---
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