~~NOTOC~~
====== Homework 09 - Metric reconstruction =======

This homework is split into two weeks.
==== Essential matrix ====

Having a fundamental matrix ''F'' and camera calibration ''K'', the essential matrix ''E'' can be computed from the equation 

<latex>F = K^{-\top} E K^{-1}\,.</latex>

First, matrix ''E1'' is computed as:

  E1 = K' * F * K;
  
Due to errors in data used to estimate ''F'' and/or ''K'', the obtained matrix ''E1'' is not a true essential matrix, its two non-zero singular values are not equal. So the matrix must be modified in order to make these singular values equal:

  [U D V] = svd( E1 ); 
  D(2,2) = D(1,1);
  E = U * D * V';
  
To summarize, starting with regular matrix ''G'' obtained from the 8-point algorithm, we apply SVD two times. First, ''G'' is decomposed, the smallest singular value zeroed and ''F'' is composed. Secondly, ''K'' is applied, ''E1'' is decomposed and the two singular values are made equal giving raise to essential matrix ''E''. Finally, ''K'' can be applied to compute a fundamental matrix ''Fe'' consistent with ''K'' from ''E''.

=== Steps ===

Find two essential matrices. A possibly bad ''Ex'' and the best ''E''
  - Compute essential matrix ''Ex'' using your best fundamental matrix ''F'' estimated in HW-09 and internal calibration [[http://cw.felk.cvut.cz/courses/GVG/2013/HomeWorks/K.mat|K.mat]] from HW-04. Compute also the fundamental matrix ''Fx'' consistent with ''K'' from ''Ex'' and ''K''
   - Draw the 12 corresponding points (from HW-09) in different colour in the two images. Using ''Fx'', compute the corresponding epipolar lines and draw them into the images in corresponding colours. Export as ''09_egx.pdf''.
   - Draw graphs of epipolar errors ''d1_i'' and ''d2_i'' w.r.t ''Fx'' for all points. Draw both graphs into single figure (different colours) and export as ''09_errorsx.pdf''.
  - Find essential matrix ''E'' by minimizing the maximum epipolar error of the respective fundamental matrix ''Fe'' consistent with ''K'' using the correspondences from HW-09:
    - Generate all (495) 8-tuples from the set of 12 correspondences and estimate fundamental matrix ''F'' (via ''G'') for each of them.
    - For each tested ''F'', compute essential matrix ''E'' using internal calibration [[http://cw.felk.cvut.cz/courses/GVG/2012/HomeWorks/HW-04/K.mat|K.mat]] from HW-04. 
    - Compute fundamental matrix ''Fe'' consistent with ''K'' from ''E'' and ''K'' and its epipolar error over all matches.
    - Choose the ''Fe'' and ''E'' that minimize maximal epipolar error over all (i.e. the 12 + the mesh) matches. 
  - Draw the 12 corresponding points in different colour in the two images. Using ''Fe'', compute the corresponding epipolar lines and draw them into the images in corresponding colours. Export as ''09_eg.pdf''.
  - Draw graphs of epipolar errors ''d1_i'' and ''d2_i'' w.r.t ''Fe'' for all points. Draw both graphs into single figure (different colours) and export as ''09_errors.pdf''.
  - Save ''F'', ''Ex'', ''Fx'', ''E'', ''Fe'' and ''u1'', ''u2'', ''point_sel'' (same as in HW-09) as ''09a_data.mat''.

==== Cameras and reconstruction ====
  - Decompose  the best ''E'' into relative rotation ''R'' and translation ''C'' (four solutions). Choose such a solution that reconstructs (most of) the points in front of both (computed) cameras.
  - Construct projective matrices ''P1'', ''P2'' (including ''K'').
  - Compute scene points ''X''.
  - Manually create set of at least 30 edges connecting the points. The edges should correspond to real edges in the scene.
  - Display the images, draw the input points as blue dots and the scene points ''X'' projected by appropriate ''P_i'' as red circles. Draw also the edges, connecting the **reprojected** points as yellow lines.  Export as ''09_reprojection.pdf''.
  - Draw graph of reprojection errors and export as ''09_errorsr.pdf''.
  - Draw the 3D point set connected by the edges as a wire-frame model. From the top, from the side, and from some general view. Export as  ''09_view1.pdf'', ''09_view2.pdf'', and ''09_view3.pdf''.
  - Save ''Fe'', ''E'', ''R'', ''C'', ''P1'', ''P2'', ''X'', and ''u1'', ''u2'', ''point_sel'' as ''09b_data.mat''. 

^  Example of the corresponding points and edges   ^  Example of a wire-frame model reconstruction  ^
|  {{courses:gvg:labs:09_mesh.jpg?direct&400|}}  |  {{courses:gvg:labs:11_projection.gif|}}  |

==== Upload ====

The first part: upload an archive consisting of:
  - ''09_errorsx.pdf'', ''09_errors.pdf''
  - ''09_egx.pdf'', ''09_eg.pdf''
  - ''09a_data.mat''
  - ''hw09a.m'' – your Matlab implementation entry point.
  - other files required by hw09a.m

The second part: upload an archive consisting of:
  - ''09b_data.mat''
  - ''09_reprojection.pdf''
  - ''09_errorsr.pdf''
  - ''09_view1.pdf'', ''09_view2.pdf'', ''09_view3.pdf''
  - ''hw09b.m'' – your Matlab implementation entry point.
  - other files required by hw09.m