One of the key properties of many surface reconstruction techniques is that they represent the volume in front of and behind the surface, e.g., using a variant of signed distance functions. This creates significant problems when reconstructing thin areas of an object since the backside interferes with the reconstruction of the front. We present a two-step technique that avoids this interference and thus imposes no constraints on object thickness. Our method first extracts an approximate surface crust and then iteratively refines the crust to yield the final surface mesh. To extract the crust, we use a novel observation-dependent kernel density estimation to robustly estimate the approximate surface location from the samples. Free space is similarly estimated from the samples' visibility information. In the following refinement, we determine the remaining error using a surface-based kernel interpolation that limits the samples' influence to nearby surface regions with similar orientation and iteratively move the surface towards its true location. We demonstrate our results on synthetic as well as real datasets reconstructed using multi-view stereo techniques or consumer depth sensors.