//Red-black Gauss-Seidel relaxation for model problem. Updates the current value of the solution
//u[1..n][1..n], using the right-hand side function rhs[1..n][1..n].
// Red-black, Gauss-Seidel relaxation, Neumann forced, pointwise coupled
void relax(double **u, double **v, double **J[JROWS][JCOLS], int rows, int cols, double alpha) {
int i, ipass, isw, j, jsw = 1;
double hx, hy, hx2, hy2, discr, detU, detV;
hx = 1.0 / (cols-1);
hy = hx; //
hx2 = hx * hx;
hy2 = hy * hy;
for (ipass=1; ipass<= 2; ipass++, jsw = 3-jsw) { //Red and black sweeps.
isw = jsw;
for (j=1; j<=cols; j++, isw=3-isw)
for (i=isw; i<=rows; i+=2) { //Gauss-Seidel formula.
if (i>1 && i<rows) {
if (j>1 && j<cols) {
discr = lin2by2det(alpha/hx2*2.0 + alpha/hy2*2.0 + J[1][1][i][j],
J[1][2][i][j],
J[1][2][i][j],
alpha/hx2*2.0 + alpha/hy2*2.0 + J[2][2][i][j]);
if (abs((double)discr)>EPS) {
detU = lin2by2det(alpha/hx2*(u[i-1][j]+u[i+1][j]) + alpha/hy2*(u[i][j-1]+u[i][j+1]) - J[1][3][i][j],
J[1][2][i][j],
alpha/hx2*(v[i-1][j]+v[i+1][j]) + alpha/hy2*(v[i][j-1]+v[i][j+1]) - J[2][3][i][j],
alpha/hx2*2.0 + alpha/hy2*2.0 + J[2][2][i][j]);
u[i][j]= detU / discr;
detV = lin2by2det(alpha/hx2*2.0 + alpha/hy2*2.0 + J[1][1][i][j],
alpha/hx2 * (u[i-1][j]+u[i+1][j]) + alpha/hy2*(u[i][j-1]+u[i][j+1]) - J[1][3][i][j],
J[1][2][i][j],
alpha/hx2 * (v[i-1][j]+v[i+1][j]) + alpha/hy2*(v[i][j-1]+v[i][j+1]) - J[2][3][i][j]);
v[i][j] = detV / discr;
} else {
u[i][j] = (alpha/hx2*(u[i-1][j]+u[i+1][j]) + alpha/hy2*(u[i][j-1]+u[i][j+1]) - (J[1][2][i][j]*v[i][j]+J[1][3][i][j])) / (alpha/hx2*2.0 + alpha/hy2*2.0 + J[1][1][i][j]);
v[i][j] = (alpha/hx2*(v[i-1][j]+v[i+1][j]) + alpha/hy2*(v[i][j-1]+v[i][j+1]) - (J[1][2][i][j]*u[i][j]+J[2][3][i][j])) / (alpha/hx2*2.0 + alpha/hy2*2.0 + J[2][2][i][j]);
}
} else if (j==1) {
u[i][j] = u[i][j+1];
v[i][j] = v[i][j+1];
} else if (j==cols) {
u[i][j] = u[i][j-1];
v[i][j] = v[i][j-1];
} else {
printf("relax: Bug: Index error(2).\n");
return;
}
} else if (i==1) {
if(j>1 && j<cols) {
u[i][j] = u[i+1][j];
v[i][j] = v[i+1][j];
} else if (j==1) {
u[i][j] = u[i+1][j+1];
v[i][j] = v[i+1][j+1];
} else if (j==cols) {
u[i][j] = u[i+1][j-1];
v[i][j] = v[i+1][j-1];
} else {
printf("relax: Bug: Index error.\n");
return;
}
} else if (i==rows) {
if(j>1 && j<cols) {
u[i][j] = u[i-1][j];
v[i][j] = v[i-1][j];
} else if (j==1) {
u[i][j] = u[i-1][j+1];
v[i][j] = v[i-1][j+1];
} else if (j==cols) {
u[i][j] = u[i-1][j-1];
v[i][j] = v[i-1][j-1];
} else {
printf("relax: Bug: Index error(3).\n");
return;
}
}
}
}
}
/**
* relaxPointwiseCoupledGaussSeidel
*
* Pointwise coupled Gauss-Seidel relaxation iteration for CLG-OF equations.
* Each call to this function updates the current value of the solution,
* u[1..m][1..n], v[1..m][1..n], using the motion tensor
* J[JROWS][JCOLS][1..m][1..n].
* Neumann boundary conditions are used (derivatives are set to zero).
* The return value is the total error of the current iteration.
*
* Parameters:
*
* u Pointer to the optical flow horizontal component
* matrix/array.
* v Pointer to the optical flow vertical component matrix/array.
* J[JROWS][JCOLS] Pointer to the array that stores the computed (and possibly
* smoothed) derivatives for each image/optical flow pixel.
* nRows Number of rows of the optical flow arrrays.
* nCols Number of columns of the optical flow arays.
* alpha Optical flow global smoothing coefficient.
*
*/
double relaxPointwiseCoupledGaussSeidel(double **u,
double **v,
double **J[JROWS][JCOLS],
int nRows,
int nCols,
double alpha) {
int i, j;
double hx, hy, hx2, hy2, discr, detU, detV, error, ulocal, vlocal;
// h, could be less than 1.0.
hx = 1.0;
hy = hx;
hx2 = hx * hx;
hy2 = hy * hy;
error = 0.0;
for (i=1; i<nRows-1; i++) {
for (j=1; j<nCols-1; j++) {
// Point-wise Coupled Gauss-Seidel formula.
// Coupled system for two variables solved by Cramer's rule.
// This is the discriminant of that system
discr = lin2by2det(alpha/hx2*2.0 + alpha/hy2*2.0 + J[0][0][i][j],
J[0][1][i][j],
J[0][1][i][j],
alpha/hx2*2.0 + alpha/hy2*2.0 + J[1][1][i][j]);
if (abs(discr) > EPS) {
// Determinant replacing first column to solve
// the 2x2 system by using Cramer's rule.
detU = lin2by2det(alpha / hx2 * (u[i-1][j] + u[i+1][j])
+ alpha / hy2 * (u[i][j-1] + u[i][j+1])
- J[0][2][i][j],
J[0][1][i][j],
alpha / hx2 * (v[i-1][j] + v[i+1][j])
+ alpha / hy2 * (v[i][j-1] + v[i][j+1])
- J[1][2][i][j],
alpha / hx2 * 2.0 + alpha / hy2 * 2.0
+ J[1][1][i][j]);
// Determinant replacing second column to solve
// the 2x2 system by using Cramer's rule.
detV = lin2by2det(alpha / hx2 * 2.0 + alpha / hy2 * 2.0
+ J[0][0][i][j],
alpha / hx2 * (u[i-1][j] + u[i+1][j])
+ alpha / hy2 * (u[i][j-1] + u[i][j+1])
- J[0][2][i][j],
J[0][1][i][j],
alpha / hx2 * (v[i-1][j] + v[i+1][j])
+ alpha / hy2 * (v[i][j-1] + v[i][j+1])
- J[1][2][i][j]);
// Division of two discriminants (Cramer's rule).
ulocal = detU / discr;
vlocal = detV / discr;
error += (ulocal-u[i][j])*(ulocal-u[i][j]);
error += (vlocal-v[i][j])*(vlocal-v[i][j]);
u[i][j]=ulocal;
v[i][j]=vlocal;
// Normal Gauss-Seidel iteration.
} else {
// w_Factor=1.0, thus a Gauss-Seidel iteration.
error += SOR_at(u, v, J, i, j, alpha, 1.0);
} // End if.
} // End columns.
} // End rows.
error += boundaryCondition(u, v, nRows, nCols);
return sqrt(error / (nRows*nCols));;
}