Briefing Eight-point algorithm Lecture

Additional Reading Chapter 9 in OpenCV 3 Computer Vision with Python Cookbook by Alexey Spizhevoy (author). Search for it in Oria. There is an e-book available.

Learning Outcome Understand how the eight-point algorithm works on given points.

Exercise

In this exercise we will implement and test the eight-point algorithm using a synthetic dataset. In a real world application you will have to identify and match features in the image, and be prepared for mismatches which require a lot of error-handling. This would be too much for one exercise.

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

Part 1 makes the synthetic data set and is not part of the solutions for the reconstruction problem.
Part 2 solves the reconstruction problem. Because you use a synthetic dataset, Part 1 provides you with ground truth, which can be used to validate the solutions in Steps 6-7. Such use of synthetic data is important to test solutions also in real life development work.

Part 1. Making the synthetic dataset.

Step 1-1.

Select eight points in general position in 3D. For instance \[(-20,0,25), (20,0,25), (0,20,25), (-10,10,50), (10,10,50), (0,-10,50), (0,0,60), (-7,7,70), \] or generate some at random if you prefer.

Note
The exercise has been updated to move the points further from the camera (increasing their \(z\)-coordinates). It seems that at least one point in the first version fell behind the image plane in the rotated camera frame.

Step 1-2.

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 1-3.

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.

Part 2. The Eight-Point Algorithm

Step 2-1.

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 2-2. 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 2-3. 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?

Remember there are four solutions for \((R,T)\), three of which produce objects point behind either or both cameras. Remember to check both \(E\) and \(-E\), in addition to the two solutions for \((R,T)\) from one choice for \(E\).

Note If the transformation you find seem off, typically with errors of sign, there are at least two plausible explanations.

1. You have not checked all four solutions (see above).
2. Some object point from Step 1 actually does end up behind the image plane in the second camera frame. If this happens, you should try to choose different object points.

Step 2-4. 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

Exercise using real images.

In this exercise, we shall try to determine the relative pose of two cameras, using the eight-point algorithm (or a variant thereof).

Step 1. Make a Data Set

1. Take two images of the same scene, using different camera poses.
   • the difference between the poses should be significant, but small enough to recognise the same feature points.
   • i.e. two consecutive frames from a video will probably be too similar.
2. Run the Harris Detector on both images, and identify at least eight features which you can pair between the images.
   • if you do not find eight, you need to use more similar poses.

Note 1 It may be useful to calibrate the camera(s) and undistort the images before starting. It is ok to try without calibration first, for the sake of simplicity.

Note 2 you should pair the feature points manually in this exercise, to make sure that no mismatches ruin your results. When you have the first prototype working, you can try to pair feature points programmatically, using SIFT or other methods to match features.

Step 2. Essential matrix

Each pair \((\mathbf{x}_i,\mathbf{x}_i')\) of corresponding features from the two images give rise to a Kronecker product. \[\mathbf{a}_i=\mathbf{x}_i\otimes\mathbf{x}_i'\] Using these vectors as columns, we get the \(8\times n\) matrix \(\chi\), where \(n\) is the number of feature pairs.

1. Calculate \(\chi\) from your dataset.
2. Compute the singular value decomposition of \(\chi=U_\chi\Sigma_\chi V_\chi^T\) 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 fundamental matrix \(E\).
   • you can do this with the numpy reshape function, but test it. You may have to transpose the resulting matrix.

Step 3. Projection onto the Essential Space

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

Step 4. Rotation and Translation

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

What do the two solutions represent?

Does it seem reasonable compared to how you moved the cameras?

Debrief