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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

# 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