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homography

$\left(\begin{array}{ccc} {\mathrm{q1}}^2 + {\mathrm{q2}}^2 - {\mathrm{q3}}^2 - {\mathrm{q4}}^2 & 2\, \mathrm{q2}\, \mathrm{q3} - 2\, \mathrm{q1}\, \mathrm{q4} & 2\, \mathrm{q1}\, \mathrm{q3} + 2\, \mathrm{q2}\, \mathrm{q4}\\ 2\, \mathrm{q1}\, \mathrm{q4} + 2\, \mathrm{q2}\, \mathrm{q3} & {\mathrm{q1}}^2 - {\mathrm{q2}}^2 + {\mathrm{q3}}^2 - {\mathrm{q4}}^2 & 2\, \mathrm{q3}\, \mathrm{q4} - 2\, \mathrm{q1}\, \mathrm{q2}\\ 2\, \mathrm{q2}\, \mathrm{q4} - 2\, \mathrm{q1}\, \mathrm{q3} & 2\, \mathrm{q1}\, \mathrm{q2} + 2\, \mathrm{q3}\, \mathrm{q4} & {\mathrm{q1}}^2 - {\mathrm{q2}}^2 - {\mathrm{q3}}^2 + {\mathrm{q4}}^2 \end{array}\right)
$

$\left(\begin{array}{ccccccccccc} 2\, \mathrm{q1} & 2\, \mathrm{q2} & - 2\, \mathrm{q3} & - 2\, \mathrm{q4} & -\frac{\mathrm{nx}}{d} & 0 & 0 & -\frac{\mathrm{tx}}{d} & 0 & 0 & \frac{\mathrm{nx}\, \mathrm{tx}}{d^2}\\ - 2\, \mathrm{q4} & 2\, \mathrm{q3} & 2\, \mathrm{q2} & - 2\, \mathrm{q1} & -\frac{\mathrm{ny}}{d} & 0 & 0 & 0 & -\frac{\mathrm{tx}}{d} & 0 & \frac{\mathrm{ny}\, \mathrm{tx}}{d^2}\\ 2\, \mathrm{q3} & 2\, \mathrm{q4} & 2\, \mathrm{q1} & 2\, \mathrm{q2} & -\frac{\mathrm{nz}}{d} & 0 & 0 & 0 & 0 & -\frac{\mathrm{tx}}{d} & \frac{\mathrm{nz}\, \mathrm{tx}}{d^2}\\ 2\, \mathrm{q4} & 2\, \mathrm{q3} & 2\, \mathrm{q2} & 2\, \mathrm{q1} & 0 & -\frac{\mathrm{nx}}{d} & 0 & -\frac{\mathrm{ty}}{d} & 0 & 0 & \frac{\mathrm{nx}\, \mathrm{ty}}{d^2}\\ 2\, \mathrm{q1} & - 2\, \mathrm{q2} & 2\, \mathrm{q3} & - 2\, \mathrm{q4} & 0 & -\frac{\mathrm{ny}}{d} & 0 & 0 & -\frac{\mathrm{ty}}{d} & 0 & \frac{\mathrm{ny}\, \mathrm{ty}}{d^2}\\ - 2\, \mathrm{q2} & - 2\, \mathrm{q1} & 2\, \mathrm{q4} & 2\, \mathrm{q3} & 0 & -\frac{\mathrm{nz}}{d} & 0 & 0 & 0 & -\frac{\mathrm{ty}}{d} & \frac{\mathrm{nz}\, \mathrm{ty}}{d^2}\\ - 2\, \mathrm{q3} & 2\, \mathrm{q4} & - 2\, \mathrm{q1} & 2\, \mathrm{q2} & 0 & 0 & -\frac{\mathrm{nx}}{d} & -\frac{\mathrm{tz}}{d} & 0 & 0 & \frac{\mathrm{nx}\, \mathrm{tz}}{d^2}\\ 2\, \mathrm{q2} & 2\, \mathrm{q1} & 2\, \mathrm{q4} & 2\, \mathrm{q3} & 0 & 0 & -\frac{\mathrm{ny}}{d} & 0 & -\frac{\mathrm{tz}}{d} & 0 & \frac{\mathrm{ny}\, \mathrm{tz}}{d^2}\\ 2\, \mathrm{q1} & - 2\, \mathrm{q2} & - 2\, \mathrm{q3} & 2\, \mathrm{q4} & 0 & 0 & -\frac{\mathrm{nz}}{d} & 0 & 0 & -\frac{\mathrm{tz}}{d} & \frac{\mathrm{nz}\, \mathrm{tz}}{d^2} \end{array}\right)$

$\left(\begin{array}{cccccccc} \frac{u_{1}1}{\mathrm{h31}\, u_{1}1 + \mathrm{h32}\, \mathrm{v11} + 1} & \frac{\mathrm{v11}}{\mathrm{h31}\, u_{1}1 + \mathrm{h32}\, \mathrm{v11} + 1} & \frac{1}{\mathrm{h31}\, u_{1}1 + \mathrm{h32}\, \mathrm{v11} + 1} & 0 & 0 & 0 & -\frac{u_{1}1\, \left(\mathrm{h13} + \mathrm{h11}\, u_{1}1 + \mathrm{h12}\, \mathrm{v11}\right)}{{\left(\mathrm{h31}\, u_{1}1 + \mathrm{h32}\, \mathrm{v11} + 1\right)}^2} & -\frac{\mathrm{v11}\, \left(\mathrm{h13} + \mathrm{h11}\, u_{1}1 + \mathrm{h12}\, \mathrm{v11}\right)}{{\left(\mathrm{h31}\, u_{1}1 + \mathrm{h32}\, \mathrm{v11} + 1\right)}^2}\\ 0 & 0 & 0 & \frac{u_{1}1}{\mathrm{h31}\, u_{1}1 + \mathrm{h32}\, \mathrm{v11} + 1} & \frac{\mathrm{v11}}{\mathrm{h31}\, u_{1}1 + \mathrm{h32}\, \mathrm{v11} + 1} & \frac{1}{\mathrm{h31}\, u_{1}1 + \mathrm{h32}\, \mathrm{v11} + 1} & -\frac{u_{1}1\, \left(\mathrm{h23} + \mathrm{h21}\, u_{1}1 + \mathrm{h22}\, \mathrm{v11}\right)}{{\left(\mathrm{h31}\, u_{1}1 + \mathrm{h32}\, \mathrm{v11} + 1\right)}^2} & -\frac{\mathrm{v11}\, \left(\mathrm{h23} + \mathrm{h21}\, u_{1}1 + \mathrm{h22}\, \mathrm{v11}\right)}{{\left(\mathrm{h31}\, u_{1}1 + \mathrm{h32}\, \mathrm{v11} + 1\right)}^2}\\ \frac{\mathrm{u12}}{\mathrm{h31}\, \mathrm{u12} + \mathrm{h32}\, \mathrm{v12} + 1} & \frac{\mathrm{v12}}{\mathrm{h31}\, \mathrm{u12} + \mathrm{h32}\, \mathrm{v12} + 1} & \frac{1}{\mathrm{h31}\, \mathrm{u12} + \mathrm{h32}\, \mathrm{v12} + 1} & 0 & 0 & 0 & -\frac{\mathrm{u12}\, \left(\mathrm{h13} + \mathrm{h11}\, \mathrm{u12} + \mathrm{h12}\, \mathrm{v12}\right)}{{\left(\mathrm{h31}\, \mathrm{u12} + \mathrm{h32}\, \mathrm{v12} + 1\right)}^2} & -\frac{\mathrm{v12}\, \left(\mathrm{h13} + \mathrm{h11}\, \mathrm{u12} + \mathrm{h12}\, \mathrm{v12}\right)}{{\left(\mathrm{h31}\, \mathrm{u12} + \mathrm{h32}\, \mathrm{v12} + 1\right)}^2}\\ 0 & 0 & 0 & \frac{\mathrm{u12}}{\mathrm{h31}\, \mathrm{u12} + \mathrm{h32}\, \mathrm{v12} + 1} & \frac{\mathrm{v12}}{\mathrm{h31}\, \mathrm{u12} + \mathrm{h32}\, \mathrm{v12} + 1} & \frac{1}{\mathrm{h31}\, \mathrm{u12} + \mathrm{h32}\, \mathrm{v12} + 1} & -\frac{\mathrm{u12}\, \left(\mathrm{h23} + \mathrm{h21}\, \mathrm{u12} + \mathrm{h22}\, \mathrm{v12}\right)}{{\left(\mathrm{h31}\, \mathrm{u12} + \mathrm{h32}\, \mathrm{v12} + 1\right)}^2} & -\frac{\mathrm{v12}\, \left(\mathrm{h23} + \mathrm{h21}\, \mathrm{u12} + \mathrm{h22}\, \mathrm{v12}\right)}{{\left(\mathrm{h31}\, \mathrm{u12} + \mathrm{h32}\, \mathrm{v12} + 1\right)}^2}\\ \frac{\mathrm{u13}}{\mathrm{h31}\, \mathrm{u13} + \mathrm{h32}\, \mathrm{v13} + 1} & \frac{\mathrm{v13}}{\mathrm{h31}\, \mathrm{u13} + \mathrm{h32}\, \mathrm{v13} + 1} & \frac{1}{\mathrm{h31}\, \mathrm{u13} + \mathrm{h32}\, \mathrm{v13} + 1} & 0 & 0 & 0 & -\frac{\mathrm{u13}\, \left(\mathrm{h13} + \mathrm{h11}\, \mathrm{u13} + \mathrm{h12}\, \mathrm{v13}\right)}{{\left(\mathrm{h31}\, \mathrm{u13} + \mathrm{h32}\, \mathrm{v13} + 1\right)}^2} & -\frac{\mathrm{v13}\, \left(\mathrm{h13} + \mathrm{h11}\, \mathrm{u13} + \mathrm{h12}\, \mathrm{v13}\right)}{{\left(\mathrm{h31}\, \mathrm{u13} + \mathrm{h32}\, \mathrm{v13} + 1\right)}^2}\\ 0 & 0 & 0 & \frac{\mathrm{u13}}{\mathrm{h31}\, \mathrm{u13} + \mathrm{h32}\, \mathrm{v13} + 1} & \frac{\mathrm{v13}}{\mathrm{h31}\, \mathrm{u13} + \mathrm{h32}\, \mathrm{v13} + 1} & \frac{1}{\mathrm{h31}\, \mathrm{u13} + \mathrm{h32}\, \mathrm{v13} + 1} & -\frac{\mathrm{u13}\, \left(\mathrm{h23} + \mathrm{h21}\, \mathrm{u13} + \mathrm{h22}\, \mathrm{v13}\right)}{{\left(\mathrm{h31}\, \mathrm{u13} + \mathrm{h32}\, \mathrm{v13} + 1\right)}^2} & -\frac{\mathrm{v13}\, \left(\mathrm{h23} + \mathrm{h21}\, \mathrm{u13} + \mathrm{h22}\, \mathrm{v13}\right)}{{\left(\mathrm{h31}\, \mathrm{u13} + \mathrm{h32}\, \mathrm{v13} + 1\right)}^2}\\ \frac{\mathrm{u14}}{\mathrm{h31}\, \mathrm{u14} + \mathrm{h32}\, \mathrm{v14} + 1} & \frac{\mathrm{v14}}{\mathrm{h31}\, \mathrm{u14} + \mathrm{h32}\, \mathrm{v14} + 1} & \frac{1}{\mathrm{h31}\, \mathrm{u14} + \mathrm{h32}\, \mathrm{v14} + 1} & 0 & 0 & 0 & -\frac{\mathrm{u14}\, \left(\mathrm{h13} + \mathrm{h11}\, \mathrm{u14} + \mathrm{h12}\, \mathrm{v14}\right)}{{\left(\mathrm{h31}\, \mathrm{u14} + \mathrm{h32}\, \mathrm{v14} + 1\right)}^2} & -\frac{\mathrm{v14}\, \left(\mathrm{h13} + \mathrm{h11}\, \mathrm{u14} + \mathrm{h12}\, \mathrm{v14}\right)}{{\left(\mathrm{h31}\, \mathrm{u14} + \mathrm{h32}\, \mathrm{v14} + 1\right)}^2}\\ 0 & 0 & 0 & \frac{\mathrm{u14}}{\mathrm{h31}\, \mathrm{u14} + \mathrm{h32}\, \mathrm{v14} + 1} & \frac{\mathrm{v14}}{\mathrm{h31}\, \mathrm{u14} + \mathrm{h32}\, \mathrm{v14} + 1} & \frac{1}{\mathrm{h31}\, \mathrm{u14} + \mathrm{h32}\, \mathrm{v14} + 1} & -\frac{\mathrm{u14}\, \left(\mathrm{h23} + \mathrm{h21}\, \mathrm{u14} + \mathrm{h22}\, \mathrm{v14}\right)}{{\left(\mathrm{h31}\, \mathrm{u14} + \mathrm{h32}\, \mathrm{v14} + 1\right)}^2} & -\frac{\mathrm{v14}\, \left(\mathrm{h23} + \mathrm{h21}\, \mathrm{u14} + \mathrm{h22}\, \mathrm{v14}\right)}{{\left(\mathrm{h31}\, \mathrm{u14} + \mathrm{h32}\, \mathrm{v14} + 1\right)}^2} \end{array}\right)$

$\left(\begin{array}{c} \frac{\mathrm{fx}\, x + \mathrm{pu}\, z}{z} - \frac{\mathrm{fx}\, x - \mathrm{bp} + \mathrm{pu}\, z}{z}\\ 0 \end{array}\right)$

BSPLINES

$\left(\begin{array}{ccccccccc}  - \frac{u^3}{6} + \frac{u^2}{2} - \frac{u}{2} + \frac{1}{6} & 0 & \frac{u^3}{2} - u^2 + \frac{2}{3} & 0 &  - \frac{u^3}{2} + \frac{u^2}{2} + \frac{u}{2} + \frac{1}{6} & 0 & \frac{u^3}{6} & 0 &  - \left(\mathrm{p1x} - \mathrm{p2x}\right)\, \left( - u^2 + u + \frac{1}{2}\right) - \left(\mathrm{p0x} - \mathrm{p1x}\right)\, \left(\frac{u^2}{2} - u + \frac{1}{2}\right) - \frac{u^2\, \left(\mathrm{p2x} - \mathrm{p3x}\right)}{2}\\ 0 &  - \frac{u^3}{6} + \frac{u^2}{2} - \frac{u}{2} + \frac{1}{6} & 0 & \frac{u^3}{2} - u^2 + \frac{2}{3} & 0 &  - \frac{u^3}{2} + \frac{u^2}{2} + \frac{u}{2} + \frac{1}{6} & 0 & \frac{u^3}{6} &  - \left(\mathrm{p1y} - \mathrm{p2y}\right)\, \left( - u^2 + u + \frac{1}{2}\right) - \left(\mathrm{p0y} - \mathrm{p1y}\right)\, \left(\frac{u^2}{2} - u + \frac{1}{2}\right) - \frac{u^2\, \left(\mathrm{p2y} - \mathrm{p3y}\right)}{2} \end{array}\right)$

homography.txt · Dernière modification: 2016/09/06 12:51 par bvandepo