Lecture Notes on Camera Motion and Homography

Assumptions and Equations

• Equations discussed:
  • x_t = A * x_s
  • y_t = A * y_s
• Notation:
  • A represents a scaling factor related to camera motion.

Camera Motion and Plane Normal

• Explanation of camera motion with respect to a plane normal.
• Assumption: Plane is fronto-parallel (normal vector n = (0, 0, 1)).
• Camera translation:
  • T_x = 0
  • T_y = 0
  • T_z > 0 (moving along optical axis)
• Rotation matrix R = Identity (no rotation).

Intrinsic Camera Matrix

• Simplified assumption: Focal length F = 1.
• Intrinsic camera matrix (K) is more complex in reality.
• K = Identity matrix under current assumptions.

Resulting Transformation

• Resulting equations:
  • X_t = X_s / (1 + T_z / d)
  • Y_t = Y_s / (1 + T_z / d)
• A = 1 / (1 + T_z / d)
• Interpretation of T_z:
  • If T_z < 0, zooming in (A > 1).
  • If T_z > 0, zooming out (A < 1).

Homework Assignment

• Show what camera motion and plane orientation yield shear.
  • Example matrix for shear:
    • H = [1 k 0; 0 1 0; 0 0 1].
  • Example scenario: Camera moving along X-axis while observing the ground plane.

Homography and Its Estimation

• Homography relates two images taken from the same camera.
• Focus on solving for planar homography, not rotation homography.
• Intrinsic vs. Extrinsic Camera Matrices:
  • Intrinsic (K): Fixed parameters, unchanged by camera movement.
  • Extrinsic: Related to camera motion, varies with position.
• Real intrinsic matrix includes parameters for optical center, focal length, and aspect ratio.

Estimating Homography from Correspondences

• Need at least 4 feature correspondences to estimate the homography matrix (H).
• Homography matrix H has 8 unknowns (estimation up to a scale factor).
• Correspondences identified using feature detection algorithms:
  • Harris Corner Detector
  • SIFT (Scale-Invariant Feature Transform)
  • SURF (Speeded Up Robust Features)

Finding Feature Correspondences

• Feature descriptors are essential for matching points between images.
• SIFT is commonly used for its robustness to translation, rotation, scale, and illumination changes.

Final Steps in Estimation

• Construct the matrix equation A * h = 0 for solving H.
• Use Singular Value Decomposition (SVD) to find the null space of A, yielding H.
• The resulting H can then be used for image alignment and spatial transformations.*