http://mathworld.wolfram.com/JincFunction.html and page 11 of
http://www.ph.ed.ac.uk/%7ewjh/teaching/mo/slides/lens/lens.pdf
- The original "zoom" program by Paul Heckbert called this "Bessel"
- But really its is more accuritally named "Jinc".
+ The original "zoom" program by Paul Heckbert called this "Bessel".
+ But really it is more accurately named "Jinc".
*/
if (x == 0.0)
return(0.25*MagickPIL);
{ Jinc, 2.0, 1.0, 0.0, 0.0 }, /* Lanczos2D, adjusted below */
};
/*
- The known zero crossings of the Jinc() or more accuritally the Jinc(x*PI)
- function being used as a filter. It is used by the "filter:lobes" for of
- support selection, so users do not have to deal with the highly irrational
- sizes of the 'lobes' of the Jinc filter.
+ The known zero crossings of the Jinc() or more accurately the
+ Jinc(x*PI) function being used as a filter. It is used by the
+ "filter:lobes" for support selection, so users do not have to deal
+ with the highly irrational sizes of the 'lobes' of the Jinc
+ filter.
- Values were sourced from
+ Values taken from
http://cose.math.bas.bg/webMathematica/webComputing/BesselZeros.jsp
- Using Jv-function with v=1, then divided by PI.
+ using Jv-function with v=1, then dividing by PI.
*/
static MagickRealType
jinc_zeros[16] =
break;
case Lanczos2DFilter:
/* Special 2-lobed cylindrical Jinc-Jinc filter with a special
- * "blur" adjustment (actually, a shortening of the
- * support). To be used as the default filter for EWA
- * Resampling and Distorts.
+ * "blur" adjustment (actually, a "negative blur," since the
+ * support is made slightly smaller, with the consequence that
+ * the corresponding convolution is slightly sharper).
*
- * Derivation: Set the scaling s=1/blur of the Lanczos2D
- * filter function so that
+ * To be used as default filter for EWA Resampling and
+ * Distorts.
+ *
+ * Executive summary:
+ *
+ * This is the closest a two-lobe Jinc-Jinc used radially
+ * (with Clamped EWA) comes preserving straight vertical---and
+ * straight horizontal---features when the geometrical
+ * transformation does not lead to downsampling (that is, when
+ * enlarging, or rotating, or reflecting, or translating, or
+ * performing very mild shearing or warping).
+ *
+ * Details:
+ *
+ * Basically, this is the usual
+ * Lanczos2D(x) = Jinc(x)*Jinc(x*r1/r2) with support r2
+ * (where r1 is the first root of the Jinc function, and r2 is
+ * the second), rescaled in the x-direction so that images
+ * which are constant in the vertical direction, and images
+ * which are constant in the horizontal direction, are almost
+ * unchanged (the "almost" being generically as small as
+ * possible) when the geometrical transformation applied to is
+ * is the identity (a.k.a. "no-op").
+ *
+ * Specifically, we use Lanczos2D(s*x) with support r2/s,
+ * where the scaling s of the Lanczos2D filter function is
+ * chosen so that
* Lanczos2D(s)=-2*Lanczos2D(s*sqrt(2))-Lanczos2D(s*2).
*
- * This choice ensures that the height of single vertical line
- * is exactly preserved when "no-op" (the "identity
- * geometrical transformation) is applied to the image. It
- * also ensures that, with no-op, the nearest two columns (one
- * on the left and one on the right) are not changed (they are
- * kept to zero). The only side-effect of no-op on a single
- * vertical line is a "ripple" in the two columns located two
- * pixel away. The size of this ripple, for a unit column
- * (constant 1), is -5.86142e-4 (essentially non-existant).
- * In addition, with this choice of "r-scaling," the high
- * frequency checkerboard mode, as well as the high frequency
- * vertical and horizontal stripe modes are slightly damped.
+ * This choice ensures that the height of a single vertical
+ * line (equal to 1, say, on a black=0 background) is exactly
+ * preserved when no-op is applied to the image. It also
+ * ensures that---with no-op---the nearest two columns on
+ * either side are minimally changed. Specifically, nearest
+ * columns on the left and the right are raised to c from
+ * zero, and the second closest columns on the left and right
+ * are lower by c from 0. (Hence, the closest columns are made
+ * slightly positive, and the second closest are made slightly
+ * negative, equally.) All the other columns are
+ * unchanged. The size c of this blur/ripple is .002042317
+ * (about one fifth of one percent). Consequently, this choice
+ * of scaling of the support of the standard Lanczos2D
+ * preserves "contant on columns" images with values between
+ * -1 and 1 to within .00817 (within one percent).
*
- * Derived by Nicolas Robidoux of Laurentian University.
+ * This special scaling was discovered by Nicolas Robidoux of
+ * Laurentian University.
+ *
+ * resize_filter->blur, below, is 1/s.
*/
resize_filter->blur *= (MagickRealType) 0.958033808;
default:
switch (filter_type)
{
case Lanczos2DFilter:
- /* depromote to a 2-lobe Sinc-Sinc for orthoginal use */
+ /* Depromote to a 2-lobe Sinc-Sinc for orthogonal use. */
resize_filter->filter=SincFast;
break;
default:
projects / imagemagick / commitdiff