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Affine Epipolar Geometry

Figure 2: Affine and Perspective Epipolar Geometries
\begin{figure}\centerline{\psfig{figure=epipolars.ps,width=4.0in}}\end{figure}

When the perspective effects are small, the problem of locating perspective epipolar lines becomes ill-conditioned. In such cases it is convenient to assume the parallel projection model of the affine camera which explicitly models the ambiguities.

The affine epipolar constraint can be described in terms of the affine fundamental matrix ${\bf F}$ as $p'^{T}Qp = 0$, i.e.,

\begin{displaymath}
\left[\begin{array}{ccc}
x'_{i} & y'_{i} & 1
\end{array}
...
...\begin{array}{c}
x_{i} \\ y_{i} \\ 1
\end{array} \right] = 0
\end{displaymath}

where $p' = (x',y',1)^{T}$ and $p = (x,y,1)$ are homogeneous 3-vectors representing corresponding image points in two views.

(See Shapiro, Zisserman and Brady).

To derive the above, we write ${\bf M}$ as $({\bf B} \mid {\bf b})$ where ${\bf B}$ is a general (non-singular) $2 \times 2$ matrix and ${\bf b}$ is a $2 \time 1$ vector. The projection equation then gives

\begin{displaymath}
{\bf x_i} = {\bf B} \left[ \begin{array}{c} X_i \\ Y_i\end{array}\right] +
Z_i {\bf b} + {\bf t}
\end{displaymath}

Similarly, for ${\bf M'A}$, we have

\begin{displaymath}
{\bf x'_i} = {\bf B'} \left[ \begin{array}{c} X_i \\ Y_i\end{array}\right] +
Z_i {\bf b'} + {\bf M'D} + {\bf t'}
\end{displaymath}

Eliminating scene coordinates $(X_i,Y_i)$ gives

\begin{displaymath}
{\bf x'_i}= {\bf\Gamma x_i} + Z_i {\bf d} + {\bf\epsilon}
\end{displaymath}

where ${\bf\Gamma} = {\bf B'B}^{-1}$, ${\bf d} = {\bf b'} - {\bf B'B}^{-1}{\bf b}$ and ${\bf\epsilon} = {\bf t'} - {\bf\Gamma t} + {\bf M'D}$.

${\bf\Gamma}$ and ${\bf d}$ are functions only of camera parameters $\{{\bf M}, {\bf M'}\}$ and the motion transformation ${\bf A}$, while ${\bf\epsilon}$ explains the motion of the reference point (centroid) and depend on the translation of the object ${\bf D}$ and the camera origins ${\bf t}$ and ${\bf t'}$.

This equation shows that ${\bf x'_i}$ associated with ${\bf x_i}$ lies on a line (epipolar) on the second image with offset ${\bf\Gamma x_i} + {\bf\epsilon}$ and direction ${\bf d}$. The unknown depth $Z_i$ determines how far along this line does ${\bf x'_i}$ lie. Inverting the equation we obtain

\begin{displaymath}
{\bf x_i}= {\bf\Gamma^{-1} x'_i} - Z_i {\bf\Gamma^{-1} d} - {\bf\Gamma^{-1} \epsilon}
\end{displaymath}

The translation invariant versions of these equations are

\begin{displaymath}
\begin{array}{ccc}
{\bf\Delta x'_i}& =& {\bf\Gamma \Delta x_...
...a^{-1} \Delta x'_i} - \Delta Z_i {\bf\Gamma^{-1} d}
\end{array}\end{displaymath}

We can eliminate $Z_i$ from the above equations and obtain a single equation in terms of image measurables:

\begin{displaymath}
({\bf x'_i} - {\bf\Gamma x_i} - {\bf\epsilon}).{\bf d}^{\perp} = 0
\end{displaymath}

where, ${\bf d} = (d_x,d_y)$ and its perpendicular ${\bf d^{\perp}} = (d_y,-d_x)$. This equation can be written as

\begin{displaymath}
a x'_i + by'_i + cx_i+dy_i + e = 0
\end{displaymath}

where $(a,b)^{T} = {\bf d}$, $(c,d)^{T} = -{\bf\Gamma^{T} d^{\perp}}$ and $e = -{\bf\epsilon^{T}d^{\perp}}$. This gives us

\begin{displaymath}
\left[\begin{array}{ccc}
x'_{i} & y'_{i} & 1
\end{array}
...
...\begin{array}{c}
x_{i} \\ y_{i} \\ 1
\end{array} \right] = 0
\end{displaymath}



Subsections
next up previous
Next: Computation of Affine epipolar Up: Affine Multiple Views Geometry Previous: Affine Multiple Views Geometry
Subhashis Banerjee 2002-02-18