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/*M///////////////////////////////////////////////////////////////////////////////////////
//
// This is a homography decomposition implementation contributed to OpenCV
// by Samson Yilma. It implements the homography decomposition algorithm
// descriped in the research report:
// Malis, E and Vargas, M, "Deeper understanding of the homography decomposition
// for vision-based control", Research Report 6303, INRIA (2007)
//
// IMPORTANT: READ BEFORE DOWNLOADING, COPYING, INSTALLING OR USING.
//
// By downloading, copying, installing or using the software you agree to this license.
// If you do not agree to this license, do not download, install,
// copy or use the software.
//
//
// License Agreement
// For Open Source Computer Vision Library
//
// Copyright (C) 2014, Samson Yilma¸ (samson_yilma@yahoo.com), all rights reserved.
//
// Third party copyrights are property of their respective owners.
//
// Redistribution and use in source and binary forms, with or without modification,
// are permitted provided that the following conditions are met:
//
// * Redistribution's of source code must retain the above copyright notice,
// this list of conditions and the following disclaimer.
//
// * Redistribution's in binary form must reproduce the above copyright notice,
// this list of conditions and the following disclaimer in the documentation
// and/or other materials provided with the distribution.
//
// * The name of the copyright holders may not be used to endorse or promote products
// derived from this software without specific prior written permission.
//
// This software is provided by the copyright holders and contributors "as is" and
// any express or implied warranties, including, but not limited to, the implied
// warranties of merchantability and fitness for a particular purpose are disclaimed.
// In no event shall the Intel Corporation or contributors be liable for any direct,
// indirect, incidental, special, exemplary, or consequential damages
// (including, but not limited to, procurement of substitute goods or services;
// loss of use, data, or profits; or business interruption) however caused
// and on any theory of liability, whether in contract, strict liability,
// or tort (including negligence or otherwise) arising in any way out of
// the use of this software, even if advised of the possibility of such damage.
//
//M*/
#include "precomp.hpp"
#include <memory>
namespace cv
{
namespace HomographyDecomposition
{
//struct to hold solutions of homography decomposition
typedef struct _CameraMotion {
cv::Matx33d R; //!< rotation matrix
cv::Vec3d n; //!< normal of the plane the camera is looking at
cv::Vec3d t; //!< translation vector
} CameraMotion;
inline int signd(const double x)
{
return ( x >= 0 ? 1 : -1 );
}
class HomographyDecomp {
public:
HomographyDecomp() {}
virtual ~HomographyDecomp() {}
virtual void decomposeHomography(const cv::Matx33d& H, const cv::Matx33d& K,
std::vector<CameraMotion>& camMotions);
bool isRotationValid(const cv::Matx33d& R, const double epsilon=0.01);
protected:
bool passesSameSideOfPlaneConstraint(CameraMotion& motion);
virtual void decompose(std::vector<CameraMotion>& camMotions) = 0;
const cv::Matx33d& getHnorm() const {
return _Hnorm;
}
private:
cv::Matx33d normalize(const cv::Matx33d& H, const cv::Matx33d& K);
void removeScale();
cv::Matx33d _Hnorm;
};
class HomographyDecompZhang : public HomographyDecomp {
public:
HomographyDecompZhang():HomographyDecomp() {}
virtual ~HomographyDecompZhang() {}
private:
virtual void decompose(std::vector<CameraMotion>& camMotions);
bool findMotionFrom_tstar_n(const cv::Vec3d& tstar, const cv::Vec3d& n, CameraMotion& motion);
};
class HomographyDecompInria : public HomographyDecomp {
public:
HomographyDecompInria():HomographyDecomp() {}
virtual ~HomographyDecompInria() {}
private:
virtual void decompose(std::vector<CameraMotion>& camMotions);
double oppositeOfMinor(const cv::Matx33d& M, const int row, const int col);
void findRmatFrom_tstar_n(const cv::Vec3d& tstar, const cv::Vec3d& n, const double v, cv::Matx33d& R);
};
// normalizes homography with intrinsic camera parameters
Matx33d HomographyDecomp::normalize(const Matx33d& H, const Matx33d& K)
{
return K.inv() * H * K;
}
void HomographyDecomp::removeScale()
{
Mat W;
SVD::compute(_Hnorm, W);
_Hnorm = _Hnorm * (1.0/W.at<double>(1));
}
/*! This checks that the input is a pure rotation matrix 'm'.
* The conditions for this are: R' * R = I and det(R) = 1 (proper rotation matrix)
*/
bool HomographyDecomp::isRotationValid(const Matx33d& R, const double epsilon)
{
Matx33d RtR = R.t() * R;
Matx33d I(1,0,0, 0,1,0, 0,0,1);
if (norm(RtR, I, NORM_INF) > epsilon)
return false;
return (fabs(determinant(R) - 1.0) < epsilon);
}
bool HomographyDecomp::passesSameSideOfPlaneConstraint(CameraMotion& motion)
{
typedef Matx<double, 1, 1> Matx11d;
Matx31d t = Matx31d(motion.t);
Matx31d n = Matx31d(motion.n);
Matx11d proj = n.t() * motion.R.t() * t;
if ( (1 + proj(0, 0) ) <= 0 )
return false;
return true;
}
//!main routine to decompose homography
void HomographyDecomp::decomposeHomography(const Matx33d& H, const cv::Matx33d& K,
std::vector<CameraMotion>& camMotions)
{
//normalize homography matrix with intrinsic camera matrix
_Hnorm = normalize(H, K);
//remove scale of the normalized homography
removeScale();
//apply decomposition
decompose(camMotions);
}
/* function computes R&t from tstar, and plane normal(n) using
R = H * inv(I + tstar*transpose(n) );
t = R * tstar;
returns true if computed R&t is a valid solution
*/
bool HomographyDecompZhang::findMotionFrom_tstar_n(const cv::Vec3d& tstar, const cv::Vec3d& n, CameraMotion& motion)
{
Matx31d tstar_m = Mat(tstar);
Matx31d n_m = Mat(n);
Matx33d temp = tstar_m * n_m.t();
temp(0, 0) += 1.0;
temp(1, 1) += 1.0;
temp(2, 2) += 1.0;
motion.R = getHnorm() * temp.inv();
motion.t = motion.R * tstar;
motion.n = n;
return passesSameSideOfPlaneConstraint(motion);
}
void HomographyDecompZhang::decompose(std::vector<CameraMotion>& camMotions)
{
Mat W, U, Vt;
SVD::compute(getHnorm(), W, U, Vt);
double lambda1=W.at<double>(0);
double lambda3=W.at<double>(2);
double lambda1m3 = (lambda1-lambda3);
double lambda1m3_2 = lambda1m3*lambda1m3;
double lambda1t3 = lambda1*lambda3;
double t1 = 1.0/(2.0*lambda1t3);
double t2 = sqrt(1.0+4.0*lambda1t3/lambda1m3_2);
double t12 = t1*t2;
double e1 = -t1 + t12; //t1*(-1.0f + t2 );
double e3 = -t1 - t12; //t1*(-1.0f - t2);
double e1_2 = e1*e1;
double e3_2 = e3*e3;
double nv1p = sqrt(e1_2*lambda1m3_2 + 2*e1*(lambda1t3-1) + 1.0);
double nv3p = sqrt(e3_2*lambda1m3_2 + 2*e3*(lambda1t3-1) + 1.0);
double v1p[3], v3p[3];
v1p[0]=Vt.at<double>(0)*nv1p, v1p[1]=Vt.at<double>(1)*nv1p, v1p[2]=Vt.at<double>(2)*nv1p;
v3p[0]=Vt.at<double>(6)*nv3p, v3p[1]=Vt.at<double>(7)*nv3p, v3p[2]=Vt.at<double>(8)*nv3p;
/*The eight solutions are
(A): tstar = +- (v1p - v3p)/(e1 -e3), n = +- (e1*v3p - e3*v1p)/(e1-e3)
(B): tstar = +- (v1p + v3p)/(e1 -e3), n = +- (e1*v3p + e3*v1p)/(e1-e3)
*/
double v1pmv3p[3], v1ppv3p[3];
double e1v3me3v1[3], e1v3pe3v1[3];
double inv_e1me3 = 1.0/(e1-e3);
for(int kk=0;kk<3;++kk){
v1pmv3p[kk] = v1p[kk]-v3p[kk];
v1ppv3p[kk] = v1p[kk]+v3p[kk];
}
for(int kk=0; kk<3; ++kk){
double e1v3 = e1*v3p[kk];
double e3v1=e3*v1p[kk];
e1v3me3v1[kk] = e1v3-e3v1;
e1v3pe3v1[kk] = e1v3+e3v1;
}
Vec3d tstar_p, tstar_n;
Vec3d n_p, n_n;
///Solution group A
for(int kk=0; kk<3; ++kk) {
tstar_p[kk] = v1pmv3p[kk]*inv_e1me3;
tstar_n[kk] = -tstar_p[kk];
n_p[kk] = e1v3me3v1[kk]*inv_e1me3;
n_n[kk] = -n_p[kk];
}
CameraMotion cmotion;
//(A) Four different combinations for solution A
// (i) (+, +)
if (findMotionFrom_tstar_n(tstar_p, n_p, cmotion))
camMotions.push_back(cmotion);
// (ii) (+, -)
if (findMotionFrom_tstar_n(tstar_p, n_n, cmotion))
camMotions.push_back(cmotion);
// (iii) (-, +)
if (findMotionFrom_tstar_n(tstar_n, n_p, cmotion))
camMotions.push_back(cmotion);
// (iv) (-, -)
if (findMotionFrom_tstar_n(tstar_n, n_n, cmotion))
camMotions.push_back(cmotion);
//////////////////////////////////////////////////////////////////
///Solution group B
for(int kk=0;kk<3;++kk){
tstar_p[kk] = v1ppv3p[kk]*inv_e1me3;
tstar_n[kk] = -tstar_p[kk];
n_p[kk] = e1v3pe3v1[kk]*inv_e1me3;
n_n[kk] = -n_p[kk];
}
//(B) Four different combinations for solution B
// (i) (+, +)
if (findMotionFrom_tstar_n(tstar_p, n_p, cmotion))
camMotions.push_back(cmotion);
// (ii) (+, -)
if (findMotionFrom_tstar_n(tstar_p, n_n, cmotion))
camMotions.push_back(cmotion);
// (iii) (-, +)
if (findMotionFrom_tstar_n(tstar_n, n_p, cmotion))
camMotions.push_back(cmotion);
// (iv) (-, -)
if (findMotionFrom_tstar_n(tstar_n, n_n, cmotion))
camMotions.push_back(cmotion);
}
double HomographyDecompInria::oppositeOfMinor(const Matx33d& M, const int row, const int col)
{
int x1 = col == 0 ? 1 : 0;
int x2 = col == 2 ? 1 : 2;
int y1 = row == 0 ? 1 : 0;
int y2 = row == 2 ? 1 : 2;
return (M(y1, x2) * M(y2, x1) - M(y1, x1) * M(y2, x2));
}
//computes R = H( I - (2/v)*te_star*ne_t )
void HomographyDecompInria::findRmatFrom_tstar_n(const cv::Vec3d& tstar, const cv::Vec3d& n, const double v, cv::Matx33d& R)
{
Matx31d tstar_m = Matx31d(tstar);
Matx31d n_m = Matx31d(n);
Matx33d I(1.0, 0.0, 0.0,
0.0, 1.0, 0.0,
0.0, 0.0, 1.0);
R = getHnorm() * (I - (2/v) * tstar_m * n_m.t() );
}
void HomographyDecompInria::decompose(std::vector<CameraMotion>& camMotions)
{
const double epsilon = 0.001;
Matx33d S;
//S = H'H - I
S = getHnorm().t() * getHnorm();
S(0, 0) -= 1.0;
S(1, 1) -= 1.0;
S(2, 2) -= 1.0;
//check if H is rotation matrix
if( norm(S, NORM_INF) < epsilon) {
CameraMotion motion;
motion.R = Matx33d(getHnorm());
motion.t = Vec3d(0, 0, 0);
motion.n = Vec3d(0, 0, 0);
camMotions.push_back(motion);
return;
}
//! Compute nvectors
Vec3d npa, npb;
double M00 = oppositeOfMinor(S, 0, 0);
double M11 = oppositeOfMinor(S, 1, 1);
double M22 = oppositeOfMinor(S, 2, 2);
double rtM00 = sqrt(M00);
double rtM11 = sqrt(M11);
double rtM22 = sqrt(M22);
double M01 = oppositeOfMinor(S, 0, 1);
double M12 = oppositeOfMinor(S, 1, 2);
double M02 = oppositeOfMinor(S, 0, 2);
int e12 = signd(M12);
int e02 = signd(M02);
int e01 = signd(M01);
double nS00 = abs(S(0, 0));
double nS11 = abs(S(1, 1));
double nS22 = abs(S(2, 2));
//find max( |Sii| ), i=0, 1, 2
int indx = 0;
if(nS00 < nS11){
indx = 1;
if( nS11 < nS22 )
indx = 2;
}
else {
if(nS00 < nS22 )
indx = 2;
}
switch (indx) {
case 0:
npa[0] = S(0, 0), npb[0] = S(0, 0);
npa[1] = S(0, 1) + rtM22, npb[1] = S(0, 1) - rtM22;
npa[2] = S(0, 2) + e12 * rtM11, npb[2] = S(0, 2) - e12 * rtM11;
break;
case 1:
npa[0] = S(0, 1) + rtM22, npb[0] = S(0, 1) - rtM22;
npa[1] = S(1, 1), npb[1] = S(1, 1);
npa[2] = S(1, 2) - e02 * rtM00, npb[2] = S(1, 2) + e02 * rtM00;
break;
case 2:
npa[0] = S(0, 2) + e01 * rtM11, npb[0] = S(0, 2) - e01 * rtM11;
npa[1] = S(1, 2) + rtM00, npb[1] = S(1, 2) - rtM00;
npa[2] = S(2, 2), npb[2] = S(2, 2);
break;
default:
break;
}
double traceS = S(0, 0) + S(1, 1) + S(2, 2);
double v = 2.0 * sqrt(1 + traceS - M00 - M11 - M22);
double ESii = signd(S(indx, indx)) ;
double r_2 = 2 + traceS + v;
double nt_2 = 2 + traceS - v;
double r = sqrt(r_2);
double n_t = sqrt(nt_2);
Vec3d na = npa / norm(npa);
Vec3d nb = npb / norm(npb);
double half_nt = 0.5 * n_t;
double esii_t_r = ESii * r;
Vec3d ta_star = half_nt * (esii_t_r * nb - n_t * na);
Vec3d tb_star = half_nt * (esii_t_r * na - n_t * nb);
camMotions.resize(4);
Matx33d Ra, Rb;
Vec3d ta, tb;
//Ra, ta, na
findRmatFrom_tstar_n(ta_star, na, v, Ra);
ta = Ra * ta_star;
camMotions[0].R = Ra;
camMotions[0].t = ta;
camMotions[0].n = na;
//Ra, -ta, -na
camMotions[1].R = Ra;
camMotions[1].t = -ta;
camMotions[1].n = -na;
//Rb, tb, nb
findRmatFrom_tstar_n(tb_star, nb, v, Rb);
tb = Rb * tb_star;
camMotions[2].R = Rb;
camMotions[2].t = tb;
camMotions[2].n = nb;
//Rb, -tb, -nb
camMotions[3].R = Rb;
camMotions[3].t = -tb;
camMotions[3].n = -nb;
}
} //namespace HomographyDecomposition
// function decomposes image-to-image homography to rotation and translation matrices
int decomposeHomographyMat(InputArray _H,
InputArray _K,
OutputArrayOfArrays _rotations,
OutputArrayOfArrays _translations,
OutputArrayOfArrays _normals)
{
using namespace std;
using namespace HomographyDecomposition;
Mat H = _H.getMat().reshape(1, 3);
CV_Assert(H.cols == 3 && H.rows == 3);
Mat K = _K.getMat().reshape(1, 3);
CV_Assert(K.cols == 3 && K.rows == 3);
auto_ptr<HomographyDecomp> hdecomp(new HomographyDecompInria);
vector<CameraMotion> motions;
hdecomp->decomposeHomography(H, K, motions);
int nsols = static_cast<int>(motions.size());
int depth = CV_64F; //double precision matrices used in CameraMotion struct
if (_rotations.needed()) {
_rotations.create(nsols, 1, depth);
for (int k = 0; k < nsols; ++k ) {
_rotations.getMatRef(k) = Mat(motions[k].R);
}
}
if (_translations.needed()) {
_translations.create(nsols, 1, depth);
for (int k = 0; k < nsols; ++k ) {
_translations.getMatRef(k) = Mat(motions[k].t);
}
}
if (_normals.needed()) {
_normals.create(nsols, 1, depth);
for (int k = 0; k < nsols; ++k ) {
_normals.getMatRef(k) = Mat(motions[k].n);
}
}
return nsols;
}
} //namespace cv