Enforcing Integrability by Error Correction Using L1-Minimization
Abstract
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 <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> - ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> equivalence. We propose to obtain the surface by finding the gradient field which best fits the corrupted gradient field in ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> sense. We present an exhaustive analysis of the properties of ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> 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 <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> solution performs as well as least squares in the absence of outliers. While previous ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> - ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> 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 <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> solution both in terms of location and number of outliers, and outline scenarios where ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> solution is equivalent to ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> solution. We also show that when ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> 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 <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> solution performs well across all scenarios without the need for any tunable parameter adjustments.
Cite
Text
Reddy et al. "Enforcing Integrability by Error Correction Using L1-Minimization." IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2009. doi:10.1109/CVPR.2009.5206603Markdown
[Reddy et al. "Enforcing Integrability by Error Correction Using L1-Minimization." IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2009.](https://mlanthology.org/cvpr/2009/reddy2009cvpr-enforcing/) doi:10.1109/CVPR.2009.5206603BibTeX
@inproceedings{reddy2009cvpr-enforcing,
title = {{Enforcing Integrability by Error Correction Using L1-Minimization}},
author = {Reddy, Dikpal and Agrawal, Amit K. and Chellappa, Rama},
booktitle = {IEEE/CVF Conference on Computer Vision and Pattern Recognition},
year = {2009},
pages = {2350-2357},
doi = {10.1109/CVPR.2009.5206603},
url = {https://mlanthology.org/cvpr/2009/reddy2009cvpr-enforcing/}
}