We present a simple framework for solving different ill-posed inverse problems in image processing by means of constrained total variation minimizations. We argue that drawbacks commonly attributed to total variation algorithms (slowness and incomplete fit to the image model) can be easily bypassed by performing only a few number of iterations in our optimization process. We illustrate this approach in the context of computerized tomography, that comes down to inverse a Radon transform obtained by illuminating an object by straight and parallel beams of x-rays. This problem is ill-posed because only a finite number of line integrals can be measured, resulting in an incomplete coverage of the frequency plane and requiring, for a direct Fourier reconstruction, frequencies interpolation from a polar to a Cartesian grid. We introduce a new method of interpolation based on a total variation minimization constrained by the knowledge of frequency coefficients in the polar grid, subject to a Lipschitz regularity assumption. The experiments show that our algorithm is able to avoid Gibbs and noise oscillations associated to the direct Fourier method, and that it outperforms classical reconstruction methods such as filtered backprojection and Rudin-Osher-Fatemi total variation restoration, in terms of both PSNR and visual quality.