Constructing the rectified image

The rectified images are built up row by row. Each row corresponds to a certain angular sector. The length along the epipolar line is preserved. Figure 7.10 clarifies these concepts. The coordinates of every epipolar line are saved in a list for later reference (i.e. transformation back to original images). The distance of the first and the last pixels are remembered for every epipolar line. This information allows a simple inverse transformation through the constructed look-up table.

Figure 7.10: The image is transformed from (x,y)-space to (r,$\theta$)-space. Note that the $\theta$-axis is non-uniform so that every epipolar line has an optimal width (this width is determined over the two images).

Note that an upper bound for the image size is easily obtained. The height is bound by the contour of the image $2 \times (W + H)$. The width is bound by the diagonal $\sqrt{W^2+H^2}$. Note that the image size is uniquely determined with our procedure and that it is the minimum that can be achieved without pixel compression.