We propose an energy-based framework for approximating surfaces from a cloud of point measurements corrupted by noise and outliers. Our energy assigns a tangent plane to each (noisy) data point by minimizing the squared distances to the points and the irregularity of the surface implicitly defined by the tangent planes. In order to avoid the well-known "shrinking" bias associated with first-order surface regularization, we choose a robust smoothing term that approximates curvature of the underlying surface. In contrast to a number of recent publications estimating curvature using discrete (e.g. binary) labellings with triple-cliques we use higher-dimensional labels that allows modeling curvature with only pair-wise interactions. Hence, many standard optimization algorithms (e.g. message passing, graph cut, etc) can minimize the proposed curvature-based regularization functional. The accuracy of our approach for representing curvature is demonstrated by theoretical and empirical results on synthetic and real data sets from multiview reconstruction and stereo.