Recontruction variance on top of GRIC

This is WIP of recontruction variance criteria applied on top of GRIC.

[sergey.vfx, 2013-02-27 09:00:52]

WIP patch, stage 2

[sergey.vfx, 2013-02-27 09:04:00]

Added second stage of WIP patch.

Changes:

- Switched H, F refining to normalized space. Pixel space gave issues with F.
Didn't have time to fully solve this, but wanted to move on, so TEMPORARY
switched back to refining in normalzied space
- Added base skeleton of reconstruction variance, currently done:
  * Camera matricies estimations
  * Points reconstruction
  * Bundling
  * Jacobian computation

The part with Jacobian is REALLY temporary, just was studying how evaluate API
works. And this is actually incorrect to do (J^T*J)+ here, we need to estimate
Sigma_P. However, pseudoInverse seems to be working, so if we'll need it here it
is :)

What to do:
- Estimate covariance matrix
- Extract Sigma_P from covariance matrix and J
- ...

[sergey.vfx, 2013-03-05 18:15:32]

WIP patch, stage 3

[sergey.vfx, 2013-03-05 18:17:08]

Made it full patch against trunk.

Since last patch i've ordered parameters and tried to compute trace of P part of
Sigma_P.

[sergey.vfx, 2013-03-06 18:26:39]

Patch Set 3 : WIP patch, stage 4

[sergey.vfx, 2013-03-06 18:30:25]

Changes:
- Made it more generic way of getting minimizer's information from BA (like
number of cameras, points, jacobians and so)
- Moved Sigma_P computation to keyframe_selection
- In fact, we need to use num_points*3 x num_poitns*3 right bottom corner of
J^T*J
- Also inversion of this matrix is not needed. I've wrote small unittest for
libmv which demonstrates this.

Basically, for simple cases this criteria seems to be working. However, there're
some fuzziness:
- Error expecation is REALLY high. Like, error expectation of
573570874672.050293 does not look healthy.
- Error expectation could be negative..
- Currently using dense matrix for jacobian, we'd better switch to sparse before
final commit.

[sergey.vfx, 2013-03-06 18:44:00]

On 2013/03/06 18:30:25, sergey.vfx wrote:
> Changes:
> - Made it more generic way of getting minimizer's information from BA (like
> number of cameras, points, jacobians and so)
> - Moved Sigma_P computation to keyframe_selection
> - In fact, we need to use num_points*3 x num_poitns*3 right bottom corner of
> J^T*J
> - Also inversion of this matrix is not needed. I've wrote small unittest for
> libmv which demonstrates this.
> 
> Basically, for simple cases this criteria seems to be working. However,
there're
> some fuzziness:
> - Error expecation is REALLY high. Like, error expectation of
> 573570874672.050293 does not look healthy.
> - Error expectation could be negative..
> - Currently using dense matrix for jacobian, we'd better switch to sparse
before
> final commit.

One more thing to be mentioned. I've currently switched from ITERATIVE_SCHUR to
DENSE_SCHUR because iterative one seemed to behaving fuzzy. But that could have
been a result of experiments while debugging isApprox crash..

[mierle, 2013-03-13 07:17:43]

https://codereview.appspot.com/7306106/diff/1/extern/libmv/libmv/simple_pipel...
File extern/libmv/libmv/simple_pipeline/intersect.cc (right):

https://codereview.appspot.com/7306106/diff/1/extern/libmv/libmv/simple_pipel...
extern/libmv/libmv/simple_pipeline/intersect.cc:68: 
80 columns

https://codereview.appspot.com/7306106/diff/1/extern/libmv/libmv/simple_pipel...
File extern/libmv/libmv/simple_pipeline/keyframe_selection.cc (right):

https://codereview.appspot.com/7306106/diff/1/extern/libmv/libmv/simple_pipel...
extern/libmv/libmv/simple_pipeline/keyframe_selection.cc:350: +      // Uses
paper Keyframe Selection for Camera Motion and Structure Estimation from
Multiple Views
80 columns.

https://codereview.appspot.com/7306106/diff/27001/extern/libmv/libmv/simple_p...
File extern/libmv/libmv/simple_pipeline/keyframe_selection.cc (right):

https://codereview.appspot.com/7306106/diff/27001/extern/libmv/libmv/simple_p...
extern/libmv/libmv/simple_pipeline/keyframe_selection.cc:255: Mat
pseudoInverse(Mat &matrix,
This is actually a regular inverse?

https://codereview.appspot.com/7306106/diff/27001/extern/libmv/libmv/simple_p...
extern/libmv/libmv/simple_pipeline/keyframe_selection.cc:514: next_keyframe =
candidate_image;
We may want to store a set of ~3 or ~4 candidate pairs. We could have a larger
automatic loop that just tries all of them and picks the best one.

[sergey.vfx, 2013-04-23 20:31:59]

WIP patch, stage 5

[sergey.vfx, 2013-04-23 20:38:30]

Comparing to previous patchset:

- Better epsilon handing in pseudoInverse function. Now it uses the same
tolerance as NumPy does.
- Made it better covariance matrix estimation, using generalized inverse with N
eigen values zeroed (N=7 in our case).
- Made it return single keyframe pair, which is detecting based on
reconstruction estimation. Previous code to do full reconstruction of 2 frames
didn't work well. Also, apparently average reprojection error could be the same
for bad and good keyframe pair.
- Made it so EuclideanIntersect returns false in case point is behind the
camera.

Seems to behave better than bare GRIC, but still need to do:
- Make solution "permanent" size -- that thing to eliminate scale indeterminacy.
- Use sparse matrix for bundler evaluation and pseudo-inverse. Shall give some
boost.

I'm actually pretty much close to commit this stuff to tomato branch.