Remove the 8e extra safety margin in COVER_CLIP analysis

As discussed in
http://lists.freedesktop.org/archives/pixman/2015-August/003905.html

the 8 * pixman_fixed_e (8e) adjustment which was applied to the transformed
coordinates is a legacy of rounding errors which used to occur in old
versions of Pixman, but which no longer apply. For any affine transform,
you are now guaranteed to get the same result by transforming the upper
coordinate as though you transform the lower coordinate and add (size-1)
steps of the increment in source coordinate space. No projective
transform routines use the COVER_CLIP flags, so they cannot be affected.

Proof by Siarhei Siamashka:

Let's take a look at the following affine transformation matrix (with 16.16
fixed point values) and two vectors:

         | a   b     c    |
M      = | d   e     f    |
         | 0   0  0x10000 |

         |  x_dst  |
P     =  |  y_dst  |
         | 0x10000 |

         | 0x10000 |
ONE_X  = |    0    |
         |    0    |

The current matrix multiplication code does the following calculations:

             | (a * x_dst + b * y_dst + 0x8000) / 0x10000 + c |
    M * P =  | (d * x_dst + e * y_dst + 0x8000) / 0x10000 + f |
             |                   0x10000                      |

These calculations are not perfectly exact and we may get rounding
because the integer coordinates are adjusted by 0.5 (or 0x8000 in the
16.16 fixed point format) before doing matrix multiplication. For
example, if the 'a' coefficient is an odd number and 'b' is zero,
then we are losing some of the least significant bits when dividing by
0x10000.

So we need to strictly prove that the following expression is always
true even though we have to deal with rounding:

                                          | a |
    M * (P + ONE_X) - M * P = M * ONE_X = | d |
                                          | 0 |

or

   ((a * (x_dst + 0x10000) + b * y_dst + 0x8000) / 0x10000 + c)
  -
   ((a * x_dst             + b * y_dst + 0x8000) / 0x10000 + c)
  =
    a

It's easy to see that this is equivalent to

    a + ((a * x_dst + b * y_dst + 0x8000) / 0x10000 + c)
      - ((a * x_dst + b * y_dst + 0x8000) / 0x10000 + c)
  =
    a

Which means that stepping exactly by one pixel horizontally in the
destination image space (advancing 'x_dst' by 0x10000) is the same as
changing the transformed 'x_src' coordinate in the source image space
exactly by 'a'. The same applies to the vertical direction too.
Repeating these steps, we can reach any pixel in the source image
space and get exactly the same fixed point coordinates as doing
matrix multiplications per each pixel.

By the way, the older matrix multiplication implementation, which was
relying on less accurate calculations with three intermediate roundings
"((a + 0x8000) >> 16) + ((b + 0x8000) >> 16) + ((c + 0x8000) >> 16)",
also has the same properties. However reverting
    http://cgit.freedesktop.org/pixman/commit/?id=ed39992564beefe6b12f81e842caba11aff98a9c
and applying this "Remove the 8e extra safety margin in COVER_CLIP
analysis" patch makes the cover test fail. The real reason why it fails
is that the old pixman code was using "pixman_transform_point_3d()"
function
    http://cgit.freedesktop.org/pixman/tree/pixman/pixman-matrix.c?id=pixman-0.28.2#n49
for getting the transformed coordinate of the top left corner pixel
in the image scaling code, but at the same time using a different
"pixman_transform_point()" function
    http://cgit.freedesktop.org/pixman/tree/pixman/pixman-matrix.c?id=pixman-0.28.2#n82
in the extents calculation code for setting the cover flag. And these
functions did the intermediate rounding differently. That's why the 8e
safety margin was needed.

** proof ends

However, for COVER_CLIP_NEAREST, the actual margins added were not 8e.
Because the half-way cases round down, that is, coordinate 0 hits pixel
index -1 while coordinate e hits pixel index 0, the extra safety margins
were actually 7e to the left and up, and 9e to the right and down. This
patch removes the 7e and 9e margins and restores the -e adjustment
required for NEAREST sampling in Pixman. For reference, see
pixman/rounding.txt.

For COVER_CLIP_BILINEAR, the margins were exactly 8e as there are no
additional offsets to be restored, so simply removing the 8e additions
is enough.

Proof:

All implementations must give the same numerical results as
bits_image_fetch_pixel_nearest() / bits_image_fetch_pixel_bilinear().

The former does
    int x0 = pixman_fixed_to_int (x - pixman_fixed_e);
which maps directly to the new test for the nearest flag, when you consider
that x0 must fall in the interval [0,width).

The latter does
    x1 = x - pixman_fixed_1 / 2;
    x1 = pixman_fixed_to_int (x1);
    x2 = x1 + 1;
When you write a COVER path, you take advantage of the assumption that
both x1 and x2 fall in the interval [0, width).

As samplers are allowed to fetch the pixel at x2 unconditionally, we
require
    x1 >= 0
    x2 < width
so
    x - pixman_fixed_1 / 2 >= 0
    x - pixman_fixed_1 / 2 + pixman_fixed_1 < width * pixman_fixed_1
so
    pixman_fixed_to_int (x - pixman_fixed_1 / 2) >= 0
    pixman_fixed_to_int (x + pixman_fixed_1 / 2) < width
which matches the source code lines for the bilinear case, once you delete
the lines that add the 8e margin.

Signed-off-by: Ben Avison <bavison@riscosopen.org>
[Pekka: adjusted commit message, left affine-bench changes for another patch]
[Pekka: add commit message parts from Siarhei]
Signed-off-by: Pekka Paalanen <pekka.paalanen@collabora.co.uk>
Reviewed-by: Siarhei Siamashka <siarhei.siamashka@gmail.com>
Reviewed-by: Ben Avison <bavison@riscosopen.org>

1 files changed, 4 insertions, 13 deletions

diff --git a/pixman/pixman.c b/pixman/pixman.c
index a07c577..f932eac 100644
--- a/pixman/pixman.c
+++ b/pixman/pixman.c
@@ -497,21 +497,12 @@ analyze_extent (pixman_image_t       *image,
     if (!compute_transformed_extents (transform, extents, &transformed))
 	return FALSE;
 
-    /* Expand the source area by a tiny bit so account of different rounding that
-     * may happen during sampling. Note that (8 * pixman_fixed_e) is very far from
-     * 0.5 so this won't cause the area computed to be overly pessimistic.
-     */
-    transformed.x1 -= 8 * pixman_fixed_e;
-    transformed.y1 -= 8 * pixman_fixed_e;
-    transformed.x2 += 8 * pixman_fixed_e;
-    transformed.y2 += 8 * pixman_fixed_e;
-
     if (image->common.type == BITS)
     {
-	if (pixman_fixed_to_int (transformed.x1) >= 0			&&
-	    pixman_fixed_to_int (transformed.y1) >= 0			&&
-	    pixman_fixed_to_int (transformed.x2) < image->bits.width	&&
-	    pixman_fixed_to_int (transformed.y2) < image->bits.height)
+	if (pixman_fixed_to_int (transformed.x1 - pixman_fixed_e) >= 0                &&
+	    pixman_fixed_to_int (transformed.y1 - pixman_fixed_e) >= 0                &&
+	    pixman_fixed_to_int (transformed.x2 - pixman_fixed_e) < image->bits.width &&
+	    pixman_fixed_to_int (transformed.y2 - pixman_fixed_e) < image->bits.height)
 	{
 	    *flags |= FAST_PATH_SAMPLES_COVER_CLIP_NEAREST;
 	}