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Tomasi and Kanade factorization

In case $n$ point correspondences $(n \geq 4)$ over $k$ views $(k \geq 2)$ are available, we use the factorization procedure of Tomasi and Kanade to obtain the bases and structure. Their formulation can be written as an extension of the above equation as

\begin{displaymath}
\left[
\begin{array}{cccc}
{\bf\Delta x_{1}} & {\bf\Delta ...
...} & \gamma_{2} & \ldots & \gamma_{n-1} \\
\end{array}\right]
\end{displaymath}

where the left measurement matrix ${\bf W}$ represents the $n$ point correspondences in $k$ views and has dimensions $2k$ x $(n-1)$. The matrices on the right, $\tilde{\bf M}$ ($2k$ x $3$) and $\tilde{\bf S}$ ($3$ x $(n-1)$), are called motion and structure matrices respectively. The matrix $\tilde{\bf S}$ gives the invariant affine structure of the $n$ points in motion, and the $i^th$ row of $\tilde{\bf M}$, $\tilde{\bf M}(i)$, along with the corresponding image center ${\bf x_{0}(i)}$, gives the projection parameters for the $i^th$ view $\{\tilde{\bf M}(i),{\bf x_{0}(i)}\}$.

Clearly, in the absence of noise, ${\bf W}$ must have a rank at-most 3. Tomasi and Kanade perform a singular value decomposition of ${\bf W}$ and use the 3 largest eigenvalues to construct $\tilde{\bf M}$ and $\tilde{\bf S}$. If the SVD returns a rank greater than 3, then the affine projection model is invalid and we use this as a check. The rank 2 case signifies either a planar object (which is not possible for facial images!) or degenerate motion. In such a case, the 3D affine structure cannot be determined and the views are related by 2D affine transformations. The 2D affine structure can then be recovered in only two axes using the same formalism.


next up previous
Next: Image transfer and linear Up: Affine Structure Previous: Affine Structure
Subhashis Banerjee 2002-02-18