Wavelets, local tomography and interventional X-Ray imaging

Interventional reconstruction of tomographic sections is still a challenging task: interventional scanners obstruct the operative field, and CT-assisted surgical acts are known to be irradiant. An alternative consists in using a C-Arm fitted out with a digital detector for X-Ray data acquisition, but one has then to cope with the limited size of the existing detectors: they are not wide enough to collect the whole beam emerging from a section, and thus prevent from applying conventional reconstruction techniques. The work we present here relies on such an architecture, and aims at applying local tomography principles to this context: when only X-Ray projections through a central section of a larger one are available, one refers to the interior problem, for which it is known that the density of the central section is then not recoverable, but surfaces of discontinuity between anatomical surfaces are theoretically accessible. We propose an implementation of a wavelet-based approach of this problem inspired by a work by Holschneider, where it is proved that the Radon Transform is nothing but a wavelet transform for a well chosen analysis "wavelet". Numerical tests show that our approach supplies reconstruction results from local measurements preserving discontinuities.