---
title: Pre- and co-image (Lecture)
categories: lecture
---

# Lecture Notes

See also [Notes from 2021](/2021/Pre- and co-image Lecture).

## Linear objects in 2D

+ The most important linear object is the line through the origin.
    + These are subspaces of dimension one.
+ The object is a set $\ell\subset\mathbb{R}^2$
+ Three descriptions
    + **functions**
      $$\ell = \{ \vec{x}=(x,y) | y = a\cdot x, x\in\mathbb{R} \}$$
      for some $a\in\mathbb{R}$
        + Exception: The vertical line would have $a=\infty$,
          for infinitely steep
    + **equations**
      $$\ell = \{ \vec{x}=(x,y) | \vec{x}\cdot\vec{x}^\bot \}$$
      for some $\vec{x}^\bot\in\mathbb{R}^2$
        + Note that for $c\neq0$, $\vec{x}^\bot$ and $c\vec{x}^\bot$
          define the same line.
    + **span**
      $$\ell = \{ \vec{x}=(x,y) | a\cdot \vec{x}_0, a\in\mathbb{R} \}$$
      for some $\vec{x}_0\in\mathbb{R}^2$
        + Exception: The vertical line would have $a=\infty$,
          for infinitely steep

If we normalise $\vec{x}^\bot$, we can write $\vec{x}^\bot=(a,1)$
for $a\in\mathbb{R}$ unless we describe the vertical line, which has
$\vec{x}^\bot=(1,0)$, which we could imagine writing $(\infty,1)$.

+ We can normalise $\vec{x}_0$ in the same way.
+ The set of lines through origo is equivalent to $\mathbb{R}\cup\{\infty\}$,
  which can be seen in either representation.
  
## Linear objects in 3D

We have the same situation in 3D, but we have more objects of interest.

+ In 2D, the line is defined by **one** function or **one** equation.
+ In 3D we have
    + the line $\ell= \{(x,y,z) | z = ax + by, (x,y)\in\mathbb{R}\}$
    + the plane $\mathcal{P}= \{(x,y,z) | z = ax, y = bx, x\in\mathbb{R}^2\}$
      (two function)
+ Using equations to define it
    + The plane needs **one** equation
       $$\mathcal{P}=\{\vec{x} | \vec{x}\cdot\vec{x}^\bot=0 \}$$
        + $\vec{x}^\bot$ is the dual space $\mathcal{P}$
    + The line needs **two** equation
       $$\ell=\{\vec{x} | \vec{x}\cdot\vec{y}_1=0, \vec{x}\cdot\vec{y}_1=0\}$$
        + The space spanned by  $\vec{y}_1$ and $\vec{y}_2$ is the
          dual space $\ell^\bot$
+ What does it look like as **spans**?
+ An object needs
    + one function per dimension; or
        + Each adds one degree of freedom 
    + one equation per *codimension* 
        + Each equation removes one degree of freedom
    
## Projections from 3D to 2D

+ Recall that each point $x$ in the image plane is the image of any point on
  a line through $O$ 
    + Correspondence between lines through $O$ and point in the image.
    + This line is called the **pre-image** of $x$.

Draw frontal model with image at $Z=1$.  This gives projective
image co-ordinage $(x,y,1)$ embedded in 3D.

+ What about a line $l$ in the image plane?  What is the pre-image?
    + Plane $P$ through the origin.  The line $l$ is the intersection
      of $P$ and the image plane
+ What is the image of a line $L$ in 3D?
    + if $O\in L$ we have a point, whose pre-image is $L$
    + if $P\not\in L$, we have a line $l$ whose pre-image is a plane $P\ni O$
    + $P$ is described by an orthogonal vector, the dual space $P^\bot$,  
      which we call the **co-image** of $l$