Computation of pi in CUDA

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
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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 10.0sqrt(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(RR-(startthreadIdx.x
-1.0)(startthreadIdx.x-1.0)))) int d 0
int flag 0 for (int kstartthreadIdx.x k
lt endthreadIdx.x k) int ynew
floor(0.5(sqrt(RR-kk))) if (ynew yold)
d d 1.0 else if (ynew lt
yold) d d 1.41421356 else
flag 1 yold ynew
distthreadIdx.x d statusthreadIdx.x
flag
Kernel function The function computes contour
length of a circular segment with radius R
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Some important parts of the CUDA code
float sum 0 for (int i0 iltNt i)
sum sum dist_hi if (status_hi 1)
printf("Error has occured",'\n') sum
sum4.0/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
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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 R100xsqrt(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 3.7712 for R3xsqrt(2)
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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