Homography-Based Dynamic Calibration

Abstract

The relative pose problem is to find the possible solutions for relative camera pose between two calibrated views given several corresponding coplanar or non-coplanar points. In this chapter, we will talk about an algorithm for determining the relative pose in a structured light system using coplanar points, supposing that its intrinsic parameters have been obtained by static calibration as discussed in the previous chapter. The image-to-image Homographic matrix is extensively explored with the assumption of one arbitrary plane in the scene. A closed-form solution is provided to obtain computational efficiency. Redundancy in the data is easily incorporated to improve the reliability of the estimations in the presence of noise. By using the matrix perturbation theory, we give a typical sensitivity analysis on the estimated pose parameters with respect to the noise in the image points. Finally, some experimental results are shown to valid this technique.

Appendices

Appendix 1

In this appendix, we show that three linear equations can be obtained from the six quadratic equations by polynomial eliminations.

From ${{w}_{23}}\text{*(4}\text{4.22a}\text{)-}{{w}_{33}}\text{*(4}\text{4.22e}\text{)}$, we get

$$ {{w}_{33}}{{w}_{13}}{{t}_{1}}{{t}_{2}}+( {{w}_{33}}{{w}_{12}}-2{{w}_{23}}{{w}_{13}} ){{t}_{1}}-{{w}_{33}}{{w}_{11}}{{t}_{2}}+{{w}_{23}}{{w}_{11}}=0 $$

(4.54)

From ${{w}_{22}}\text{*(4}\text{4.22a}\text{)-}{{w}_{33}}\text{*(4.22f}\text{}\text{)}$, we get

$$ 2{{w}_{33}}{{w}_{12}}{{t}_{1}}{{t}_{2}}-{{w}_{33}}{{w}_{11}}t_{2}^{2}-2{{w}_{22}}{{w}_{13}}{{t}_{1}}+{{w}_{22}}{{w}_{11}}=0 $$

(4.55)

From ${{w}_{13}}\text{*(4.22b}\text{}\text{)-}{{w}_{33}}\text{*(4.22d}\text{}\text{)}$, we have

$$ {{w}_{23}}{{w}_{33}}{{t}_{1}}{{t}_{2}}-{{w}_{22}}{{w}_{33}}t_{1}^{{}}+( {{w}_{12}}{{w}_{33}}-2{{w}_{23}}{{w}_{13}} ){{t}_{2}}+{{w}_{22}}{{w}_{13}}=0 $$

(4.56)

From $(\text{4}\text{.55)}+{{w}_{11}}\text{*(4.22b}\text{}\text{)}$, we have

$$ {{w}_{12}}{{w}_{33}}{{t}_{1}}{{t}_{2}}-{{w}_{22}}{{w}_{13}}t_{1}^{{}}-{{w}_{11}}{{w}_{23}}{{t}_{2}}+{{w}_{11}}{{w}_{22}}=0 $$

(4.57)

From ${{w}_{13}}\text{*(4.22c}\text{}\text{)-}(\text{4}\text{.54)}$, we have

$$ ( {{w}_{13}}{{w}_{23}}-{{w}_{12}}{{w}_{33}} ){{t}_{1}}+\left( {{w}_{11}}{{w}_{33}}-w_{13}^{2} \right)t_{2}^{{}}+{{w}_{12}}{{w}_{13}}-{{w}_{11}}{{w}_{23}}=0 $$

(4.58)

From ${{w}_{23}}\text{*(4.22c}\text{}\text{)-}(\text{4}\text{.56)}$, we have

$$ \left( {{w}_{22}}{{w}_{33}}-w_{23}^{2} \right){{t}_{1}}+( {{w}_{13}}{{w}_{23}}-w_{12}^{{}}w_{33}^{{}} )t_{2}^{{}}+{{w}_{12}}{{w}_{23}}-{{w}_{13}}{{w}_{22}}=0 $$

(4.59)

From ${{w}_{12}}\text{*(4}\text{4.22c}\text{)-}(\text{4}\text{.57)}$, we have

$$ ( {{w}_{22}}{{w}_{13}}-w_{12}^{{}}w_{23}^{{}} ){{t}_{1}}+( {{w}_{11}}{{w}_{23}}-w_{12}^{{}}w_{13}^{{}} )t_{2}^{{}}+w_{12}^{2}-{{w}_{11}}{{w}_{22}}=0 $$

(4.60)

We then have obtained the three linear equations in (4.58), (4.59) and (4.60).

Appendix 2

Let c _ij and d _ij be the (i, j) elements of matrix C and D respectively.

Let ${{\boldsymbol D}_{\boldsymbol {B}}}=[\begin{matrix} {{\boldsymbol D}_{\lambda }} & {{\boldsymbol D}_{{{t}_{1}}}} & {{\boldsymbol D}_{{{t}_{2}}}} & {{\boldsymbol D}_{{{h}_{L}}}} & {{\boldsymbol D}_{{{h}_{R}}}}\end{matrix}]$. Then its five elements can be formulated as follows.

$$ {{\boldsymbol D}_{\lambda }}=\left[ \begin{matrix} {{d}_{31}}{{t}_{2}}-{{d}_{32}}{{t}_{1}}+2{{c}_{13}}{{t}_{2}}+{{d}_{23}}{{t}_{1}}-2{{c}_{12}}-{{d}_{21}}-2{{c}_{23}}{{t}_{1}}+{{d}_{12}}-{{d}_{13}}{{t}_{2}}\\[5pt]{{d}_{22}}{{t}_{1}}-{{d}_{31}}-{{d}_{21}}{{t}_{2}}+{{d}_{33}}{{t}_{1}}\\ [5pt]-{{d}_{12}}{{t}_{1}}+{{d}_{11}}{{t}_{2}}+{{d}_{33}}{{t}_{2}}-{{d}_{32}}\\ [5pt]-{{d}_{13}}{{t}_{1}}-{{d}_{23}}{{t}_{2}}+{{d}_{11}}+{{d}_{22}}\\ [5pt]-{{d}_{31}}{{t}_{2}}+{{d}_{32}}{{t}_{1}}+2{{c}_{13}}{{t}_{2}}-{{d}_{23}}{{t}_{1}}-2{{c}_{12}}+{{d}_{21}}-2{{c}_{23}}{{t}_{1}}+{{d}_{12}}-{{d}_{13}}{{t}_{2}}\\ [5pt]{{d}_{13}}{{t}_{1}}-{{d}_{23}}{{t}_{2}}-{{d}_{11}}+{{d}_{22}}\\ [5pt]-{{d}_{12}}{{t}_{1}}+{{d}_{11}}{{t}_{2}}-{{d}_{33}}{{t}_{2}}+{{d}_{32}}\\ [5pt]-{{d}_{31}}{{t}_{2}}+{{d}_{32}}{{t}_{1}}+2{{c}_{13}}{{t}_{2}}+{{d}_{23}}{{t}_{1}}-2{{c}_{12}}-{{d}_{21}}-2{{c}_{23}}{{t}_{1}}-{{d}_{12}}+{{d}_{13}}{{t}_{2}}\\ [5pt]-{{d}_{22}}{{t}_{1}}-{{d}_{31}}+{{d}_{21}}{{t}_{2}}+{{d}_{33}}{{t}_{1}}\\ [5pt]{{d}_{31}}{{t}_{2}}-{{d}_{32}}{{t}_{1}}+2{{c}_{13}}{{t}_{2}}-{{d}_{23}}{{t}_{1}}-2{{c}_{12}}+{{d}_{21}}-2{{c}_{23}}{{t}_{1}}-{{d}_{12}}+{{d}_{13}}{{t}_{2}}\\[5pt]\end{matrix} \right], $$

$${D_{{t_1}}} = \left[{\begin{array}{*{20}{c}}\begin{array}{l}{c_{13}}{h_4} +{c_{23}}{h_5} - {d_{13}}{h_4} - 2{c_{23}}\lambda - {d_{23}}{h_5}- {d_{33}}{h_6} - {c_{12}}{h_7} + {c_{23}}{h_9} \\[5pt]\quad +{d_{22}}{h_8} + {d_{12}}{h_7} + {d_{32}}{h_9} + {d_{23}}\lambda - {d_{32}}\lambda \\ [5pt]\end{array} \\ [5pt] { - {c_{23}}{h_8} + {d_{22}}\lambda + {c_{23}}{h_6} + {d_{33}}\lambda } \\ [5pt] { - {c_{13}}{h_6} - {d_{12}}\lambda + {c_{12}}{h_9} + {c_{23}}{h_7}} \\ [5pt] { - {c_{23}}{h_4} + {c_{13}}{h_5} - {d_{13}}\lambda - {c_{12}}{h_8}} \\ [5pt] \begin{array}{l} {c_{13}}{h_4} - {c_{23}}{h_5} - {d_{13}}{h_4} - 2{c_{23}}\lambda - {d_{23}}{h_5} - {d_{33}}{h_6} - {c_{12}}{h_7} - {c_{23}}{h_9} \\ [5pt]\quad+ {d_{22}}{h_8} + {d_{12}}{h_7} + {d_{32}}{h_9} - {d_{23}}\lambda + {d_{32}}\lambda \\ [5pt]\end{array} \\ [5pt] {{c_{23}}{h_4} + {c_{13}}{h_5} + {d_{13}}\lambda - {c_{12}}{h_8}} \\ [5pt]{{c_{13}}{h_6} - {d_{12}}\lambda - {c_{12}}{h_9} + {c_{23}}{h_7}} \\ [5pt] \begin{array}{l} - {c_{13}}{h_4} + {c_{23}}{h_5} - {d_{13}}{h_4} - 2{c_{23}}\lambda - {d_{23}}{h_5} - {d_{33}}{h_6} + {c_{12}}{h_7} - {c_{23}}{h_9} \\[5pt]\quad \; +{d_{22}}{h_8} + {d_{12}}{h_7} + {d_{32}}{h_9} + {d_{23}}\lambda + {d_{32}}\lambda \\ [5pt]\end{array} \\ [5pt] {{c_{23}}{h_8} - {d_{22}}\lambda + {c_{23}}{h_6} + {d_{33}}\lambda } \\ [5pt] \begin{array}{l} - {c_{13}}{h_4} - {c_{23}}{h_5} - {d_{13}}{h_4} - 2{c_{23}}\lambda - {d_{23}}{h_5} - {d_{33}}{h_6} + {c_{12}}{h_7} + {c_{23}}{h_9} \\[5pt]\quad\; + {d_{22}}{h_8} + {d_{12}}{h_7} + {d_{32}}{h_9} - {d_{23}}\lambda - {d_{32}}\lambda \\ [5pt]\end{array} \\[5pt]\end{array}} \right],$$

$$ {{\boldsymbol D}_{{t_2}}} = \left[ {\begin{array}{*{20}{c}} \begin{array}{l} - {c_{13}}{h_1} + {d_{33}}{h_3} + {d_{23}}{h_2} + 2{c_{13}}\lambda - {d_{11}}{h_7} - {c_{13}}{h_9} - {d_{21}}{h_8} + {d_{13}}{h_1} \\[5pt]\;\quad- {c_{12}}{h_8} - {c_{23}}{h_2} - {d_{31}}{h_9} - {d_{13}}\lambda + {d_{31}}\lambda \\ [5pt]\end{array} \\ [5pt]{{c_{13}}{h_8} - {c_{23}}{h_3} - {c_{12}}{h_9} - {d_{21}}\lambda } \\ [5pt]{{d_{33}}\lambda + {c_{13}}{h_3} + {c_{11}}\lambda - {c_{13}}{h_7}} \\ [5pt]{ - {c_{13}}{h_2} + {c_{12}}{h_7} - {d_{23}}\lambda + {c_{23}}{h_1}} \\ [5pt]\begin{array}{l} - {c_{13}}{h_1} + {d_{33}}{h_3} + {d_{23}}{h_2} + 2{c_{13}}\lambda - {d_{11}}{h_7} + {c_{13}}{h_9} - {d_{21}}{h_8} + {d_{13}}{h_1} \\[5pt]\;\quad + {c_{12}}{h_8} + {c_{23}}{h_2} - {d_{31}}{h_9} - {d_{13}}\lambda - {d_{31}}\lambda \\ [5pt]\end{array} \\ [5pt] { - {c_{13}}{h_2} - {c_{12}}{h_7} - {d_{23}}\lambda - {c_{23}}{h_1}} \\ [5pt] { - {d_{33}}\lambda - {c_{13}}{h_3} + {c_{11}}\lambda - {c_{13}}{h_7}} \\ [5pt] \begin{array}{l} {c_{13}}{h_1} + {d_{33}}{h_3} + {d_{23}}{h_2} + 2{c_{13}}\lambda - {d_{11}}{h_7} + {c_{13}}{h_9} - {d_{21}}{h_8} + {d_{13}}{h_1} \\ [5pt]\quad- {c_{12}}{h_8} - {c_{23}}{h_2} - {d_{31}}{h_9} + {d_{13}}\lambda - {d_{31}}\lambda \\ [5pt]\end{array} \\ [5pt]{ - {c_{13}}{h_8} - {c_{23}}{h_3} - {c_{12}}{h_9} + {d_{21}}\lambda } \\ [5pt] \begin{array}{l} {c_{13}}{h_1} + {d_{33}}{h_3} + {d_{23}}{h_2} + 2{c_{13}}\lambda - {d_{11}}{h_7} - {c_{13}}{h_9} - {d_{21}}{h_8} + {d_{13}}{h_1} \\[5pt]\quad + {c_{12}}{h_8} + {c_{23}}{h_2} - {d_{31}}{h_9} + {d_{13}}\lambda + {d_{31}}\lambda \\ [5pt]\end{array} \\ [5pt]\end{array}} \right], $$

$$ {{\mathbf{\boldsymbol D}}_{{{h}_{L}}}}=\left[ \begin{matrix} ({{d}_{13}}-{{c}_{13}}){{t}_{2}}-{{d}_{12}}+{{c}_{12}} & ({{d}_{23}}-{{c}_{23}}){{t}_{2}}-{{d}_{22}} & {{d}_{33}}{{t}_{2}}-{{d}_{32}}-{{c}_{23}} & ({{c}_{13}}-{{d}_{13}}){{t}_{1}}+{{d}_{11}} & (-{{d}_{23}}+{{c}_{23}}){{t}_{1}}+{{d}_{21}}+{{c}_{12}}\\ [5pt]0 & {{c}_{23}} & -{{c}_{23}}{{t}_{2}} & 0 & -{{c}_{13}}\\ [5pt]-{{c}_{23}} & 0 & {{c}_{13}}{{t}_{2}}-{{c}_{12}} & {{c}_{13}} & 0\\ [5pt]{{c}_{23}}{{t}_{2}} & {{c}_{12}}-{{c}_{13}}{{t}_{2}} & 0 & -{{c}_{23}}{{t}_{1}}-{{c}_{12}} & {{c}_{13}}{{t}_{1}}\\ [5pt]({{d}_{13}}-{{c}_{13}}){{t}_{2}}-{{d}_{12}}+{{c}_{12}} & ({{d}_{23}}+{{c}_{23}}){{t}_{2}}-{{d}_{22}} & {{d}_{33}}{{t}_{2}}-{{d}_{32}}+{{c}_{23}} & ({{c}_{13}}-{{d}_{13}}){{t}_{1}}+{{d}_{11}} & (-{{d}_{23}}-{{c}_{23}}){{t}_{1}}+{{d}_{21}}-{{c}_{12}}\\ [5pt]-{{c}_{23}}{{t}_{2}} & {{c}_{12}}-{{c}_{13}}{{t}_{2}} & 0 & {{c}_{23}}{{t}_{1}}+{{c}_{12}} & {{c}_{13}}{{t}_{1}}\\ [5pt]-{{c}_{23}} & 0 & -{{c}_{13}}{{t}_{2}}+{{c}_{12}} & {{c}_{13}} & 0\\ [5pt]({{d}_{13}}+{{c}_{13}}){{t}_{2}}-{{d}_{12}}-{{c}_{12}} & ({{d}_{23}}-{{c}_{23}}){{t}_{2}}-{{d}_{22}} & {{d}_{33}}{{t}_{2}}-{{d}_{32}}+{{c}_{23}} & (-{{c}_{13}}-{{d}_{13}}){{t}_{1}}+{{d}_{11}} & (-{{d}_{23}}+{{c}_{23}}){{t}_{1}}+{{d}_{21}}+{{c}_{12}}\\ [5pt]0 & -{{c}_{23}} & -{{c}_{23}}{{t}_{2}} & 0 & {{c}_{13}}\\ [5pt]({{d}_{13}}+{{c}_{13}}){{t}_{2}}-{{d}_{12}}-{{c}_{12}} & ({{d}_{23}}+{{c}_{23}}){{t}_{2}}-{{d}_{22}} & {{d}_{33}}{{t}_{2}}-{{d}_{32}}-{{c}_{23}} & (-{{c}_{13}}-{{d}_{13}}){{t}_{1}}+{{d}_{11}} & (-{{d}_{23}}-{{c}_{23}}){{t}_{1}}+{{d}_{21}}-{{c}_{12}}\\[5pt]\end{matrix} \right], $$

$$ {{\boldsymbol D}_{{{h}_{R}}}}=\left[ \begin{matrix} -{{d}_{33}}{{t}_{1}}+{{d}_{31}}+{{c}_{13}} & {{d}_{12}}{{t}_{1}}-{{d}_{11}}{{t}_{2}}-{{c}_{12}}{{t}_{1}} & {{d}_{22}}{{t}_{1}}-{{d}_{21}}{{t}_{2}}-{{c}_{12}}{{t}_{2}} & ({{d}_{32}}+{{c}_{23}}){{t}_{1}}-({{d}_{31}}+{{c}_{13}}){{t}_{2}}\\ [5pt]{{c}_{23}}{{t}_{1}}+{{c}_{12}} & 0 & -{{c}_{23}}{{t}_{1}}+{{c}_{13}}{{t}_{2}} & -{{c}_{12}}{{t}_{2}}\\ [5pt]-{{c}_{13}}{{t}_{1}} & {{c}_{23}}{{t}_{1}}-{{c}_{13}}{{t}_{2}} & 0 & {{c}_{12}}{{t}_{1}}\\ [5pt]0 & {{c}_{12}}{{t}_{2}} & -{{c}_{12}}{{t}_{1}} & 0\\ [5pt]-{{d}_{33}}{{t}_{1}}+{{d}_{31}}-{{c}_{13}} & {{d}_{12}}{{t}_{1}}-{{d}_{11}}{{t}_{2}}-{{c}_{12}}{{t}_{1}} & {{d}_{22}}{{t}_{1}}-{{d}_{21}}{{t}_{2}}+{{c}_{12}}{{t}_{2}} & ({{d}_{32}}-{{c}_{23}}){{t}_{1}}-({{d}_{31}}-{{c}_{13}}){{t}_{2}}\\ [5pt]0 & -{{c}_{12}}{{t}_{2}} & -{{c}_{12}}{{t}_{1}} & 0\\ [5pt]{{c}_{13}}{{t}_{1}} & {{c}_{23}}{{t}_{1}}-{{c}_{13}}{{t}_{2}} & 0 & -{{c}_{12}}{{t}_{1}}\\ [5pt]-{{d}_{33}}{{t}_{1}}+{{d}_{31}}-{{c}_{13}} & {{d}_{12}}{{t}_{1}}-{{d}_{11}}{{t}_{2}}+{{c}_{12}}{{t}_{1}} & {{d}_{22}}{{t}_{1}}-{{d}_{21}}{{t}_{2}}-{{c}_{12}}{{t}_{2}} & ({{d}_{32}}-{{c}_{23}}){{t}_{1}}-({{d}_{31}}-{{c}_{13}}){{t}_{2}}\\ [5pt]{{c}_{23}}{{t}_{1}}+{{c}_{12}} & 0 & {{c}_{23}}{{t}_{1}}-{{c}_{13}}{{t}_{2}} & -{{c}_{12}}{{t}_{2}}\\ [5pt]-{{d}_{33}}{{t}_{1}}+{{d}_{31}}+{{c}_{13}} & {{d}_{12}}{{t}_{1}}-{{d}_{11}}{{t}_{2}}+{{c}_{12}}{{t}_{1}} & {{d}_{22}}{{t}_{1}}-{{d}_{21}}{{t}_{2}}+{{c}_{12}}{{t}_{2}} & ({{d}_{32}}+{{c}_{23}}){{t}_{1}}-({{d}_{31}}+{{c}_{13}}){{t}_{2}}\\[5pt]\end{matrix} \right]. $$