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 estimation. Typically, the resulting gradient field is non-integrable 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 ℓ<inf>0</inf> - ℓ<inf>1</inf> equivalence. We propose to obtain the surface by finding the gradient field which best fits the corrupted gradient field in ℓ<inf>1</inf> sense. We present an exhaustive analysis of the properties of ℓ<inf>1</inf> 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 ℓ<inf>1</inf> solution performs as well as least squares in the absence of outliers. While previous ℓ<inf>0</inf> - ℓ<inf>1</inf> 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 ℓ<inf>1</inf> solution both in terms of location and number of outliers, and outline scenarios where ℓ<inf>1</inf> solution is equivalent to ℓ<inf>0</inf> solution. We also show that when ℓ<inf>1</inf> 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 ℓ<inf>1</inf> solution performs well across all scenarios without the need for any tunable parameter adjustments.