Image rotation by shear. More...

#include <stdio.h>
#include <string.h>
#include <stdlib.h>
#include <math.h>
#include "allheaders.h"

Functions
PIX *	pixRotateShear (PIX *pixs, l_int32 xcen, l_int32 ycen, l_float32 angle, l_int32 incolor)
PIX *	pixRotate2Shear (PIX *pixs, l_int32 xcen, l_int32 ycen, l_float32 angle, l_int32 incolor)
PIX *	pixRotate3Shear (PIX *pixs, l_int32 xcen, l_int32 ycen, l_float32 angle, l_int32 incolor)
l_int32	pixRotateShearIP (PIX *pixs, l_int32 xcen, l_int32 ycen, l_float32 angle, l_int32 incolor)
PIX *	pixRotateShearCenter (PIX *pixs, l_float32 angle, l_int32 incolor)
l_int32	pixRotateShearCenterIP (PIX *pixs, l_float32 angle, l_int32 incolor)
Variables
static const l_float32	VERY_SMALL_ANGLE = 0.001
static const l_float32	MAX_2_SHEAR_ANGLE = 0.05

Detailed Description

Image rotation by shear.

    Shear rotation about arbitrary point using 2 and 3 shears

            PIX      *pixRotateShear()
            PIX      *pixRotate2Shear()
            PIX      *pixRotate3Shear()

    Shear rotation in-place about arbitrary point using 3 shears
            l_int32   pixRotateShearIP()

    Shear rotation around the image center
            PIX      *pixRotateShearCenter()    (2 or 3 shears)
            l_int32   pixRotateShearCenterIP()  (3 shears)

Rotation is measured in radians; clockwise rotations are positive.

Rotation by shear works on images of any depth, 
including 8 bpp color paletted images and 32 bpp
rgb images.  It works by translating each src pixel
value to the appropriate pixel in the rotated dest.
For 8 bpp grayscale images, it is about 10-15x faster
than rotation by area-mapping.

This speed and flexibility comes at the following cost,
relative to area-mapped rotation:

  -  Jaggies are created on edges of straight lines

  -  For large angles, where you must use 3 shears,
     there is some extra clipping from the shears.

For small angles, typically less than 0.05 radians,
rotation can be done with 2 orthogonal shears.
Two such continuous shears (as opposed to the discrete
shears on a pixel lattice that we have here) give
a rotated image that has a distortion in the lengths
of the two rotated and still-perpendicular axes.  The
length/width ratio changes by a fraction 

     0.5 * (angle)**2

For an angle of 0.05 radians, this is about 1 part in
a thousand.  This distortion is absent when you use
3 continuous shears with the correct angles (see below).

Of course, the image is on a discrete pixel lattice.
Rotation by shear gives an approximation to a continuous
rotation, leaving pixel jaggies at sharp boundaries.
For very small rotations, rotating from a corner gives
better sensitivity than rotating from the image center.
Here's why.  Define the shear "center" to be the line such
that the image is sheared in opposite directions on
each side of and parallel to the line.  For small
rotations there is a "dead space" on each side of the
shear center of width equal to half the shear angle,
in radians.  Thus, when the image is sheared about the center,
the dead space width equals the shear angle, but when
the image is sheared from a corner, the dead space
width is only half the shear angle.

All horizontal and vertical shears are implemented by
rasterop.  The in-place rotation uses special in-place
shears that copy rows sideways or columns vertically
without buffering, and then rewrite old pixels that are
no longer covered by sheared pixels.  For that rewriting,
you have the choice of using white or black pixels.
(Note that this may give undesirable results for colormapped
images, where the white and black values are arbitrary
indexes into the colormap, and may not even exist.)

Rotation by shear is fast and depth-independent.  However, it
does not work well for large rotation angles.  In fact, for
rotation angles greater than about 7 degrees, more pixels are
lost at the edges than when using pixRotationBySampling(), which
only loses pixels because they are rotated out of the image. 
For large rotations, use pixRotationBySampling() or, for
more accuracy when d > 1 bpp, pixRotateAM().

For small angles, when comparing the quality of rotation by
sampling and by shear, you can see that rotation by sampling
is slightly more accurate.  However, the difference in
accuracy of rotation by sampling when compared to 3-shear and
(for angles less than 2 degrees, when compared to 2-shear) is
less than 1 pixel at any point.  For very small angles, rotation by
sampling is slower than rotation by shear.  The speed difference
depends on the pixel depth and the rotation angle.  Rotation
by shear is very fast for small angles and for small depth (esp. 1 bpp).
Rotation by sampling speed is independent of angle and relatively
more efficient for 8 and 32 bpp images.  Here are some timings
for the ratio of rotation times: (time for sampling)/ (time for shear)

     depth (bpp)       ratio (2 deg)       ratio (10 deg)
     -----------------------------------------------------
        1                  25                  6
        8                   5                  2.6
        32                  1.6                1.0

Consequently, for small angles and low bit depth, use rotation by shear.
For large angles or large bit depth, use rotation by sampling.

There has been some work on what is called a "quasishear
rotation" ("The Quasi-Shear Rotation, Eric Andres,
DGCI 1996, pp. 307-314).  I believe they use a 3-shear
approximation to the continuous rotation, exactly as
we do here.  The approximation is due to being on
a square pixel lattice.  They also use integers to specify
the rotation angle and center offset, but that makes
little sense on a machine where you have a few GFLOPS
and only a few hundred floating point operations to do (!) 
They also allow subpixel specification of the center of
rotation, which I haven't bothered with, and claim that
better results are possible if each of the 4 quadrants is
handled separately.
 
But the bottom line is that for binary images, the quality
of the simple 3-shear rotation is about as good as you can do,
visually, without dithering the result.  The effect of dither
is to break up the horizontal and vertical shear lines.
It's a bit tricky to dither with block shears -- you have to
dither the pixels on the block boundaries!

Definition in file rotateshear.c.

Function Documentation

PIX* pixRotateShear	(	PIX *	pixs,
		l_int32	xcen,
		l_int32	ycen,
		l_float32	angle,
		l_int32	incolor
	)

pixRotateShear()

Input: pixs xcen (x value for which there is no horizontal shear) ycen (y value for which there is no vertical shear) angle (radians) incolor (L_BRING_IN_WHITE, L_BRING_IN_BLACK); Return: pixd, or null on error.

Notes: (1) This rotates an image about the given point, using either 2 or 3 shears. (2) A positive angle gives a clockwise rotation. (3) This brings in 'incolor' pixels from outside the image.

Definition at line 170 of file rotateshear.c.

References ERROR_PTR, L_ABS, L_BRING_IN_BLACK, L_BRING_IN_WHITE, MAX_2_SHEAR_ANGLE, NULL, pixClone(), pixRotate2Shear(), pixRotate3Shear(), PROCNAME, and VERY_SMALL_ANGLE.

Referenced by main(), and pixRotateShearCenter().

PIX* pixRotate2Shear	(	PIX *	pixs,
		l_int32	xcen,
		l_int32	ycen,
		l_float32	angle,
		l_int32	incolor
	)

pixRotate2Shear()

Input: pixs xcen, ycen (center of rotation) angle (radians) incolor (L_BRING_IN_WHITE, L_BRING_IN_BLACK); Return: pixd, or null on error.

Notes: (1) This rotates the image about the given point, using the 2-shear method. It should only be used for angles smaller than MAX_2_SHEAR_ANGLE. (2) A positive angle gives a clockwise rotation. (3) 2-shear rotation by a specified angle is equivalent to the sequential transformations x' = x + tan(angle) * (y - ycen) for x-shear y' = y + tan(angle) * (x - xcen) for y-shear (4) Computation of tan(angle) is performed within the shear operation. (5) This brings in 'incolor' pixels from outside the image.

Definition at line 216 of file rotateshear.c.

References ERROR_PTR, L_ABS, L_BRING_IN_BLACK, L_BRING_IN_WHITE, NULL, pixClone(), pixDestroy(), pixHShear(), pixVShear(), PROCNAME, and VERY_SMALL_ANGLE.

Referenced by pixRotateShear().

PIX* pixRotate3Shear	(	PIX *	pixs,
		l_int32	xcen,
		l_int32	ycen,
		l_float32	angle,
		l_int32	incolor
	)

pixRotate3Shear()

Input: pixs xcen, ycen (center of rotation) angle (radians) incolor (L_BRING_IN_WHITE, L_BRING_IN_BLACK); Return: pixd, or null on error.

Notes: (1) This rotates the image about the image center, using the 3-shear method. It can be used for any angle, and should be used for angles larger than MAX_2_SHEAR_ANGLE. (2) A positive angle gives a clockwise rotation. (3) 3-shear rotation by a specified angle is equivalent to the sequential transformations y' = y + tan(angle/2) * (x - xcen) for first y-shear x' = x + sin(angle) * (y - ycen) for x-shear y' = y + tan(angle/2) * (x - xcen) for second y-shear (4) Computation of tan(angle) is performed in the shear operations. (5) This brings in 'incolor' pixels from outside the image. (6) The algorithm was published by Alan Paeth: "A Fast Algorithm for General Raster Rotation," Graphics Interface '86, pp. 77-81, May 1986. A description of the method, along with an implementation, can be found in Graphics Gems, p. 179, edited by Andrew Glassner, published by Academic Press, 1990.

Definition at line 272 of file rotateshear.c.

References ERROR_PTR, L_ABS, L_BRING_IN_BLACK, L_BRING_IN_WHITE, NULL, pixClone(), pixDestroy(), pixHShear(), pixVShear(), PROCNAME, and VERY_SMALL_ANGLE.

Referenced by main(), and pixRotateShear().

l_int32 pixRotateShearIP	(	PIX *	pixs,
		l_int32	xcen,
		l_int32	ycen,
		l_float32	angle,
		l_int32	incolor
	)

pixRotateShearIP()

Input: pixs (any depth; not colormapped) xcen, ycen (center of rotation) angle (radians) incolor (L_BRING_IN_WHITE, L_BRING_IN_BLACK) Return: 0 if OK; 1 on error

Notes: (1) This does an in-place rotation of the image about the image center, using the 3-shear method. (2) A positive angle gives a clockwise rotation. (3) 3-shear rotation by a specified angle is equivalent to the sequential transformations y' = y + tan(angle/2) * (x - xcen) for first y-shear x' = x + sin(angle) * (y - ycen) for x-shear y' = y + tan(angle/2) * (x - xcen) for second y-shear (4) Computation of tan(angle) is performed in the shear operations. (5) This brings in 'incolor' pixels from outside the image. (6) The pix cannot be colormapped, because the in-place operation only blits in 0 or 1 bits, not an arbitrary colormap index.

Definition at line 330 of file rotateshear.c.

References ERROR_INT, L_BRING_IN_BLACK, L_BRING_IN_WHITE, NULL, pixGetColormap(), pixHShearIP(), pixVShearIP(), and PROCNAME.

Referenced by main(), and pixRotateShearCenterIP().