Implementing a Fast Cartesian-Polar Matrix Interpolator - MIT

outperformed the hardware implementation. We attribute this
difference to the superior memory systems of modern general
purpose processors.
A. Improving the Algorithm
As in hardware, using fixed point values for computation
resulted in a substantial speedup. To further improve performance,
we need to reduce the number of trigonometric functions.
This is accomplished by leveraging the sum-to-product
formulas for cosine and sine to derive a fast computation for
generating the sine-cosine pair for one ray from the sine-cosine
pair of the previous ray.
cos(θ + ∆θ) = cos(θ) cos(∆θ) − sin(θ) sin(∆θ)
sin(θ + ∆θ) = sin(θ) cos(∆θ) + sin(θ) cos(∆θ)
Since we scale cosine and sine by dx and dy we need to
rescale sin(∆θ) by dy
dx to obtain the correct results. Our final
single-threaded version can be found in Figure 7.
N1=N-1; RN=R*N1;
dx=((R+1)-(R*cos(theta)))/N1;
dy=((R+1)*sin(theta))/N1;
xoffset=R*cos(theta)/dx;
sin_dt_dy_dx=sin(theta/N1)*dy/dx;
sin_dt_dx_dy=sin(theta/N1)*dx/dy;
cos_dt=cos(theta/N1);
scaledcos_t=(1.0/(R+1 - (R*cos(theta));
scaledsin_t=0.0;
for(t=0;t