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[F, G] = u2FG( u1, u2 )

The function `u2FG`

computes the 'fundamental matrix' `G`

using the eight-point algorithm from 8 euclidean correspondences `u1`

, `u2`

, measured in two images. Then the true fundamental matrix `F`

is found, which is close to `G`

and `rank(F) = 2`

.

Inside this function, prior to constructing a matrix for SVD, the points must be normalized. This can be done by subtracting a centroid of the 8 points and dividing by it's standard deviation; in each image separately. Express the normalization as homographies `H1`

and `H2`

, the points are modified by these.

u1n = H1 * u1p; % u1p are homogeneous u2n = H2 * u2p; % u2p are homogeneous

Then the computed `Gn`

and `Fn`

relate the normalized correspondences `u1n`

and `u2n`

and must be back-normalized.

G = H2' * Gn * H1 F = H2' * Fn * H1

The matrix `Gn`

computed by SVD has usually rank = 3 due to noise in data. Therefore modification to rank = 2 is needed. Compute SVD of the `Gn`

, nullify the D(3,3) and compose the matrix back.

**1. Correspondences between the images**

- Manually obtain a set of 12 image correspondences (matches) between the images above. These correspondences must sufficiently cover whole area of both images as well as the 3D space.
- Manually obtain another set of at least 40 correspondences. Choose reasonable vertices (e.g., roof corners) that can be later connected by edges into a wire-frame mesh (for visualization). The edges (together with the vertices) will be used in the next homework. (You do not need to set-up edges now.)

**2. Epipolar geometry**

- Find the fundamental matrix
`F`

relating the images above: generate all (495) 8-tuples from the set of 12 correspondences, estimate (the true)`F`

for each of them and chose the one, that minimizes maximal epipolar error over all (i.e. the 12 + the mesh) matches.- For a particular tested fundamental matrix
`F`

, compute epipolar lines`l1_i = F'*x2_i`

and`l2_i = F*x1_i`

(provided that points are homogeneous) for each point`i`

. Then evaluate the Euclidean distances`d1_i = d(x1_i,l1_i)`

and`d2_i = d(x2_i,l2_i)`

in images. The epipolar error for the i-th match is defined as`d1_i + d2_i`

.

- Draw the 12 corresponding points in different colour in the two images. Using the best
`F`

, compute the corresponding epipolar lines and draw them into the images in corresponding colours (a line segment given by the intersection of the image area and a line must me computed). Export as`08_eg.pdf`

. - Draw graphs of epipolar errors
`d1_i`

and`d2_i`

for all points (point index on horizontal axis, the error on vertical axis). Draw both graphs into single figure (different colours) and export as`08_errors.pdf`

. - Save all the data into
`08_data.mat`

: all correspondences as`u1`

,`u2`

, the indices of the 12 selected correspondences as`point_sel`

, the 'best' matrices`F`

and`G`

.

Hand an archive consisting of:

`08_eg.pdf`

`08_errors.pdf`

`08_data.mat`

`u2FG.m`

`hw08.m`

โ your Matlab implementation entry point.

any other files required by hw08.m (including data and files from the repository).

courses/gvg/labs/gvg-2017-hw-08.txt ยท Last modified: 2019/01/19 18:22 (external edit)