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---
author:
- Maksym Ivashechkin
bibliography: 'bibs.bib'
csl: 'acm-sigchi-proceedings.csl'
date: August 2020
title: 'Google Summer of Code: Improvement of Random Sample Consensus in OpenCV'
...
Contribution
============
The integrated part to OpenCV `calib3d` module is RANSAC-based universal
framework USAC (`namespace usac`) written in C++. The framework includes
different state-of-the-arts methods for sampling, verification or local
optimization. The main advantage of the framework is its independence to
any estimation problem and modular structure. Therefore, new solvers or
methods can be added/removed easily. So far it includes the following
components:
1. Sampling method:
1. Uniform – standard RANSAC sampling proposed in \[8\] which draw
minimal subset independently uniformly at random. *The default
option in proposed framework*.
2. PROSAC – method \[4\] that assumes input data points sorted by
quality so sampling can start from the most promising points.
Correspondences for this method can be sorted e.g., by ratio of
descriptor distances of the best to second match obtained from
SIFT detector. *This is method is recommended to use because it
can find good model and terminate much earlier*.
3. NAPSAC – sampling method \[10\] which takes initial point
uniformly at random and the rest of points for minimal sample in
the neighborhood of initial point. This is method can be
potentially useful when models are localized. For example, for
plane fitting. However, in practise struggles from degenerate
issues and defining optimal neighborhood size.
4. Progressive-NAPSAC – sampler \[2\] which is similar to NAPSAC,
although it starts from local and gradually converges to
global sampling. This method can be quite useful if local models
are expected but distribution of data can be arbitrary. The
implemented version assumes data points to be sorted by quality
as in PROSAC.
2. Score Method. USAC as well as standard RANSAC finds model which
minimizes total loss. Loss can be represented by following
functions:
1. RANSAC – binary 0 / 1 loss. 1 for outlier, 0 for inlier. *Good
option if the goal is to find as many inliers as possible.*
2. MSAC – truncated squared error distance of point to model. *The
default option in framework*. The model might not have as many
inliers as using RANSAC score, however will be more accurate.
3. MAGSAC – threshold-free method \[3\] to compute score. Using,
although, maximum sigma (standard deviation of noise) level to
marginalize residual of point over sigma. Score of the point
represents likelihood of point being inlier. *Recommended option
when image noise is unknown since method does not require
threshold*. However, it is still recommended to provide at least
approximated threshold, because termination itself is based on
number of points which error is less than threshold. By giving 0
threshold the method will output model after maximum number of
iterations reached.
4. LMeds – the least median of squared error distances. In the
framework finding median is efficiently implement with $O(n)$
complexity using quick-sort algorithm. Note, LMeds does not have
to work properly when inlier ratio is less than 50%, in other
cases this method is robust and does not require threshold.
3. Error metric which describes error distance of point to
estimated model.
1. Re-projection distance – used for affine, homography and
projection matrices. For homography also symmetric re-projection
distance can be used.
2. Sampson distance – used for Fundamental matrix.
3. Symmetric Geometric distance – used for Essential matrix.
4. Degeneracy:
1. DEGENSAC – method \[7\] which for Fundamental matrix estimation
efficiently verifies and recovers model which has at least 5
points in minimal sample lying on the dominant plane.
2. Collinearity test – for affine and homography matrix estimation
checks if no 3 points lying on the line. For homography matrix
since points are planar is applied test which checks if points
in minimal sample lie on the same side w.r.t. to any line
crossing any two points in sample (does not assume reflection).
3. Oriented epipolar constraint – method \[6\] for epipolar
geometry which verifies model (fundamental and essential matrix)
to have points visible in the front of the camera.
5. SPRT verification – method \[9\] which verifies model by its
evaluation on randomly shuffled points using statistical properties
given by probability of inlier, relative time for estimation,
average number of output models etc. Significantly speeding up
framework, because bad model can be rejected very quickly without
explicitly computing error for every point.
6. Local Optimization:
1. Locally Optimized RANSAC – method \[5\] that iteratively
improves so-far-the-best model by non-minimal estimation. *The
default option in framework. This procedure is the fastest and
not worse than others local optimization methods.*
2. Graph-Cut RANSAC – method \[1\] that refine so-far-the-best
model, however, it exploits spatial coherence of the
data points. *This procedure is quite precise however
computationally slower.*
3. Sigma Consensus – method \[3\] which improves model by applying
non-minimal weighted estimation, where weights are computed with
the same logic as in MAGSAC score. This method is better to use
together with MAGSAC score.
7. Termination:
1. Standard – standard equation for independent and
uniform sampling.
2. PROSAC – termination for PROSAC.
3. SPRT – termination for SPRT.
8. Solver. In the framework there are minimal and non-minimal solvers.
In minimal solver standard methods for estimation is applied. In
non-minimal solver usually the covariance matrix is built and the
model is found as the eigen vector corresponding to the highest
eigen value.
1. Affine2D matrix
2. Homography matrix – for minimal solver is used RHO
(Gaussian elimination) algorithm from OpenCV.
3. Fundamental matrix – for 7-points algorithm two null vectors are
found using Gaussian elimination (eliminating to upper
triangular matrix and back-substitution) instead of SVD and then
solving 3-degrees polynomial. For 8-points solver Gaussian
elimination is used too.
4. Essential matrix – 4 null vectors are found using
Gaussian elimination. Then the solver based on Gröbner basis
described in \[11\] is used. Essential matrix can be computed
only if <span style="font-variant:small-caps;">LAPACK</span> or
<span style="font-variant:small-caps;">Eigen</span> are
installed as it requires eigen decomposition with complex
eigen values.
5. Perspective-n-Point – the minimal solver is classical 3 points
with up to 4 solutions. For RANSAC the low number of sample size
plays significant role as it requires less iterations,
furthermore in average P3P solver has around 1.39
estimated models. Also, in new version of `solvePnPRansac(...)`
with `UsacParams` there is an options to pass empty intrinsic
matrix `InputOutputArray cameraMatrix`. If matrix is empty than
using Direct Linear Transformation algorithm (PnP with 6 points)
framework outputs not only rotation and translation vector but
also calibration matrix.
Also, the framework can be run in parallel. The parallelization is done
in the way that multiple RANSACs are created and they share two atomic
variables `bool success` and `int num_hypothesis_tested` which
determines when all RANSACs must terminate. If one of RANSAC terminated
successfully then all other RANSAC will terminate as well. In the end
the best model is synchronized from all threads. If PROSAC sampler is
used then threads must share the same sampler since sampling is done
sequentially. However, using default options of framework parallel
RANSAC is not deterministic since it depends on how often each thread is
running. The easiest way to make it deterministic is using PROSAC
sampler without SPRT and Local Optimization and not for Fundamental
matrix, because they internally use random generators.\
\
For NAPSAC, Progressive NAPSAC or Graph-Cut methods is required to build
a neighborhood graph. In framework there are 3 options to do it:
1. `NEIGH_FLANN_KNN` – estimate neighborhood graph using OpenCV FLANN
K nearest-neighbors. The default value for KNN is 7. KNN method may
work good for sampling but not good for GC-RANSAC.
2. `NEIGH_FLANN_RADIUS` – similarly as in previous case finds neighbor
points which distance is less than 20 pixels.
3. `NEIGH_GRID` – for finding points’ neighborhood tiles points in
cells using hash-table. The method is described in \[2\]. Less
accurate than `NEIGH_FLANN_RADIUS`, although significantly faster.
Note, `NEIGH_FLANN_RADIUS` and `NEIGH_FLANN_RADIUS` are not able to PnP
solver, since there are 3D object points.\
\
New flags:
1. `USAC_DEFAULT` – has standard LO-RANSAC.
2. `USAC_PARALLEL` – has LO-RANSAC and RANSACs run in parallel.
3. `USAC_ACCURATE` – has GC-RANSAC.
4. `USAC_FAST` – has LO-RANSAC with smaller number iterations in local
optimization step. Uses RANSAC score to maximize number of inliers
and terminate earlier.
5. `USAC_PROSAC` – has PROSAC sampling. Note, points must be sorted.
6. `USAC_FM_8PTS` – has LO-RANSAC. Only valid for Fundamental matrix
with 8-points solver.
7. `USAC_MAGSAC` – has MAGSAC++.
Every flag uses SPRT verification. And in the end the final
so-far-the-best model is polished by non minimal estimation of all found
inliers.\
\
A few other important parameters:
1. `randomGeneratorState` – since every USAC solver is deterministic in
OpenCV (i.e., for the same points and parameters returns the
same result) by providing new state it will output new model.
2. `loIterations` – number of iterations for Local Optimization method.
*The default value is 10*. By increasing `loIterations` the output
model could be more accurate, however, the computationial time may
also increase.
3. `loSampleSize` – maximum sample number for Local Optimization. *The
default value is 14*. Note, that by increasing `loSampleSize` the
accuracy of model can increase as well as the computational time.
However, it is recommended to keep value less than 100, because
estimation on low number of points is faster and more robust.
Samples:
There are three new sample files in opencv/samples directory.
1. `epipolar_lines.cpp` – input arguments of `main` function are two
pathes to images. Then correspondences are found using
SIFT detector. Fundamental matrix is found using RANSAC from
tentaive correspondences and epipolar lines are plot.
2. `essential_mat_reconstr.cpp` – input arguments are path to data file
containing image names and single intrinsic matrix and directory
where these images located. Correspondences are found using SIFT.
The essential matrix is estimated using RANSAC and decomposed to
rotation and translation. Then by building two relative poses with
projection matrices image points are triangulated to object points.
By running RANSAC with 3D plane fitting object points as well as
correspondences are clustered into planes.
3. `essential_mat_reconstr.py` – the same functionality as in .cpp
file, however instead of clustering points to plane the 3D map of
object points is plot.
References:
1\. Daniel Barath and Jiří Matas. 2018. Graph-Cut RANSAC. In *Proceedings
of the iEEE conference on computer vision and pattern recognition*,
6733–6741.
2\. Daniel Barath, Maksym Ivashechkin, and Jiri Matas. 2019. Progressive
NAPSAC: Sampling from gradually growing neighborhoods. *arXiv preprint
arXiv:1906.02295*.
3\. Daniel Barath, Jana Noskova, Maksym Ivashechkin, and Jiri Matas.
2020. MAGSAC++, a fast, reliable and accurate robust estimator. In
*Proceedings of the iEEE/CVF conference on computer vision and pattern
recognition (cVPR)*.
4\. O. Chum and J. Matas. 2005. Matching with PROSAC-progressive sample
consensus. In *Computer vision and pattern recognition*.
5\. O. Chum, J. Matas, and J. Kittler. 2003. Locally optimized RANSAC. In
*Joint pattern recognition symposium*.
6\. O. Chum, T. Werner, and J. Matas. 2004. Epipolar geometry estimation
via RANSAC benefits from the oriented epipolar constraint. In
*International conference on pattern recognition*.
7\. Ondrej Chum, Tomas Werner, and Jiri Matas. 2005. Two-view geometry
estimation unaffected by a dominant plane. In *2005 iEEE computer
society conference on computer vision and pattern recognition
(cVPR’05)*, 772–779.
8\. M. A. Fischler and R. C. Bolles. 1981. Random sample consensus: A
paradigm for model fitting with applications to image analysis and
automated cartography. *Communications of the ACM*.
9\. Jiri Matas and Ondrej Chum. 2005. Randomized RANSAC with sequential
probability ratio test. In *Tenth iEEE international conference on
computer vision (iCCV’05) volume 1*, 1727–1732.
10\. D. R. Myatt, P. H. S. Torr, S. J. Nasuto, J. M. Bishop, and R.
Craddock. 2002. NAPSAC: High noise, high dimensional robust estimation.
In *In bMVC02*, 458–467.
11\. Henrik Stewénius, Christopher Engels, and David Nistér. 2006. Recent
developments on direct relative orientation.