Atlanta World Assumption; refresh on the L1 norm's other names.15 Mar 2022
I saw a paper on arXiv, Lim2022, that used the ‘Atlanta world assumption,’ and since I didn’t know what that was, I learned about it today.
From SD2004, the Atlanta world assumes “multiple pairs of orthogonal vanishing directions.” This assumption is in contrast to the Manhattan world assumption, where the “scene is built on a Cartesian grid,” CY2000. In other words, under the Manhattan world assumptions planes are axis-aligned.
The world assumptions are used in robotics.
Then I got thinking about Manhattan distance. The Manhattan distance, or taxicab distance, are different names for the same distance. Also the same distance is the Minkowski distance, when p = 1. The Manhattan distance is also analogous to the L1-norm. Or, one could say that the Manhattan distance is a p-norm with p = 1 TB1997, lecture 3, norms and a metric. To conclude, all of these distances / norms are equivalent:
- Manhattan distance,
- Taxicab distance,
- Minkowski distance with p = 1,
- L1 norm,
- p-norm with p = 1.
[Lim2022] H. Lim et al., “A Single Correspondence Is Enough: Robust Global Registration to Avoid Degeneracy in Urban Environments,” arXiv:2203.06612 [cs], Mar. 2022, Accessed: Mar. 15, 2022. [Online]. Available: http://arxiv.org/abs/2203.06612.
[SD2004] G. Schindler and F. Dellaert, “Atlanta world: an expectation maximization framework for simultaneous low-level edge grouping and camera calibration in complex man-made environments,” in Proceedings of the 2004 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2004. CVPR 2004., Washington, DC, USA, 2004, vol. 1, pp. 203–209. doi: 10.1109/CVPR.2004.1315033. Available: Georgia Tech.
[Minkowski] “Minkowski distance,” Wikipedia. Jan. 14, 2022. Accessed: Mar. 15, 2022. [Online]. Available: Wikipedia.
[Taxicab] “Taxicab geometry,” Wikipedia. Jan. 21, 2022. Accessed: Mar. 15, 2022. [Online]. Available: Wikipedia.
[CY2000] J. Coughlan and A. L. Yuille, “The Manhattan World Assumption: Regularities in Scene Statistics which Enable Bayesian Inference,” in Advances in Neural Information Processing Systems, 2000, vol. 13. [Online]. Available: NeurIPS.
[TB1997] Lloyd N. Trefethen; David Bau III. 1997. Numerical Linear Algebra. ISBN 9780898713619.