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Computation of Affine epipolar Geometry

Given correspondences in two views the affine fundamental matrix can be computed using orthogonal regression by minimizing

\begin{displaymath}
\frac{1}{{\bf\mid n \mid }^{2}} \sum_{i=0}^{n-1} ({\bf r_{i} \cdot n} + e)^{2}
\end{displaymath}

Here ${\bf r_{i}} = (x'_{i},y'_{i},x_{i},y_{i})^{T}$ and ${\bf n} = (a,b,c,d)^{T}$ . The minimization finds a hyper-plane that globally minimizes the sum of the squared perpendicular distances between ${\bf r_i}$ and the hyper-plane.

Defining

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{\bf v_i} = {\bf r_i} - {\bf\bar{r}}
\end{displaymath}

and

\begin{displaymath}
{\bf W} = \sum_{i=0}^{n-1}{\bf v_i}{\bf v_i}^{T}
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it can be shown that the solution satisfies the eigenvector equation

\begin{displaymath}
{\bf W n} = \lambda_i {\bf n}, \mid {\bf n} \mid^2 = 1
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Subhashis Banerjee 2002-02-18