math.CA — Classical Analysis and ODEs
Image Reconstruction from Projections using Fourier Slice Theorem
We study the problem of image reconstruction from projection data by combining the Fourier Slice Theorem with scattered-data approximation in the Fourier domain. The unknown image is assumed to belong to the space of functions of bounded variation (BV), allowing discontinuities that naturally arise in practical tomography. In contrast to conventional band-limited formulations, no assumption is imposed that the image itself is band-limited. Instead, the finite bandwidth is attributed to the image acquisition system through the detector response, finite sampling resolution, and associated measurement process. Consequently, the measured Radon projections contain only a finite range of spatial frequencies, and the reconstruction objective is the recovery of the instrument-limited image determined by the measurable Fourier bandwidth. To exploit the structure of the Fourier Slice Theorem, we introduce a symmetric embedding of the image on the torus together with the corresponding symmetrization of the Radon data. The resulting Fourier transform becomes real-valued and even, allowing the Fourier-transformed projection data to be interpreted as real-valued scattered samples of the two-dimensional Fourier transform on a fixed bounded frequency domain. This reformulates image reconstruction as a scattered-data interpolation problem in the Fourier domain. The interpolation problem is analyzed using the recently developed approximation theory for multivariate BV functions from scattered data. The resulting convergence error is expressed in terms of the total reconstruction error separating into an instrument error, determined solely by the finite bandwidth of the acquisition system, and a reconstruction error arising from interpolation of the measured Fourier data. The proposed framework provides a mathematically rigorous and physically realistic formulation of Fourier-slice tomography in which the limitations of the imaging hardware are distinguished from the accuracy of the reconstruction algorithm. Numerical experiments demonstrate the applicability of the method to image reconstruction from limited projection data and validate the predicted convergence behaviour.