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# Pre- and co-image

# 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 \(\ell^\bot\)

- The plane needs
- 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

- one function per dimension; or

## 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

## 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