Lecture 8 Problems - CSCI 476/576

Consider the following geometric properties. The first several questions ask which of them are preserved by a given class of geometric transformations. For each question, give a list of letters corresponding to which properties are preserved (i.e., left unchanged) by the given class of geometric transformations.

Note: Feel free to use this online demo to gain intuition and try things out as you work through these, keeping in mind that affine transformations are only those where the last row remains \(\begin{bmatrix}0 & 0 & 1\end{bmatrix}\).

  1. Which are preserved under translation?
  2. Which are preserved under rigid transformation (i.e., affine but with an orthonormal linear component; translation, rotation, reflection only)?
  3. Which are preserved under affine transformations?
  4. Which are preserved under projective transformations?

A. Line straightness

B. Line lengths

C. Ratios of lengths along a line

D. Parallelism of lines

E. Angles

F. Locations of points

G. Location of the origin

In the next two problems, we’ll show that a homography has only 8 degrees of freedom. In other words, for any homography \(H\) that is not all zeros, there is a homography \(H' = H / h_{33}\)​ that has the same effect on homogeneous coordinates.

  1. Let $ H = \[\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & k \\ \end{bmatrix}\] $ and \(\mathbf{x} = \begin{bmatrix}x\\ y \\ 1\end{bmatrix}\). Compute \(H\mathbf{x}\)​ and normalize the resulting homogeneous point.
  2. Compute \(\frac{1}{k}H\mathbf{x}\) and normalize the resulting homogeneous point.
  3. If 5 and 6 came out the same, then we’ve proved that \(H \approx \frac{1}{k}H\), which implies that homographies, like homogeneous points, are equivalent up to normalization, making the number of degrees of freedom 8 intstead of 9. Would the this proof change if we’d changed \(\mathbf{x}\)’s third coordinate from 1 to \(w\)?