Combining the Radon, Markov, and Stieltjes Transforms for Object Reconstruction

In shape reconstruction, the celebrated Fourier slice theorem plays an essential role. By virtue of the relation between the Radon transform, the Fourier transform and the 2-dimensional inverse Fourier transform, the shape of an object can be reconstructed from the knowledge of the object’s Radon transform. Unfortunately, a discrete implementation requires the use of interpolation techniques, such as in the filtered back projection. We show how the need for interpolation can be overcome by using the relationship between the Radon transform, the Markov transform and the 2-dimensional Stieltjes transform. When combining the knowledge of an object’s Radon transform for discrete angles θ, with the less well-known Padé slice theorem, the object under consideration can be reconstructed from the solution of a linear least squares problem. 1 The Radon, Markov and Stieltjes integral transforms The Radon transform R ξ (u) of a square-integrable n-variate function f( x) with x = (x1, . . . , xn) is defined as