Direct Fourier methods in 3D-reconstruction from cone-beam data

Sammanfattning: The problem of 3D-reconstruction is encountered in both medical and industrial applications of X-ray tomography. It would seem natural to recover volume data by just repeating 2D-reconstructions slice-by-slice. However, 2D fan-beam exposure is wasting most of the photons on the collimator and such a procedure is inefficient in terms of data acquisition. Fast scanning with acceptable signal-to-noise ratio requires cone-beam exposure.A method able to utilize a complete set of projections complying with Tuy's condition was proposed by Grangeat. His method is mathematically exact and consists of two distinct phases. In phase 1 cone-beam projection data are used to produce the derivative of the Radon transform. In phase 2, after interpolation, the Radon transform data are used to reconstruct the three-dimensional object function.To a large extent our method is an extension of the Grangeat method. Our aim is to reduce the computational complexity, i.e. to produce a faster method. The most taxing procedure during phase 1 is computation of line-integrals in the detector plane. By applying the direct Fourier method in reverse for this computation, we reduce the complexity of phase 1 from O(N4 ) to O(N3logN). Phase 2 can be performed either as a straight 3D-reconstruction or as a sequence of two 2D-reconstructions in vertical and horizontal planes, respectively. Direct Fourier methods can be applied for the 2D- and for the 3D-reconstruction, which reduces the complexity of phase 2 from O(N4) to O(N31ogN) as well. In both cases, linogram techniques are applied.For 3D-reconstruction the inversion formula contains the second derivative filter instead of the well-known ramp-filter employed in the 2D-case. The derivative filter is more well-behaved than the 2D ramp-filter. This implies that less zeropadding is necessary which brings about a further reduction of the computational efforts.The method has been verified by experiments on simulated data. The image quality is satisfactory and independent of cone-beam angles. For a 5123 volume we estimate that our method is ten times faster than Grangeat's method.