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Computation of pi in CUDA

Computation of pi in CUDA. Measure circumference of the circle by counting pixels on the edge of the circle. Compute value of pi using this circumference. Motivation. Say I have a digital camera and magnification system which gives me pixel size of 1 unit on object plane

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Computation of pi in CUDA

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  1. Computation of pi in CUDA • Measure circumference of the circle by counting pixels on the edge of the circle. • Compute value of pi using this circumference

  2. Motivation Say I have a digital camera and magnification system which gives me pixel size of 1 unit on object plane The image obtained is binary i.e. ‘1’ inside the circle and ‘0’ elsewhere. What are my limitations in measurement of local radius of curvature ? My hypothesis is that if I can measure circumference or compute ‘pi’. I can measure that radius of curvature.

  3. Algorithm Pixel counting performed in region 0≤ x ≤ R/sqrt(2). In this region, Not Possible state Possible states ‘d’ is the counter which keep tracks of contour length pi = 4 x d/R Change ‘R’ and see the error in pi

  4. Implementation in CUDA The region from 0 ≤ x ≤ R/sqrt(2) is further divided into segments. The contour length of each segment is computed independently by separate threads. The sum of these contour lengths gives us 1/8th of the circumference of the circle. ‘pi’ value and error value are computed and outputted at the end of the program.

  5. Some important parts of the CUDA code dist_h = (float*)malloc(fsize); status_h = (int*)malloc(isize); start_h = (int*)malloc(isize); end_h = (int*)malloc(isize); float R = 10.0*sqrt(2.0); int Nt = 8; Variables for inputting Radius And # of threads required for computation Variables for tracking contour length, Errors, start and end ‘x’ value for each thread /* Kernel */ __global__ void distance(int *start, int *end, float *dist, int *status, float R) { int yold = floor(0.5+(sqrt(R*R-(start[threadIdx.x]-1.0)*(start[threadIdx.x]-1.0)))); int d = 0; int flag = 0; for (int k=start[threadIdx.x]; k <= end[threadIdx.x]; k++) { int ynew = floor(0.5+(sqrt(R*R-k*k))); if (ynew == yold) { d = d + 1.0; } else { if (ynew < yold) { d = d + 1.41421356; } else { flag = 1; } } yold = ynew; } dist[threadIdx.x] = d; status[threadIdx.x] = flag;} Kernel function The function computes contour length of a circular segment with radius ‘R’

  6. Some important parts of the CUDA code float sum = 0; for (int i=0; i<Nt; i++) { sum = sum + dist_h[i]; if (status_h[i] == 1) printf("Error has occured",'\n'); } sum = sum*4.0/R; float error; error = sum - 3.141592654; printf(" value of pi = %2.15f\n ",sum); printf(" error = %2.15f\n ",error); Computation of ‘pi’ and the error Printing of the values obtained Result (for R = 10xsqrt(2)) value of pi =3.11126995 error = -0.0303227

  7. Sources of error Curvature change is not picked up because of poorer pixel resolution. Causes the circumference to be underestimated Large ‘R’ compared to pixel resolution Computed pi = 2.8567 for R=100xsqrt(2) Highly curved boundaries are not captured because of poorer pixel resolution. Causes the circumference to be overestimated Small ‘R’ compared to pixel resolution Computed pi = 3.7712for R=3xsqrt(2)

  8. Reducing uncertainty in ‘R’ estimation • Camera pixels record light intensity. If we can predict the intensity distribution close to edge and if that intensity distribution spreads over 3 or more pixels, we can possibly get subpixel resolution. • If R = f(x) is known, we can use that information to reduce the uncertainty in ‘R’

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