1. Theory and Practice of Projective Rectification
Ko Dae-Won

2. Theory and Practice of Projective Rectification
1. Preliminaries
𝐴 :𝑆𝑞𝑢𝑎𝑟𝑒 𝑚𝑎𝑡𝑟𝑖𝑥 𝐴 ∗ :𝑖𝑡𝑠 𝑚𝑎𝑡𝑟𝑖𝑥 𝑜𝑓 𝑐𝑜𝑓𝑎𝑐𝑡𝑜𝑟𝑠 The following identities are well known: 𝐴 ∗ 𝐴=𝐴 𝐴 ∗ = det(A)I where I is the identity matrix. In particular, if A is an invertible matrix, then 𝐴 ∗ ≈ 𝐴 𝑇 −1 .

3. Theory and Practice of Projective Rectification
1. Preliminaries
Given a vector it is convenient to introduce the skew-symmetric matrix

4. Theory and Practice of Projective Rectification
1. Preliminaries
The matrix [𝑡] 𝑥 is closely related to the cross-product of vectors in that for any vectors 𝑠 and t, we have 𝑠 𝑇 [𝑡] 𝑥 =𝑠×𝑡 and [𝑡] 𝑥 𝑠=𝑡 ×𝑠. Proposition 1. For any 3 x 3 matrix M and vector t, 𝑀 ∗ [𝑡] 𝑥 = [𝑀𝑡] 𝑥 𝑀

5. Theory and Practice of Projective Rectification
1. Preliminaries
If A is a 3 x 3 non-singular matrix representing a projective transformation of 𝑃 2 , then 𝐴 ∗ is the corresponding line map. In other words, if 𝑢 1 and 𝑢 2 line on a line L, then 𝐴 𝑢 1 and 𝐴 𝑢 2 : in symbols 𝐴 ∗ 𝑢 1 × 𝑢 2 =(A 𝑢 1 ) × (A 𝑢 2 ). This formula is derived from Proposition 1.

6. 2. Property of the fundamental matrix
Theory and Practice of Projective Rectification
2. Property of the fundamental matrix

7. 2. Property of the fundamental matrix
Theory and Practice of Projective Rectification
2. Property of the fundamental matrix
According to Proposition2, the matrix F determines the epipoles in both images. Furthermore, F provides the map between points in one image and epipolar lines in the other image.

8. 3. Mapping the Epipole to Infinity
Theory and Practice of Projective Rectification
3. Mapping the Epipole to Infinity
Goal: Finding a projective transformation H of an image mapping an epipole to a point at infinity. In fact, if epipolar lines are to be transformed to lines parrallel with x axis, then the epipole should be mapped to the infinite point If inappropriate H is chosen, severe projective distortion of the image can take place.

9. 3. Mapping the Epipole to Infinity
Theory and Practice of Projective Rectification
3. Mapping the Epipole to Infinity
In order that the resampled image should look somewhat like the original image, we may put closer Restrictions on the choice of H. One condition that leads to good results is to insist that The transformation H should act as far as possible as a Rigid transformation in the neighborhood of a given selected point of the image.

10. 3. Mapping the Epipole to Infinity
Theory and Practice of Projective Rectification
3. Mapping the Epipole to Infinity
By this is meant that to first order neighborhood of may undergo rotation and translation only. An appropriate choice of point may be the centre of the image.

11. 3. Mapping the Epipole to Infinity
Theory and Practice of Projective Rectification
3. Mapping the Epipole to Infinity

12. 3. Mapping the Epipole to Infinity
Theory and Practice of Projective Rectification
3. Mapping the Epipole to Infinity