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. 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.
v7: Merged with patch to reduce from 128 samples to 16
v9: Changed samples from 16 to 12
v10: Fixed rebase error that made it not compile
v11: minor whitespace change
Signed-off-by: Bill Spitzak <spit...@gmail.com>
Reviewed-by: Oded Gabbay <oded.gab...@gmail.com>
---
pixman/pixman-filter.c | 23 ++++++++++++++++-------
1 file changed, 16 insertions(+), 7 deletions(-)
diff --git a/pixman/pixman-filter.c b/pixman/pixman-filter.c
index 69ac3ab..8b8fb82 100644
--- a/pixman/pixman-filter.c
+++ b/pixman/pixman-filter.c
@@ -189,13 +189,19 @@ integral (pixman_kernel_t reconstruct, double x1,
}
else
{
- /* Integration via Simpson's rule */
-#define N_SEGMENTS 128
+ /* Integration via Simpson's rule
+ * See http://www.intmath.com/integration/6-simpsons-rule.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(a1, a2) \
(filters[reconstruct].func ((a1)) * filters[sample].func ((a2) / size))
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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