Abstract
We study the accuracy of triangulation in multi-camera systems with respect to the number of cameras. We show that, under certain conditions, the optimal achievable reconstruction error decays quadratically as more cameras are added to the system. Furthermore, we analyze the error decay-rate of major state-of-the-art algorithms with respect to the number of cameras. To this end, we introduce the notion of consistency for triangulation, and show that consistent reconstruction algorithms achieve the optimal quadratic decay, which is asymptotically faster than some other methods. Finally, we present simulations results supporting our findings. Our simulations have been implemented in MATLAB and the resulting code is available in the supplementary material, which can be found on the Computer Society Digital Library at http://doi.ieeecomputersociety.org/10.1109/TPAMI.2019.2939530.
Highlights
CAMERAS are finite; yet, we commonly assume Gaussian noise models with infinite tails
After introducing major state-of-the-art techniques, we present our main results on the error decay rate of consistent triangulation algorithms and the best possible performance of multi-camera systems
This paper focuses on triangulation—a fundamental problem in multiple-view geometry, which, as well as being interesting in its own right, provides a basic building block for many higher-level computer vision tasks, such as visual metrology, Simulataneous Localization And Mapping (SLAM), and Structure from Motion (SfM)
Summary
CAMERAS are finite; yet, we commonly assume Gaussian noise models with infinite tails. We can attempt to minimize the ‘2-norm of the reprojection error between the prospective 3-D points and the known image locations, which results in the maximum-likelihood estimator if we assume that the projected points are subjected to i.i.d. zero-mean Gaussian noise in the image plane This assumption is very likely a much better approximation of the underlying uncertainty, the resulting cost function is non-convex and extremely difficult to solve. The ‘1-norm has recently been considered as a measure of the reprojection error [3], [4], [17] This may not correspond to the best model of the underlying uncertainty, the resulting cost is a quasi-convex function and efficient algorithms can find the global optimum [5]. After introducing major state-of-the-art techniques, we present our main results on the error decay rate of consistent triangulation algorithms and the best possible performance of multi-camera systems.
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