Feature preserving variational smoothing of terrain data

In this paper, we present a novel two-step, variational and feature preserving smoothing method for terrain data. The first step computes the field of 3D normal vectors from the height map and smoothes them by minimizing a robust penalty function of curvature. This penalty function favors piecewise planar surfaces; therefore, it is better suited for processing terrain data then previous methods which operate on intensity images. We formulate the total curvature of a height map as a function of its normals. Then, the gradient descent minimization is implemented with a second-order partial differential equation (PDE) on the field of normals. For the second step, we define another penalty function that measures the mismatch between the the 3D normals of a height map model and the field of smoothed normals from the first step. Then, starting with the original height map as the initialization, we fit a non-parametric terrain model to the smoothed normals minimizing this penalty function. This gradient descent minimization is also implemented with a second-order PDE. We demonstrate the effectiveness of our approach with a ridge/gully detection application.