HW1: Projective Geometry and Homography

Q1: Affine Rectification (30 points)

Brief Description:

• User Annotation: Select two pairs of parallel lines in the image.
• Compute Vanishing Points: For each pair, compute the vanishing point \( \text{vp} = l_1 \times l_2 \).
• Compute Line at Infinity: \( l_{\infty} = \text{vp}_1 \times \text{vp}_2 \).
• Affine Homography: Construct \( H_{\text{affine}} \) using \( l_{\infty} \):
  \[ H_{\text{affine}} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ l_{\infty}[0] & l_{\infty}[1] & l_{\infty}[2] \end{bmatrix} \]
• Warp Image: Apply \( H_{\text{affine}} \) to rectify the image.
• Evaluate Angles: Calculate cosine similarities before and after rectification using:
  \[ \cos(\theta) = \frac{l_1^\top l_2}{\|l_1\| \|l_2\|} \]

The 5 images for each sample are: Input Image, Annotated parallel lines on input image, Affine-Rectified Image, Test lines on Input Image, and Test lines on Affine-Rectified Image.

Input 1: checker1.jpg

Input 2: facade.jpg

Input 3: tiles5.JPG

My Input 1: my1.jpg

My Input 2: my2.jpg

Q2: Metric Rectification (40 points)

Brief Description:

• Affine Rectification: First, perform affine rectification as in Q1.
• User Annotation: Select pairs of perpendicular lines on the affine-rectified image.
• Compute Constraints: For each pair of perpendicular lines \( l_1 \) and \( l_2 \), the constraint is:
  \[ l_1^\top S l_2 = 0 \]
  where \( S \) is a symmetric matrix representing the image of the absolute conic.
• Solve for \( S \): Collect constraints and solve for \( S \) using SVD.
• Compute Metric Homography: Decompose \( S \) to find a transformation \( A \) such that \( A^\top A = S \), and construct \( H_{\text{metric}} \):
  \[ H_{\text{metric}} = \begin{bmatrix} A^{-1} & 0 \\ 0 & 1 \end{bmatrix} \]
• Warp Image: Apply \( H_{\text{metric}} \) to achieve metric rectification.
• Evaluate Angles: Calculate cosine similarities before and after rectification to verify orthogonality.

The 6 images for each sample are: Input Image, Annotated perpendicular lines on input image, Annotated perpendicular lines on Affine-Rectified Image, Rectified Image, Test lines on Input Image, and Test lines on Metric-Rectified Image.

Input 1: checker1.jpg

Input 2: chess1.jpg

Input 3: tiles3.jpg

My Input 1: my3.jpg

My Input 2: my4.jpg

Q3: Planar Homography from Point Correspondences (30 points)

Brief Description:

• User Annotation: Select corresponding points between the normal image and the perspective image.
• Compute Homography: Set up equations based on point correspondences and solve for \( H \). For each pair of corresponding points \( p \) and \( p' \), the relationship is:
  \[ p' = H p \]
  In terms of coordinates, this leads to the equations:
  \[ \begin{cases} x' = \frac{h_{11}x + h_{12}y + h_{13}}{h_{31}x + h_{32}y + h_{33}} \\ y' = \frac{h_{21}x + h_{22}y + h_{23}}{h_{31}x + h_{32}y + h_{33}} \end{cases} \]
  These equations can be rearranged into a linear system and solved using SVD.
• Warp Image: Apply \( H \) to warp the normal image to the perspective view using functions like cv2.warpPerspective.
• Overlay Images: Blend the warped image onto the perspective image using masking techniques.

The 4 images for each sample are: Normal Image, Perspective Image, Annotated corners in Perspective Image, and Warped and Overlaid Image.

Input: desk

My Input: pathfinder

Q4: Bonus: Metric Rectification from Perpendicular Lines (10 points)

Brief Description:

• User Annotation: Directly select multiple pairs of perpendicular lines on the original image.
• Compute Constraints: Use the orthogonality constraint \( l_1^\top S l_2 = 0 \) for each pair.
• Solve for Homography: Compute \( H_{\text{metric}} \) without prior affine rectification.
• Warp Image: Apply \( H_{\text{metric}} \) to rectify the image.

The 3 images for each sample are: Input Image, Annotated perpendicular lines, and Rectified Image.

Input: checker1.jpg

My Input: my3.jpg

Q5: Bonus: More Planar Homography from Point Correspondences (10 points)

Brief Description:

• User Annotation: For each image, select corresponding points on the background image.
• Compute Homographies: Compute \( H_i \) for each image \( i \).
• Warp and Overlay: Warp each image using \( H_i \) and overlay onto the background.

The 5 images are: Normal Image 1, Normal Image 2, Normal Image 3, Perspective Image, and Output Image.

Usage Instructions

You can run each question's code using the following commands:

• Q1: python main.py --question q1
• Q2: python main.py --question q2
• Q3: python main.py --question q3
• Q4: python main.py --question q4
• Q5: python main.py --question q5