This paper presents a novel method for 3D surface reconstruction based on a sparse set of 3D control points. For object classes such as human heads, prior information about the class is used in order to constrain the results. A common strategy to represent object classes for a reconstruction application is to build holistic models, such as PCA models. Using holistic models involves a trade-off between reconstruction of the measured points and plausibility of the result. We introduce a novel object representation that provides local adaptation of the surface, able to fit 3D control points exactly without affecting areas of the surface distant from the control points. The method is based on an interpolation scheme, opposed to approximation schemes generally used for surface reconstruction. Our interpolation method reduces the Euclidean distance between a reconstruction and its ground truth while preserving its smoothness and increasing its perceptual quality.