mypapers/drafts/cam-calibr/cam-calibr.tex

   1 \documentclass[a4paper,10pt]{article}
   2 \usepackage{amsmath}
   3 \title{Camera calibration}
   4 \author{Petter Reinholdtsen $<$pere@td.org.uit.no$>$}
   5 \date{2000-02-01}
   6
   7 \begin{document}
   8 \maketitle
   9
  10 \section{Classical camera calibration theory}
  11
  12 In classical camera calibration theory, one needs to solve the
  13 following matrix equation with 11 unknowns and 2N x 11 knowns.
  14
  15 \begin{equation*}
  16 \setcounter{MaxMatrixCols}{20}
  17 \begin{bmatrix}
  18 X_1 & Y_1 & Z_1 & 1 & 0   &   0 &   0 & 0 & -u_1 X_1 & -u_1 Y_1 & -u_1 Z_1 \\
  19   0 &   0 &   0 & 0 & X_1 & Y_1 & Z_1 & 1 & -v_1 X_1 & -v_1 Y_1 & -v_1 Z_1 \\
  20 X_2 & Y_2 & Z_2 & 1 & 0   &   0 &   0 & 0 & -u_2 X_2 & -u_2 Y_2 & -u_2 Z_2 \\
  21   0 &   0 &   0 & 0 & X_2 & Y_2 & Z_2 & 1 & -v_2 X_2 & -v_2 Y_2 & -v_2 Z_2 \\
  22   . &   . &   . & . &   . &   . &   . & . &        . &        . &        . \\
  23   . &   . &   . & . &   . &   . &   . & . &        . &        . &        . \\
  24   . &   . &   . & . &   . &   . &   . & . &        . &        . &        . \\
  25   . &   . &   . & . &   . &   . &   . & . &        . &        . &        . \\
  26   . &   . &   . & . &   . &   . &   . & . &        . &        . &        . \\
  27 X_N & Y_N & Z_N & 1 & 0   &   0 &   0 & 0 & -u_N X_N & -u_N Y_N & -u_N Z_N \\
  28   0 &   0 &   0 & 0 & X_N & Y_N & Z_N & 1 & -v_N X_N & -v_N Y_N & -v_N Z_N
  29 \end{bmatrix}
  30 \cdot
  31 \begin{bmatrix}
  32 q_{11} \\ q_{12} \\ q_{13} \\ q_{14} \\
  33 q_{21} \\ q_{22} \\ q_{23} \\ q_{24} \\
  34 q_{31} \\ q_{32} \\ q_{33}
  35 \end{bmatrix}
  36 =
  37 \begin{bmatrix}
  38 u_1 \\ v_1 \\
  39 u_2 \\ v_2 \\
  40 . \\ . \\ . \\ . \\ . \\
  41 u_N \\ v_N
  42 \end{bmatrix}
  43 \end{equation*}
  44
  45 To solve this equation, the pixel and real world coordinates of at
  46 least six points must be known\footnote{Source: UWA Computer Vision
  47   IT412 lecture notes} ($N >= 6$).  $(X_i,Y_i,Z_i)$ is the world
  48 coordinates with $(u_i,v_i)$ image coordinates. $q_{ij}$ is the unknown
  49 camera calibration constants.
  50
  51
  52 \section{Reconstruction of 3D coordinates}
  53
  54 In stereo vision, when the camera calibration for both cameras are
  55 known ($C = [q_{ij}]$ and $C' = [q'_{ij}]$), the following equation
  56 will give the real world coordinates $(X,Y,Z)$ of a common scene point
  57 in projected into camera coordinates $(u,v)$ and $(u',v')$.
  58
  59 \begin{equation*}
  60 \setcounter{MaxMatrixCols}{20}
  61 \begin{bmatrix}
  62 q_{11} - uq_{31}    & q_{12} - uq_{32}    & q_{13} - uq_{33} \\
  63 q_{21} - vq_{31}    & q_{22} - vq_{32}    & q_{23} - vq_{33} \\
  64 q'_{11} - u'q'_{31} & q'_{12} - u'q'_{32} & q'_{13} - u'q'_{33} \\
  65 q'_{21} - v'q'_{31} & q'_{22} - v'q'_{32} & q'_{23} - v'q'_{33}
  66 \end{bmatrix}
  67 \begin{bmatrix}
  68 X \\ Y \\ Z
  69 \end{bmatrix}
  70 =
  71 \begin{bmatrix}
  72 u - q_{14} \\
  73 v - q_{24} \\
  74 u' - q'_{14} \\
  75 v' - q'_{24}
  76 \end{bmatrix}
  77 \end{equation*}
  78 \end{document}