CSE 383 Computer Vision: Lecture 12 Part 2

Geometric Transformations & Projective Geometry

The Camera as a Coordinate Transformation

• A camera is a mapping from 3D object to 2D image, involving 3D to 2D transform and 2D to 2D transform (image warping)
• Camera transformation can be viewed as a coordinate transformation from 3D world to 2D image

Homogeneous Coordinates

• Homogeneous coordinates allow translations to be handled as multiplications, enabling uniform treatment of scaling, rotations, and translations
• Each point is given a third coordinate (X, Y, W), allowing translations to be handled as multiplications
• In practice, W = 1, so this step can be considered as mapping the point from 3D space onto the plane W = 1
• Homogeneous coordinates can be converted to Cartesian coordinates by dividing the triple by W

Projective Transformation

• A projective transformation is a transformation from 3D world to 2D image
• The image plane is a projection of a point P on the image plane

Pinhole Camera Matrix

• The pinhole camera matrix P is a 3x4 matrix that transforms 3D world coordinates to 2D image coordinates
• When Z = 1 and f = 1, the perspective projection matrix P is given by:
  • 1 0 0 0
  • 0 1 0 0
  • 0 0 1 0
• The pinhole camera matrix can be generalized to arbitrary focal length using the equation:
  • x = PX, where X is the 3D world coordinate and x is the 2D image coordinate

General Pinhole Camera Matrix

• The general pinhole camera matrix is a 3x4 matrix that transforms 3D world coordinates to 2D image coordinates
• The matrix can be written in terms of homogeneous coordinates as:
  • x = [I I 0] [X Y Z W]^T
• The camera matrix can be generalized to arbitrary focal length using the equation:
  • x = (fX + px) / (Z + pz)