sup>x</sup>*

Recent Developments on Compton Scatter Tomography: Theory and Numerical Simulations 119

(*<sup>l</sup>* − <sup>2</sup>)*Il*(*x*) = <sup>2</sup> (tan *<sup>x</sup>* sin(*<sup>l</sup>* − <sup>2</sup>)*<sup>x</sup>* − cos(*<sup>l</sup>* − <sup>2</sup>)*x*) − *l Il*−2(*x*), (44)

<sup>2</sup> tan−<sup>1</sup> *<sup>x</sup>* and *<sup>J</sup>*1(*x*) = ln � *<sup>x</sup>*

*<sup>δ</sup>* <sup>=</sup> *<sup>n</sup>*��

*K*−1 ∑ *k*=*j n*�� *dz* sin *lz* cos2 *z*

*<sup>x</sup>l*(<sup>1</sup> <sup>+</sup> *<sup>x</sup>*<sup>2</sup> <sup>−</sup> (*<sup>l</sup>* <sup>+</sup> <sup>2</sup>)*Jl*−1(*x*), (45)

� +

*<sup>l</sup>*(*k*)(*Jl*((*k* + 1)*δ*) − *Jl*(*kδ*))

1 2(1 + *x*2)

*<sup>l</sup>*(*k*), *i.e.*

⎞


{I*o*(*i*, *<sup>j</sup>*)} ,

⎠ . (48)

*<sup>l</sup>*(*k*). (47)

. (46)

√ 1 + *x*<sup>2</sup> , (43)

*dz <sup>e</sup>*−*lz* cosh2 *z*

*l Jl*<sup>+</sup>1(*x*) = <sup>−</sup> <sup>1</sup>

The derivatives of *dn*�*l*(*p*)/*dp* are replaced by a simple linear interpolation *<sup>n</sup>*��

*<sup>l</sup>*(*k*)(*Il*((*k* + 1)*δ*) − *Il*(*kδ*)) −

*dp* � *nl*(*<sup>k</sup>* <sup>+</sup> <sup>1</sup>) <sup>−</sup> *nl*(*k*)

Then the discretized value of the reconstructed angular component of *nl*(*r*), with *r* ∼ *jδ*, is

In this subsection, we present numerical simulations on the first CST modality applied to the Shepp-Logan medical phantom, see [50]. The source **S** is placed below on the left of the image and the detector moves along the line *SD*. The relevant space is represented by 256 × 256 (length unit)2. Let *N<sup>φ</sup>* be the number of angular steps and let *Np* be the number of radii for circles going through the origin *O*. Then the following sampling steps *δφ* = 2*π*/*N<sup>φ</sup>* and *δ* = 4 256/*Np* are taken. Since there is no rotation around the object, the (*φ*, *p*)-space is very large in comparison to the studied image. To have a good representation of the object, we need a

To estimate the reconstruction quality, we use the normalized mean square error (*NMSE*) and

{I*o*(*i*, *<sup>j</sup>*)}<sup>2</sup> and *NMAE* <sup>=</sup> <sup>100</sup>

*N*<sup>2</sup>

∑ (*i*,*j*)∈[1,*N*]<sup>2</sup>

> max (*i*,*j*)∈[1,*N*]<sup>2</sup>

the normalized mean absolute error (*NMAE*) (expressed as a percentage), defined by

2

we see that they obey the following recursion relations, see [50]

1

*dn*�*l*(*p*)

Equation 48 shall be used in numerical simulations3.

large maximum value of *p*, (*e.g.* four times the image size).


<sup>3</sup> For difficulties on the handling of the inversion formula of S. J. Norton, see [55]

*Il*(*x*) = � *<sup>x</sup>*

with *I*0(*x*) = 1/ cos *x* and *I*1(*x*) = −2 ln(cos *x*), and

<sup>2</sup>(<sup>1</sup> <sup>+</sup> *<sup>x</sup>*2) <sup>+</sup>

with

*<sup>J</sup>*0(*x*) = *<sup>x</sup>*

*nl*(*jδ*) = <sup>1</sup>

*NMSE* <sup>=</sup> <sup>100</sup>

*N*<sup>2</sup>

∑ (*i*,*j*)∈[1,*N*]<sup>2</sup>

> max (*i*,*j*)∈[1,*N*]<sup>2</sup>

*π*

**5.2. Simulation results**

⎛ ⎝ *j*−1 ∑ *k*=1 *n*��

Now redefining the functions by

$$
\hat{N}\_l(\tau) = \frac{\tau \hat{n}\_l(\tau)}{\sqrt{\tau^2 - 1}} \quad \text{and} \quad N\_l(\mathbf{g''}) = \frac{p(\sqrt{\mathbf{g''} + \mathbf{g''}^2 - 1})}{\sqrt{\mathbf{g''}^2 - 1}} \ n\_l(p(\mathbf{g''} + \sqrt{\mathbf{g''}^2 - 1})), \tag{38}
$$

we obtain the form of equation 10, and surprisingly the same equation as for internal scanning. Consequently one end up with the same reconstruction formula (34), which is a remarkable advantage for this modality. This is due to the fact that the two arcs are on the same circle.
