2D affine reconstruction

The 2D affine projection from scene to image can then be written in its most general form as

Here is the image point, is a 2D scene point, and M , constitute the camera matrix parameters, which we will term the motion parameters because they represent the camera/scene motion over time. The 2D affine reconstruction problem for point features is: given for multiple features i in multiple images j , determine the motion , and structure . We shall add the subscripts l and r to refer to left/right image quantities, so that we have camera matrix parameters , , and . Stereo 2D affine projection is illustrated in figure  1 .

   
Figure 1: Stereo affine 2D projection from planar scene to image planes.

Given complete data for n features over k images, Tomasi & Kanade's algorithm [ 18 ] achieves the optimal solution for a 3D affine reconstruction. It also generalizes simply to the 2D problem by taking the biggest two singular vectors instead of the biggest three. Thus and , and . may computed simply using the SVD, This is quite efficient for a small number of images; however it is a batch algorithm, and the speed degrades quickly as more image data is added. As we have previously demonstrated [ 12 ], the variable state dimension filter algorithm achieves virtually the same accuracy, but has the advantages of being recursive, not requiring complete data, and allowing new features to be added to the reconstruction as they appear and discarded features to be removed. We thus employ the Tomasi/Kanade algorithm on the first stereopair of our image sequence, and thereafter use the simplest version of the VSDF as presented in [ 12 ]. Details may be found in [ 11 ].