Zakariyya Mughal

# NAME

Image::Leptonica::Func::projective

version 0.04

# `projective.c`

``````  projective.c

Projective (4 pt) image transformation using a sampled
(to nearest integer) transform on each dest point
PIX      *pixProjectiveSampledPta()
PIX      *pixProjectiveSampled()

Projective (4 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      *pixProjectivePta()
PIX      *pixProjective()
PIX      *pixProjectivePtaColor()
PIX      *pixProjectiveColor()
PIX      *pixProjectivePtaGray()
PIX      *pixProjectiveGray()

Projective transform including alpha (blend) component
PIX      *pixProjectivePtaWithAlpha()

Projective coordinate transformation
l_int32   getProjectiveXformCoeffs()
l_int32   projectiveXformSampledPt()
l_int32   projectiveXformPt()

A projective transform can be specified as a specific functional
mapping between 4 points in the source and 4 points in the dest.
It preserves straight lines, but is less stable than a bilinear
transform, because it contains a division by a quantity that
can get arbitrarily small.)

We give both a projective coordinate transformation and
two projective image transformations.

For the former, we ask for the coordinate value (x',y')
in the transformed space for any point (x,y) in the original
space.  The coefficients of the transformation are found by
solving 8 simultaneous equations for the 8 coordinates of
the 4 points in src and dest.  The transformation can then
be used to compute the associated image transform, by
computing, for each dest pixel, the relevant pixel(s) in
the source.  This can be done either by taking the closest
src pixel to each transformed dest pixel ("sampling") or
by doing an interpolation and averaging over 4 source
pixels with appropriate weightings ("interpolated").

A typical application would be to remove keystoning
due to a projective transform in the imaging system.

The projective transform is given by specifying two equations:

x' = (ax + by + c) / (gx + hy + 1)
y' = (dx + ey + f) / (gx + hy + 1)

where the eight coefficients have been computed from four
sets of these equations, each for two corresponding data pts.
In practice, for each point (x,y) in the dest image, this
equation is used to compute the corresponding point (x',y')
in the src.  That computed point in the src is then used
to determine the dest value in one of two ways:

- sampling: take the value of the src pixel in which this
point falls
- interpolation: take appropriate linear combinations of the
four src pixels that this dest pixel would
overlap, with the coefficients proportional
to the amount of overlap

For small warp where there is little scale change, (e.g.,
for rotation) area mapping is nearly equivalent to interpolation.

Typical relative timing of pointwise transforms (sampled = 1.0):
8 bpp:   sampled        1.0
interpolated   1.5
32 bpp:  sampled        1.0
interpolated   1.6
Additionally, the computation time/pixel is nearly the same
for 8 bpp and 32 bpp, for both sampled and interpolated.``````

# FUNCTIONS

## getProjectiveXformCoeffs

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

``````  getProjectiveXformCoeffs()

Input:  ptas  (source 4 points; unprimed)
&vc   (<return> vector of coefficients of transform)
Return: 0 if OK; 1 on error

We have a set of 8 equations, describing the projective
transformation that takes 4 points (ptas) into 4 other

x1' = (c[0]*x1 + c[1]*y1 + c[2]) / (c[6]*x1 + c[7]*y1 + 1)
y1' = (c[3]*x1 + c[4]*y1 + c[5]) / (c[6]*x1 + c[7]*y1 + 1)
x2' = (c[0]*x2 + c[1]*y2 + c[2]) / (c[6]*x2 + c[7]*y2 + 1)
y2' = (c[3]*x2 + c[4]*y2 + c[5]) / (c[6]*x2 + c[7]*y2 + 1)
x3' = (c[0]*x3 + c[1]*y3 + c[2]) / (c[6]*x3 + c[7]*y3 + 1)
y3' = (c[3]*x3 + c[4]*y3 + c[5]) / (c[6]*x3 + c[7]*y3 + 1)
x4' = (c[0]*x4 + c[1]*y4 + c[2]) / (c[6]*x4 + c[7]*y4 + 1)
y4' = (c[3]*x4 + c[4]*y4 + c[5]) / (c[6]*x4 + c[7]*y4 + 1)

Multiplying both sides of each eqn by the denominator, we get

AC = B

where B and C are column vectors

B = [ x1' y1' x2' y2' x3' y3' x4' y4' ]
C = [ c[0] c[1] c[2] c[3] c[4] c[5] c[6] c[7] ]

and A is the 8x8 matrix

x1   y1     1     0   0    0   -x1*x1'  -y1*x1'
0    0     0    x1   y1   1   -x1*y1'  -y1*y1'
x2   y2     1     0   0    0   -x2*x2'  -y2*x2'
0    0     0    x2   y2   1   -x2*y2'  -y2*y2'
x3   y3     1     0   0    0   -x3*x3'  -y3*x3'
0    0     0    x3   y3   1   -x3*y3'  -y3*y3'
x4   y4     1     0   0    0   -x4*x4'  -y4*x4'
0    0     0    x4   y4   1   -x4*y4'  -y4*y4'

These eight equations are solved here for the coefficients C.

These eight coefficients can then be used to find the mapping
(x,y) --> (x',y'):

x' = (c[0]x + c[1]y + c[2]) / (c[6]x + c[7]y + 1)
y' = (c[3]x + c[4]y + c[5]) / (c[6]x + c[7]y + 1)

that is implemented in projectiveXformSampled() and
projectiveXFormInterpolated().``````

## pixProjective

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

``````  pixProjective()

Input:  pixs (all depths; colormap ok)
vc  (vector of 8 coefficients for projective 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``````

## pixProjectiveColor

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

``````  pixProjectiveColor()

Input:  pixs (32 bpp)
vc  (vector of 8 coefficients for projective transformation)
colorval (e.g., 0 to bring in BLACK, 0xffffff00 for WHITE)
Return: pixd, or null on error``````

## pixProjectiveGray

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

``````  pixProjectiveGray()

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

## pixProjectivePta

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

``````  pixProjectivePta()

Input:  pixs (all depths; colormap ok)
ptad  (4 pts of final coordinate space)
ptas  (4 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``````

## pixProjectivePtaColor

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

``````  pixProjectivePtaColor()

Input:  pixs (32 bpp)
ptad  (4 pts of final coordinate space)
ptas  (4 pts of initial coordinate space)
colorval (e.g., 0 to bring in BLACK, 0xffffff00 for WHITE)
Return: pixd, or null on error``````

## pixProjectivePtaGray

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

``````  pixProjectivePtaGray()

Input:  pixs (8 bpp)
ptad  (4 pts of final coordinate space)
ptas  (4 pts of initial coordinate space)
grayval (0 to bring in BLACK, 255 for WHITE)
Return: pixd, or null on error``````

## pixProjectivePtaWithAlpha

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

``````  pixProjectivePtaWithAlpha()

Input:  pixs (32 bpp rgb)
ptad  (4 pts of final coordinate space)
ptas  (4 pts of initial coordinate space)
pixg (<optional> 8 bpp, for alpha channel, 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 projective 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 projective transform on the image after
(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);
NULL, fract, border);
pixGammaTRCWithAlpha(pixd, pixd, gamma, 0, 255);
pixDestroy(&pixt);
This has the side-effect of producing artifacts in the very
dark regions.``````

## pixProjectiveSampled

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

``````  pixProjectiveSampled()

Input:  pixs (all depths)
vc  (vector of 8 coefficients for projective 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 pixProjective().  See that function
for relative timings between sampled and interpolated.``````

## pixProjectiveSampledPta

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

``````  pixProjectiveSampledPta()

Input:  pixs (all depths)
ptad  (4 pts of final coordinate space)
ptas  (4 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) No 3 of the 4 points may be collinear.
(4) For 8 and 32 bpp pix, better quality is obtained by the
somewhat slower pixProjectivePta().  See that
function for relative timings between sampled and interpolated.``````

## projectiveXformPt

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

``````  projectiveXformPt()

Input:  vc (vector of 8 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!``````

## projectiveXformSampledPt

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

``````  projectiveXformSampledPt()

Input:  vc (vector of 8 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!``````

# AUTHOR

Zakariyya Mughal <zmughal@cpan.org>