/*
prosac.c
version 1.3 (Sep. 22, 2011)
Author: Frederic Devernay
Description: a sample implementation of the PROSAC sampling algorithm, derived from RANSAC.
Reference:
O. Chum and J. Matas.
Matching with PROSAC - progressive sample consensus.
Proc. of Conference on Computer Vision and Pattern Recognition (CVPR), volume 1, pages 220-226,
Los Alamitos, California, USA, June 2005.
ftp://cmp.felk.cvut.cz/pub/cmp/articles/matas/chum-prosac-cvpr05.pdf
Note:
In the above article, the test is step 2 of Algorithm 1 seem to be reversed, and the test in step 1
is not consistent with the text. They were corrected in this implementation.
History:
version 1.0 (Apr. 14, 2009): initial version
version 1.1 (Apr. 22, 2009): fix support computation, "The hypotheses are veriﬁed against all data"
version 1.2 (Mar. 16, 2011): Add comments about beta, psi, and set eta0 to its original value (0.05 rather than 0.01)
version 1.3 (Sep. 22, 2011): Check that k_n_star is never nore than T_N
version 1.4 (Sep. 24, 2011): Don't stop until we have found at least the expected number of inliers (improvement over original PROSAC).
version 1.5 (Oct. 10, 2011): Also stop if t > T_N (maximum number of iterations given the apriori proportion of outliers).
*/
#include
#include
#include
#include
#define CORRESPONDENCES 1000 // N: number of correspondences
#define SAMPLE_SIZE 7 // m: minimum sample size to compute model (7 for 7-pt F-matrix)
/* NOTE: the PROSAC article sets T_N (the number of iterations before PROSAC becomes RANSAC) to 200000,
but that means :
- only 535 correspondences out of 1000 will be used after 2808 iterations (60% outliers)
- 395 588 (50%)
- 170 163 (40%)
(the # of iterations is the # of RANSAC draws for a 0.99 confidence
of finding the right model given the percentage of outliers)
QUESTION: Is it more reasonable to set it to the maximum number of iterations we plan to
do given the percentage of outlier?
MY ANSWER: If you know that you won't draw more than XX samples (say 2808, because you only
accept 60% outliers), then it's more reasonable to set N to that value, in order to give
all correspondences a chance to be drawn (even if that chance is very small for the last ones).
Anyway, PROSAC should find many inliers in the first rounds and stop right away.
T_N=2808 gives:
- only 961 correspondences out of 1000 will be used after 2808 iterations (60% outliers)
- 595 588 (50%)
- 177 163 (40%)
*/
// Note on MAX_OUTLIERS_PROPORTION: in this implementation, PROSAC won't stop before having reached the
// corresponding inliers rate on the complete data set.
#define MAX_OUTLIERS_PROPORTION 0.8 // maximum allowed outliers proportion in the input data: used to compute T_N (can be as high as 0.95)
#define P_GOOD_SAMPLE 0.99 // probability that at least one of the random samples picked up by RANSAC is free of outliers
#define TEST_NB_OF_DRAWS 60000 // the max number of draws performed by this test
#define TEST_INLIERS_RATIO 0.5 // The ratio of inliers found by model verification (for the stopping condition)
// beta is the probability that a match is declared inlier by mistake, i.e. the ratio of the "inlier"
// surface by the total surface. The inlier surface is a disc with radius 1.96s for
// homography/displacement computation, or a band with width 1.96*s*2 for epipolar geometry (s is the
// detection noise), and the total surface is the surface of the image.
// YOU MUST ADJUST THIS VALUE, DEPENDING ON YOUR PROBLEM!
#define BETA 0.01
// eta0 is the maximum probability that a solution with more than In_star inliers in Un_star exists and was not found
// after k samples (typically set to 5%, see Sec. 2.2 of [Chum-Matas-05]).
#define ETA0 0.05
/// Computation of the Maximum number of iterations for Ransac
/// with the formula from [HZ] Section: "How many samples" p.119
static inline
int niter_RANSAC(double p, // probability that at least one of the random samples picked up by RANSAC is free of outliers
double outliersProportion,
int sampleSize)
{
double n = log(1. - p) / log(1. - exp(log(1.-outliersProportion) * sampleSize));
// C++/boost version:
// double n = boost::math::log1p(-p) / boost::math::log1p(-std::pow(1-outliersProportion, sampleSize));
return (n < INT_MAX) ? ceil(n) : INT_MAX;
}
// * Non-randomness: eq. (7) states that i-m (where i is the cardinal of the set of inliers for a wrong
// model) follows the binomial distribution B(n,beta). http://en.wikipedia.org/wiki/Binomial_distribution
// For n big enough, B(n,beta) ~ N(mu,sigma^2) (central limit theorem),
// with mu = n*beta and sigma = sqrt(n*beta*(1-beta)).
// psi, the probability that In_star out of n_star data points are by chance inliers to an arbitrary
// incorrect model, is set to 0.05 (5%, as in the original paper), and you must change the Chi2 value if
// you chose a different value for psi.
static inline
int Imin(int m, int n, double beta) {
const double mu = n*beta;
const double sigma = sqrt(n*beta*(1-beta));
// Imin(n) (equation (8) can then be obtained with the Chi-squared test with P=2*psi=0.10 (Chi2=2.706)
return (int)ceil(m + mu + sigma*sqrt(2.706));
}
/*
find the support of a given model, computed from a given sample.
the first n values of the isInlier array will be filled with 0's (outlier) and 1's (inlier).
*/
static inline
int findSupport(/* correspondences, sample, model, */ int n, int *isInlier)
{
int i, total = 0;
#if 1
/* dummy function:
- the first 1/8 of the correspondences have 60% inliers
- the second 1/8 of the correspondences have 40% inliers
- the three other quarters have 20% inliers
- that makes a total of 27.5% of inliers
*/
assert(n <= CORRESPONDENCES);
for(i=0; i T'n, not if t = T'n as written in the algorithm 1
if ((t > T_n_prime) && (n < n_star)) {
double T_nplus1 = (T_n * (n+1)) / (n+1-m);
n = n+1;
T_n_prime = T_n_prime + ceil(T_nplus1 - T_n);
printf("g(t)=n=%d, n_star=%d, T_n-1>=%d, T_n>=%d, T_n'=%d...",
n, n_star, (int)ceil(T_n), (int)ceil(T_nplus1), T_n_prime);
T_n = T_nplus1;
}
else {
printf("g(t)=n=%d, n_star=%d, T_n>=%d, T_n'=%d: ",
n, n_star, (int)ceil(T_n), T_n_prime);
}
// Draw semi-random sample (note that the test condition from Algorithm1 in the paper is reversed):
if (t > T_n_prime) {
// during the finishing stage (n== n_star && t > T_n_prime), draw a standard RANSAC sample
// The sample contains m points selected from U_n at random
printf("Draw %d points from U_%d... ", m, n);
}
else {
// The sample contains m-1 points selected from U_{n−1} at random and u_n
printf("Draw %d points from U_%d and point u_%d... ", m-1, n-1, n);
}
// INSERT Compute model parameters p_t from the sample M_t
//printf("Model parameter estimation... ");
// INSERT (OPTIONAL): Test for degenerate model configuration (DEGENSAC)
// (i.e. discard the sample if more than 1 model is consistent with the sample)
// ftp://cmp.felk.cvut.cz/pub/cmp/articles/matas/chum-degen-cvpr05.pdf
// Find support of the model with parameters p_t
// From first paragraph of section 2: "The hypotheses are veriﬁed against all data"
I_N = findSupport(/* model, sample, */ N, isInlier);
printf("found %d inliers!\n", I_N);
if (I_N > I_N_best) {
int n_best; // best value found so far in terms of inliers ratio
int I_n_best; // number of inliers for n_best
// INSERT (OPTIONAL): Do local optimization, and recompute the support (LO-RANSAC)
// http://cmp.felk.cvut.cz/~matas/papers/chum-dagm03.pdf
// for the fundamental matrix, the normalized 8-points algorithm performs very well:
// http://axiom.anu.edu.au/~hartley/Papers/fundamental/ICCV-final/fundamental.pdf
// ...
// I_N = findSupport(/* model, sample, */ N, isInlier);
I_N_best = I_N;
// INSERT: Store the best model
// Select new termination length n_star if possible, according to Sec. 2.2.
// Note: the original paper seems to do it each time a new sample is drawn,
// but this really makes sense only if the new sample is better than the previous ones.
n_best = N;
I_n_best = I_N;
if (1) { // change to if(0) to disable n_star optimization (i.e. draw the same # of samples as RANSAC)
int n_test; // test value for the termination length
int I_n_test; // number of inliers for that test value
double epsilon_n_best = (double)I_n_best/n_best;
for(n_test = N, I_n_test = I_N; n_test > m; n_test--) {
// Loop invariants:
// - I_n_test is the number of inliers for the n_test first correspondences
// - n_best is the value between n_test+1 and N that maximizes the ratio I_n_best/n_best
assert(n_test >= I_n_test);
// * Non-randomness : In >= Imin(n*) (eq. (9))
// * Maximality: the number of samples that were drawn so far must be enough
// so that the probability of having missed a set of inliers is below eta=0.01.
// This is the classical RANSAC termination criterion (HZ 4.7.1.2, eq. (4.18)),
// except that it takes into account only the n first samples (not the total number of samples).
// kn_star = log(eta0)/log(1-(In_star/n_star)^m) (eq. (12))
// We have to minimize kn_star, e.g. maximize I_n_star/n_star
//printf("n_best=%d, I_n_best=%d, n_test=%d, I_n_test=%d\n",
// n_best, I_n_best, n_test, I_n_test);
// a straightforward implementation would use the following test:
//if (I_n_test > epsilon_n_best*n_test) {
// However, since In is binomial, and in the case of evenly distributed inliers,
// a better test would be to reduce n_star only if there's a significant improvement in
// epsilon. Thus we use a Chi-squared test (P=0.10), together with the normal approximation
// to the binomial (mu = epsilon_n_star*n_test, sigma=sqrt(n_test*epsilon_n_star*(1-epsilon_n_star)).
// There is a significant difference between the two tests (e.g. with the findSupport
// functions provided above).
// We do the cheap test first, and the expensive test only if the cheap one passes.
if (( I_n_test * n_best > I_n_best * n_test ) &&
( I_n_test > epsilon_n_best*n_test + sqrt(n_test*epsilon_n_best*(1.-epsilon_n_best)*2.706) )) {
if (I_n_test < Imin(m,n_test,beta)) {
// equation 9 not satisfied: no need to test for smaller n_test values anyway
break; // jump out of the for(n_test) loop
}
n_best = n_test;
I_n_best = I_n_test;
epsilon_n_best = (double)I_n_best/n_best;
}
// prepare for next loop iteration
I_n_test -= isInlier[n_test-1];
} // for(n_test ...
} // n_star optimization
// is the best one we found even better than n_star?
if ( I_n_best * n_star > I_n_star * n_best ) {
double logarg;
assert(n_best >= I_n_best);
// update all values
n_star = n_best;
I_n_star = I_n_best;
// note: if you port this to ISO C++, you should use std::pow(double,int)
logarg = -exp(log(I_n_star/(double)n_star) * m); // -std::pow(I_n_star/(double)n_star, m);
if (logarg <= -1.) {
k_n_star = 1;
} else {
// use lop1p(lorgarg) if it is available
double logval = log(1. + logarg); // boost::math::log1p(logarg);
if (logval < 0. && (logeta0 / logval) < T_N) {
k_n_star = (int)ceil(logeta0 / logval);
} else {
k_n_star = T_N;
}
}
printf("new values: n_star=%d, k_n_star=%d, I_n_star=%d, I_min=%d\n", n_star, k_n_star, I_n_star, Imin(m,n_best,beta));
}
} // if (I_N > I_N_best)
} // while(t <= k_n_star ...
printf("PROSAC finished, reason:\n");
if (t > TEST_NB_OF_DRAWS)
printf("t=%d > max_t=%d (k_n_star=%d, T_N=%d)\n", t, TEST_NB_OF_DRAWS, k_n_star, T_N);
else if (t > T_N)
printf("t=%d > T_N=%d (k_n_star=%d)\n", t, T_N, k_n_star);
else if (t > k_n_star)
printf("t=%d > k_n_star=%d (T_N=%d)\n", t, k_n_star, T_N);
return 0;
}