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Pre- and co-image

Lecture Notes

Image and Image Plane

  • Image Plane is the universe where the image lives

\[ \text{image}\subset\text{image plane} \]

  • The Image Plane is a 2D World
  • The Image Plane exists in a 3D World

Pre-image

  • Preimage is the set of points in 3D projecting onto the Image Plane
  • What is projection?
    • draw a line through the 3D point and origo (the pinhole)
    • the projection is the intersection with the image plane.
  • Thus
    • \(\text{preimage} = \mathsf{span}(\text{image})\)
    • \(\text{image} = \text{preimage}\cap\text{image plane}\)
  • The span of a set of points is the smallest linear subspace containing all the points

Points and Lines

Image object Pre-Image
Point (dimension 0) Line through origo (dimension 1)
Line (dimension 1) Plane through origo (dimension 2)
  • Preimage is a linear subspaces, i.e.¬†includes origo
  • A single point projects onto a point
    • any other point on the same line through origo projects onto the same point
  • A line projects onto a line if it does not pass through origo

Co-image

  • Coimage is the set of points (space) orthogonal on the preimage

\[\text{coimage} = \text{preimage}^\bot\]

\[\text{preimage} = \text{coimage}^\bot\]

Points and Lines

Image object Pre-Image Co-Image
Point (dimension 0) Line through origo (dimension 1) Plane (co-dimension 1)
Line (dimension 1) Plane through origo (dimension 2) Line (co-dimension 2)
  • Preimage and coimage are linear subspaces
    • origo is in both the pre- and co-image