Computing Memoir: July 2021

Rigid Transformation

A point P in the world can be expressed as Body coordinate as well as World coordinates, and the relationship between the two are rigid transformation - rotation and translation - like below.

$P_{W} = \begin{bmatrix}P_{Wx}\\P_{Wy} \\1\end{bmatrix} = \begin{bmatrix}cos \phi & -sin \phi & d_{x} \\sin \phi & cos \phi & d_{y} \\0 & 0 & 1 \\\end{bmatrix} \begin{bmatrix} P_{Bx}\\ P_{By} \\ 1 \end{bmatrix} = T_{B}^{W} P_{B}$

Refer Wiki - Kinematics : Matrix representation

Image resolution and dimension

A point at an image can be expressed on world coordinates using Image resolution - i.e. mm / pixel - and dimension. At below, resolution is r. W is width and H is Height. Point at center (0,0) will be (320,240) of 640x480 image.

$P_{P} = \begin{bmatrix}\frac{1}{r_{x}} & 0 & \frac{W}{2} \\0 & -\frac{1}{r_{y}} & \frac{H}{2} \\0 & 0 & 1 \\\end{bmatrix}P_{C} = T_{C}^{P}P_{C}$

Geometric Model

For example, there is a pixel point (P) on a camera ( C ), that is offseted ( CO ) and assmebled to a machine (W).

$P_{W} = T_{CO}^{W} \,\, T_{C}^{CO} \,\, T_{P}^{C} \,\, P_{P}$

And the machine has a stage (S) and it is moved by motor (M) and the stage has a point.

$P_{W} = T_{M}^{W} \,\, T_{S}^{M} \,\, P_{S}$

Then any point that camera see can be matched to a point on the stage.

$T_{CO}^{W} \,\, T_{C}^{CO} \,\, T_{P}^{C} \,\, P_{P} = T_{M}^{W} \,\, T_{S}^{M} \,\, P_{S}$

Least square method on a linearized transformation matrix

If we know all other transformation except camera offset, then we can rearrange above equation like below for an i-th point.

$T_{C}^{CO} \,\, T_{P}^{C} \,\, P_{Pi} = T_{CO}^{W-1} \,\, T_{M}^{W} \,\, T_{S}^{M} \,\, P_{Si}$
$T_{C}^{CO} \,\, U_{i} = V_{i}$

The unknown CO transformation matrix can be linearized in assumption that the angle is small.

$\begin{bmatrix}cos \phi & -sin \phi & d_{x} \\sin \phi & cos \phi & d_{y} \\0 & 0 & 1 \\\end{bmatrix} \begin{bmatrix} U_{io}\\ U_{i1} \\ 1 \end{bmatrix} \approx \begin{bmatrix} 1 & -\phi & d_{x} \\ \phi & 1 & d_{y} \\0 & 0 & 1 \\\end{bmatrix} \begin{bmatrix} U_{i1}\\ U_{i2} \\ 1 \end{bmatrix} = \begin{bmatrix} V_{i1}\\ V_{i2} \\ 1 \end{bmatrix}$

Then the equation can be rearranged putting the unknown X on right. Then the unknown can be solved by matrix inversion. Refer Wiki - Least Square : Linear least squares

$\begin{bmatrix}1 & 0 & -U_{i2} & U_{i1} \\0 & 1 & U_{i1} & U_{i2} \\\end{bmatrix} \begin{bmatrix}d_{x} \\d_{y} \\\phi \\ 1 \end{bmatrix} = U \,\, X = V = \begin{bmatrix}V_{i1} \\ V_{i2}\end{bmatrix}$
$X = (U^{T}U)^{-1}U^{T}V$

Weighted least square method

If there is an uncertainty on each measurment of pixel point, the uncertainty can be put as weight (W) on each pixel point.

$W_{i} P_{Pi} = \begin{bmatrix}W_{i1} & 0 \\0 & W_{i2} \\\end{bmatrix} P_{Pi} = W_{i} \,\, (T_{CO}^{W} \,\, T_{C}^{CO} \,\, T_{P}^{C})^{-1}\,\, T_{M}^{W} \,\, T_{S}^{M} \,\, P_{Si}$
$= W_{i}\,\, R \,\, T_{C}^{CO-1} \,\, U_{i} \approx W_{i} \begin{bmatrix}r_{1} & r_{2} & r_{3} \\r_{4} & r_{5} & r_{6} \\r_{7} & r_{8} & r_{9} \\\end{bmatrix} \begin{bmatrix}1 \ & -\phi & d_{x} \\\phi & 1 & d_{y} \\0 & 0 & 1 \\ \end{bmatrix}\begin{bmatrix}U_{i1} \\U_{i2} \\1 \end{bmatrix}$

Then the equation can be rearranged for the unknown X. Note that element of X changed sign for convenience.

$W_{i}\begin{bmatrix}r_{1} & r_{2} & -r_{1} U_{i2} + r_{2} U_{i1} & r_{1} U_{i1} + r_{2}U_{i2}+ r_{3} \\r_{4} & r_{5} & -r_{4} U_{i2} + r_{5} U_{i1} & r_{4}U_{i1} + r_{5} U_{i2} + r_{6}\end{bmatrix} \begin{bmatrix}d_{x} \\ d_{y} \\ \phi \\ 1\end{bmatrix} = W_{i}\,\, U_{i} \,\, X = W_{i}\,\, V_{i}$

$W \,\, U \,\, X = W \,\, V$

$X = ( U^{T}\,\,W\,\,U)^{-1}\,\,U^{T}\,\,W\,\,V$

Computing Memoir

Wednesday, July 7, 2021

Finding planar rigid tranformation - linearized and weighted least squares

Rigid Transformation

Image resolution and dimension

Geometric Model

Least square method on a linearized transformation matrix

Weighted least square method

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