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

• 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

• 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