# [Pixman] [PATCH 05/15] pixman-filter: Correct Simpsons integration

Bill Spitzak spitzak at gmail.com
Thu Dec 17 13:59:44 PST 2015

```Best one I think:

http://www.intmath.com/integration/6-simpsons-rule.php

On Thu, Dec 17, 2015 at 1:50 PM, Bill Spitzak <spitzak at gmail.com> wrote:

> This was based on looking up Simpson's integration on the web, from the
>
> 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.gabbay at gmail.com>
> wrote:
>
>> On Thu, Dec 17, 2015 at 8:20 PM, Oded Gabbay <oded.gabbay at gmail.com>
>> wrote:
>> > On Sat, Dec 12, 2015 at 8:06 PM,  <spitzak at gmail.com> wrote:
>> >> From: Bill Spitzak <spitzak at 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
>> >> 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 at 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
>>
>
>
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