Proposed is a theoretically exact formula for inversion of data obtained by a spiral computed tomography (CT) scan with a two-dimensional detector array. The detector array is supposed to be of limited extent in the axial direction. The main property of the formula is that it can be implemented in a truly filtered backprojection fashion. First, one performs shift-invariant filtering of a derivative of the cone beam projections, and, second, the result is backprojected in order to form an image. Another property is that the formula solves the so-called long object problem. Limitations of the algorithm are discussed. Results of numerical experiments are presented.

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Given a rather general weight function n0, we derive a new cone beam transform inversion formula. The derivation is explicitly based on Grangeat's formula (1990) and the classical 3D Radon transform inversion. The new formula is theoretically exact and is represented by a 2D integral. We show that if the source trajectory C is complete in the sense of Tuy (1983) (and satisfies two other very mild assumptions), then substituting the simplest weight n0≡1 gives a convolution-based FBP algorithm. However, this easy choice is not always optimal from the point of view of practical applications. The weight n0≡1 works well for closed trajectories, but the resulting algorithm does not solve the long object problem if C is not closed. In the latter case one has to use the flexibility in choosing n0 and find the weight that gives an inversion formula with the desired properties. We show how this can be done for spiral CT. It turns out that the two inversion algorithms for spiral CT proposed earlier by the author are particular cases of the new formula. For general trajectories the choice of weight should be done on a case-by-case basis

In this paper we continue studying a theoretically exact filtered backprojection inversion formula for cone beam spiral CT proposed earlier by the author. Our results show that if the phantom f is constant along the axial direction, the formula is equivalent to the 2D Radon transform inversion. Also, the inversion formula remains exact as spiral pitch goes to zero and in the limit becomes again the 2D Radon transform inversion formula. Finally, we show that according to the formula the processed cone beam projections should be backprojected using both the inverse distance squared law and the inverse distance law.

We study the asymptotics of singular values and singular functions of a Finite Hilbert transform (FHT), which is defined on several intervals. Transforms of this kind arise in the study of the interior problem of tomography. We suggest a novel approach based on the technique of the matrix RiemannHilbert problem and the steepest descent method of Deift-Zhou. We obtain a family of matrix RHPs depending on the spectral parameter λ and show that the singular values of the FHT coincide with the values of λ for which the RHP is not solvable. Expressing the leading order solution as λ → 0 of the RHP in terms of the Riemann Theta functions, we prove that the asymptotics of the singular values can be obtained by studying the intersections of the locus of zeroes of a certain Theta function with a straight line. This line can be calculated explicitly, and it depends on the geometry of the intervals that define the FHT. The leading order asymptotics of the singular functions and singular values are explicitly expressed in terms of the Riemann Theta functions and of the period matrix of the corresponding normalized differentials, respectively. We also obtain the error estimates for our asymptotic results.