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 and supervised segmentation by Sapiro 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.
As in most works we adopt the Lambertian hypothesis. That is, a uniform surface (in the model) with a constant angle will be seen with a constant intensity in the image. This assumption allows us to extrapolate information across uniform regions of the image. Clearly not all uniform regions will have some data sample in it, especially if we consider textured surfaces. In those cases we extrapolate from the "nearest" sample, where nearest refers to the geodesic distance that takes into account the radiometric and edge information of the image.
We define geodesic dinstance between two points in the image as the the lenght of the shortest path between them, where the lenth accounts for graylevel changes along the path. Therefore, the distance of two points along the same isophote is 0, while the distance of two points at both sides of a discontinuity of the image will be proportional to the graylevel "jump".
Geodesic Neighborhod (GN) of a point p is the set formed by the K-nearest (in the geodesic sense) samples to it. Most of the samples in the GN of a point are likely to belong to the same structure, therefore are used to fit a model and interpolate the surface at the point.
| Experiment | Rererence image | Available disparity | GN affine interpolation | Affine region merging | GN display |
| Toulouse | |
|
|
|
VIS |
| Map | |
|
|
|
VIS |
| Venus | |
|
|
|
VIS half res |
| Sawthoot | |
|
|
|
VIS half res |
| Teddy (simulated random sampling) | |
|
|
|
VIS half res |