*** General notes about rounding Suppose a function is sampled at positions [k + o] where k is an integer and o is a fractional offset 0 <= o < 1. To round a value to the nearest sample, breaking ties by rounding up, we can do this: round(x) = floor(x - o + 0.5) + o That is, first subtract o to let us pretend that the samples are at integer coordinates, then add 0.5 and floor to round to nearest integer, then add the offset back in. To break ties by rounding down: round(x) = ceil(x - o - 0.5) + o or if we have an epsilon value: round(x) = floor(x - o + 0.5 - e) + o To always round *up* to the next sample: round_up(x) = ceil(x - o) + o To always round *down* to the previous sample: round_down(x) = floor(x - o) + o If a set of samples is stored in an array, you get from the sample position to an index by subtracting the position of the first sample in the array: index(s) = s - first_sample *** Application to pixman In pixman, images are sampled with o = 0.5, that is, pixels are located midways between integers. We usually break ties by rounding down (i.e., "round towards north-west"). -- NEAREST filtering: The NEAREST filter simply picks the closest pixel to the given position: round(x) = floor(x - 0.5 + 0.5 - e) + 0.5 = floor (x - e) + 0.5 The first sample of a pixman image has position 0.5, so to find the index in the pixel array, we have to subtract 0.5: floor (x - e) + 0.5 - 0.5 = floor (x - e). Therefore a 16.16 fixed-point image location is turned into a pixel value with NEAREST filtering by doing this: pixels[((y - e) >> 16) * stride + ((x - e) >> 16)] where stride is the number of pixels allocated per scanline and e = 0x0001. -- CONVOLUTION filtering: A convolution matrix is considered a sampling of a function f at values surrounding 0. For example, this convolution matrix: [a, b, c, d] is interpreted as the values of a function f: a = f(-1.5) b = f(-0.5) c = f(0.5) d = f(1.5) The sample offset in this case is o = 0.5 and the first sample has position s0 = -1.5. If the matrix is: [a, b, c, d, e] the sample offset is o = 0 and the first sample has position s0 = -2.0. In general we have s0 = (- width / 2.0 + 0.5). and o = frac (s0) To evaluate f at a position between the samples, we round to the closest sample, and then we subtract the position of the first sample to get the index in the matrix: f(t) = matrix[floor(t - o + 0.5) + o - s0] Note that in this case we break ties by rounding up. If we write s0 = m + o, where m is an integer, this is equivalent to f(t) = matrix[floor(t - o + 0.5) + o - (m + o)] = matrix[floor(t - o + 0.5 - m) + o - o] = matrix[floor(t - s0 + 0.5)] The convolution filter in pixman positions f such that 0 aligns with the given position x. For a given pixel x0 in the image, the closest sample of f is then computed by taking (x - x0) and rounding that to the closest index: i = floor ((x0 - x) - s0 + 0.5) To perform the convolution, we have to find the first pixel x0 whose corresponding sample has index 0. We can write x0 = k + 0.5, where k is an integer: 0 = floor(k + 0.5 - x - s0 + 0.5) = k + floor(1 - x - s0) = k - ceil(x + s0 - 1) = k - floor(x + s0 - e) = k - floor(x - (width - 1) / 2.0 - e) And so the final formula for the index k of x0 in the image is: k = floor(x - (width - 1) / 2.0 - e) Computing the result is then simply a matter of convolving all the pixels starting at k with all the samples in the matrix. --- SEPARABLE_CONVOLUTION For this filter, x is first rounded to one of n regularly spaced subpixel positions. This subpixel position determines which of n convolution matrices is being used. Then, as in a regular convolution filter, the first pixel to be used is determined: k = floor (x - (width - 1) / 2.0 - e) and then the image pixels starting there are convolved with the chosen matrix. If we write x = xi + frac, where xi is an integer, we get k = xi + floor (frac - (width - 1) / 2.0 - e) so the location of k relative to x is given by: (k + 0.5 - x) = xi + floor (frac - (width - 1) / 2.0 - e) + 0.5 - x = floor (frac - (width - 1) / 2.0 - e) + 0.5 - frac which means the contents of the matrix corresponding to (frac) should contain width samplings of the function, with the first sample at: floor (frac - (width - 1) / 2.0 - e) + 0.5 - frac This filter is called separable because each of the k x k convolution matrices is specified with two k-wide vectors, one for each dimension, where each entry in the matrix is computed as the product of the corresponding entries in the vectors.