14.4 Image reconstruction process

As the experimental frequency list is finite, and in addition the experimental data blurred, the object representation that can be obtained from these data is of course incomplete. This simple remark shows that the inverse problems of Fourier synthesis must be regularized: the high-frequency components of the image to be reconstructed must be negligible.

The central problem is to specify in which conditions it is possible to extrapolate or interpolate, in some region of the Fourier domain, the Fourier transform of a function $\phi$ whose support is contained in some finite region of H_o. It is now well established that extrapolation is forbidden, and interpolation allowed to a certain extent. The corresponding regularization principle is then intimately related to the concept of resolution: the interpolation is performed in the frequency gaps of the frequency coverage to be synthesized.