Enforcing integrability by error correction using l1-minimization

Surface reconstruction from gradient fields is an important final step in several applications involving gradient manipulations and estimations. Typically, the resulting gradient field is nonintegrable due to linear/non-linear gradient manipulations, or due to presence of noise/outliers in gradient estimation. In this paper, we analyze integrability as error correction, inspired from recent work in compressed sensing, particulary l0-l1. We propose to obtain the surface by finding the gradient field which best fits the corrupted gradient field in l1 sense. We present an exhaustive analysis of the properties of l1 solution for gradient field integration using linear algebra and graph analogy. We consider three cases: (a)noise, but no outliers (b) no-noise but outliers and (c) presence of both noise and outliers in the given gradient field. We show that l1 solution performs as well as least squares in the absence of outliers. While previous l0 -l1 equivalence work has focused on the number of errors (outliers), we show that the location of errors is equally important for gradient field integration. We characterize the l1 solution both in terms of location and number of outliers, and ouline scenarios where l1 solution is equivalent to l0 solution. We also show that when l1 solution is not able to remove outliers, the property of local error confinement holds: i.e., the errors do not propagate to the entire surface as in least squares. We compare with previous techniques and show that l1 solution performs well across all scenarios without the need for any tunable parameter adjustments.