After searching almost every topic on Catmull-Rom splines and finally implementing it successfully I'm now stuck at a point where I don't know if I made a logical mistake or if my code is simply wrong.
My problem is, that my code won't accept the alpha parameter for the centripetal catmull-rom spline interpolation. The results with different alpha values are always the same.
My main goal is to interpolate 4 points (6 when I double the first and last points) with the centripetal catmull-rom algorithm. I implemented basically the C# version which can be found here (Centripetal Catmull-Rom Spline) or also in a couple of other threads throughout SO (Catmull-rom curve with no cusps and no self-intersections, Catmull-Rom interpolation on SVG Paths).
My implementation:
First I have an 3D image with 4 different voxels set to one. I try to identify those voxels and save the coordinates. For example z=250, y=100, x=323. I save those values and cast them as double to use them as control points for my CR-algorithm.
for(int nx=0; nx<Nx; ++nx)
for(int ny=0; ny<Ny; ++ny)
for(int nz=0; nz<Nz; ++nz)
{
if(TempArray[nz*Ny*Nx+ny*Nx+nx] > 0)
{
Coordinates[counter*3+0] = nz; // +0 = z0
Coordinates[counter*3+1] = ny; // +1 = y0
Coordinates[counter*3+2] = nx; // +2 = x0
++counter;
}
}
Just to clearify. The arrays have the following layout: Coordinates[z0,y0,x0,z1,y1,x1,...,zn,yn,xn].
My implementation of the catamull-rom interpolation is as follows:
double P0z, P0y, P0x, P1z, P1y, P1x, P2z, P2y, P2x, P3z, P3y, P3x;
P0z = Coordinates[n*3+0];
P1z = Coordinates[n*3+3];
P2z = Coordinates[n*3+6];
P3z = Coordinates[n*3+9];
P0y = Coordinates[n*3+1];
P1y = Coordinates[n*3+4];
P2y = Coordinates[n*3+7];
P3y = Coordinates[n*3+10];
P0x = Coordinates[n*3+2];
P1x = Coordinates[n*3+5];
P2x = Coordinates[n*3+8];
P3x = Coordinates[n*3+11];
double disAxBx, disAyBy, disAzBz; disAzBz=disAyBy=disAxBx=0.0;
disAzBz = pow( P1z - P0z, 2 );
disAyBy = pow( P1y - P0y, 2 );
disAxBx = pow( P1x - P0x, 2 );
double distanceVec = pow( disAzBz + disAyBy + disAxBx, 0.5 );
t0 = 0.0;
t1 = pow( distanceVec, alpha) + t0; // This is the part where the alpha comes in
t2 = pow( distanceVec, alpha) + t1;
t3 = pow( distanceVec, alpha) + t2;
for ( double t=t1; t<t2; t+= (t2-t1)/(NumberOfPoints) ) // t starts at t1 and runs till t2 (like in the examples)
{
z=y=x=0;
A1[0] = ( ( (t1-t)/(t1-t0) ) * P0z ) + ( ( (t-t0)/(t1-t0) ) * P1z );
A1[1] = ( ( (t1-t)/(t1-t0) ) * P0y ) + ( ( (t-t0)/(t1-t0) ) * P1y );
A1[2] = ( ( (t1-t)/(t1-t0) ) * P0x ) + ( ( (t-t0)/(t1-t0) ) * P1x );
A2[0] = ( ( (t2-t)/(t2-t1) ) * P1z ) + ( ( (t-t1)/(t2-t1) ) * P2z );
A2[1] = ( ( (t2-t)/(t2-t1) ) * P1y ) + ( ( (t-t1)/(t2-t1) ) * P2y );
A2[2] = ( ( (t2-t)/(t2-t1) ) * P1x ) + ( ( (t-t1)/(t2-t1) ) * P2x );
A3[0] = ( ( (t3-t)/(t3-t2) ) * P2z ) + ( ( (t-t2)/(t3-t2) ) * P3z );
A3[1] = ( ( (t3-t)/(t3-t2) ) * P2y ) + ( ( (t-t2)/(t3-t2) ) * P3y );
A3[2] = ( ( (t3-t)/(t3-t2) ) * P2x ) + ( ( (t-t2)/(t3-t2) ) * P3x );
B1[0] = ( ( (t2-t)/(t2-t0) ) * A1[0] ) + ( ( (t-t0)/(t2-t0) ) * A2[0] );
B1[1] = ( ( (t2-t)/(t2-t0) ) * A1[1] ) + ( ( (t-t0)/(t2-t0) ) * A2[1] );
B1[2] = ( ( (t2-t)/(t2-t0) ) * A1[2] ) + ( ( (t-t0)/(t2-t0) ) * A2[2] );
B2[0] = ( ( (t3-t)/(t3-t1) ) * A2[0] ) + ( ( (t-t1)/(t3-t1) ) * A3[0] );
B2[1] = ( ( (t3-t)/(t3-t1) ) * A2[1] ) + ( ( (t-t1)/(t3-t1) ) * A3[1] );
B2[2] = ( ( (t3-t)/(t3-t1) ) * A2[2] ) + ( ( (t-t1)/(t3-t1) ) * A3[2] );
C[0] =int ( ( ( (t2-t)/(t2-t1) ) * B1[0] ) + ( ( (t-t1)/(t2-t1) ) * B2[0] ) +0.5 );
C[1] =int ( ( ( (t2-t)/(t2-t1) ) * B1[1] ) + ( ( (t-t1)/(t2-t1) ) * B2[1] ) +0.5 );
C[2] =int ( ( ( (t2-t)/(t2-t1) ) * B1[2] ) + ( ( (t-t1)/(t2-t1) ) * B2[2] ) +0.5 );
z=C[0]; y=C[1]; x=C[2];
OutputArray[z*Nx*Ny+y*Nx+x] = 1.f;
}
} // Loop over 4 consecutive points
Now what I try to do is, to set every voxel (or pixel in 2D) to 1, which is calculated by the CR-interpolation method. I basically want to calculate the coordinates with the CR algorithm. If I set alpha=0, the results are as expected (I used similar points to the wikipedia example). I get a nice self-intersection at the top. But if I change the value to 0.5 or 1, I still get the same results.
Now I suspect that there is something wrong with the types I use. Probably it is not wise to cast the integer coordinates as double or to cast them back to integer (with +0.5). But that would not explain the self-intersection I get. I actually did not provide an image, because it is very similar to the images we have in 1 and 2. Thanks to everyone who even considers to read this.
