Reconstructing a surface/image from corrupted gradient fields is a crucial step in many imaging applications where a gradient field is subject to both noise and unlocalized outliers, resulting typically in a non-integrable field. We present in this paper a new optimization method for robust surface reconstruction. The proposed formulation is based on a triple sparsity prior : a sparse prior on the residual gradient field and a double sparse prior on the surface it- self. We develop an efficient alternate minimization strategy to solve the proposed optimization problem. The method is able to recover a good quality surface from severely cor- rupted gradients thanks to its ability to handle both noise and outliers. We demonstrate the performance of the pro- posed method on synthetic and real data. Experiments show that the proposed solution outperforms some existing meth- ods in the three possible cases : noise only, outliers only and mixed noise/outliers.