**Author details**

Truong T. T.

24 Will-be-set-by-IN-TECH

(or *MD*−1,−2) does not spoiled the inversion procedure. One may redefine an "effective" electron density and an "effective" Radon transform. This is due to the fact that the product (*SM* <sup>×</sup> *MD*)−1,−<sup>2</sup> is of the form of a product of a function of *<sup>τ</sup>* and a function of *<sup>r</sup>*, see [42, 54]. 3) *Multiple scattering.* This degrading factor is due to the use of wide angle collimator and is difficult to be removed by theoretical means. As in other gamma-ray imaging systems, such as Single Photon Emission Computed Tomography (SPECT) and Positron Emission Tomography (PET), this problem can be only approximatively fixed by corrective measures adapted to each

4) *Noise.* Radiation processes carry fluctuations as noise which could drastically affect numerical simulations in image reconstruction. So algorithm robustness should be tested against noise systematically. There is no universal receipt and an adapted de-noising procedure is to be set up for each scanner type. An example of such de-noising procedure

Compton scatter tomography (CST) is an outstanding example of the use of Compton scattered radiation for non-invasive imaging. CST provides high resolution and high sensitivity imaging with virtually any material, including lightweight structures and organic matter, which normally pose problems in conventional x-ray computed tomography because of low contrast. However it also suffers from the usual degrading factors of radiation imaging, such as multiple scattering, attenuation, noise and fluctuations, etc., which have to be treated by appropriate (existing or to be developed) methods. Yet it has undergone a very impressive theoretical evolution. In this chapter, we have shown how the expression of the differential cross-section of the Compton effect has led to the idea of measuring the value of matter electron density point by point. The ensuing progress brought to light the successive line by line and plane by plane scanning procedures, however with limited success. A quantum leap in redefining CST was made in the 80's by N. N. Kondic, who advocated the use or wide angle collimators and pointed out the occurrence of integral measurements along circular arcs corresponding to a definite measured photon energy. Then a concept of integral transform of Radon type has emerged from this type of integral measurement in [9]. Later in the 90's, elaborating on a proposal of N. N. Kondic, T. H. Prettyman has constructed a scanner on this principle and called an energy dispersive projective scatterometer. He also realized that a relative motion between this apparatus and the object is necessary to generate the data needed for image reconstruction. This is the starting point for recent CST modalities which are now based on Radon transform on Kondic isogonic circular arcs. We have presented and discussed three existing CST modalities based on three "circular-arc" Radon transforms. Two of them are of recent origin and have arisen as new elements of integral geometry. The remarkable aspect is that they share a common inversion method based on the inversion of the Radon transform on circles intersecting a fixed point. As numerical simulations show their feasibility and viability as imaging systems, CST appears as a promising area of research and development which could bring interesting advances in the field of non-invasive imaging for medicine and industry. Work towards more efficient ways for data processing, *e.g.* setting

faster back-projection types of reconstruction methods, is underway in this context.

particular scanner.

is given in [51].

**6. Conclusion and perspectives**

*Laboratoire de Physique Théorique et Modélisation, University of Cergy-Pontoise/CNRS UMR 8089, F-95302 Cergy-Pontoise Cedex, France*

Nguyen M. K.

*Laboratoire Équipes Traitement de l'Information et Systèmes, ETIS-ENSEA/University of Cergy-Pontoise/CNRS UMR 8051, F-95014 Cergy-Pontoise Cedex, France*
