[Pixman] [PATCH v9 06/15] pixmanfilter: Correct Simpsons integration
spitzak at gmail.com
spitzak at gmail.com
Fri Jan 22 01:42:04 PST 2016
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. Since it divided by
3 this produced about 2x the desired value (the normalization fixed this).
Also this is effectively a linear interpolation, not Simpsons integration.
With this fix the integration is accurate enough that the number of samples
could be reduced a lot. Multiples of 12 seem to work best.
v9: Changed samples from 16 to 12
v7: Merged with patch to reduce from 128 samples to 16
Signedoffby: Bill Spitzak <spitzak at gmail.com>

pixman/pixmanfilter.c  29 +++++++++++++++++++
1 file changed, 19 insertions(+), 10 deletions()
diff git a/pixman/pixmanfilter.c b/pixman/pixmanfilter.c
index 55073c4..718649a 100644
 a/pixman/pixmanfilter.c
+++ b/pixman/pixmanfilter.c
@@ 189,13 +189,19 @@ integral (pixman_kernel_t reconstruct, double x1,
}
else
{
 /* Integration via Simpson's rule */
#define N_SEGMENTS 128
#define SAMPLE(a1, a2) \
 (filters[reconstruct].func ((a1)) * filters[sample].func ((a2) / scale))

+ /* Integration via Simpson's rule
+ * See http://www.intmath.com/integration/6simpsonsrule.php
+ * 12 segments (6 cubic approximations) seems to produce best
+ * result for lanczos3.linear, which was the combination that
+ * showed the most errors. This makes sense as the lanczos3
+ * filter is 6 wide.
+ */
+#define N_SEGMENTS 12
+#define SAMPLE(a) \
+ (filters[reconstruct].func ((a)) * filters[sample].func (((a)  pos) / scale))
+
double s = 0.0;
 double h = width / (double)N_SEGMENTS;
+ double h = width / N_SEGMENTS;
int i;
s = SAMPLE (x1, x2);
@@ 204,11 +210,14 @@ 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
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