Q1: Affine Rectification

Input images

Annotated images

Output images

Implementation

Find $l_\infty^\prime$ by finding intersection of parallel lines on line at infinity and restoring to its canonical position:

• Compute parallel lines based on input points: $l=x_1 \times x_2$
• Compute homography $H$ s.t. $l_\infty = H^{-\intercal}l_\infty^\prime$

  $$H = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ l_1 & l_2 & l_3\end{bmatrix}$$

• Apply $H$ as a perspective warp to rectify parallel lines in the image

Q2: Metric Rectification

Input images

Same as for Q1

Annotated images

Output images

Implementation

Find image of conic dual to circular points $C^{\star\prime}_\infty$ s.t. for images of perpendicular lines $l^\prime$ and $m^\prime$:

$$l^{\prime\intercal} C^{\star\prime}_\infty m^{\prime} = 0$$

Where

$$C^{\star\prime}_\infty = \begin{bmatrix} a & b/2 & d/2 \\ b/2 & c & e/2 \\ d/2 & e/2 & f \end{bmatrix}$$

Using the affine rectification result from question 1, $d=e=f=0$:

$$C^{\star\prime}_\infty = \begin{bmatrix} a & b/2 & 0 \\ b/2 & c & 0 \\ 0 & 0 & 0 \end{bmatrix}$$

Expanding the polynomial represented by $C^{\star\prime}_\infty$:

$$a{x_1}^2 + bx_1x_2 + c{x_2}^2 + dx_1x_3 + ex_2x_3 + f{x_3}^2 = 0 \\ a{x_1}^2 + bx_1x_2 + c{x_2}^2 = 0 \\$$

Solve for vector of coefficients based on constraints from perpendicular lines using SVD, then solve for homography from $C^{\star\prime}_\infty$ using SVD.

Q3: Planar Homography from Point Correspondences

Input images

Annotated images

Output images

Implementation

• Find homography $H$ gives the following "projectively equal" relationship:

$$Hx_i \times x_i^\prime = 0$$

• Reconstruct cross product as matrix multiplication for each of 4 point correspondences:

$$\begin{bmatrix} 0 & -w^\prime x^\intercal \ & y^\prime x^\intercal \\ -w^\prime x^\intercal & 0 & -x^\prime x^\intercal \\ \end{bmatrix} h = 0$$

• Solve for $h$ using SVD.