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

• 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

• Preimage and coimage are linear subspaces
  • origo is in both the pre- and co-image

Notation

• Recall \(\hat u\) is a skew-symmetric matrix associated with \(u\)
• $(u) = u^$
• Associate an image point \(x\) with either its pre-image or co-image

Systems of Equations and Orthogonal Vectors

• \(\ell^Tx=0\) is an equation in three unknowns
  • This defines a plane (two unknowns)
  • e.g. \(x_1+ax_2+bx_3\)
• If you have two points, say, \(\ell^TL=0\), you have two equations
  • This defines a line (one unknowns)
  • e.g. \(x_1+ax_2\)
• If you have two points \(x_1\) and \(x_2\) on a line
  • \(x_1\times x_2\) is orthogonal on both of them
  • and on any other point on the line