We introduce a variational approach to image segmentation based on sparse coverings of image domains by shape templates. The objective function combines a data term that achieves robustness by tolerating overlapping templates with a regularizer enforcing sparsity. A suitable convex relaxation leads to the variational approach that is amenable to large-scale convex programming. Our approach takes implicitly into account prior knowledge about the shape of objects and their parts, without resorting to combinatorially difficult problems of variational inference. We illustrate our approach by numerical examples and indicate how prior knowledge acquisition may be achieved by learning from examples.