An Exact Formula for Calculating Inverse Radial Lens Distortions

. 2016 Jun 1;16(6):807. doi: 10.3390/s16060807

\begin{matrix} b_{1} & = - k_{1} \end{matrix}

\begin{matrix} b_{2} & = 3 k_{1}^{2} - k_{2} \end{matrix}

\begin{matrix} b_{3} & = - 12 k_{1}^{3} + 8 k_{1} k_{2} - k_{3} \end{matrix}

\begin{matrix} b_{4} & = 55 k_{1}^{4} - 55 k_{1}^{2} k_{2} + 5 k_{2}^{2} + 10 k_{1} k_{3} - k_{4} \end{matrix}

\begin{matrix} b_{5} & = - 273 k_{1}^{5} + 364 k_{1}^{3} k_{2} - 78 k_{1} k_{2}^{2} - 78 k_{1}^{2} k_{3} + 12 k_{2} k_{3} + 12 k_{1} k_{4} \end{matrix}

\begin{matrix} b_{6} & = 1428 k_{1}^{6} - 2380 k_{1}^{4} k_{2} + 840 k_{1}^{2} k_{2}^{2} - 35 k_{2}^{3} + 560 k_{1}^{3} k_{3} - 210 k_{1} k_{2} k_{3} + 7 k_{3}^{2} - 105 k_{1}^{2} k_{4} + 14 k_{2} k_{4} \end{matrix}

\begin{matrix} b_{7} & = - 7752 k_{1}^{7} + 15504 k_{1}^{5} k_{2} - 7752 k_{1}^{3} k_{2}^{2} + 816 k_{1} k_{2}^{3} - 3876 k_{1}^{4} k_{3} + 2448 k_{1}^{2} k_{2} k_{3} - 136 k_{2}^{2} k_{3} - 136 k_{1} k_{3}^{2} \\ + 816 k_{1}^{3} k_{4} - 272 k_{1} k_{2} k_{4} + 16 k_{3} k_{4} \end{matrix}

\begin{matrix} b_{8} = & 43263 k_{1}^{8} - 100947 k_{1}^{6} k_{2} + 65835 k_{1}^{4} k_{2}^{2} - 11970 k_{1}^{2} k_{2}^{3} + 285 k_{2}^{4} + 26334 k_{1}^{5} k_{3} - 23940 k_{1}^{3} k_{2} k_{3} \\ + 3420 k_{1} k_{2}^{2} k_{3} + 1710 k_{1}^{2} k_{3}^{2} - 171 k_{2} k_{3}^{2} - 5985 k_{1}^{4} k_{4} + 3420 k_{1}^{2} k_{2} k_{4} - 171 k_{2}^{2} k_{4} - 342 k_{1} k_{3} k_{4} + 9 k_{4}^{2} \end{matrix}

\begin{matrix} b_{9} & = - 246675 k_{1}^{9} + 657800 k_{1}^{7} k_{2} - 531300 k_{1}^{5} k_{2}^{2} + 141680 k_{1}^{3} k_{2}^{3} - 8855 k_{1} k_{2}^{4} - 177100 k_{1}^{6} k_{3} \\ + 212520 k_{1}^{4} k_{2} k_{3} - 53130 k_{1}^{2} k_{2}^{2} k_{3} + 1540 k_{2}^{3} k_{3} - 17710 k_{1}^{3} k_{3}^{2} + 4620 k_{1} k_{2} k_{3}^{2} - 70 k_{3}^{3} + 42504 k_{1}^{5} k_{4} \\ - 35420 k_{1}^{3} k_{2} k_{4} + 4620 k_{1} k_{2}^{2} k_{4} + 4620 k_{1}^{2} k_{3} k_{4} - 420 k_{2} k_{3} k_{4} - 210 k_{1} k_{4}^{2} \end{matrix}