Geodesic neighborhoods for piecewise affine interpolation of sparse data

We propose an interpolation method for sparse data that incorporates the geometric information of a reference image. The idea consists in defining for each sample a geodesic neighborhood and then fit a model (affine for instance) to interpolate at the current point. In the field of remote sensing for urban areas, two widely used techniques are laser range scanning (LIDAR) and stereo photogrammetry. Both techniques have a common drawback, for a variety of reasons the information they provide is sparse or incomplete. But in both cases it is fair to assume that a high resolution image of the scene is available, and we propose in this paper a diffusion algorithm that takes into account the geometry of the image u to refine the range data. This allows us to interpolate the data set while respecting the edges of u. The core of the algorithm is a fast method for computing geodesic distances between image points, which has been successfully applied to colorization by Yatziv et al. and supervised segmentation by Bai et al. The geodesic distance is used to find the set of points that are used to interpolate a piecewise affine model in the current sample. This first interpolation result is refined by merging the obtained affine patches using a greedy Mumford-Shah like algorithm. The output is a piecewise affine interplation of the data set that respects both the given data and the radiometric information provided by u.