Briefing

Sorry for the change of programme. I found, last minute, that doing the eight-point algorithm with uncalibrated views will generate too many problems. We should therefore do a little more with the uncalibrated algorithm.

Reflection

(This is the same exercise as given yesterday)

Consider the material covered in the sessions 21-22 October.

Do you know,

1. how to calculate 3D co-ordinates of an object point \(p\), given image points \(x_1\) and \(x_2\) in two separate camera frames? What do you require to do so?
2. recover the relative pose of two cameras, given matching image points in the two frames? How many such image points do you require?
3. If the answer is no to either 1 or 2,
   • what knowledge do you have to build on?
   • what do you need to find out?

Exercise

Today’s exercise is a repeat of the Eight-point algorithm, but now working off a synthetic data set.

Note. You can choose, in this exercise, how much you want to compute by hand, and what you want to implement in python.

Step 1.

Select eight points in general position in 3D. For instance \[(-20,0,10), (20,0,10), (0,20,10), (-10,10,20), (10,10,20), (0,-10,20), (0,0,25), (-7,7,30), \] or generate some at random if you prefer.

Step 2.

Consider a camera posed in the world frame.
I.e. the selected points have co-ordinates in the camera frame. Assume the image plane has co-ordinates \(z=1\).

• Find the image point for each world point, using an ideal camera projection.

Step 3.

Consider a second camere which is translated ten units along the \(x\)-axis and rotated \(30^\circ\) degrees around the \(y\)-axis. Again the image plane is \(z=1\).

• Find the transformation (relative pose) \((R,T)\).
• Find (3D) co-ordinates in the second camera frame for each world point.
• Find the image points in the second camera frame for each world point.

Step 4.

Imagine now that we do not know the world points. We are going to try to reconstruct them using the Eight-point algorithm.

1. Construct the matrix \(\chi\) with the Kronecker products of the eight image pairs.
2. Compute the singular value decomposition of \(\chi=U_\chi\Sigma_\chi V_\chi^T\) (probably using numpy.linalg.svd)
   • Inspect the three components. What do you see?
   • The middle component should be a diagonal matrix containing the singular values. How are they ordered?
3. According to Ma (2004:121) the serialised fundamental matrix \(E^s\) is the ninth column of \(V_\chi\).
   • this should be wrig
4. Unserialise \(E^s\) to get the preliminary matrix \(E\).
   • you can do this with the numpy reshape function, but test it. You may have to transpose the resulting matrix.

Step 5. Projection onto the Essential Space

If \(E=U\Sigma V^T\) is our preliminary matrix as found above, we approximate the essential matrix by replacing \(\Sigma\) with \(\mathsf{diag}(1,1,0)\) (cf. Ma (2004:121)).

Step 6. Rotation and Translation

Recover Translation and Rotation matrices from the essential matrix, relying on Ma (2004:121).

• What do the two solutions represent?
• Do they match the actual transformation?

Step 7. Reconstruction in 3D

Consider one pair of image points together with the transformation \((R,T)\).

• Calculating the \(z\)-coordinate \(\lambda\) of the 3D point as we did in Relative Pose. You know that the distance between the camera origins is \(||T||=10\).
• Invert the projection to recover the complete coordinates of the 3D point.
• Does the recovered point match the original point?
• If you have time, repeat with a second and possibly a third point. You probably do not want to do all eight, though.

Debrief