Reconstruction (Points) from Symmetry

Download the file lab3.zip (36KB) under a local directory of yours.

lab3.m  The main code of this exercise. You may not need to work on it. But you can read the comments in the code to better understand how it works.

solvelyapnov.m An empty function. You need to add contents into this function to find the R₀ from R', R, N.

1.jpg - Image for reconstruction

displayresult.m Display the result. You do not need to work on it.

hat.m Calculate the skew matrix of a vector.

The 12 feature points on the image are stored in 3 by 12 matrices x10. The same points stored in x20 with a different order represent the feature points in the "hidden" view.

The points x10 in the oringinal image are order as following:
1- 9-10- 2
5             7
6             8
3-11-12-4

The points x20 in the hidden image are order by their correspondence to the x1.
2-10- 9- 1
7             5
8             6
4-12-11-3

Each column of the two matrices is a point in image coordinate (using homogenous coordinate representation). The corresponding two columns in two matrices are corresponding points, i.e, x20=g(x10).

The 3x3 calibration matrix is stored in the matrix called Calib.

You need to finish the function solvelyapnov.m to find the R₀ from R', R, N (denoted as R1,R,N1 in lab3.m) which are found by the 4-point homography algorithm.

Use the R₀, T₀ found in the algorithm to verify the rank of M_s(x). You must choose more than 2 different symmetries g. There are many different symmetries you can use. The one we used in the codes is g=[R T] where R=diag[-1,1,1] which is reflective symmetry along x-axis. You can also choose the reflective symmetry along y-axis, along the diagonal or rotational symmetry along z-axis for 90 or 180 degrees.

Due to the noise of the data, the rank of the matrix M_s(x) can be found by watching the singular values of M_s(x).

The result should be similar like this.