Discretization schemes commonly used for total variation regularization lead to images that are difficult to interpolate, which is a real issue for applications requiring subpixel accuracy and aliasing control. In the present work, we reconciliate total variation with Shannon interpolation and study a Fourier-based estimate that behaves much better in terms of grid invariance, isotropy, artifact removal, and sub-pixel accuracy. We show that this new variant (called Shannon total variation) can be easily handled with classical primal-dual formulations, and illustrate its efficiency on several image processing tasks, including deblurring, spectrum extrapolation, and a new aliasing reduction algorithm.