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Notations

To enhance the readability the notations used throughout the text are summarized here.

For matrices bold face fonts are used (i.e. ${\bf A}$). 4-vectors are represented by ${\tt A}$ and 3-vectors by ${\tt a}$. Scalar values will be represented as $a$.

Unless stated differently the indices $i$, $j$ and $k$ are used for views, while $l$ and $m$ are used for indexing points, lines or planes. The notation ${\bf A}_{ij}$ indicates the entity ${\bf A}$ which relates view $i$ to view $j$ (or going from view $i$ to view $j$). The indices $i$,$j$ and $k$ will also be used to indicate the entries of vectors, matrices and tensors. The subscripts $P$, $A$, $M$ and $E$ will refer to projective, affine, metric and Euclidean entities respectively

${\bf P}$ camera projection matrix ($3 \times 4$ matrix)
${\tt M}$ world point (4-vector)
${\tt\Pi }$ world plane (4-vector)
${\tt m}$ image point (3-vector)
${\tt l}$ image line (3-vector)
${\bf H}^{\tt\Pi}_{i j }$ homography for plane ${\tt\Pi }$ from view $i$ to view $j$ ($3 \times 3$ matrix)
${\bf H}_{{\tt\Pi} i}$ homography from plane ${\tt\Pi }$ to image $i$ ($3 \times 3$ matrix)
${\bf F}$ fundamental matrix ($3 \times 3$ rank 2 matrix)
${\tt e}_{ij}$ epipole (projection of projection center of viewpoint $i$ into image $j$)
${\bf T}$ trifocal tensor ( $3 \times 3 \times 3$ tensor)
${\bf K}$ calibration matrix ($3 \times 3$ upper triangular matrix)
$ {\bf R} $ rotation matrix
${\tt\Pi }_\infty $ plane at infinity (canonical representation: $W=0$)
$\Omega $ absolute conic
  (canonical representation: $X^2+Y^2+Z^2=0$ and $W=0$)
$\Omega ^*$ absolute dual quadric ($4 \times 4$ rank 3 matrix)
$\omega _\infty $ absolute conic embedded in the plane at infinity ($3 \times 3$ matrix)
$\omega^*_\infty$ dual absolute conic embedded in the plane at infinity ($3 \times 3$ matrix)
$\omega$ image of the absolute conic ($3 \times 3$ matrices)
$\omega^*$ dual image of the absolute conic ($3 \times 3$ matrices)
$ \sim $ equivalence up to scale ( $A \sim B \Leftrightarrow \exists \lambda \neq 0 : A = \lambda B$)
$ \Vert {\bf A} \Vert _F$ indicates the Frobenius norm of ${\bf A}$ (i.e. $\sum_{ij} {a_{ij}^2}$)
$ {\bf F}( {\bf A} )$ indicates the matrix ${\bf A}$ scaled to have unit Frobenius norm
  (i.e. $\frac{\bf A}{\Vert {\bf A} \Vert}_F$)
$ {\bf A}^\top$ is the transpose of ${\bf A}$
$ {\bf A}^{-1}$ is the inverse of ${\bf A}$ (i.e. ${\bf AA}^{-1} = {\bf A}^{-1}{\bf A} = {\bf I}$)
$ {\bf A}^\dagger$ is the Moore-Penrose pseudo inverse of ${\bf A}$


next up previous contents
Next: Contents Up: Visual 3D Modeling from Previous: Acknowledgments   Contents
Marc Pollefeys 2002-11-22