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 \(a\in\mathbb{R}\) 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 \(l\)

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