Example: Plane Fitting

The method described here was used in [#!McLauchlan_91!#]. A pair of images is rectified so that their epipolar lines are horizontal and parallel. Edges are detected in each image. A region of the left image is then matched to the right image using a planar fit in $(x,y,d)$ space where $(x,y)$ are the coordinates of edges in the left image and $d$ is the disparity, such that if $(x_r,y_r)$ are the coordinates of the corresponding edge in the right image,

$$x_r = x + d$$

and because of the rectification procedure, $y_r=y$.

A planar fit is attempted to the disparity points from candidate edge matches between the image, in the knowledge that planes in disparity space $(x,y,d)$ correspond to planes in the world.

A large number of the disparity points in the box will be incorrect matches, so direct fitting, for example for least squares, would not work. However the Hough transform is well suited to such a problem. It is used to select a large subset of the disparity points that lie near a plane. The Fast Hough transform gives a large increase in speed and decrease in storage requirement over the standard Hough transform method. It is applicable because, with an appropriate parametrisation, plane fitting is a linear Hough transform problem, to which the FHT is restricted.