# 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 $$u$$
• $(u) = u^$
• 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