Mimas Camera Calibration
|Table of contents|
Let be the homogeneous coordinate (https://en.wikipedia.org/wiki/Homogeneous_coordinates#Use_in_computer_graphics) of the ith point on the planar calibration-object and let be the homogeneous coordinate of the corresponding pixel in the camera-image. Further let be an equivalence relation (https://en.wikipedia.org/wiki/Equivalence_relation) defined by
If the camera-system does an ideal central projection (e.g. no distortion), the projective transformation (the Homography (https://en.wikipedia.org/wiki/Homography)) can be modelled using as follows
or more elaborately where is the zero-mean error-vector in the observation of the ith point in the camera-image.
Using and the model can be reformulated to or with and using
It is assumed, that the vectors have equal variances (i.e. ) so that the Gauss-Markov theorem (https://en.wikipedia.org/wiki/Gauss-Markov_theorem) can be applied using as error-vectors. In this case the miminum least-squares estimator (https://en.wikipedia.org/wiki/Least_squares) is the best linear estimator.
Each point-pair yields the following system of two linear equations
Isolating the elements of the unknown matrix gives
The combined system of all linear equations is
To avoid the trivial solution the constraint is introduced without loss of generality.
The calibration problem now has been reduced to the problem of finding such that
- is minimal and
can be computed using the Singular value decomposition (https://en.wikipedia.org/wiki/Singular_value_decomposition) , because these are the properties of the right handed singular vector with the smallest singular value σ1 (where ). I.e. .
Knowing the homography already is sufficient for the Interactive Camera-Projector System.
Intrinsic and extrinsic camera parameters
Please see references below ...
- Bright : A project using the camera calibration to build a laser-line based 3-D scanner.
- Interactive Camera-Projector System
- Julien Faucher's report (https://vision.eng.shu.ac.uk/manuel/bright/report_faucher.pdf)
- Zhengyou Zhang: A Flexible New Technique for Camera Calibration (https://research.microsoft.com/%7Ezhang/Calib/)
- Ballard and Brown: Computer vision (chapter A1.8.1 on camera calibration) (https://homepages.inf.ed.ac.uk/rbf/BOOKS/BANDB/bandb.htm)
- Augmented Reality with Sony Playstation (google video) (https://video.google.co.uk/videoplay?docid=-6642378035529869995#9m43s)
- Camera calibration toolbox for Matlab (https://www.vision.caltech.edu/bouguetj/calib_doc/)
- GML Camera Calibration toolbox (https://research.graphicon.ru/calibration/gml-c-camera-calibration-toolbox.html)
- EPFL camera calibration software (https://cvlab.epfl.ch/software/bazar/calib.php)
- MRPT camera calibration software (https://www.youtube.com/watch?v=BkZkq6zPwQM&feature=player_embedded#!)
- OpenCV / WillowGarage camera calibration (https://opencv.willowgarage.com/documentation/python/camera_calibration_and_3d_reconstruction.html)