/* Use Newton's method to find the root of a polynomial. The
* polynomial is represented by n coefficients. The coefficients are
* in order from low power to high power */
double poly_find_root(poly_t *p, double x0, double thres) {
/* Evaluate the polynomial at the current x */
int trials = 0;
double x_curr = x0;
double p_x;
// double roots[10];
poly_t *p_deriv = poly_deriv(p);
p_x = poly_eval(p, x_curr);
#define MAX_TRIALS 1024
/* Iterate until convergence */
while (fabs(p_x) > thres && trials < MAX_TRIALS) {
double dp_dx = poly_eval(p_deriv, x_curr);
x_curr = x_curr - p_x / dp_dx;
p_x = poly_eval(p, x_curr);
trials++;
}
#if 0
if (trials >= MAX_TRIALS) {
printf("[poly_find_root] Maximum number of trials exceeded.\n");
}
#endif
poly_free(p_deriv);
// find_roots(p->n, p->coeffs, roots);
return x_curr;
}
static quirc_decode_error_t correct_format(uint16_t *f_ret)
{
uint16_t u = *f_ret;
int i;
uint8_t s[MAX_POLY];
uint8_t sigma[MAX_POLY];
/* Evaluate U (received codeword) at each of alpha_1 .. alpha_6
* to get S_1 .. S_6 (but we index them from 0).
*/
if (!format_syndromes(u, s))
return QUIRC_SUCCESS;
berlekamp_massey(s, FORMAT_SYNDROMES, &gf16, sigma);
/* Now, find the roots of the polynomial */
for (i = 0; i < 15; i++)
if (!poly_eval(sigma, gf16_exp[15 - i], &gf16))
u ^= (1 << i);
if (format_syndromes(u, s))
return QUIRC_ERROR_FORMAT_ECC;
*f_ret = u;
return QUIRC_SUCCESS;
}
int main(int argc, char* argv[])
{
double coef[5];
assert(argc == 6);
int i;
for (i=1; i<=5; ++i)
coef[i-1] = atof(argv[i]);
double invA = 1/coef[4];
double roots[4];
int nr = real_quartic_roots(invA * coef[3],
invA * coef[2],
invA * coef[1],
invA * coef[0],
roots);
for (i=0; i<nr; ++i) {
double x = roots[i];
double y = poly_eval(coef, 4, x);
printf("p(%.16g) = %.16g\n", x, y);
}
return 0;
}
MAT key_genmat(gf_t *L, poly_t g)
{
//L- Support
//t- Number of errors, i.e.=30.
//n- Length of the Goppa code, i.e.=2^11
//m- The extension degree of the GF, i.e. =11
//g- The generator polynomial.
gf_t x,y;
MAT H,R;
int i,j,k,r,n;
int * perm, Laux[LENGTH];
n=LENGTH;//2^11=2048
r=NB_ERRORS*gf_extd();//32 x 11=352
H=mat_ini(r,n);//initialize matrix with actual no. of bits.
mat_set_to_zero(H); //set the matrix with all 0's.
for(i=0;i< n;i++)
{
x = poly_eval(g,L[i]);//evaluate the polynomial at the point L[i].
x = gf_inv(x);
y = x;
for(j=0;j<NB_ERRORS;j++)
{
for(k=0;k<gf_extd();k++)
{
if(y & (1<<k))//if((y>>k) & 1)
mat_set_coeff_to_one(H,j*gf_extd()+k,i);//the co-eff. are set in 2^0,...,2^11 ; 2^0,...,2^11 format along the rows/cols?
}
y = gf_mul(y,L[i]);
}
}//The H matrix is fed.
perm = mat_rref(H);
if (perm == NULL) {
mat_free(H);
return NULL;
}
R = mat_ini(n-r,r);
mat_set_to_zero(R); //set the matrix with all 0's.
for (i = 0; i < R->rown; ++i)
for (j = 0; j < R->coln; ++j)
if (mat_coeff(H,j,perm[i]))
mat_change_coeff(R,i,j);
for (i = 0; i < LENGTH; ++i)
Laux[i] = L[perm[i]];
for (i = 0; i < LENGTH; ++i)
L[i] = Laux[i];
mat_free(H);
free(perm);
return (R);
}
static quirc_decode_error_t correct_block(uint8_t *data, const struct quirc_rs_params *ecc)
{
int npar = ecc->ce;
uint8_t s[MAX_POLY];
uint8_t sigma[MAX_POLY];
uint8_t sigma_deriv[MAX_POLY];
uint8_t omega[MAX_POLY];
int i;
/* Compute syndrome vector */
if (!block_syndromes(data, ecc->bs, npar, s))
return QUIRC_SUCCESS;
berlekamp_massey(s, npar, &gf256, sigma);
/* Compute derivative of sigma */
memset(sigma_deriv, 0, MAX_POLY);
for (i = 0; i + 1 < MAX_POLY; i += 2)
sigma_deriv[i] = sigma[i + 1];
/* Compute error evaluator polynomial */
poly_mult(omega, sigma, s, &gf256);
memset(omega + npar, 0, MAX_POLY - npar);
/* Find error locations and magnitudes */
for (i = 0; i < ecc->bs; i++) {
uint8_t xinv = gf256_exp[255 - i];
if (!poly_eval(sigma, xinv, &gf256)) {
uint8_t sd_x = poly_eval(sigma_deriv, xinv, &gf256);
uint8_t omega_x = poly_eval(omega, xinv, &gf256);
uint8_t error = gf256_exp[(255 - gf256_log[sd_x] +
gf256_log[omega_x]) % 255];
data[ecc->bs - i - 1] ^= error;
}
}
if (block_syndromes(data, ecc->bs, npar, s))
return QUIRC_ERROR_DATA_ECC;
return QUIRC_SUCCESS;
}
double
poly_maximum_point(poly *p, double left, double right) {
double max;
double max_point;
double temp;
assert(p);
assert(left < right);
max = -INFINITY;
max_point = left;
/* Brute force check. */
for (; left < right; left += 0.0001) {
if ((temp = poly_eval(p, left)) > max) {
max = temp;
max_point = left;
}
}
/* Check the right-bound */
return (poly_eval(p, right) > max) ? right : max_point;
}
/*-----------------------------------------------------------------------*/
static int
find_roots_core(int deg, double *poly, double d0, double d1, double *dr)
{
double p0, p1, db, de, pm;
volatile double dm; /* important for some compilers */
p0 = poly_eval(deg, poly, d0);
p1 = poly_eval(deg, poly, d1);
if (p1 == 0.) {
*dr = d1;
return 1;
}
if (p0 == 0.)
return 0;
if (p0 * p1 > 0.)
return 1;
db = d0;
de = d1;
while (de > db) {
dm = (db + de) / 2.;
if ((dm == db) || (dm == de))
break;
pm = poly_eval(deg, poly, dm);
if (pm == 0.)
break;
if (p0 * pm > 0.) {
db = dm;
p0 = pm;
}
else {
de = dm;
p1 = pm;
}
}
*dr = dm;
return 1;
}
/* Functions */
int main(void) {
/* Variables */
int i;
int j;
double intervalStart, intervalEnd;
double x[M];
double y[M];
double coefficients[N] = {4.5, -3, 6, -6.5, 5, 1.5, 2};
printf("The polynomial has the follow coefficients: \n");
for (i=0; i < N; i++)
printf("%f ", coefficients[i]);
printf(" \n");
/* Estimated interval */
intervalStart = -4.0;
intervalEnd = 4.0;
/* Calculate a vector */
linspace(x, intervalStart, intervalEnd, M);
poly_eval(coefficients, x, y);
printf("\nPlotting Points \n");
for(j=0; j < M; j++)
printf("%+.18f %+.18f \n", x[j], y[j]);
return 0;
}
int root_laguerre(double tol, int maxit, const polyr *p, double x0, double *res) {
polyr pd, pdd;
int deg = poly_deg(p);
double x = x0;
double y = poly_eval(p, x);
int n = 0;
poly_diff(p, &pd);
poly_diff(&pd, &pdd);
#ifdef DEBUG
fprintf(stderr, "tol: %g maxit: %d x0: %g\n", tol, maxit, x0);
poly_print(stderr, p);
poly_print(stderr, &pd);
poly_print(stderr, &pdd);
#endif
while (fabs(y) > tol && n < maxit) {
double g = poly_eval(&pd, x) / poly_eval(p, x);
double h = g*g - poly_eval(&pdd, x) / poly_eval(p, x);
double sqrtarg = (deg-1)*(deg*h - g*g);
double sq = sqrt(sqrtarg);
double denom = (g > 0 ? g+sq : g-sq);
x = x - deg/denom;
y = poly_eval(p, x);
n++;
#ifdef DEBUG
fprintf(stderr, "g: %g h: %g sqa: %g sq: %g x: %g y: %g\n",
g, h, sqrtarg, sq, x, y);
#endif
/* complex roots => EDOM, divide by zero => NAN etc */
if (sqrtarg < 0 || isinf(x) || isnan(x)
|| errno == EDOM || errno == ERANGE)
return 0;
}
if (fabs(y) > tol)
return 0;
*res = x;
return 1;
}
KNDdeTorusCollocation::KNDdeTorusCollocation(KNExprSystem& sys_, size_t ndeg1_, size_t ndeg2_, size_t nint1_, size_t nint2_) :
sys(&sys_),
ndim(sys_.ndim()), ntau(sys_.ntau()), npar(sys_.npar()),
ndeg1(ndeg1_), ndeg2(ndeg2_), nint1(nint1_), nint2(nint2_),
col1(ndeg1_), col2(ndeg2_),
mesh1(ndeg1_ + 1), mesh2(ndeg2_ + 1),
lgr1(ndeg1_ + 1, ndeg1_ + 1), lgr2(ndeg2_ + 1, ndeg2_ + 1),
dlg1(ndeg1_ + 1, ndeg1_ + 1), dlg2(ndeg2_ + 1, ndeg2_ + 1),
I1((ndeg1_ + 1), (ndeg1_ + 1)),
ID1((ndeg1_ + 1), (ndeg1_ + 1)),
I2((ndeg2_ + 1), (ndeg2_ + 1)),
ID2((ndeg2_ + 1), (ndeg2_ + 1)),
mlg1((ndeg1_ + 1)*(ndeg1_ + 1)),
mlg2((ndeg2_ + 1)*(ndeg2_ + 1)),
mlgd1((ndeg1_ + 1)*(ndeg1_ + 1)),
mlgd2((ndeg2_ + 1)*(ndeg2_ + 1)),
ilg1((ndeg1_ + 1)*(ndeg1_ + 1) + 1),
ilg2((ndeg2_ + 1)*(ndeg2_ + 1) + 1),
ilgd1((ndeg1_ + 1)*(ndeg1_ + 1) + 1),
ilgd2((ndeg2_ + 1)*(ndeg2_ + 1) + 1),
time1(ndeg1*ndeg2*nint1*nint2), time2(ndeg1*ndeg2*nint1*nint2),
kk((ntau+1)*(ndeg1+1)*(ndeg2+1), ndeg1*ndeg2*nint1*nint2),
ee((ntau+1)*(ndeg1+1)*(ndeg2+1), ndeg1*ndeg2*nint1*nint2),
rr((ntau+1)*(ndeg1+1)*(ndeg2+1), ndeg1*ndeg2*nint1*nint2),
p_tau(ntau, ndeg1*ndeg2*nint1*nint2), p_dtau(ntau, ndeg1*ndeg2*nint1*nint2),
p_xx(ndim, ntau+2*(ntau+1), ndeg1*ndeg2*nint1*nint2),
p_fx(ndim, ndeg1*ndeg2*nint1*nint2),
p_dfp(ndim, 1, ndeg1*ndeg2*nint1*nint2),
p_dfx(ndim, ndim, ndeg1*ndeg2*nint1*nint2),
p_dummy(0, 0, ndeg1*ndeg2*nint1*nint2)
{
lobatto(mesh1);
lobatto(mesh2);
gauss(col1);
gauss(col2);
for (size_t i = 0; i < mesh1.size(); i++)
{
poly_coeff_lgr(lgr1(i), mesh1, i);
poly_coeff_diff(dlg1(i), lgr1(i));
}
for (size_t i = 0; i < mesh2.size(); i++)
{
poly_coeff_lgr(lgr2(i), mesh2, i);
poly_coeff_diff(dlg2(i), lgr2(i));
}
// here comes the phase condition for par(OMEGA0) and par(OMEGA1)
// the integration in the bottom border
// construct the diffint matrix
for (size_t i = 0; i < ndeg1 + 1; i++)
{
for (size_t k = 0; k < ndeg1 + 1; k++)
{
mlg1.clear();
mlgd1.clear();
ilg1.clear();
ilgd1.clear();
poly_coeff_mul(mlg1, lgr1(i), lgr1(k));
poly_coeff_mul(mlgd1, dlg1(i), lgr1(k));
poly_coeff_int(ilg1, mlg1);
poly_coeff_int(ilgd1, mlgd1);
I1(i, k) = (poly_eval(ilg1, 1.0)-poly_eval(ilg1, 0.0)) / nint1;
ID1(i, k) = poly_eval(ilgd1, 1.0);
}
}
for (size_t j = 0; j < ndeg2 + 1; j++)
{
for (size_t l = 0; l < ndeg2 + 1; l++)
{
mlg2.clear();
mlgd2.clear();
ilg2.clear();
ilgd2.clear();
poly_coeff_mul(mlg2, lgr2(j), lgr2(l));
poly_coeff_mul(mlgd2, dlg2(j), lgr2(l));
poly_coeff_int(ilg2, mlg2);
poly_coeff_int(ilgd2, mlgd2);
I2(j, l) = (poly_eval(ilg2, 1.0) - poly_eval(ilg2, 0.0))/ nint2;
ID2(j, l) = poly_eval(ilgd2, 1.0);
}
}
}
void pose_estimation_from_line_correspondence(Eigen::MatrixXf start_points,
Eigen::MatrixXf end_points,
Eigen::MatrixXf directions,
Eigen::MatrixXf points,
Eigen::MatrixXf &rot_cw,
Eigen::VectorXf &pos_cw)
{
int n = start_points.cols();
if(n != directions.cols())
{
return;
}
if(n<4)
{
return;
}
float condition_err_threshold = 1e-3;
Eigen::VectorXf cosAngleThreshold(3);
cosAngleThreshold << 1.1, 0.9659, 0.8660;
Eigen::MatrixXf optimumrot_cw(3,3);
Eigen::VectorXf optimumpos_cw(3);
std::vector<float> lineLenVec(n,1);
vfloat3 l1;
vfloat3 l2;
vfloat3 nc1;
vfloat3 Vw1;
vfloat3 Xm;
vfloat3 Ym;
vfloat3 Zm;
Eigen::MatrixXf Rot(3,3);
std::vector<vfloat3> nc_bar(n,vfloat3(0,0,0));
std::vector<vfloat3> Vw_bar(n,vfloat3(0,0,0));
std::vector<vfloat3> n_c(n,vfloat3(0,0,0));
Eigen::MatrixXf Rx(3,3);
int line_id;
for(int HowToChooseFixedTwoLines = 1 ; HowToChooseFixedTwoLines <=3 ; HowToChooseFixedTwoLines++)
{
if(HowToChooseFixedTwoLines==1)
{
#pragma omp parallel for
for(int i = 0; i < n ; i++ )
{
// to correct
float lineLen = 10;
lineLenVec[i] = lineLen;
}
std::vector<float>::iterator result;
result = std::max_element(lineLenVec.begin(), lineLenVec.end());
line_id = std::distance(lineLenVec.begin(), result);
vfloat3 temp;
temp = start_points.col(0);
start_points.col(0) = start_points.col(line_id);
start_points.col(line_id) = temp;
temp = end_points.col(0);
end_points.col(0) = end_points.col(line_id);
end_points.col(line_id) = temp;
temp = directions.col(line_id);
directions.col(0) = directions.col(line_id);
directions.col(line_id) = temp;
temp = points.col(0);
points.col(0) = points.col(line_id);
points.col(line_id) = temp;
lineLenVec[line_id] = lineLenVec[1];
lineLenVec[1] = 0;
l1 = start_points.col(0) - end_points.col(0);
l1 = l1/l1.norm();
}
for(int i = 1; i < n; i++)
{
std::vector<float>::iterator result;
result = std::max_element(lineLenVec.begin(), lineLenVec.end());
line_id = std::distance(lineLenVec.begin(), result);
l2 = start_points.col(line_id) - end_points.col(line_id);
l2 = l2/l2.norm();
lineLenVec[line_id] = 0;
MatrixXf cosAngle(3,3);
cosAngle = (l1.transpose()*l2).cwiseAbs();
if(cosAngle.maxCoeff() < cosAngleThreshold[HowToChooseFixedTwoLines])
{
break;
}
}
vfloat3 temp;
temp = start_points.col(1);
start_points.col(1) = start_points.col(line_id);
start_points.col(line_id) = temp;
temp = end_points.col(1);
end_points.col(1) = end_points.col(line_id);
end_points.col(line_id) = temp;
temp = directions.col(1);
directions.col(1) = directions.col(line_id);
directions.col(line_id) = temp;
temp = points.col(1);
points.col(1) = points.col(line_id);
points.col(line_id) = temp;
lineLenVec[line_id] = lineLenVec[1];
lineLenVec[1] = 0;
// The rotation matrix R_wc is decomposed in way which is slightly different from the description in the paper,
// but the framework is the same.
// R_wc = (Rot') * R * Rot = (Rot') * (Ry(theta) * Rz(phi) * Rx(psi)) * Rot
nc1 = x_cross(start_points.col(1),end_points.col(1));
nc1 = nc1/nc1.norm();
Vw1 = directions.col(1);
Vw1 = Vw1/Vw1.norm();
//the X axis of Model frame
Xm = x_cross(nc1,Vw1);
Xm = Xm/Xm.norm();
//the Y axis of Model frame
Ym = nc1;
//the Z axis of Model frame
Zm = x_cross(Xm,Zm);
Zm = Zm/Zm.norm();
//Rot * [Xm, Ym, Zm] = I.
Rot.col(0) = Xm;
Rot.col(1) = Ym;
Rot.col(2) = Zm;
Rot = Rot.transpose();
//rotate all the vector by Rot.
//nc_bar(:,i) = Rot * nc(:,i)
//Vw_bar(:,i) = Rot * Vw(:,i)
#pragma omp parallel for
for(int i = 0 ; i < n ; i++)
{
vfloat3 nc = x_cross(start_points.col(1),end_points.col(1));
nc = nc/nc.norm();
n_c[i] = nc;
nc_bar[i] = Rot * nc;
Vw_bar[i] = Rot * directions.col(i);
}
//Determine the angle psi, it is the angle between z axis and Vw_bar(:,1).
//The rotation matrix Rx(psi) rotates Vw_bar(:,1) to z axis
float cospsi = (Vw_bar[1])[2];//the angle between z axis and Vw_bar(:,1); cospsi=[0,0,1] * Vw_bar(:,1);.
float sinpsi= sqrt(1 - cospsi*cospsi);
Rx.row(0) = vfloat3(1,0,0);
Rx.row(1) = vfloat3(0,cospsi,-sinpsi);
Rx.row(2) = vfloat3(0,sinpsi,cospsi);
vfloat3 Zaxis = Rx * Vw_bar[1];
if(1-fabs(Zaxis[3]) > 1e-5)
{
Rx = Rx.transpose();
}
//estimate the rotation angle phi by least square residual.
//i.e the rotation matrix Rz(phi)
vfloat3 Vm2 = Rx * Vw_bar[1];
float A2 = Vm2[0];
float B2 = Vm2[1];
float C2 = Vm2[2];
float x2 = (nc_bar[1])[0];
float y2 = (nc_bar[1])[1];
float z2 = (nc_bar[1])[2];
Eigen::PolynomialSolver<double, Eigen::Dynamic> solver;
Eigen::VectorXf coeff(9);
std::vector<float> coef(9,0); //coefficients of equation (7)
Eigen::VectorXf polyDF = VectorXf::Zero(16); //%dF = ployDF(1) * t^15 + ployDF(2) * t^14 + ... + ployDF(15) * t + ployDF(16);
//construct the polynomial F'
#pragma omp parallel for
for(int i = 3 ; i < n ; i++)
{
vfloat3 Vm3 = Rx*Vw_bar[i];
float A3 = Vm3[0];
float B3 = Vm3[1];
float C3 = Vm3[2];
float x3 = (nc_bar[i])[0];
float y3 = (nc_bar[i])[1];
float z3 = (nc_bar[i])[2];
float u11 = -z2*A2*y3*B3 + y2*B2*z3*A3;
float u12 = -y2*A2*z3*B3 + z2*B2*y3*A3;
float u13 = -y2*B2*z3*B3 + z2*B2*y3*B3 + y2*A2*z3*A3 - z2*A2*y3*A3;
float u14 = -y2*B2*x3*C3 + x2*C2*y3*B3;
float u15 = x2*C2*y3*A3 - y2*A2*x3*C3;
float u21 = -x2*A2*y3*B3 + y2*B2*x3*A3;
float u22 = -y2*A2*x3*B3 + x2*B2*y3*A3;
float u23 = x2*B2*y3*B3 - y2*B2*x3*B3 - x2*A2*y3*A3 + y2*A2*x3*A3;
float u24 = y2*B2*z3*C3 - z2*C2*y3*B3;
float u25 = y2*A2*z3*C3 - z2*C2*y3*A3;
float u31 = -x2*A2*z3*A3 + z2*A2*x3*A3;
float u32 = -x2*B2*z3*B3 + z2*B2*x3*B3;
float u33 = x2*A2*z3*B3 - z2*A2*x3*B3 + x2*B2*z3*A3 - z2*B2*x3*A3;
float u34 = z2*A2*z3*C3 + x2*A2*x3*C3 - z2*C2*z3*A3 - x2*C2*x3*A3;
float u35 = -z2*B2*z3*C3 - x2*B2*x3*C3 + z2*C2*z3*B3 + x2*C2*x3*B3;
float u36 = -x2*C2*z3*C3 + z2*C2*x3*C3;
float a4 = u11*u11 + u12*u12 - u13*u13 - 2*u11*u12 + u21*u21 + u22*u22 - u23*u23
-2*u21*u22 - u31*u31 - u32*u32 + u33*u33 + 2*u31*u32;
float a3 =2*(u11*u14 - u13*u15 - u12*u14 + u21*u24 - u23*u25
- u22*u24 - u31*u34 + u33*u35 + u32*u34);
float a2 =-2*u12*u12 + u13*u13 + u14*u14 - u15*u15 + 2*u11*u12 - 2*u22*u22 + u23*u23
+ u24*u24 - u25*u25 +2*u21*u22+ 2*u32*u32 - u33*u33
- u34*u34 + u35*u35 -2*u31*u32- 2*u31*u36 + 2*u32*u36;
float a1 =2*(u12*u14 + u13*u15 + u22*u24 + u23*u25 - u32*u34 - u33*u35 - u34*u36);
float a0 = u12*u12 + u15*u15+ u22*u22 + u25*u25 - u32*u32 - u35*u35 - u36*u36 - 2*u32*u36;
float b3 =2*(u11*u13 - u12*u13 + u21*u23 - u22*u23 - u31*u33 + u32*u33);
float b2 =2*(u11*u15 - u12*u15 + u13*u14 + u21*u25 - u22*u25 + u23*u24 - u31*u35 + u32*u35 - u33*u34);
float b1 =2*(u12*u13 + u14*u15 + u22*u23 + u24*u25 - u32*u33 - u34*u35 - u33*u36);
float b0 =2*(u12*u15 + u22*u25 - u32*u35 - u35*u36);
float d0 = a0*a0 - b0*b0;
float d1 = 2*(a0*a1 - b0*b1);
float d2 = a1*a1 + 2*a0*a2 + b0*b0 - b1*b1 - 2*b0*b2;
float d3 = 2*(a0*a3 + a1*a2 + b0*b1 - b1*b2 - b0*b3);
float d4 = a2*a2 + 2*a0*a4 + 2*a1*a3 + b1*b1 + 2*b0*b2 - b2*b2 - 2*b1*b3;
float d5 = 2*(a1*a4 + a2*a3 + b1*b2 + b0*b3 - b2*b3);
float d6 = a3*a3 + 2*a2*a4 + b2*b2 - b3*b3 + 2*b1*b3;
float d7 = 2*(a3*a4 + b2*b3);
float d8 = a4*a4 + b3*b3;
std::vector<float> v { a4, a3, a2, a1, a0, b3, b2, b1, b0 };
Eigen::VectorXf vp;
vp << a4, a3, a2, a1, a0, b3, b2, b1, b0 ;
//coef = coef + v;
coeff = coeff + vp;
polyDF[0] = polyDF[0] + 8*d8*d8;
polyDF[1] = polyDF[1] + 15* d7*d8;
polyDF[2] = polyDF[2] + 14* d6*d8 + 7*d7*d7;
polyDF[3] = polyDF[3] + 13*(d5*d8 + d6*d7);
polyDF[4] = polyDF[4] + 12*(d4*d8 + d5*d7) + 6*d6*d6;
polyDF[5] = polyDF[5] + 11*(d3*d8 + d4*d7 + d5*d6);
polyDF[6] = polyDF[6] + 10*(d2*d8 + d3*d7 + d4*d6) + 5*d5*d5;
polyDF[7] = polyDF[7] + 9 *(d1*d8 + d2*d7 + d3*d6 + d4*d5);
polyDF[8] = polyDF[8] + 8 *(d1*d7 + d2*d6 + d3*d5) + 4*d4*d4 + 8*d0*d8;
polyDF[9] = polyDF[9] + 7 *(d1*d6 + d2*d5 + d3*d4) + 7*d0*d7;
polyDF[10] = polyDF[10] + 6 *(d1*d5 + d2*d4) + 3*d3*d3 + 6*d0*d6;
polyDF[11] = polyDF[11] + 5 *(d1*d4 + d2*d3)+ 5*d0*d5;
polyDF[12] = polyDF[12] + 4 * d1*d3 + 2*d2*d2 + 4*d0*d4;
polyDF[13] = polyDF[13] + 3 * d1*d2 + 3*d0*d3;
polyDF[14] = polyDF[14] + d1*d1 + 2*d0*d2;
polyDF[15] = polyDF[15] + d0*d1;
}
Eigen::VectorXd coefficientPoly = VectorXd::Zero(16);
for(int j =0; j < 16 ; j++)
{
coefficientPoly[j] = polyDF[15-j];
}
//solve polyDF
solver.compute(coefficientPoly);
const Eigen::PolynomialSolver<double, Eigen::Dynamic>::RootsType & r = solver.roots();
Eigen::VectorXd rs(r.rows());
Eigen::VectorXd is(r.rows());
std::vector<float> min_roots;
for(int j =0;j<r.rows();j++)
{
rs[j] = fabs(r[j].real());
is[j] = fabs(r[j].imag());
}
float maxreal = rs.maxCoeff();
for(int j = 0 ; j < rs.rows() ; j++ )
{
if(is[j]/maxreal <= 0.001)
{
min_roots.push_back(rs[j]);
}
}
std::vector<float> temp_v(15);
std::vector<float> poly(15);
for(int j = 0 ; j < 15 ; j++)
{
temp_v[j] = j+1;
}
for(int j = 0 ; j < 15 ; j++)
{
poly[j] = polyDF[j]*temp_v[j];
}
Eigen::Matrix<double,16,1> polynomial;
Eigen::VectorXd evaluation(min_roots.size());
for( int j = 0; j < min_roots.size(); j++ )
{
evaluation[j] = poly_eval( polynomial, min_roots[j] );
}
std::vector<float> minRoots;
for( int j = 0; j < min_roots.size(); j++ )
{
if(!evaluation[j]<=0)
{
minRoots.push_back(min_roots[j]);
}
}
if(minRoots.size()==0)
{
cout << "No solution" << endl;
return;
}
int numOfRoots = minRoots.size();
//for each minimum, we try to find a solution of the camera pose, then
//choose the one with the least reprojection residual as the optimum of the solution.
float minimalReprojectionError = 100;
// In general, there are two solutions which yields small re-projection error
// or condition error:"n_c * R_wc * V_w=0". One of the solution transforms the
// world scene behind the camera center, the other solution transforms the world
// scene in front of camera center. While only the latter one is correct.
// This can easily be checked by verifying their Z coordinates in the camera frame.
// P_c(Z) must be larger than 0 if it's in front of the camera.
for(int rootId = 0 ; rootId < numOfRoots ; rootId++)
{
float cosphi = minRoots[rootId];
float sign1 = sign_of_number(coeff[0] * pow(cosphi,4)
+ coeff[1] * pow(cosphi,3) + coeff[2] * pow(cosphi,2)
+ coeff[3] * cosphi + coeff[4]);
float sign2 = sign_of_number(coeff[5] * pow(cosphi,3)
+ coeff[6] * pow(cosphi,2) + coeff[7] * cosphi + coeff[8]);
float sinphi= -sign1*sign2*sqrt(abs(1-cosphi*cosphi));
Eigen::MatrixXf Rz(3,3);
Rz.row(0) = vfloat3(cosphi,-sinphi,0);
Rz.row(1) = vfloat3(sinphi,cosphi,0);
Rz.row(2) = vfloat3(0,0,1);
//now, according to Sec4.3, we estimate the rotation angle theta
//and the translation vector at a time.
Eigen::MatrixXf RzRxRot(3,3);
RzRxRot = Rz*Rx*Rot;
//According to the fact that n_i^C should be orthogonal to Pi^c and Vi^c, we
//have: scalarproduct(Vi^c, ni^c) = 0 and scalarproduct(Pi^c, ni^c) = 0.
//where Vi^c = Rwc * Vi^w, Pi^c = Rwc *(Pi^w - pos_cw) = Rwc * Pi^w - pos;
//Using the above two constraints to construct linear equation system Mat about
//[costheta, sintheta, tx, ty, tz, 1].
Eigen::MatrixXf Matrice(2*n-1,6);
#pragma omp parallel for
for(int i = 0 ; i < n ; i++)
{
float nxi = (nc_bar[i])[0];
float nyi = (nc_bar[i])[1];
float nzi = (nc_bar[i])[2];
Eigen::VectorXf Vm(3);
Vm = RzRxRot * directions.col(i);
float Vxi = Vm[0];
float Vyi = Vm[1];
float Vzi = Vm[2];
Eigen::VectorXf Pm(3);
Pm = RzRxRot * points.col(i);
float Pxi = Pm(1);
float Pyi = Pm(2);
float Pzi = Pm(3);
//apply the constraint scalarproduct(Vi^c, ni^c) = 0
//if i=1, then scalarproduct(Vi^c, ni^c) always be 0
if(i>1)
{
Matrice(2*i-3, 0) = nxi * Vxi + nzi * Vzi;
Matrice(2*i-3, 1) = nxi * Vzi - nzi * Vxi;
Matrice(2*i-3, 5) = nyi * Vyi;
}
//apply the constraint scalarproduct(Pi^c, ni^c) = 0
Matrice(2*i-2, 0) = nxi * Pxi + nzi * Pzi;
Matrice(2*i-2, 1) = nxi * Pzi - nzi * Pxi;
Matrice(2*i-2, 2) = -nxi;
Matrice(2*i-2, 3) = -nyi;
Matrice(2*i-2, 4) = -nzi;
Matrice(2*i-2, 5) = nyi * Pyi;
}
//solve the linear system Mat * [costheta, sintheta, tx, ty, tz, 1]' = 0 using SVD,
JacobiSVD<MatrixXf> svd(Matrice, ComputeThinU | ComputeThinV);
Eigen::MatrixXf VMat = svd.matrixV();
Eigen::VectorXf vec(2*n-1);
//the last column of Vmat;
vec = VMat.col(5);
//the condition that the last element of vec should be 1.
vec = vec/vec[5];
//the condition costheta^2+sintheta^2 = 1;
float normalizeTheta = 1/sqrt(vec[0]*vec[1]+vec[1]*vec[1]);
float costheta = vec[0]*normalizeTheta;
float sintheta = vec[1]*normalizeTheta;
Eigen::MatrixXf Ry(3,3);
Ry << costheta, 0, sintheta , 0, 1, 0 , -sintheta, 0, costheta;
//now, we get the rotation matrix rot_wc and translation pos_wc
Eigen::MatrixXf rot_wc(3,3);
rot_wc = (Rot.transpose()) * (Ry * Rz * Rx) * Rot;
Eigen::VectorXf pos_wc(3);
pos_wc = - Rot.transpose() * vec.segment(2,4);
//now normalize the camera pose by 3D alignment. We first translate the points
//on line in the world frame Pw to points in the camera frame Pc. Then we project
//Pc onto the line interpretation plane as Pc_new. So we could call the point
//alignment algorithm to normalize the camera by aligning Pc_new and Pw.
//In order to improve the accuracy of the aligment step, we choose two points for each
//lines. The first point is Pwi, the second point is the closest point on line i to camera center.
//(Pw2i = Pwi - (Pwi'*Vwi)*Vwi.)
Eigen::MatrixXf Pw2(3,n);
Pw2.setZero();
Eigen::MatrixXf Pc_new(3,n);
Pc_new.setZero();
Eigen::MatrixXf Pc2_new(3,n);
Pc2_new.setZero();
for(int i = 0 ; i < n ; i++)
{
vfloat3 nci = n_c[i];
vfloat3 Pwi = points.col(i);
vfloat3 Vwi = directions.col(i);
//first point on line i
vfloat3 Pci;
Pci = rot_wc * Pwi + pos_wc;
Pc_new.col(i) = Pci - (Pci.transpose() * nci) * nci;
//second point is the closest point on line i to camera center.
vfloat3 Pw2i;
Pw2i = Pwi - (Pwi.transpose() * Vwi) * Vwi;
Pw2.col(i) = Pw2i;
vfloat3 Pc2i;
Pc2i = rot_wc * Pw2i + pos_wc;
Pc2_new.col(i) = Pc2i - ( Pc2i.transpose() * nci ) * nci;
}
MatrixXf XXc(Pc_new.rows(), Pc_new.cols()+Pc2_new.cols());
XXc << Pc_new, Pc2_new;
MatrixXf XXw(points.rows(), points.cols()+Pw2.cols());
XXw << points, Pw2;
int nm = points.cols()+Pw2.cols();
cal_campose(XXc,XXw,nm,rot_wc,pos_wc);
pos_cw = -rot_wc.transpose() * pos_wc;
//check the condition n_c^T * rot_wc * V_w = 0;
float conditionErr = 0;
for(int i =0 ; i < n ; i++)
{
float val = ( (n_c[i]).transpose() * rot_wc * directions.col(i) );
conditionErr = conditionErr + val*val;
}
if(conditionErr/n < condition_err_threshold || HowToChooseFixedTwoLines ==3)
{
//check whether the world scene is in front of the camera.
int numLineInFrontofCamera = 0;
if(HowToChooseFixedTwoLines<3)
{
for(int i = 0 ; i < n ; i++)
{
vfloat3 P_c = rot_wc * (points.col(i) - pos_cw);
if(P_c[2]>0)
{
numLineInFrontofCamera++;
}
}
}
else
{
numLineInFrontofCamera = n;
}
if(numLineInFrontofCamera > 0.5*n)
{
//most of the lines are in front of camera, then check the reprojection error.
int reprojectionError = 0;
for(int i =0; i < n ; i++)
{
//line projection function
vfloat3 nc = rot_wc * x_cross(points.col(i) - pos_cw , directions.col(i));
float h1 = nc.transpose() * start_points.col(i);
float h2 = nc.transpose() * end_points.col(i);
float lineLen = (start_points.col(i) - end_points.col(i)).norm()/3;
reprojectionError += lineLen * (h1*h1 + h1*h2 + h2*h2) / (nc[0]*nc[0]+nc[1]*nc[1]);
}
if(reprojectionError < minimalReprojectionError)
{
optimumrot_cw = rot_wc.transpose();
optimumpos_cw = pos_cw;
minimalReprojectionError = reprojectionError;
}
}
}
}
if(optimumrot_cw.rows()>0)
{
break;
}
}
pos_cw = optimumpos_cw;
rot_cw = optimumrot_cw;
}
double
poly_maxima(poly *p, double left, double right) {
assert(p);
assert(left < right);
return poly_eval(p, poly_maximum_point(p, left, right));
}
static double
cornish_fisher (double t, double n)
{
const double coeffs6[10] = {
0.265974025974025974026,
5.449696969696969696970,
122.20294372294372294372,
2354.7298701298701298701,
37625.00902597402597403,
486996.1392857142857143,
4960870.65,
37978595.55,
201505390.875,
622437908.625
};
const double coeffs5[8] = {
0.2742857142857142857142,
4.499047619047619047619,
78.45142857142857142857,
1118.710714285714285714,
12387.6,
101024.55,
559494.0,
1764959.625
};
const double coeffs4[6] = {
0.3047619047619047619048,
3.752380952380952380952,
46.67142857142857142857,
427.5,
2587.5,
8518.5
};
const double coeffs3[4] = {
0.4,
3.3,
24.0,
85.5
};
double a = n - 0.5;
double b = 48.0 * a * a;
double z2 = a * log1p (t * t / n);
double z = sqrt (z2);
double p5 = z * poly_eval (coeffs6, 9, z2);
double p4 = -z * poly_eval (coeffs5, 7, z2);
double p3 = z * poly_eval (coeffs4, 5, z2);
double p2 = -z * poly_eval (coeffs3, 3, z2);
double p1 = z * (z2 + 3.0);
double p0 = z;
double y = p5;
y = (y / b) + p4;
y = (y / b) + p3;
y = (y / b) + p2;
y = (y / b) + p1;
y = (y / b) + p0;
if (t < 0)
y *= -1;
return y;
}
