This paper presents a new approach to estimate two- and three-dimensional affine transformations from tomographic projections. Instead of estimating the deformation from the reconstructed data, we introduce a method which works directly in the projection domain, using parallel and fan beam projection geometries. We show that any affine deformation can be analytically compensated, and we develop an efficient multiscale estimation framework based on the normalized cross correlation. The accuracy of the approach is verified using simulated and experimental data, and we demonstrate that the new method needs less projection angles and has a much lower computational complexity as compared to approaches based on the standard reconstruction techniques.