Fault and gradient fault detection and reconstruction from scattered data

We suggest an algorithm for fault detection from scattered data that relies on new fault indicators to detect clouds of points enclosing the faults, and a reconstruction phase that includes a narrowing step to thin the detected point clouds, the construction of ordered sequences of points representing all fault curves, and localization of possible areas of intersection among different faults. The fault indicators are based on recently introduced minimal numerical differentiation formulas for gradient or Laplacian on irregular centers, which bypasses any intermediate gridding of the data. We show that our indicators provide lower bounds for local H"older norms of any function the data may have been sampled from, and investigate their asymptotic behavior when the spacing between the data sites goes to zero. An application to edge detection in 3D surfaces is also proposed and a selection of numerical examples illustrates the performances of the method, including the identification and reconstruction of multi-branch faults and applications to terrain investigation and edge detection.