Zakariyya Mughal

# NAME

Image::Leptonica::Func::affine

version 0.04

# `affine.c`

``````  affine.c

Affine (3 pt) image transformation using a sampled
(to nearest integer) transform on each dest point
PIX        *pixAffineSampledPta()
PIX        *pixAffineSampled()

Affine (3 pt) image transformation using interpolation
(or area mapping) for anti-aliasing images that are
2, 4, or 8 bpp gray, or colormapped, or 32 bpp RGB
PIX        *pixAffinePta()
PIX        *pixAffine()
PIX        *pixAffinePtaColor()
PIX        *pixAffineColor()
PIX        *pixAffinePtaGray()
PIX        *pixAffineGray()

Affine transform including alpha (blend) component
PIX        *pixAffinePtaWithAlpha()

Affine coordinate transformation
l_int32     getAffineXformCoeffs()
l_int32     affineInvertXform()
l_int32     affineXformSampledPt()
l_int32     affineXformPt()

Interpolation helper functions
l_int32     linearInterpolatePixelGray()
l_int32     linearInterpolatePixelColor()

Gauss-jordan linear equation solver
l_int32     gaussjordan()

Affine image transformation using a sequence of
shear/scale/translation operations
PIX        *pixAffineSequential()

One can define a coordinate space by the location of the origin,
the orientation of x and y axes, and the unit scaling along
each axis.  An affine transform is a general linear
transformation from one coordinate space to another.

For the general case, we can define the affine transform using
two sets of three (noncollinear) points in a plane.  One set
corresponds to the input (src) coordinate space; the other to the
transformed (dest) coordinate space.  Each point in the
src corresponds to one of the points in the dest.  With two
sets of three points, we get a set of 6 equations in 6 unknowns
that specifies the mapping between the coordinate spaces.
The interface here allows you to specify either the corresponding
sets of 3 points, or the transform itself (as a vector of 6
coefficients).

Given the transform as a vector of 6 coefficients, we can compute
both a a pointwise affine coordinate transformation and an
affine image transformation.

To compute the coordinate transform, we need the coordinate
value (x',y') in the transformed space for any point (x,y)
in the original space.  To derive this transform from the
three corresponding points, it is convenient to express the affine
coordinate transformation using an LU decomposition of
a set of six linear equations that express the six coordinates
of the three points in the transformed space as a function of
the six coordinates in the original space.  Once we have
this transform matrix , we can transform an image by
finding, for each destination pixel, the pixel (or pixels)
in the source that give rise to it.

This 'pointwise' transformation can be done either by sampling
and picking a single pixel in the src to replicate into the dest,
or by interpolating (or averaging) over four src pixels to
determine the value of the dest pixel.  The first method is
implemented by pixAffineSampled() and the second method by
pixAffine().  The interpolated method can only be used for
images with more than 1 bpp, but for these, the image quality
is significantly better than the sampled method, due to
the 'antialiasing' effect of weighting the src pixels.

Interpolation works well when there is relatively little scaling,
or if there is image expansion in general.  However, if there
is significant image reduction, one should apply a low-pass
filter before subsampling to avoid aliasing the high frequencies.

A typical application might be to align two images, which
may be scaled, rotated and translated versions of each other.
Through some pre-processing, three corresponding points are
located in each of the two images.  One of the images is
then to be (affine) transformed to align with the other.
As mentioned, the standard way to do this is to use three
sets of points, compute the 6 transformation coefficients
from these points that describe the linear transformation,

x' = ax + by + c
y' = dx + ey + f

and use this in a pointwise manner to transform the image.

N.B.  Be sure to see the comment in getAffineXformCoeffs(),
regarding using the inverse of the affine transform for points
to transform images.

There is another way to do this transformation; namely,
by doing a sequence of simple affine transforms, without
computing directly the affine coordinate transformation.
We have at our disposal (1) translations (using rasterop),
(2) horizontal and vertical shear about any horizontal and vertical
line, respectively, and (3) non-isotropic scaling by two
arbitrary x and y scaling factors.  We also have rotation
about an arbitrary point, but this is equivalent to a set
of three shears so we do not need to use it.

Why might we do this?  For binary images, it is usually
more efficient to do such transformations by a sequence
of word parallel operations.  Shear and translation can be
done in-place and word parallel; arbitrary scaling is
mostly pixel-wise.

Suppose that we are tranforming image 1 to correspond to image 2.
We have a set of three points, describing the coordinate space
embedded in image 1, and we need to transform image 1 until
those three points exactly correspond to the new coordinate space
defined by the second set of three points.  In our image
matching application, the latter set of three points was
found to be the corresponding points in image 2.

The most elegant way I can think of to do such a sequential
implementation is to imagine that we're going to transform
BOTH images until they're aligned.  (We don't really want
to transform both, because in fact we may only have one image
that is undergoing a general affine transformation.)

Choose the 3 corresponding points as follows:
- The 1st point is an origin
- The 2nd point gives the orientation and scaling of the
"x" axis with respect to the origin
- The 3rd point does likewise for the "y" axis.
These "axes" must not be collinear; otherwise they are
arbitrary (although some strange things will happen if
the handedness sweeping through the minimum angle between
the axes is opposite).

An important constraint is that we have shear operations
about an arbitrary horizontal or vertical line, but always
parallel to the x or y axis.  If we continue to pretend that
we have an unprimed coordinate space embedded in image 1 and
a primed coordinate space embedded in image 2, we imagine
(a) transforming image 1 by horizontal and vertical shears about
point 1 to align points 3 and 2 along the y and x axes,
respectively, and (b) transforming image 2 by horizontal and
vertical shears about point 1' to align points 3' and 2' along
the y and x axes.  Then we scale image 1 so that the distances
from 1 to 2 and from 1 to 3 are equal to the distances in
image 2 from 1' to 2' and from 1' to 3'.  This scaling operation
leaves the true image origin, at (0,0) invariant, and will in
general translate point 1.  The original points 1 and 1' will
typically not coincide in any event, so we must translate
the origin of image 1, at its current point 1, to the origin
of image 2 at 1'.  The images should now be aligned.  But
because we never really transformed image 2 (and image 2 may
not even exist), we now perform  on image 1 the reverse of
the shear transforms that we imagined doing on image 2;
namely, the negative vertical shear followed by the negative
horizontal shear.  Image 1 should now have its transformed
unprimed coordinates aligned with the original primed
coordinates.  In all this, it is only necessary to keep track
of the shear angles and translations of points during the shears.
What has been accomplished is a general affine transformation
on image 1.

Having described all this, if you are going to use an
affine transformation in an application, this is what you
need to know:

(1) You should NEVER use the sequential method, because
the image quality for 1 bpp text is much poorer
(even though it is about 2x faster than the pointwise sampled
method), and for images with depth greater than 1, it is
nearly 20x slower than the pointwise sampled method
and over 10x slower than the pointwise interpolated method!
The sequential method is given here for purely
pedagogical reasons.

(2) For 1 bpp images, use the pointwise sampled function
pixAffineSampled().  For all other images, the best
quality results result from using the pointwise
interpolated function pixAffinePta() or pixAffine();
the cost is less than a doubling of the computation time
with respect to the sampled function.  If you use
interpolation on colormapped images, the colormap will
be removed, resulting in either a grayscale or color
image, depending on the values in the colormap.
If you want to retain the colormap, use pixAffineSampled().

Typical relative timing of pointwise transforms (sampled = 1.0):
8 bpp:   sampled        1.0
interpolated   1.6
32 bpp:  sampled        1.0
interpolated   1.8
Additionally, the computation time/pixel is nearly the same
for 8 bpp and 32 bpp, for both sampled and interpolated.``````

# FUNCTIONS

## affineInvertXform

l_int32 affineInvertXform ( l_float32 *vc, l_float32 **pvci )

``````  affineInvertXform()

Input:  vc (vector of 6 coefficients)
*vci (<return> inverted transform)
Return: 0 if OK; 1 on error

Notes:
(1) The 6 affine transform coefficients are the first
two rows of a 3x3 matrix where the last row has
only a 1 in the third column.  We invert this
using gaussjordan(), and select the first 2 rows
as the coefficients of the inverse affine transform.
(2) Alternatively, we can find the inverse transform
coefficients by inverting the 2x2 submatrix,
and treating the top 2 coefficients in the 3rd column as
a RHS vector for that 2x2 submatrix.  Then the
6 inverted transform coefficients are composed of
the inverted 2x2 submatrix and the negative of the
transformed RHS vector.  Why is this so?  We have
Y = AX + R  (2 equations in 6 unknowns)
Then
X = A'Y - A'R
Gauss-jordan solves
AF = R
and puts the solution for F, which is A'R,
into the input R vector.``````

## affineXformPt

l_int32 affineXformPt ( l_float32 *vc, l_int32 x, l_int32 y, l_float32 *pxp, l_float32 *pyp )

``````  affineXformPt()

Input:  vc (vector of 6 coefficients)
(x, y)  (initial point)
(&xp, &yp)   (<return> transformed point)
Return: 0 if OK; 1 on error

Notes:
(1) This computes the floating point location of the transformed point.
(2) It does not check ptrs for returned data!``````

## affineXformSampledPt

l_int32 affineXformSampledPt ( l_float32 *vc, l_int32 x, l_int32 y, l_int32 *pxp, l_int32 *pyp )

``````  affineXformSampledPt()

Input:  vc (vector of 6 coefficients)
(x, y)  (initial point)
(&xp, &yp)   (<return> transformed point)
Return: 0 if OK; 1 on error

Notes:
(1) This finds the nearest pixel coordinates of the transformed point.
(2) It does not check ptrs for returned data!``````

## gaussjordan

l_int32 gaussjordan ( l_float32 **a, l_float32 *b, l_int32 n )

``````  gaussjordan()

Input:   a  (n x n matrix)
b  (rhs column vector)
n  (dimension)
Return:  0 if ok, 1 on error

Note side effects:
(1) the matrix a is transformed to its inverse
(2) the vector b is transformed to the solution X to the
linear equation AX = B

Adapted from "Numerical Recipes in C, Second Edition", 1992
pp. 36-41 (gauss-jordan elimination)``````

## getAffineXformCoeffs

l_int32 getAffineXformCoeffs ( PTA *ptas, PTA *ptad, l_float32 **pvc )

``````  getAffineXformCoeffs()

Input:  ptas  (source 3 points; unprimed)
ptad  (transformed 3 points; primed)
&vc   (<return> vector of coefficients of transform)
Return: 0 if OK; 1 on error

We have a set of six equations, describing the affine
transformation that takes 3 points (ptas) into 3 other
points (ptad).  These equations are:

x1' = c[0]*x1 + c[1]*y1 + c[2]
y1' = c[3]*x1 + c[4]*y1 + c[5]
x2' = c[0]*x2 + c[1]*y2 + c[2]
y2' = c[3]*x2 + c[4]*y2 + c[5]
x3' = c[0]*x3 + c[1]*y3 + c[2]
y3' = c[3]*x3 + c[4]*y3 + c[5]

This can be represented as

AC = B

where B and C are column vectors

B = [ x1' y1' x2' y2' x3' y3' ]
C = [ c[0] c[1] c[2] c[3] c[4] c[5] c[6] ]

and A is the 6x6 matrix

x1   y1   1   0    0    0
0    0   0   x1   y1   1
x2   y2   1   0    0    0
0    0   0   x2   y2   1
x3   y3   1   0    0    0
0    0   0   x3   y3   1

These six equations are solved here for the coefficients C.

These six coefficients can then be used to find the dest
point (x',y') corresponding to any src point (x,y), according
to the equations

x' = c[0]x + c[1]y + c[2]
y' = c[3]x + c[4]y + c[5]

that are implemented in affineXformPt().

!!!!!!!!!!!!!!!!!!   Very important   !!!!!!!!!!!!!!!!!!!!!!

When the affine transform is composed from a set of simple
operations such as translation, scaling and rotation,
it is built in a form to convert from the un-transformed src
point to the transformed dest point.  However, when an
affine transform is used on images, it is used in an inverted
way: it converts from the transformed dest point to the
un-transformed src point.  So, for example, if you transform
a boxa using transform A, to transform an image in the same
way you must use the inverse of A.

For example, if you transform a boxa with a 3x3 affine matrix
'mat', the analogous image transformation must use 'matinv':

boxad = boxaAffineTransform(boxas, mat);
affineInvertXform(mat, &matinv);
pixd = pixAffine(pixs, matinv, L_BRING_IN_WHITE);

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!``````

## linearInterpolatePixelColor

l_int32 linearInterpolatePixelColor ( l_uint32 *datas, l_int32 wpls, l_int32 w, l_int32 h, l_float32 x, l_float32 y, l_uint32 colorval, l_uint32 *pval )

``````  linearInterpolatePixelColor()

Input:  datas (ptr to beginning of image data)
wpls (32-bit word/line for this data array)
w, h (of image)
x, y (floating pt location for evaluation)
colorval (color brought in from the outside when the
input x,y location is outside the image;
in 0xrrggbb00 format))
&val (<return> interpolated color value)
Return: 0 if OK, 1 on error

Notes:
(1) This is a standard linear interpolation function.  It is
equivalent to area weighting on each component, and
avoids "jaggies" when rendering sharp edges.``````

## linearInterpolatePixelGray

l_int32 linearInterpolatePixelGray ( l_uint32 *datas, l_int32 wpls, l_int32 w, l_int32 h, l_float32 x, l_float32 y, l_int32 grayval, l_int32 *pval )

``````  linearInterpolatePixelGray()

Input:  datas (ptr to beginning of image data)
wpls (32-bit word/line for this data array)
w, h (of image)
x, y (floating pt location for evaluation)
grayval (color brought in from the outside when the
input x,y location is outside the image)
&val (<return> interpolated gray value)
Return: 0 if OK, 1 on error

Notes:
(1) This is a standard linear interpolation function.  It is
equivalent to area weighting on each component, and
avoids "jaggies" when rendering sharp edges.``````

## pixAffine

PIX * pixAffine ( PIX *pixs, l_float32 *vc, l_int32 incolor )

``````  pixAffine()

Input:  pixs (all depths; colormap ok)
vc  (vector of 6 coefficients for affine transformation)
incolor (L_BRING_IN_WHITE, L_BRING_IN_BLACK)
Return: pixd, or null on error

Notes:
(1) Brings in either black or white pixels from the boundary
(2) Removes any existing colormap, if necessary, before transforming``````

## pixAffineColor

PIX * pixAffineColor ( PIX *pixs, l_float32 *vc, l_uint32 colorval )

``````  pixAffineColor()

Input:  pixs (32 bpp)
vc  (vector of 6 coefficients for affine transformation)
colorval (e.g., 0 to bring in BLACK, 0xffffff00 for WHITE)
Return: pixd, or null on error``````

## pixAffineGray

PIX * pixAffineGray ( PIX *pixs, l_float32 *vc, l_uint8 grayval )

``````  pixAffineGray()

Input:  pixs (8 bpp)
vc  (vector of 6 coefficients for affine transformation)
grayval (0 to bring in BLACK, 255 for WHITE)
Return: pixd, or null on error``````

## pixAffinePta

PIX * pixAffinePta ( PIX *pixs, PTA *ptad, PTA *ptas, l_int32 incolor )

``````  pixAffinePta()

Input:  pixs (all depths; colormap ok)
ptad  (3 pts of final coordinate space)
ptas  (3 pts of initial coordinate space)
incolor (L_BRING_IN_WHITE, L_BRING_IN_BLACK)
Return: pixd, or null on error

Notes:
(1) Brings in either black or white pixels from the boundary
(2) Removes any existing colormap, if necessary, before transforming``````

## pixAffinePtaColor

PIX * pixAffinePtaColor ( PIX *pixs, PTA *ptad, PTA *ptas, l_uint32 colorval )

``````  pixAffinePtaColor()

Input:  pixs (32 bpp)
ptad  (3 pts of final coordinate space)
ptas  (3 pts of initial coordinate space)
colorval (e.g., 0 to bring in BLACK, 0xffffff00 for WHITE)
Return: pixd, or null on error``````

## pixAffinePtaGray

PIX * pixAffinePtaGray ( PIX *pixs, PTA *ptad, PTA *ptas, l_uint8 grayval )

``````  pixAffinePtaGray()

Input:  pixs (8 bpp)
ptad  (3 pts of final coordinate space)
ptas  (3 pts of initial coordinate space)
grayval (0 to bring in BLACK, 255 for WHITE)
Return: pixd, or null on error``````

## pixAffinePtaWithAlpha

PIX * pixAffinePtaWithAlpha ( PIX *pixs, PTA *ptad, PTA *ptas, PIX *pixg, l_float32 fract, l_int32 border )

``````  pixAffinePtaWithAlpha()

Input:  pixs (32 bpp rgb)
ptad  (3 pts of final coordinate space)
ptas  (3 pts of initial coordinate space)
pixg (<optional> 8 bpp, can be null)
fract (between 0.0 and 1.0, with 0.0 fully transparent
and 1.0 fully opaque)
border (of pixels added to capture transformed source pixels)
Return: pixd, or null on error

Notes:
(1) The alpha channel is transformed separately from pixs,
and aligns with it, being fully transparent outside the
boundary of the transformed pixs.  For pixels that are fully
transparent, a blending function like pixBlendWithGrayMask()
will give zero weight to corresponding pixels in pixs.
(2) If pixg is NULL, it is generated as an alpha layer that is
partially opaque, using @fract.  Otherwise, it is cropped
to pixs if required and @fract is ignored.  The alpha channel
in pixs is never used.
(3) Colormaps are removed.
(4) When pixs is transformed, it doesn't matter what color is brought
in because the alpha channel will be transparent (0) there.
(5) To avoid losing source pixels in the destination, it may be
necessary to add a border to the source pix before doing
the affine transformation.  This can be any non-negative number.
(6) The input @ptad and @ptas are in a coordinate space before
the border is added.  Internally, we compensate for this
before doing the affine transform on the image after the border
is added.
(7) The default setting for the border values in the alpha channel
is 0 (transparent) for the outermost ring of pixels and
(0.5 * fract * 255) for the second ring.  When blended over
a second image, this
(a) shrinks the visible image to make a clean overlap edge
with an image below, and
(b) softens the edges by weakening the aliasing there.
Use l_setAlphaMaskBorder() to change these values.
(8) A subtle use of gamma correction is to remove gamma correction
before scaling and restore it afterwards.  This is done
by sandwiching this function between a gamma/inverse-gamma
photometric transform:
pixt = pixGammaTRCWithAlpha(NULL, pixs, 1.0 / gamma, 0, 255);
pixd = pixAffinePtaWithAlpha(pixg, ptad, ptas, NULL,
fract, border);
pixGammaTRCWithAlpha(pixd, pixd, gamma, 0, 255);
pixDestroy(&pixt);
This has the side-effect of producing artifacts in the very
dark regions.``````

## pixAffineSampled

PIX * pixAffineSampled ( PIX *pixs, l_float32 *vc, l_int32 incolor )

``````  pixAffineSampled()

Input:  pixs (all depths)
vc  (vector of 6 coefficients for affine transformation)
incolor (L_BRING_IN_WHITE, L_BRING_IN_BLACK)
Return: pixd, or null on error

Notes:
(1) Brings in either black or white pixels from the boundary.
(2) Retains colormap, which you can do for a sampled transform..
(3) For 8 or 32 bpp, much better quality is obtained by the
somewhat slower pixAffine().  See that function
for relative timings between sampled and interpolated.``````

## pixAffineSampledPta

PIX * pixAffineSampledPta ( PIX *pixs, PTA *ptad, PTA *ptas, l_int32 incolor )

``````  pixAffineSampledPta()

Input:  pixs (all depths)
ptad  (3 pts of final coordinate space)
ptas  (3 pts of initial coordinate space)
incolor (L_BRING_IN_WHITE, L_BRING_IN_BLACK)
Return: pixd, or null on error

Notes:
(1) Brings in either black or white pixels from the boundary.
(2) Retains colormap, which you can do for a sampled transform..
(3) The 3 points must not be collinear.
(4) The order of the 3 points is arbitrary; however, to compare
with the sequential transform they must be in these locations
and in this order: origin, x-axis, y-axis.
(5) For 1 bpp images, this has much better quality results
than pixAffineSequential(), particularly for text.
It is about 3x slower, but does not require additional
border pixels.  The poor quality of pixAffineSequential()
is due to repeated quantized transforms.  It is strongly
recommended that pixAffineSampled() be used for 1 bpp images.
(6) For 8 or 32 bpp, much better quality is obtained by the
somewhat slower pixAffinePta().  See that function
for relative timings between sampled and interpolated.
(7) To repeat, use of the sequential transform,
pixAffineSequential(), for any images, is discouraged.``````

## pixAffineSequential

PIX * pixAffineSequential ( PIX *pixs, PTA *ptad, PTA *ptas, l_int32 bw, l_int32 bh )

``````  pixAffineSequential()

Input:  pixs
ptad  (3 pts of final coordinate space)
ptas  (3 pts of initial coordinate space)
bw    (pixels of additional border width during computation)
bh    (pixels of additional border height during computation)
Return: pixd, or null on error

Notes:
(1) The 3 pts must not be collinear.
(2) The 3 pts must be given in this order:
- origin
- a location along the x-axis
- a location along the y-axis.
(3) You must guess how much border must be added so that no
pixels are lost in the transformations from src to
dest coordinate space.  (This can be calculated but it
is a lot of work!)  For coordinate spaces that are nearly
at right angles, on a 300 ppi scanned page, the addition
of 1000 pixels on each side is usually sufficient.
(4) This is here for pedagogical reasons.  It is about 3x faster
on 1 bpp images than pixAffineSampled(), but the results
on text are much inferior.``````

# AUTHOR

Zakariyya Mughal <zmughal@cpan.org>

# COPYRIGHT AND LICENSE

This software is copyright (c) 2014 by Zakariyya Mughal.

This is free software; you can redistribute it and/or modify it under the same terms as the Perl 5 programming language system itself.