---
title: Pre- and co-image
categories: lecture
---

# 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 $u$
+ $\mathsf{span}(\hat u) = u^\bot $
+ 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