We consider the minimization problem for a class of convex integral functionals composed of two terms:

a regularizing term of linear growth in the gradient,
and a fidelity term penalizing the distance from a noisy datum.

Such functionals attain their minima on the BV space of functions of bounded variation, in particular the minimizers may exhibit jump discontinuities. This is a desirable feature in image denoising, corresponding to sharp contours.

We show an estimate on the location and size of jumps of the minimizers in terms of data. Our method works for a large class of regularizers under a mild assumption of differentiability along inner variations, and applies in the vectorial setting, corresponding to color images.

This is joint work with Antonin Chambolle.