Relative Pose

Reading Ma 2004 Chapter 5

Briefing Relative Pose Lecture

Exercises

Exercise 5.5

Explain under what conditions the family of epipolar lines in at least one of the image planes will be parallel to each other. Where is the corresponding epipole (in terms of is homogeneous co-ordinates)?

To make sense of the question, you have two draw the situation.

  1. Draw the two origins and the two image planes. Where are the epipoles?
  2. Draw two or three separate object points (3D) and the corresponding epipolar planes. (Avoid object points in the same epipolar plane.) Where are the corresponding epipolar lines?
  3. If you have now identified different epipolar planes within one fixed camera configuration, you will have a family of epipolar lines in each image plane. Where do these lines intersect?
  4. Try to rotate one of the cameras.
    • Can you rotate it so that the epipolar lines become parallel?
    • How does this come about?
    • Where is the epipole when this happens?
  5. Review the original exercise text. Does it make sense in terms of the thought experiment that you have just conducted?

Exercise 5.8

Given two images \(x_1\), \(x_2\) of a point \(p\) together with the relative camera motion \((R,T)\), \(\mathbf{X}_2=R\mathbf{X}_1+T\):

  1. express the depth of \(p\) with respect to the first image, i.e. \(\lambda_1\) in terms of \(x_1\), \(x_2\), and \((R,T)\).
  2. express the depth of \(p\) with respect to the second image, i.e. \(\lambda_2\) in terms of \(x_1\), \(x_2\), and \((R,T)\).

  1. Again, you need to draw the situation.

    • Draw the two image planes and origins, as well as \(p\).
    • Identify the points \(x_1\) and \(x_2\).
    • Mark the lengths \(\lambda_1\) and \(\lambda_2\) (i.e. the depths or \(z\)-co-ordinates of \(p\)) in the figure. (We do not calculate them yet.)
    • Identify the epipoles and epipolar lines as usual.

  2. Imagine that you move \(p\) closer to camera 1, but keeping it on the same line so that the image point \(x_1\) remains fixed.

    • what happens to \(\lambda_1\)?
    • what happens to \(x_2\)?

  3. Imagine \(\lambda_1\to0\).

    • Where does \(p\) end up?
    • Where does \(x_2\) end up?

The task (part 1 in the exercise) is to find a formula or method to determine \(\lambda_1\), given \(x_1\), \(x_2\), and \((R,T)\). We can assume that the cameras are normalised so that the image plane has equation \(z=1\) with respect to its own basis. There are two approaches to completing the exercises. The geometric one will probably give the most insight.

Algebraic Solution

The relationship between \(x_1\), \(x_2\), \((R,T)\), \(\lambda_1\), and \(\lambda_2\) is given on page 111.

\[\lambda_2x_2 = R\lambda_1x_1+T\]

  • Does this make sense in your figure?
  • What are the unknowns?
  • Do you have enough constraints to find a unique solution?
  • Rewrite the equation as a set of scalar equations. Can you solve for \(\lambda_1\) and \(\lambda_2\)?

Geometric Solution

All the objects that we consider are situated in the same epipolar plane. Redraw the origins, epipoles, image points, and depths \(\lambda_1,\lambda_2\) in this plane.

  • What is the co-ordinates of the epipoles of camera 2?

  • Consider the triangle with corners in the origin, epipole, and image point for camera 2.

    • Which sides (lengths) are known?
    • Can you determine the angle at the origin?

  • Consider the larger triangle with corners in the two origins and \(p\).

    • Which sides and angles are known?
    • The unknown \(\lambda_1\) is one of these sides. Can you calculate this with the help of triangulation?

Part 2

We have only considered \(\lambda_1\) above. The calculation of \(\lambda_2\) is symmetric, though you may have to invert the transformation \((R,T)\).

Optional exercises

Debrief