Mimas Camera Calibration
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=Calibration= | =Calibration= | ||
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Let <math>\vec{m}_i\in\mathbb{R}^3,\,i\in\{1,2,\ldots,N\}</math> be the [http://en.wikipedia.org/wiki/Homogeneous_coordinates#Use_in_computer_graphics homogeneous coordinate] of the <math>i</math>th point on the planar calibration-object and let <math>\vec{m}^\prime_i\in\mathbb{R}^3</math> be the homogeneous coordinate of the corresponding pixel in the camera-image. Further let <math>(\mathbb{R}^n,\simeq)</math> be an [http://en.wikipedia.org/wiki/Equivalence_relation equivalence relation] defined by | Let <math>\vec{m}_i\in\mathbb{R}^3,\,i\in\{1,2,\ldots,N\}</math> be the [http://en.wikipedia.org/wiki/Homogeneous_coordinates#Use_in_computer_graphics homogeneous coordinate] of the <math>i</math>th point on the planar calibration-object and let <math>\vec{m}^\prime_i\in\mathbb{R}^3</math> be the homogeneous coordinate of the corresponding pixel in the camera-image. Further let <math>(\mathbb{R}^n,\simeq)</math> be an [http://en.wikipedia.org/wiki/Equivalence_relation equivalence relation] defined by | ||
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\begin{pmatrix}m_{i1}\\m_{i2}\\m_{i3}\end{pmatrix} | \begin{pmatrix}m_{i1}\\m_{i2}\\m_{i3}\end{pmatrix} | ||
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with <math>\vec{m}^\prime_i=\widehat{\vec{m}}_i+\vec{\epsilon}_i</math>. | with <math>\vec{m}^\prime_i=\widehat{\vec{m}}_i+\vec{\epsilon}_i</math>. | ||
| − | + | Using <math>m_{i3}=m^\prime_{i3}=\widehat{m}_{i3}=1</math> and <math>\vec{h_i}^\top=\begin{pmatrix}h_{i1}&h_{i2}&h_{i3}\end{pmatrix},\ i\in\{1,2,3\}</math> the | |
| − | + | model can be represented by | |
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| − | </math> | + | |
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| − | <math> | + | |
| − | \vec{h_i}^\top=\begin{pmatrix}h_{i1}&h_{i2}&h_{i3}\end{pmatrix},\ i\in\{1,2,3\} | + | |
| − | </math> | + | |
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<math> | <math> | ||
\begin{pmatrix}\widehat{m}_{i1}\\\widehat{m}_{i2}\end{pmatrix}= | \begin{pmatrix}\widehat{m}_{i1}\\\widehat{m}_{i2}\end{pmatrix}= | ||
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\begin{pmatrix}m_{i1}\\m_{i2}\\1\end{pmatrix}+ | \begin{pmatrix}m_{i1}\\m_{i2}\\1\end{pmatrix}+ | ||
\vec{\epsilon_i} | \vec{\epsilon_i} | ||
| − | </math> | + | </math>. |
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| + | [[Image:working.gif]] Under construction ... | ||
Other symbols: | Other symbols: | ||
Revision as of 21:34, 25 June 2006
Calibration
Let <math>\vec{m}_i\in\mathbb{R}^3,\,i\in\{1,2,\ldots,N\}</math> be the homogeneous coordinate of the <math>i</math>th point on the planar calibration-object and let <math>\vec{m}^\prime_i\in\mathbb{R}^3</math> be the homogeneous coordinate of the corresponding pixel in the camera-image. Further let <math>(\mathbb{R}^n,\simeq)</math> be an equivalence relation defined by
<math> \vec{a}\simeq\vec{b}\ :\Leftrightarrow\ \exists\lambda\in\mathbb{R}/\{0\}:\,\lambda\,\vec{a}=\vec{b} </math>
If the camera-system does an ideal central projection (e.g. no distortion), the projective transformation (the Homography) can be modelled using <math>\mathcal{H}\in\mathbb{R}^{3\times 3}</math> as follows
<math> \widehat{\vec{m}}_i\simeq\mathcal{H}\vec{m}_i </math> or more elaborately <math> \lambda\,\begin{pmatrix}\widehat{m}_{i1}\\\widehat{m}_{i2}\\\widehat{m}_{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} </math> with <math>\vec{m}^\prime_i=\widehat{\vec{m}}_i+\vec{\epsilon}_i</math>.
Using <math>m_{i3}=m^\prime_{i3}=\widehat{m}_{i3}=1</math> and <math>\vec{h_i}^\top=\begin{pmatrix}h_{i1}&h_{i2}&h_{i3}\end{pmatrix},\ i\in\{1,2,3\}</math> the model can be represented by <math> \begin{pmatrix}\widehat{m}_{i1}\\\widehat{m}_{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}+ \vec{\epsilon_i} </math>.
Under construction ...
Other symbols: <math> \vec{0}, \neq, ||\mathcal{A}||, \mathcal{A}\in\mathbb{C}^{3\times 3}, i\in\mathbb{N}_0,0\approx 1, \begin{pmatrix}1-\lambda&0\\0&1-\lambda\end{pmatrix}, \mathbb{R}^3, i\in\{1,2,\ldots,n\}</math>
<math> \begin{pmatrix}1-\lambda&0&\cdots&\cdots&0\\0&1-\lambda&\ddots&&\vdots\end{pmatrix},V^*,a*b, \sigma, \Sigma, \lambda, \Lambda</math>
<math> \sum_{i=1}^n p^i, \prod_{i=1}^{+\infty}a_i, \cos(\omega), \widehat{i}=\mathrm{argmax}_{i\in\mathbb{N}}f(i), a\simeq b </math>
