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Pre- and co-image

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title: Pre- and co-image
categories: lecture
title: Pre- and Co-Image
categories: exercises

# Lecture Notes

## 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 $a\in\R$ 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 $l$

# Notes from the text book

*The following notes were made 2021 based on a textbook.
This exposition is not recommended because it is driven
by the definitions which only gain meaning later in the course.*

## 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
+ Exercise 3.9 are from Ma 2004 page 62ff.
+ Exercise 3.10 are from Ma 2004 page 62ff.

## Pre-image
# First Exercise (3.9)

+ 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
Exercise 3.9 are from Ma 2004 page 62ff.

## Points and Lines
## Debrief Notes

| Image object | Pre-Image | 
| :- | :- | 
| Point (dimension 0) | Line through origo (dimension 1) | 
| Line (dimension 1) | Plane through origo (dimension 2) | 
### Part 1

+ 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
+ A line projects onto a line if it does not pass through origo
You should first find the pre-image of the image of $L$.

## Co-image
+ What kind of object is the pre-image?  
    + How did we describe  such an object previously?
+ What is the relationship between this pre-image and a point $x\in L$?
+ What is the relationship between the pre-image and and the vector $\ell$?

+ Coimage is the set of points (space) orthogonal on the preimage
### Part 2

$$\text{coimage} = \text{preimage}^\bot$$
+ If you read the points $x^1$ and $x^2$ as vectors in 3D, what do 
  they look like?
+ Can you describe the pre-image in terms of $x^1$ and $x^2$?
    + maybe as a span?
+ What then is the relationship between $\ell$ and $x^1,x^2\in L$?

$$\text{preimage} = \text{coimage}^\bot$$
How do you find a vector which is orthogonal on two known vectors in 3D?

## Points and Lines
### Part 3

| 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) |
+ Note that $x$ is an image point.
+ $\ell^1$ and $\ell^2$ are vectors in 3D, and co-images of two image lines 
+ If you view $x$ as a 3D vector instead of a point, what does it look like?
+ What would be the relationship between this vector $x$ and
  $\ell^1$ and $\ell^2$?
+ How do we find vector $x$ with the right relationship with $\ell^1$ and
+ How do we make sure that the vector $x$ is an image point $x$?

+ Preimage and coimage are linear subspaces
    + origo is in both the pre- and co-image
# Second Exercise (3.10)

# Notation
Exercise 3.10 are from Ma 2004 page 62ff.

+ 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
## Debrief Notes

# Systems of Equations and Orthogonal Vectors
1.  Here, it is necessary to look at the pre-images of the two lines.
    + What does the pre-images look like?
    + What is the intersection of the pre-images?  Could it be empty?
    + What is the intersection between the image plane and the pre-images?  
2.  Here, you need to look at the co-images.
    + What can you say about co-images of parallel lines?
    + What can you say about the relationship between the co-images
      and the images?  Is there are relationship between one line and the
      co-image of the other line?
    + Now return to Part 3 of the previous exercise (3.9).
3.  Because the two lines are parallel, they lie in the same plane 
    (not necessarily through the origin).
    Consider the orientation of this plane.
    + Suppose first that it intersects the image plane close to the centre
      (image origin).  Where is the vanishing point?
    + Suppose you turn the plane.  Where does the vanishing point go?
    + At the extremity, the plane is parallel to the image plane.
      Where is the vanishing point now?

+ $\ell^Tx=0$ 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
# Debrief