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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

## 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