Mimas Camera Calibration

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Prototype low-cost platform for 3D reconstruction of objects using laser-triangulation (10 MByte video (https://vision.eng.shu.ac.uk/jan/reconstruction%20platform.avi))
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Prototype low-cost platform for 3D reconstruction of objects using laser-triangulation (10 MByte video (https://vision.eng.shu.ac.uk/jan/reconstruction%20platform.avi))
Pinhole camera model
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Pinhole camera model
Calibration introduced to the by Julien Faucher, ERASMUS exchange student from France
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Calibration introduced to the MMVL by Julien Faucher, ERASMUS exchange student from France
Table of contents
1 Calibration

1.1 Plane-to-camera homography
1.2 Intrinsic and extrinsic camera parameters

2 See Also
3 External Links
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Calibration

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Plane-to-camera homography

Let \vec{m}_i\in\mathbb{R}^3,\,i\in\{1,2,\ldots,N\} 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 \vec{m}^\prime_i\in\mathbb{R}^3 be the homogeneous coordinate of the corresponding pixel in the camera-image. Further let (\mathbb{R}^n,\simeq) be an equivalence relation (https://en.wikipedia.org/wiki/Equivalence_relation) defined by

\vec{a}\simeq\vec{b}\ :\Leftrightarrow\ \exists\lambda\in\mathbb{R}/\{0\}:\,\lambda\,\vec{a}=\vec{b}

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 \mathcal{H}\in\mathbb{R}^{3\times 3} as follows

\vec{m^\prime}_i+\vec{\epsilon}_i\simeq\mathcal{H}\vec{m}_i

or more elaborately \lambda\,\begin{pmatrix}m^\prime_{i1}\\m^\prime_{i2}\\m^\prime_{i3}\end{pmatrix}+ \begin{pmatrix}\epsilon_{i1}\\\epsilon_{i2}\\\epsilon_{i3}\end{pmatrix}= \begin{pmatrix}h_{11}&h_{12}&h_{13}\\h_{21}&h_{22}&h_{23}\\h_{31}&h_{32}&h_{33}\end{pmatrix}\, \begin{pmatrix}m_{i1}\\m_{i2}\\m_{i3}\end{pmatrix} where (\epsilon_{i1},\epsilon_{i2})^\top is the zero-mean error-vector in the observation of the ith point in the camera-image.

Using m_{i3}=m^\prime_{i3}=1,\ \epsilon_{i3}=0 and \vec{h_i}^\top=\begin{pmatrix}h_{i1}&h_{i2}&h_{i3}\end{pmatrix},\ i\in\{1,2,3\} the model can be reformulated to \begin{pmatrix}m^\prime_{i1}\\m^\prime_{i2}\end{pmatrix}=\cfrac{1}{\vec{h_3}^\top\cdot\vec{m}_i}\, \begin{pmatrix}\vec{h_1}^\top\\\vec{h_2}^\top\end{pmatrix}\, \begin{pmatrix}m_{i1}\\m_{i2}\\1\end{pmatrix}- \begin{pmatrix}\epsilon_{i1}\\\epsilon_{i2}\end{pmatrix} or \begin{pmatrix}m^\prime_{i1}\\m^\prime_{i2}\end{pmatrix}\,\big(\vec{h_3}^\top\cdot\vec{m}_i\big)= \begin{pmatrix}\vec{h_1}^\top\\\vec{h_2}^\top\end{pmatrix}\, \begin{pmatrix}m_{i1}\\m_{i2}\\1\end{pmatrix}- \begin{pmatrix}\epsilon^\prime_{i1}\\\epsilon^\prime_{i2}\end{pmatrix} with \vec{\epsilon^\prime}_i=\big[\vec{h_3}^\top\cdot\vec{m}_i\big]\,\vec{\epsilon}_i and using |\mathcal{H}|\neq 0

It is assumed, that the vectors \vec{\epsilon^\prime}_i have equal variances (i.e. \vec{h_3}^\top\cdot\vec{m}_1\approx\vec{h_3}^\top\cdot\vec{m}_2\approx\ldots) so that the Gauss-Markov theorem (https://en.wikipedia.org/wiki/Gauss-Markov_theorem) can be applied using (\epsilon^\prime_{i1},\epsilon^\prime_{i2})^\top 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 \begin{pmatrix}h_{11}\,m_{i1}+h_{12}\,m_{i2}+h_{13}\\h_{21}\,m_{i1}+h_{22}\,m_{i2}+h_{23}\end{pmatrix}- \begin{pmatrix}m^\prime_{i1}\,m_{i1}\,h_{31}+m^\prime_{i1}\,m_{i2}\,h_{32}+m^\prime_{i1}\,h_{33}\\ m^\prime_{i2}\,m_{i1}\,h_{31}+m^\prime_{i2}\,m_{i2}\,h_{32}+m^\prime_{i2}\,h_{33}\end{pmatrix}= \begin{pmatrix}\epsilon^\prime_{i1}\\\epsilon^\prime_{i2}\end{pmatrix}

Isolating the elements of the unknown matrix \mathcal{H} gives \begin{pmatrix} m_{i1}&m_{i2}&1&0&0&0&-m^\prime_{i1}\,m_{i1}&-m^\prime_{i1}\,m_{i2}&-m^\prime_{i1}\\ 0&0&0&m_{i1}&m_{i2}&1&-m^\prime_{i2}\,m_{i1}&-m^\prime_{i2}\,m_{i2}&-m^\prime_{i2} \end{pmatrix}\, \begin{pmatrix}h_{11}\\h_{12}\\\vdots\\h_{33}\end{pmatrix}= \begin{pmatrix}\epsilon^\prime_{i1}\\\epsilon^\prime_{i2}\end{pmatrix}

The combined system of all linear equations is \underbrace{\begin{pmatrix} m_{11}&m_{12}&1&0&0&0&-m^\prime_{11}\,m_{11}&-m^\prime_{11}\,m_{12}&-m^\prime_{11}\\ 0&0&0&m_{11}&m_{12}&1&-m^\prime_{12}\,m_{11}&-m^\prime_{12}\,m_{12}&-m^\prime_{12}\\ m_{21}&m_{22}&1&0&0&0&-m^\prime_{21}\,m_{21}&-m^\prime_{21}\,m_{22}&-m^\prime_{21}\\ 0&0&0&m_{21}&m_{22}&1&-m^\prime_{22}\,m_{21}&-m^\prime_{22}\,m_{22}&-m^\prime_{22}\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ m_{N1}&m_{N2}&1&0&0&0&-m^\prime_{N1}\,m_{N1}&-m^\prime_{N1}\,m_{N2}&-m^\prime_{N1}\\ 0&0&0&m_{N1}&m_{N2}&1&-m^\prime_{N2}\,m_{N1}&-m^\prime_{N2}\,m_{N2}&-m^\prime_{N2} \end{pmatrix}}_{=:\mathcal{M}}\, \underbrace{\begin{pmatrix}h_{11}\\h_{12}\\\vdots\\h_{33}\end{pmatrix}}_{=:\vec{h}}= \begin{pmatrix}\epsilon^\prime_{11}\\\epsilon^\prime_{12}\\\epsilon^\prime_{21}\\\epsilon^\prime_{22}\\ \vdots\\\epsilon^\prime_{N1}\\\epsilon^\prime_{N2}\end{pmatrix}

To avoid the trivial solution \vec{h}=\vec{0} the constraint ||\vec{h}||=1 is introduced without loss of generality.

The calibration problem now has been reduced to the problem of finding \widehat{\vec{h}}\in\mathbb{R}^9 such that

  1. ||\mathcal{M}\,\widehat{\vec{h}}|| is minimal and
  2. ||\widehat{\vec{h}}||=1

\widehat{\vec{h}} can be computed using the Singular value decomposition (https://en.wikipedia.org/wiki/Singular_value_decomposition) \mathcal{M}=\mathcal{U}\,\Sigma\,\mathcal{V}^*, because these are the properties of the right handed singular vector \vec{v}_1 with the smallest singular value σ1 (where \mathcal{V}=\big(\vec{v}_1\,\vec{v}_2\,\cdots\big)). I.e. \widehat{\vec{h}}=\vec{v}_1.

Knowing the homography \mathcal{H} already is sufficient for the Interactive Camera-Projector System.

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Intrinsic and extrinsic camera parameters

Please see references below ...

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See Also

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External Links

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