16-822 Geometry-based Methods in Vision

Assignment 1: Projective Geometry and Homography

Qitao Zhao (qitaoz), Fall 2024

Q1. Affine Rectification

Input image	Annotated parallel lines on input image	Affine-rectified image

Held-out Test Lines and Metrics

Test lines on input image	Test lines on affine-rectified image	Metrics
		Before: [1.0000, 0.9600] After: [1.0000, 0.9953]
		Before: [0.9878, 0.9846] After: [0.9998, 1.0000]
		Before: [0.8964, 0.9575] After: [1.0000, 1.0000]
		Before: [0.9905, 0.9977] After: [0.9999, 0.9998]
		Before: [0.8312, 1.0000] After: [1.0000, 1.0000]

We performed affine rectification by annotating two pairs of parallel lines and calculating the line equation at infinity. We found the homography that recovers this particular line, which is given by: $\begin{array}{c}H=\left[\begin{array}{ccc}1& 0& t_x\\ 0& 1& t_y\\ a& b& c\end{array}\right]\end{array}$$H = \begin{bmatrix} 1 & 0 & t_x \\ 0 & 1 & t_y \\ a & b & c \end{bmatrix}$.

Q2. Metric Rectification

Input image	Annotated perpendicular lines on input image	Annotated perpendicular lines on affine-rectified image	Rectified image

Held-out Test Lines and Metrics

Test lines on input image	Test lines on metric-rectified image	Metrics
		Before: [-0.1187, 0.1602] After: [-0.0000, 0.0000]
		Before: [0.6686, -0.0448] After: [-0.0000, -0.0000]
		Before: [-0.2527, 0.0882] After: [-0.0000, 0.0000]

First, we apply an affine transformation to the image using a predefined homography matrix $H_a$$H_a$ (obtained from the last question) . This removes projective distortion but leaves metric distortions like scaling and shearing:

We then Identify at least two pairs of perpendicular lines in the affine rectified image. Use these line pairs to compute the constraints required for metric rectification. The lines' equations $l$$l$ and $m$$m$ are computed as:

Form the constraint system for the dual conic (which represents the circular points), solving for $C^{\prime }$$C'$:

Solve for $x$$x$, then construct $C^{\prime }$$C'$, the conic matrix.

Compute the Singular Value Decomposition (SVD) of $C^{\prime }$$C'$ to derive the final metric rectifying homography $H_m$$H_m$:

We apply the final homography $H_m$$H_m$ to the affine rectified image to achieve metric rectification:

Q3. Planar Homography from Point Correspondences

Normal image	Perspective image	Annotated corners in perspective image	Warped and overlaid image

The algorithm uses four-point correspondences between the source and target images to compute the transformation. These points are visualized with solid red circles on the target image for clarity. Each pair of corresponding points provides two equations of the form: $\begin{array}{rl}{x}^{\prime }& =H_{11}x+H_{12}y+H_{13}\\ y^{\prime }& =H_{21}x+H_{22}y+H_{23}\end{array}$$\begin{aligned} x' &= H_{11}x + H_{12}y + H_{13} \\ y' &= H_{21}x + H_{22}y + H_{23} \end{aligned}$, where x′, y' are the transformed coordinates in the target image, and x, y are the original coordinates in the source image. We use SVD to solve the linear equations by taking the last row of VT.