I have three images of the same scene with a set of triple correspondences. I can calculate the trifocal tensor and recover pairwise fundamental matrices, rotation matrices and translation vectors (up to a scale between first and second translations). This scale can be easily calculated in a noiseless case (simulation).
However, as soon as even small noise is added (Gaussian, with sigma=0.1 pixel), all estimations go seriously wrong. All triples are guaranteed inliers, but even optimization techniques fail to find correct (known in simulation) scale.
Does anyone know where the catch might be?
Hi Yuri,
Did you try using "normalized coordinates" to improve the numerical conditioning of the system?
http://en.wikipedia.org/wiki/Eight-point_algorithm#The_normalized_eight-point_algorithm
Hi Guillermo.
I did use coordinate normalization as recommended in HZ. Without that the algorithm is highly unstable.
I still don't know why linear solution is so noise-sensitive, but once I've used sparse bundle adjustment, the solution is getting much more stable - noise with sigma=1 pixel converges to practically ideal tensor - the difference is less than 1.e-6.
The (normalized) linear algorithm in the book by Hartley and Zisserman does not enforce the constraints of the Trifocal tensor. Maybe that might be one source of problems. The algebraic minimization algorithm, however, does enforce the constraints of a valid trifocal tensor. If, as you said, bundle adjustment did a very good job, I think this method should work well, too, for points and cameras in general position.
Otherwise, we might need more information to answer your question.
My paper:
Ming Liu, Bekir Tufan Alper, Roland Siegwart, An adaptive descriptor for uncalibrated omnidirectional images - towards scene reconstruction by trifocal tensor, IEEE International Conference on Robotics and Automation (ICRA), 2013
did some study on 1d trifocal tensor. Feel free to discuss.
You can get it from my page:
http://www.asl.ethz.ch/people/lium/personal
Hi Guillermo,
the algebraic minimization has been added in the processing stream as well, but it didn't do much for solution stability. Looks like only SBA is a real help.
Ming: thank you for the hint, I'll certainly carefully read your paper.
While I'm on a trifocal subject, I have another question which really puzzles me: if I a priori know projection matrices for all three cameras (first is canonical) and try to calculate trifocal tensor from them, the solution is not correct (trilinearity conditions are not satisfied for it). Knowledge of three fundamental matrices, however, allows to calculate correct tensor. Correct tensor can be decomposed into three projection matrices, which have nothing to do with matrices known a priori. I know that decomposition is not unique, but there should be a common 3D transformation which converts decomposed matrices into original, and there is none.
Can anyone comment on this mystery?
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