Warning

# Homework 04 - Calibrated camera pose

This homework is split into two weeks.

### Implementation − part a

 [N1, N2, N3] = p3p_distances( d12, d23, d31, c12, c23, c31 ) N1, N2, N3 = hw04a.p3p_distances( d12, d23, d31, c12, c23, c31 ) d12, d23, d31 .. distances between three points in the space (scalars) c12, c23, c31 .. appropriate cosines (scalars)

Create the function p3p_distances for computing distances of three spatial points from a center of calibrated camera. The function must return the distances η_i in N1, N2, N3 of the three points. Implement only the case 'A' of the computation. If there are more solutions, the returned variables are row vectors (matlab) or lists (python). If there is no solution by the case 'A', return empty vector/list ([]]). For constructing the fourth order polynomial, there is the p3p_polynom function in the tools repository, that can be used in your code.

The function p3p_distances should return only such solutions, that are consistent with input arguments, i.e., that pass verification using cosine law (see below). Use (relative) threshold 1e-4.

 e = p3p_dverify( N1, N2, N3, d12, d23, d31, c12, c23, c31 ) e = hw04a.p3p_dverify( N1, N2, N3, d12, d23, d31, c12, c23, c31 )

Create the function p3p_dverify for verification of computed camera-to-point distances using the cosine law. Use this function in p3p_distances. The function returns vector of three errors, one for each equation. Each computed error should be distance (not squared), relative to particular $d_{jk}$, i.e.

$e(1) = \frac{\sqrt{\eta_1^2 + \eta_2^2 - 2\eta_1\eta_2c_{12}} - d_{12}}{d_{12}}$

### Implementation − part b

 [R C] = p3p_RC( N, u, X, K ) R, C = hw03b.p3p_RC( N, u, X, K )

Create the function p3p_RC for computing calibrated camera centre C and orientation R from three scene-to-image correspondences (X,u), using already computed distances N = [η_1, η_2, η_3].

The function takes one configuration of η_i and returns a single R and C. Note that R must be ortho-normal with determinant equal to +1.

### Steps − part a

1. Test the p3p_distances using the following known values:
1. Construct a simple projection matrix where C=[1,2,-3]T, f=1, K = R = I (3×3 identity). Project the 3D points X1=[0,0,0]T, X2=[1,0,0]T, X3=[0,1,0]T by the P and compute the cosines c12, c23 and c31 for the projected image points. Using the 3D points and the cosines, compute the camera-points distances η and compare with correct known values (C - Xi).
2. Compute the distances η for the following configuration: X1=[1,0,0]T, X2=[0,2,0]T, X3=[0,0,3]T, c12 = 0.9037378393, c23 = 0.8269612542, c31 = 0.9090648231. There should be two solutions, N1 is [4.984934779184205, 4.123105628753765], N2 is [5.172995343542000, 5.099019520925607], N3 is [2.147077467616821, 6.403124241485243].
2. Compute the distances for all triplets of point correspondences chosen from the 10 points as in HW-02 (there is 120 such triplets). Use the matrix K from the A.E. Input data. Plot the distances in a single graph; use red for N1, blue for N2, green for N3. Since there can be more than one solution for a single triplet, horizontal axis shows just total numerical order of solutions. Export as 04_distances.pdf.

Note: The matrix Q obtained in HW-02 is neither related to this task nor is used in any way.

Example distances

### Steps − part b

1. Construct simple projection matrix P where C = [1,2,-3]T, f=1, K = R = I. Project the 3D points X1 = [0, 0, 0]T, X2 = [1, 0, 0]T, X3 = [0, 1, 0]T by the P. Compute the distances η and camera pose using your p3p_RC for all solutions. Compare with correct known values of R, C.
2. Find optimal camera pose using point correspondences by a similar way as in HW02:
3. Compute camera poses for all 120 triplets of point correspondences chosen from your 10 points (as in HW-02). Use the matrix K from the A.E. Input data. (There can be more than one solutions for each triplet.)
4. For a particular camera pose R, C, compose camera matrix and compute the reprojection errors on all 109 points and find their maximum.
5. Select the best camera pose minimising the maximum reprojection error.
6. Export the optimal R, C, and point_sel (indices [i1, i2, i3] of the three points used for computing the optimal R, C) as 04_p3p.mat.
7. Display the image (daliborka_01) and draw u as blue dots, highlight the three points used for computing the best R, C by drawing them as yellow dots, and draw the displacements of reprojected points x multiplied 100 times as red lines. Export as 04_RC_projections_errors.pdf.
8. Plot the decadic logarithm (log10()) of the maximum reprojection error of all the computed poses as the function of their trial index and export as 04_RC_maxerr.pdf. Plot the errors as points, not lines, in this case.
9. Plot the reprojection error of the best R, C on all 109 points as the function of point index and export as 04_RC_pointerr.pdf.
10. Draw the coordinate systems δ (black), ε (magenta) of the optimal R, C, draw the 3D scene points (blue), and draw centers (red) of all cameras you have tested. Export as 04_scene.pdf.
11. Compare your graphs with the graphs in HW02.
 save( '04_p3p.mat', 'R', 'C', 'point_sel' ); sio.savemat( '04_p3p.mat', { 'R':R, 'C':C, 'point_sel': point_sel } )

Note: Use your p3p_distances and plot_csystem from the previous HW.

Note: The matrix P obtained in HW02 is neither related to this task nor is not used in any way.

Example results

Upload two archives containing the following files.

The first part (04a):

matlab python
p3p_distances.m
p3p_dverify.m
04_distances.pdf
hw04a.m hw04a.py containing the required functions
any other files required by your solution

The second part (04b):

matlab python
p3p_RC.m
04_RC_projections_errors.pdf, 04_RC_maxerr.pdf, 04_RC_pointerr.pdf, 04_scene.pdf
04_p3p.mat
hw04b.m hw04b.py containing p3p_RC function
The input entry point scripts hw04a, hw04b should make all required figures, output files and prints without manual intervention.