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Next: Change of basis Up: Affine Structure Previous: Tomasi and Kanade factorization

Image transfer and linear combination of views

Once the affine structure has been computed, it can be used to generate a new view of the object (``transfer'') by simply selecting a new spanning set $\{{\bf e''_1},{\bf e''_2},{\bf e''_3}\}$. No camera calibration is needed. Note that this is same as choosing a new projection matrix ${\bf M''}$.

\begin{displaymath}
{\bf x''_i} = {\bf x''_0} + \alpha_i {\bf e''_1} + \beta_i {\bf e''_2} + \gamma_i {\bf e''_k}
\end{displaymath}

If the affine structure is not of interest (graphics), it is possible to bypass the affine coordinates and express the new image coordinates ${\bf\Delta x''}$ directly in terms of the first two sets of image coordinates ${\bf\Delta x}$ and ${\bf\Delta x'}$. One can write the projection equations in the first two views as

\begin{displaymath}
\begin{array}{ccc}
{\bf\Delta x} & = & {\bf G}{\bf\Delta X} \\
{\bf\Delta x'} & = & {\bf G'}{\bf\Delta X}
\end{array}\end{displaymath}

where ${\bf G}$ and ${\bf G'}$ are $2 \times 3$ matrices with rows $\{{\bf G_1},{\bf G_2}\}$ and $\{{\bf G'_1},{\bf G'_2}\}$ respectively. The new view can be similarly written as

\begin{displaymath}
{\bf\Delta x''} = {\bf G''}{\bf\Delta X}
\end{displaymath}

where ${\bf G''}$ has rows $\{{\bf G''_1},{\bf G''_2}\}$.

Now, any three rows of $\{{\bf G_1},{\bf G_2},{\bf G'_1},{\bf G'_2}\}$ define a linearly independent spanning set for ${\cal A}^3$, say $\{{\bf G_1},{\bf G_2},{\bf G'_1}\}$. So, there exists scalars such that

\begin{displaymath}
{\bf G''} = \left[ \begin{array}{cc}a_1 & a_2 \\ b_1 & b_2 \...
...egin{array}{cc}a_3 & 0 \\ b_3 & 0 \end{array} \right] {\bf G'}
\end{displaymath}

Then, ${\bf\Delta x''} = {\bf G''}{\bf\Delta X}$ gives

\begin{displaymath}
{\bf\Delta x''} = \left[ \begin{array}{cc}a_1 & a_2 \\ b_1 &...
...array}{c} \Delta x \\ \Delta y \\ \Delta x' \end{array}\right]
\end{displaymath}

Thus, if images of an object are obtained using affine cameras, then a novel view can be expressed as a linear combination of views (this is useful for object recognition).


next up previous
Next: Change of basis Up: Affine Structure Previous: Tomasi and Kanade factorization
Subhashis Banerjee 2002-02-18