**4.1. First CST modality (Norton 1994)**

The first CST scanner which integrates the notion of Radon transform on Kondic isogonic arcs was proposed by S. J. Norton [44]. At that time it was known that A. M. Cormack [16] had established that the Radon transform on circles intersecting a fixed point in the plane is invertible. Norton proposed to use this result in the conception of his CST scanner, which was patented in 1995, see [45]. The Norton scanner has a gamma-ray source at fixed point **S** and a detector **D** movable on a line intersecting the source site, as sketched in Fig. 8.

**Figure 8.** Fisrt CST modality

The point source **S** emits primary radiation towards an object, of which **M** is a running point. A point detector **D** moves along an *Ox*-axis and collects, at given energy *E*(*ω*), scattered radiation from the object. The physics of Compton scattering demands that the registered radiation flux energy *<sup>n</sup>*(**D**) at site **<sup>D</sup>** is due to the contribution of all scattering sites **<sup>M</sup>** lying on an arc of circle from **S** to **D** subtending an angle (*π* − *ω*), where *ω* is the scattering angle corresponding to the outgoing energy *E*(*ω*), as given by equation 1.

Mathematically, *<sup>n</sup>*(**D**) is essentially the integral of the object electron density *<sup>n</sup>*(**M**) on such arc of circles, when radiation attenuation and beam spreading effects due to radiation propagation are neglected. If polar coordinates with **S** as origin are used, then a running point **M** on a circle of diameter *p*, with center at the point of polar coordinates (*p*/2, *φ*), is given by (*r*, *θ*) with *r* = *p* cos(*θ* − *φ*). Thus, calling *γ* = (*θ* − *φ*) and recalling that the circle arc element is *ds* <sup>=</sup> *p dγ*, we have *<sup>n</sup>*�(**D**) = *<sup>n</sup>*�(*p*, *<sup>φ</sup>*) with

10 Will-be-set-by-IN-TECH

modalities, corresponding to three choices of circle families in the plane. To remain on the track of the basic ideas, attenuation and beam spreading factors shall not be taken into account at first since solving such a general problem would be out of reach. Consequently we shall first concentrate on the problem of integral transform inversion. Attenuation and propagation spreading would be dealt with later. We shall see that in some cases the beam spreading effect can be included in the exact solution but attenuation must be either corrected or compensated

The first CST scanner which integrates the notion of Radon transform on Kondic isogonic arcs was proposed by S. J. Norton [44]. At that time it was known that A. M. Cormack [16] had established that the Radon transform on circles intersecting a fixed point in the plane is invertible. Norton proposed to use this result in the conception of his CST scanner, which was patented in 1995, see [45]. The Norton scanner has a gamma-ray source at fixed point **S** and a

The point source **S** emits primary radiation towards an object, of which **M** is a running point. A point detector **D** moves along an *Ox*-axis and collects, at given energy *E*(*ω*), scattered radiation from the object. The physics of Compton scattering demands that the registered radiation flux energy *<sup>n</sup>*(**D**) at site **<sup>D</sup>** is due to the contribution of all scattering sites **<sup>M</sup>** lying on an arc of circle from **S** to **D** subtending an angle (*π* − *ω*), where *ω* is the scattering angle

Mathematically, *<sup>n</sup>*(**D**) is essentially the integral of the object electron density *<sup>n</sup>*(**M**) on such arc of circles, when radiation attenuation and beam spreading effects due to radiation propagation are neglected. If polar coordinates with **S** as origin are used, then a running point **M** on a circle of diameter *p*, with center at the point of polar coordinates (*p*/2, *φ*), is given by (*r*, *θ*) with *r* = *p* cos(*θ* − *φ*). Thus, calling *γ* = (*θ* − *φ*) and recalling that the circle

corresponding to the outgoing energy *E*(*ω*), as given by equation 1.

detector **D** movable on a line intersecting the source site, as sketched in Fig. 8.

for.

**4.1. First CST modality (Norton 1994)**

**Figure 8.** Fisrt CST modality

$$
\hat{n}(p,\phi) = \int\_{\text{arc}\,SD} ds \, n(\mathbf{r}, \gamma + \phi) = p \int\_{-\phi}^{\pi/2} d\gamma \, n(p \cos \gamma, \gamma + \phi),
\tag{7}
$$

where, for ease of notations, we have absorbed in the definition of *<sup>n</sup>*�(**D**) the Compton differential cross-section and the emitted flux density of the source and assumed no attenuation and beam spreading factors on the paths *SM* and *MD*. Equation 7 is not really the circular Radon transform of *n*(**M**) in A. M. Cormack [16], since the integral goes over only the upper circular arc, which physically corresponds to the scattering angle *ω* and the detected energy *E*(*ω*). Note that the lower arc *SD*, is related to the scattering angle (*π* − *ω*) and the scattering energy *E*(*π* − *ω*), which is according to equation 1 not equal to *E*(*ω*). Yet Norton argued that for an object situated above the line *SD*, which means that the support of *n*(*r*, *θ*) is situated in the upper half-plane, the inversion procedure of A. M. Cormack should work. A closed form inverse formula, which mathematically well defined, has been given in [17] as

$$m(r,\theta) = \frac{1}{2\pi^2 r} \int\_0^{2\pi} d\phi \int\_0^\infty dp \, \frac{\partial \hat{n}(p,\phi)}{\partial p} \frac{1}{r/p - \cos(\theta - \phi)}.\tag{8}$$

In his work of 1994 [44], Norton has also derived an alternative inversion formula *via* the radial Fourier transform of *n*(*r*, *θ*).

We now give some details on how equation 8 is derived. Assuming that both *n*(*r*, *θ*) and *<sup>n</sup>*�(*p*, *<sup>φ</sup>*) can be represented by their angular Fourier series

$$n(r,\theta) = \sum\_{l} n\_{l}(r) \, e^{il\theta}, \quad \text{with} \quad \hat{n}(p,\phi) = \sum\_{l} \hat{n}\_{l}(p) \, e^{il\phi}. \tag{9}$$

then equation 7 takes the form of a Chebyshev <sup>1</sup> transform, *i.e.*

$$\hat{m}\_l(p) = 2 \int\_0^p dr \, \frac{\cos l \left( \cos^{-1}(r/p) \right)}{\sqrt{1 - (r/p)^2}} \, n\_l(r) \, \tag{10}$$

which can be inverted using the following identity, (see [16]),

$$\int\_{s}^{t} \frac{d\mathbf{x}}{\mathbf{x}} \, \frac{\cosh\left(l \cosh^{-1}\left(\mathbf{s}/\mathbf{x}\right)\right)}{\sqrt{\left(\mathbf{s}/\mathbf{x}\right)^{2} - 1}} \, \frac{\cos l \left(\cos^{-1}(t/\mathbf{x})\right)}{\sqrt{1 - \left(t/\mathbf{x}\right)^{2}}} = \frac{\pi}{2}. \tag{11}$$

The inverse formula for *nl*(*r*) is given in [17] as

$$m\_l(r) = \frac{1}{\pi r} \left( \int\_0^r dp \, \frac{d\hat{n}\_l(p)}{dp} \, \frac{\left( (r/p) - \sqrt{(r/p)^2 - 1} \right)^l}{\sqrt{(r/p)^2 - 1}} - \int\_r^\infty dp \, \frac{d\hat{n}\_l(p)}{dp} \, \mathcal{U}\_{l-1}(r/p) \right), \tag{12}$$

where *Ul*−1(cos *<sup>x</sup>*) = sin *lx*/ sin *<sup>x</sup>*. Then *<sup>n</sup>*(*r*, *<sup>θ</sup>*) in equation 8 is obtained by resumming the series in equation 9.

<sup>1</sup> This name comes from the fact that the Chebyshev polynomial of the first kind is *Tl*(cos *x*) = cos *lx*.
