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

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

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

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

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

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

At first this Chebyshev transform looks a bit hopeless. However if one introduces a new

cos �

√ 1 + *τ*<sup>2</sup>

Recent Developments on Compton Scatter Tomography: Theory and Numerical Simulations 113

*l* cos−<sup>1</sup>

� 1 2*τ*

*nl*(*p*( �

*nl*(*p*( �

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

, (16)

*<sup>τ</sup>* sin *<sup>γ</sup>* . (14)

��� *nl*(*r*). (15)

<sup>1</sup> + *<sup>g</sup>*<sup>2</sup> − *<sup>g</sup>*)). (17)

<sup>1</sup> + *<sup>g</sup>*<sup>2</sup> − *<sup>g</sup>*)), (18)

*Nl*(*g*), (19)

*Ul*−1(*g*/*τ*)

⎞

⎟⎠ . (20)

*τ* is positive for 0 *< ω < π*/2 and the range of *θ* is (*φ* − *π*/2) *< θ <* (*φ* + *π*/2).

�

*dr*

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

> *τ* �

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

� *τ* 0

through a fixed point. Hence an inversion formula exists in the (*τ*, *g*) variables, *i.e.*

(*g*/*τ*) <sup>−</sup> �(*g*/*τ*)<sup>2</sup> <sup>−</sup> <sup>1</sup>

*p*(

*dg* cos *<sup>l</sup>* cos−<sup>1</sup> � *<sup>g</sup>*

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

which is precisely of the form of equation 10, obtained in the Radon problem on circles passing

�(*g*/*τ*)<sup>2</sup> <sup>−</sup> <sup>1</sup> <sup>−</sup>

�<sup>1</sup> <sup>+</sup> *<sup>g</sup>*<sup>2</sup> <sup>−</sup> *<sup>g</sup>*) �1 + *g*<sup>2</sup>

�<sup>1</sup> <sup>+</sup> *<sup>g</sup>*<sup>2</sup> <sup>−</sup> *<sup>g</sup>*) �1 + *g*<sup>2</sup>

> *τ* �

�*l*

� ∞ *g dτ* *dN*�*l*(*τ*) *dτ*

*ds* = *r dγ*

�

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

*dg* cos *<sup>l</sup>* cos−<sup>1</sup> � *<sup>g</sup>*

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

*N*�*l*(*τ*) = 2

�

arc element given by

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

we obtain

*Nl*(*g*) = <sup>1</sup>

*π g*

⎛

⎜⎝ � *g* 0

*dp dN*�*l*(*τ*) *dτ*

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

variable *g* defined by

one can put it under the form

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

Then defining new functions by

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

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

9, we see that they are related by

� *p p*(

<sup>√</sup>1+*τ*<sup>2</sup>−*τ*)

� *τ* 0
