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This homework is split into two weeks.

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`

.

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 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. - 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`

.

- 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`

.

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

courses/gvg/labs/gvg-2017-hw-09.txt · Last modified: 2018/02/14 16:01 by policmic