---
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