// correspondences is a list of float[4]s, consisting of the points x
// and y concatenated. We will compute a homography such that y = Hx
matd_t *homography_compute(zarray_t *correspondences, int flags)
{
// compute centroids of both sets of points (yields a better
// conditioned information matrix)
double x_cx = 0, x_cy = 0;
double y_cx = 0, y_cy = 0;
for (int i = 0; i < zarray_size(correspondences); i++) {
float *c;
zarray_get_volatile(correspondences, i, &c);
x_cx += c[0];
x_cy += c[1];
y_cx += c[2];
y_cy += c[3];
}
int sz = zarray_size(correspondences);
x_cx /= sz;
x_cy /= sz;
y_cx /= sz;
y_cy /= sz;
// NB We don't normalize scale; it seems implausible that it could
// possibly make any difference given the dynamic range of IEEE
// doubles.
matd_t *A = matd_create(9,9);
for (int i = 0; i < zarray_size(correspondences); i++) {
float *c;
zarray_get_volatile(correspondences, i, &c);
// (below world is "x", and image is "y")
double worldx = c[0] - x_cx;
double worldy = c[1] - x_cy;
double imagex = c[2] - y_cx;
double imagey = c[3] - y_cy;
double a03 = -worldx;
double a04 = -worldy;
double a05 = -1;
double a06 = worldx*imagey;
double a07 = worldy*imagey;
double a08 = imagey;
MATD_EL(A, 3, 3) += a03*a03;
MATD_EL(A, 3, 4) += a03*a04;
MATD_EL(A, 3, 5) += a03*a05;
MATD_EL(A, 3, 6) += a03*a06;
MATD_EL(A, 3, 7) += a03*a07;
MATD_EL(A, 3, 8) += a03*a08;
MATD_EL(A, 4, 4) += a04*a04;
MATD_EL(A, 4, 5) += a04*a05;
MATD_EL(A, 4, 6) += a04*a06;
MATD_EL(A, 4, 7) += a04*a07;
MATD_EL(A, 4, 8) += a04*a08;
MATD_EL(A, 5, 5) += a05*a05;
MATD_EL(A, 5, 6) += a05*a06;
MATD_EL(A, 5, 7) += a05*a07;
MATD_EL(A, 5, 8) += a05*a08;
MATD_EL(A, 6, 6) += a06*a06;
MATD_EL(A, 6, 7) += a06*a07;
MATD_EL(A, 6, 8) += a06*a08;
MATD_EL(A, 7, 7) += a07*a07;
MATD_EL(A, 7, 8) += a07*a08;
MATD_EL(A, 8, 8) += a08*a08;
double a10 = worldx;
double a11 = worldy;
double a12 = 1;
double a16 = -worldx*imagex;
double a17 = -worldy*imagex;
double a18 = -imagex;
MATD_EL(A, 0, 0) += a10*a10;
MATD_EL(A, 0, 1) += a10*a11;
MATD_EL(A, 0, 2) += a10*a12;
MATD_EL(A, 0, 6) += a10*a16;
MATD_EL(A, 0, 7) += a10*a17;
MATD_EL(A, 0, 8) += a10*a18;
MATD_EL(A, 1, 1) += a11*a11;
MATD_EL(A, 1, 2) += a11*a12;
MATD_EL(A, 1, 6) += a11*a16;
MATD_EL(A, 1, 7) += a11*a17;
MATD_EL(A, 1, 8) += a11*a18;
MATD_EL(A, 2, 2) += a12*a12;
MATD_EL(A, 2, 6) += a12*a16;
MATD_EL(A, 2, 7) += a12*a17;
MATD_EL(A, 2, 8) += a12*a18;
MATD_EL(A, 6, 6) += a16*a16;
MATD_EL(A, 6, 7) += a16*a17;
MATD_EL(A, 6, 8) += a16*a18;
MATD_EL(A, 7, 7) += a17*a17;
MATD_EL(A, 7, 8) += a17*a18;
MATD_EL(A, 8, 8) += a18*a18;
double a20 = -worldx*imagey;
double a21 = -worldy*imagey;
double a22 = -imagey;
double a23 = worldx*imagex;
double a24 = worldy*imagex;
double a25 = imagex;
MATD_EL(A, 0, 0) += a20*a20;
MATD_EL(A, 0, 1) += a20*a21;
MATD_EL(A, 0, 2) += a20*a22;
MATD_EL(A, 0, 3) += a20*a23;
MATD_EL(A, 0, 4) += a20*a24;
MATD_EL(A, 0, 5) += a20*a25;
MATD_EL(A, 1, 1) += a21*a21;
MATD_EL(A, 1, 2) += a21*a22;
MATD_EL(A, 1, 3) += a21*a23;
MATD_EL(A, 1, 4) += a21*a24;
MATD_EL(A, 1, 5) += a21*a25;
MATD_EL(A, 2, 2) += a22*a22;
MATD_EL(A, 2, 3) += a22*a23;
MATD_EL(A, 2, 4) += a22*a24;
MATD_EL(A, 2, 5) += a22*a25;
MATD_EL(A, 3, 3) += a23*a23;
MATD_EL(A, 3, 4) += a23*a24;
MATD_EL(A, 3, 5) += a23*a25;
MATD_EL(A, 4, 4) += a24*a24;
MATD_EL(A, 4, 5) += a24*a25;
MATD_EL(A, 5, 5) += a25*a25;
}
// make symmetric
for (int i = 0; i < 9; i++)
for (int j = i+1; j < 9; j++)
MATD_EL(A, j, i) = MATD_EL(A, i, j);
matd_t *H = matd_create(3,3);
if (flags & HOMOGRAPHY_COMPUTE_FLAG_INVERSE) {
// compute singular vector by (carefully) inverting the rank-deficient matrix.
if (1) {
matd_t *Ainv = matd_inverse(A);
double scale = 0;
for (int i = 0; i < 9; i++)
scale += sq(MATD_EL(Ainv, i, 0));
scale = sqrt(scale);
for (int i = 0; i < 3; i++)
for (int j = 0; j < 3; j++)
MATD_EL(H, i, j) = MATD_EL(Ainv, 3*i+j, 0) / scale;
matd_destroy(Ainv);
} else {
matd_t *b = matd_create_data(9, 1, (double[]) { 1, 0, 0, 0, 0, 0, 0, 0, 0 });
matd_t *Ainv = NULL;
if (0) {
matd_lu_t *lu = matd_lu(A);
Ainv = matd_lu_solve(lu, b);
matd_lu_destroy(lu);
} else {
matd_chol_t *chol = matd_chol(A);
Ainv = matd_chol_solve(chol, b);
matd_chol_destroy(chol);
}
double scale = 0;
for (int i = 0; i < 9; i++)
scale += sq(MATD_EL(Ainv, i, 0));
scale = sqrt(scale);
for (int i = 0; i < 3; i++)
for (int j = 0; j < 3; j++)
MATD_EL(H, i, j) = MATD_EL(Ainv, 3*i+j, 0) / scale;
matd_destroy(b);
matd_destroy(Ainv);
}
} else {
// correspondences is a list of float[4]s, consisting of the points x
// and y concatenated. We will compute a homography such that y = Hx
matd_t *homography_compute(zarray_t *correspondences)
{
// compute centroids of both sets of points (yields a better
// conditioned information matrix)
double x_cx = 0, x_cy = 0;
double y_cx = 0, y_cy = 0;
for (int i = 0; i < zarray_size(correspondences); i++) {
float *c;
zarray_get_volatile(correspondences, i, &c);
x_cx += c[0];
x_cy += c[1];
y_cx += c[2];
y_cy += c[3];
}
int sz = zarray_size(correspondences);
x_cx /= sz;
x_cy /= sz;
y_cx /= sz;
y_cy /= sz;
// NB We don't normalize scale; it seems implausible that it could
// possibly make any difference given the dynamic range of IEEE
// doubles.
matd_t *A = matd_create(9,9);
for (int i = 0; i < zarray_size(correspondences); i++) {
float *c;
zarray_get_volatile(correspondences, i, &c);
// (below world is "x", and image is "y")
double worldx = c[0] - x_cx;
double worldy = c[1] - x_cy;
double imagex = c[2] - y_cx;
double imagey = c[3] - y_cy;
double a03 = -worldx;
double a04 = -worldy;
double a05 = -1;
double a06 = worldx*imagey;
double a07 = worldy*imagey;
double a08 = imagey;
MATD_EL(A, 3, 3) += a03*a03;
MATD_EL(A, 3, 4) += a03*a04;
MATD_EL(A, 3, 5) += a03*a05;
MATD_EL(A, 3, 6) += a03*a06;
MATD_EL(A, 3, 7) += a03*a07;
MATD_EL(A, 3, 8) += a03*a08;
MATD_EL(A, 4, 4) += a04*a04;
MATD_EL(A, 4, 5) += a04*a05;
MATD_EL(A, 4, 6) += a04*a06;
MATD_EL(A, 4, 7) += a04*a07;
MATD_EL(A, 4, 8) += a04*a08;
MATD_EL(A, 5, 5) += a05*a05;
MATD_EL(A, 5, 6) += a05*a06;
MATD_EL(A, 5, 7) += a05*a07;
MATD_EL(A, 5, 8) += a05*a08;
MATD_EL(A, 6, 6) += a06*a06;
MATD_EL(A, 6, 7) += a06*a07;
MATD_EL(A, 6, 8) += a06*a08;
MATD_EL(A, 7, 7) += a07*a07;
MATD_EL(A, 7, 8) += a07*a08;
MATD_EL(A, 8, 8) += a08*a08;
double a10 = worldx;
double a11 = worldy;
double a12 = 1;
double a16 = -worldx*imagex;
double a17 = -worldy*imagex;
double a18 = -imagex;
MATD_EL(A, 0, 0) += a10*a10;
MATD_EL(A, 0, 1) += a10*a11;
MATD_EL(A, 0, 2) += a10*a12;
MATD_EL(A, 0, 6) += a10*a16;
MATD_EL(A, 0, 7) += a10*a17;
MATD_EL(A, 0, 8) += a10*a18;
MATD_EL(A, 1, 1) += a11*a11;
MATD_EL(A, 1, 2) += a11*a12;
MATD_EL(A, 1, 6) += a11*a16;
MATD_EL(A, 1, 7) += a11*a17;
MATD_EL(A, 1, 8) += a11*a18;
MATD_EL(A, 2, 2) += a12*a12;
MATD_EL(A, 2, 6) += a12*a16;
MATD_EL(A, 2, 7) += a12*a17;
MATD_EL(A, 2, 8) += a12*a18;
MATD_EL(A, 6, 6) += a16*a16;
MATD_EL(A, 6, 7) += a16*a17;
MATD_EL(A, 6, 8) += a16*a18;
MATD_EL(A, 7, 7) += a17*a17;
MATD_EL(A, 7, 8) += a17*a18;
MATD_EL(A, 8, 8) += a18*a18;
double a20 = -worldx*imagey;
double a21 = -worldy*imagey;
double a22 = -imagey;
double a23 = worldx*imagex;
double a24 = worldy*imagex;
double a25 = imagex;
MATD_EL(A, 0, 0) += a20*a20;
MATD_EL(A, 0, 1) += a20*a21;
MATD_EL(A, 0, 2) += a20*a22;
MATD_EL(A, 0, 3) += a20*a23;
MATD_EL(A, 0, 4) += a20*a24;
MATD_EL(A, 0, 5) += a20*a25;
MATD_EL(A, 1, 1) += a21*a21;
MATD_EL(A, 1, 2) += a21*a22;
MATD_EL(A, 1, 3) += a21*a23;
MATD_EL(A, 1, 4) += a21*a24;
MATD_EL(A, 1, 5) += a21*a25;
MATD_EL(A, 2, 2) += a22*a22;
MATD_EL(A, 2, 3) += a22*a23;
MATD_EL(A, 2, 4) += a22*a24;
MATD_EL(A, 2, 5) += a22*a25;
MATD_EL(A, 3, 3) += a23*a23;
MATD_EL(A, 3, 4) += a23*a24;
MATD_EL(A, 3, 5) += a23*a25;
MATD_EL(A, 4, 4) += a24*a24;
MATD_EL(A, 4, 5) += a24*a25;
MATD_EL(A, 5, 5) += a25*a25;
}
// make symmetric
for (int i = 0; i < 9; i++)
for (int j = i+1; j < 9; j++)
MATD_EL(A, j, i) = MATD_EL(A, i, j);
matd_svd_t svd = matd_svd(A);
matd_t *Ainv = matd_inverse(A);
double scale = 0;
for (int i = 0; i < 9; i++)
scale += sq(MATD_EL(Ainv, i, 0));
scale = sqrt(scale);
if (1) {
// compute singular vector using SVD. A bit slower, but more accurate.
matd_svd_t svd = matd_svd(A);
for (int i = 0; i < 3; i++)
for (int j = 0; j < 3; j++)
// MATD_EL(H, i, j) = MATD_EL(Ainv, 3*i+j, 0)/ scale;
MATD_EL(H, i, j) = MATD_EL(svd.U, 3*i+j, 8);
matd_destroy(svd.U);
matd_destroy(svd.S);
matd_destroy(svd.V);
} else {
// compute singular vector by (carefully) inverting the rank-deficient matrix.
matd_t *Ainv = matd_inverse(A);
double scale = 0;
for (int i = 0; i < 9; i++)
scale += sq(MATD_EL(Ainv, i, 0));
scale = sqrt(scale);
for (int i = 0; i < 3; i++)
for (int j = 0; j < 3; j++)
MATD_EL(H, i, j) = MATD_EL(Ainv, 3*i+j, 0)/ scale;
matd_destroy(Ainv);
}
matd_t *Tx = matd_identity(3);
MATD_EL(Tx,0,2) = -x_cx;
MATD_EL(Tx,1,2) = -x_cy;
matd_t *Ty = matd_identity(3);
MATD_EL(Ty,0,2) = y_cx;
MATD_EL(Ty,1,2) = y_cy;
matd_t *H2 = matd_op("M*M*M", Ty, H, Tx);
matd_destroy(A);
matd_destroy(Tx);
matd_destroy(Ty);
matd_destroy(H);
matd_destroy(svd.U);
matd_destroy(svd.S);
matd_destroy(svd.V);
return H2;
}
