---
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
    + $\text{preimage} = \mathsf{span}(\text{image})$
    + $\text{image} = \text{preimage}\cap\text{image plane}$

## Co-image

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

$$\text{coimage} = \text{preimage}^\bot$$

$$\text{preimage} = \text{coimage}^\bot$$