KTH/Nordita/SU seminar in Theoretical Physics

A Reconstruction Method in Transmission Tomography

by Skiff Sokolov (Stockholm)

FA31 ()


Transmission tomography seeks to reconstruct the density of an object from images made by X-rays passed through the object. In the simplest continuous 2D model of the actual discrete 3D problem, it reduces to the classical (Radon) problem to find the function from its parallel projections at all angles and has analytical solution. Passing to the discrete case leads to ill-conditioned equations which require regularization. I will present an alternative reconstruction method for the discrete 3D case. It supposes that the object is contained within a known rectangular box and limits the reconstruction goal to the estimation of a finite number of amplitudes of the expansion of the density in this box into a discrete Fourier series. As input, it uses the projections of the density on planes passing through the center of the box and parallel to its sides. The method is based on simple relations between the amplitudes of the density expansion and the amplitudes of the Fourier expansion of the 'folded' projections - the periodic functions obtained from the usual projections by adding their periodically shifted copies so that one period of the 'folded' projection fills the part of the projection plane contained within the box. If the projection is made orthogonal to the wave vector corresponding to the element of the Fourier expansion of the density, the relation between the respective amplitudes becomes trivial and the amplitude of this element can be found from the projection data immediately. Generally, the amplitudes have to be estimated from projection data interpolated between available projection directions closest to directions orthogonal to the wave vectors for corresponding harmonics. This efficient reconstruction method needs neither regularization nor matrix inversion. It is illustrated by simulated examples.