Best one I think: http://www.intmath.com/integration/6-simpsons-rule.php
On Thu, Dec 17, 2015 at 1:50 PM, Bill Spitzak <spit...@gmail.com> wrote: > This was based on looking up Simpson's integration on the web, from the > wikipedia page and another page I found. > > It cuts the samples into sets of 3, with an overlap of 1. Each set then > weighs 1,4,1 in the average, to simulate the weight of the control points > of a cubic curve. Since the overlapping samples of 1 add to 2 this results > in 1,4,2,4,2,...4,1 as the weights. As there are two points per set and > the total weight is 1+4+1=6, you divide the full sum by 6/2 = 3. > > It appears this implementation attempted to overlap them by 2, resulting > in weights of 1,5,6,...6,5,1. However this is very close to a flat average > of all the points. Also this is a total of 6 for every point so the divisor > should be 6, but it was left at 3. > > Based on my reading the new version is correct. However I have not been > able to see any visible difference in the filtering even if I reduce the > number of samples to 3. > > > On Thu, Dec 17, 2015 at 10:21 AM, Oded Gabbay <oded.gab...@gmail.com> > wrote: > >> On Thu, Dec 17, 2015 at 8:20 PM, Oded Gabbay <oded.gab...@gmail.com> >> wrote: >> > On Sat, Dec 12, 2015 at 8:06 PM, <spit...@gmail.com> wrote: >> >> From: Bill Spitzak <spit...@gmail.com> >> >> >> >> Simpsons uses cubic curve fitting, with 3 samples defining each cubic. >> This >> >> makes the weights of the samples be in a pattern of 1,4,2,4,2...4,1, >> and then >> >> dividing the result by 3. >> >> >> >> The previous code was using weights of 1,2,6,6...6,2,1 which produced >> about 2x >> >> the correct value, as it was still dividing by 3. The filter >> normalization >> >> removed this error. Also this is effectively a linear interpolation >> except for >> >> the ends. >> >> --- >> >> pixman/pixman-filter.c | 11 +++++++---- >> >> 1 file changed, 7 insertions(+), 4 deletions(-) >> >> >> >> diff --git a/pixman/pixman-filter.c b/pixman/pixman-filter.c >> >> index 15f9069..7c1da0d 100644 >> >> --- a/pixman/pixman-filter.c >> >> +++ b/pixman/pixman-filter.c >> >> @@ -204,11 +204,14 @@ integral (pixman_kernel_t reconstruct, double x1, >> >> { >> >> double a1 = x1 + h * i; >> >> double a2 = x2 + h * i; >> >> + s += 4 * SAMPLE(a1, a2); >> >> + } >> >> >> >> - s += 2 * SAMPLE (a1, a2); >> >> - >> >> - if (i >= 2 && i < N_SEGMENTS - 1) >> >> - s += 4 * SAMPLE (a1, a2); >> >> + for (i = 2; i < N_SEGMENTS; i += 2) >> >> + { >> >> + double a1 = x1 + h * i; >> >> + double a2 = x2 + h * i; >> >> + s += 2 * SAMPLE(a1, a2); >> >> } >> >> >> >> s += SAMPLE (x1 + width, x2 + width); >> >> -- >> >> 1.9.1 >> >> >> >> _______________________________________________ >> >> Pixman mailing list >> >> Pixman@lists.freedesktop.org >> >> http://lists.freedesktop.org/mailman/listinfo/pixman >> > >> > You say: >> > >> > "The filter normalization removed this error. Also this is effectively >> > a linear interpolation except for the ends." >> > >> > So if the error was removed, why is this change needed ? I can see it >> > is more accurate (similar to the Simpson equation), but it also causes >> > the code to run over the loop twice. >> > >> > Do you have some example we can see the difference ? >> > >> > >> > Oded >> >> OK, now I see that in the next patch, you reduce the samples from 128 >> to 16, so we are now running less iterations. >> I still would be happy to see an example with my own eyes where this >> makes a difference. >> >> Oded >> > >
_______________________________________________ Pixman mailing list Pixman@lists.freedesktop.org http://lists.freedesktop.org/mailman/listinfo/pixman