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rectification_key - ECE 181b Homework 3 Image Rectication...

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ECE 181b Homework 3 Image Rectification April 20, 2006 The goal of this project is to explore some fundamental concepts of projective geometry related to the problem of image rectification. We will rectify the image of one of the facades of the Bren School of Environmental Science & Management at UCSB (see Figure 1) and verify how points at infinity can be mapped to finite points via homographies. Caveat. To receive full credit your answers must be clearly justified. Make sure to include the relevant steps that were required to obtain the numerical answer. 1 Calculating the Homography A transformation from the projective space P 2 to itself is called homography. A homography is represented by a 3 × 3 matrix with 8 degrees of freedom (scale, as usual, does not matter): x y w = h 1 h 4 h 7 h 2 h 5 h 8 h 3 h 6 h 9 x y w To estimate the coefficients of the homography H we will use the approach described in class (see the notes that can be found at http://www.ece.ucsb.edu/ ~ manj/ece181b/ homography_zuliani.pdf ). The test images can be downloaded from the course website. For sake of convenience we will adopt the coordinate system convention displayed in Figure 2 (for a more detailed image go to http://vision.ece.ucsb.edu/ ~ zuliani/Code/lattice. png ). Question 1 How many point correspondences are necessary to compute the homography that relates Image 1(a) to Image 1(b) under the constrain that the vector h obtained stacking the components of H one on top of the other has unit norm (i.e. h = 1 )? Why? Answer 1 We need at least 4 point correspondences. In fact each point correspondence provides two equations (for a total of 8 constraints) and we have 9 unknowns. The extra degree of freedom is fixed by imposing h = 1 .
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