Transform(3) User Contributed Perl Documentation Transform(3)NAMEPDL::Transform - Coordinate transforms, image warping, and N-D
functions
SYNOPSIS
use PDL::Transform;
my $t = new PDL::Transform::<type>(<opt>)
$out = $t->apply($in) # Apply transform to some N-vectors (Transform method)
$out = $in->apply($t) # Apply transform to some N-vectors (PDL method)
$im1 = $t->map($im); # Transform image coordinates (Transform method)
$im1 = $im->map($t); # Transform image coordinates (PDL method)
$t2 = $t->compose($t1); # compose two transforms
$t2 = $t x $t1; # compose two transforms (by analogy to matrix mult.)
$t3 = $t2->inverse(); # invert a transform
$t3 = !$t2; # invert a transform (by analogy to logical "not")
DESCRIPTIONPDL::Transform is a convenient way to represent coordinate
transformations and resample images. It embodies functions mapping R^N
-> R^M, both with and without inverses. Provision exists for
parametrizing functions, and for composing them. You can use this part
of the Transform object to keep track of arbitrary functions mapping
R^N -> R^M with or without inverses.
The simplest way to use a Transform object is to transform vector data
between coordinate systems. The apply method accepts a PDL whose 0th
dimension is coordinate index (all other dimensions are threaded over)
and transforms the vectors into the new coordinate system.
Transform also includes image resampling, via the map method. You
define a coordinate transform using a Transform object, then use it to
remap an image PDL. The output is a remapped, resampled image.
You can define and compose several transformations, then apply them all
at once to an image. The image is interpolated only once, when all the
composed transformations are applied.
In keeping with standard practice, but somewhat counterintuitively, the
map engine uses the inverse transform to map coordinates FROM the
destination dataspace (or image plane) TO the source dataspace; hence
PDL::Transform keeps track of both the forward and inverse transform.
For terseness and convenience, most of the constructors are exported
into the current package with the name "t_<transform>", so the
following (for example) are synonyms:
$t = new PDL::Transform::Radial(); # Long way
$t = t_radial(); # Short way
Several math operators are overloaded, so that you can compose and
invert functions with expression syntax instead of method syntax (see
below).
EXAMPLE
Coordinate transformations and mappings are a little counterintuitive
at first. Here are some examples of transforms in action:
use PDL::Transform;
$a = rfits('m51.fits'); # Substitute path if necessary!
$ts = t_linear(Scale=>3); # Scaling transform
$w = pgwin(xs);
$w->imag($a);
## Grow m51 by a factor of 3; origin is at lower left.
$b = $ts->map($a,{pix=>1}); # pix option uses direct pixel coord system
$w->imag($b);
## Shrink m51 by a factor of 3; origin still at lower left.
$c = $ts->unmap($a, {pix=>1});
$w->imag($c);
## Grow m51 by a factor of 3; origin is at scientific origin.
$d = $ts->map($a,$a->hdr); # FITS hdr template prevents autoscaling
$w->imag($d);
## Shrink m51 by a factor of 3; origin is still at sci. origin.
$e = $ts->unmap($a,$a->hdr);
$w->imag($e);
## A no-op: shrink m51 by a factor of 3, then autoscale back to size
$f = $ts->map($a); # No template causes autoscaling of output
OPERATOR OVERLOADS
'!'
The bang is a unary inversion operator. It binds exactly as tightly
as the normal bang operator.
'x'
By analogy to matrix multiplication, 'x' is the compose operator, so
these two expressions are equivalent:
$f->inverse()->compose($g)->compose($f) # long way
!$f x $g x $f # short way
Both of those expressions are equivalent to the mathematical
expression f^-1 o g o f, or f^-1(g(f(x))).
'**'
By analogy to numeric powers, you can apply an operator a positive
integer number of times with the ** operator:
$f->compose($f)->compose($f) # long way
$f**3 # short way
INTERNALS
Transforms are perl hashes. Here's a list of the meaning of each key:
func
Ref to a subroutine that evaluates the transformed coordinates.
It's called with the input coordinate, and the "params" hash. This
springboarding is done via explicit ref rather than by subclassing,
for convenience both in coding new transforms (just add the
appropriate sub to the module) and in adding custom transforms at
run-time. Note that, if possible, new "func"s should support inplace
operation to save memory when the data are flagged inplace. But
"func" should always return its result even when flagged to compute
in-place.
"func" should treat the 0th dimension of its input as a dimensional
index (running 0..N-1 for R^N operation) and thread over all other
input dimensions.
inv
Ref to an inverse method that reverses the transformation. It must
accept the same "params" hash that the forward method accepts. This
key can be left undefined in cases where there is no inverse.
idim, odim
Number of useful dimensions for indexing on the input and output
sides (ie the order of the 0th dimension of the coordinates to be
fed in or that come out). If this is set to 0, then as many are
allocated as needed.
name
A shorthand name for the transformation (convenient for debugging).
You should plan on using UNIVERAL::isa to identify classes of
transformation, e.g. all linear transformations should be subclasses
of PDL::Transform::Linear. That makes it easier to add smarts to,
e.g., the compose() method.
itype
An array containing the name of the quantity that is expected from
the input piddle for the transform, for each dimension. This field
is advisory, and can be left blank if there's no obvious quantity
associated with the transform. This is analogous to the CTYPEn
field used in FITS headers.
oname
Same as itype, but reporting what quantity is delivered for each
dimension.
iunit
The units expected on input, if a specific unit (e.g. degrees) is
expected. This field is advisory, and can be left blank if there's
no obvious unit associated with the transform.
ounit
Same as iunit, but reporting what quantity is delivered for each
dimension.
params
Hash ref containing relevant parameters or anything else the func
needs to work right.
is_inverse
Bit indicating whether the transform has been inverted. That is
useful for some stringifications (see the PDL::Transform::Linear
stringifier), and may be useful for other things.
Transforms should be inplace-aware where possible, to prevent excessive
memory usage.
If you define a new type of transform, consider generating a new
stringify method for it. Just define the sub "stringify" in the
subclass package. It should call SUPER::stringify to generate the
first line (though the PDL::Transform::Composition bends this rule by
tweaking the top-level line), then output (indented) additional lines
as necessary to fully describe the transformation.
NOTES
Transforms have a mechanism for labeling the units and type of each
coordinate, but it is just advisory. A routine to identify and, if
necessary, modify units by scaling would be a good idea. Currently, it
just assumes that the coordinates are correct for (e.g.) FITS
scientific-to-pixel transformations.
Composition works OK but should probably be done in a more
sophisticated way so that, for example, linear transformations are
combined at the matrix level instead of just strung together pixel-to-
pixel.
FUNCTIONS
There are both operators and constructors. The constructors are all
exported, all begin with "t_", and all return objects that are
subclasses of PDL::Transform.
The apply, invert, map, and unmap methods are also exported to the
"PDL" package: they are both Transform methods and PDL methods.
FUNCTIONS
apply
Signature: (data(); PDL::Transform t)
$out = $data->apply($t);
$out = $t->apply($data);
Apply a transformation to some input coordinates.
In the example, $t is a PDL::Transform and $data is a PDL to be
interpreted as a collection of N-vectors (with index in the 0th
dimension). The output is a similar but transformed PDL.
For convenience, this is both a PDL method and a Transform method.
invert
Signature: (data(); PDL::Transform t)
$out = $t->invert($data);
$out = $data->invert($t);
Apply an inverse transformation to some input coordinates.
In the example, $t is a PDL::Transform and $data is a piddle to be
interpreted as a collection of N-vectors (with index in the 0th
dimension). The output is a similar piddle.
For convenience this is both a PDL method and a PDL::Transform method.
map
Signature: (k0(); SV *in; SV *out; SV *map; SV *boundary; SV *method;
SV *big; SV *blur; SV *sv_min; SV *flux; SV *bv)
PDL::match
$b = $a->match($c); # Match $c's header and size
$b = $a->match([100,200]); # Rescale to 100x200 pixels
$b = $a->match([100,200],{rect=>1}); # Rescale and remove rotation/skew.
Resample a scientific image to the same coordinate system as another.
The example above is syntactic sugar for
$b = $a->map(t_identity, $c, ...);
it resamples the input PDL with the identity transformation in
scientific coordinates, and matches the pixel coordinate system to $c's
FITS header.
There is one difference between match and map: match makes the
"rectify" option to "map" default to 0, not 1. This only affects
matching where autoscaling is required (i.e. the array ref example
above). By default, that example simply scales $a to the new size and
maintains any rotation or skew in its scientiic-to-pixel coordinate
transform.
map
$b = $a->map($xform,[<template>],[\%opt]); # Distort $a with tranform $xform
$b = $a->map(t_identity,[$pdl],[\%opt]); # rescale $a to match $pdl's dims.
Resample an image or N-D dataset using a coordinate transform.
The data are resampled so that the new pixel indices are proportional
to the transformed coordinates rather than the original ones.
The operation uses the inverse transform: each output pixel location is
inverse-transformed back to a location in the original dataset, and the
value is interpolated or sampled appropriately and copied into the
output domain. A variety of sampling options are available, trading
off speed and mathematical correctness.
For convenience, this is both a PDL method and a PDL::Transform method.
"map" is FITS-aware: if there is a FITS header in the input data, then
the coordinate transform acts on the scientific coordinate system
rather than the pixel coordinate system.
By default, the output coordinates are separated from pixel coordinates
by a single layer of indirection. You can specify the mapping between
output transform (scientific) coordinates to pixel coordinates using
the "orange" and "irange" options (see below), or by supplying a FITS
header in the template.
If you don't specify an output transform, then the output is
autoscaled: "map" transforms a few vectors in the forward direction to
generate a mapping that will put most of the data on the image plane,
for most transformations. The calculated mapping gets stuck in the
output's FITS header.
Autoscaling is especially useful for rescaling images -- if you specify
the identity transform and allow autoscaling, you duplicate the
functionality of rescale2d, but with more options for interpolation.
You can operate in pixel space, and avoid autoscaling of the output, by
setting the "nofits" option (see below).
The output has the same data type as the input. This is a feature, but
it can lead to strange-looking banding behaviors if you use
interpolation on an integer input variable.
The "template" can be one of:
· a PDL
The PDL and its header are copied to the output array, which is then
populated with data. If the PDL has a FITS header, then the FITS
transform is automatically applied so that $t applies to the output
scientific coordinates and not to the output pixel coordinates. In
this case the NAXIS fields of the FITS header are ignored.
· a FITS header stored as a hash ref
The FITS NAXIS fields are used to define the output array, and the
FITS transformation is applied to the coordinates so that $t applies
to the output scientific coordinates.
· a list ref
This is a list of dimensions for the output array. The code
estimates appropriate pixel scaling factors to fill the available
space. The scaling factors are placed in the output FITS header.
· nothing
In this case, the input image size is used as a template, and
scaling is done as with the list ref case (above).
OPTIONS:
The following options are interpreted:
b, bound, boundary, Boundary (default = 'truncate')
This is the boundary condition to be applied to the input image; it
is passed verbatim to range or interpND in the sampling or
interpolating stage. Other values are 'forbid','extend', and
'periodic'. You can abbreviate this to a single letter. The
default 'truncate' causes the entire notional space outside the
original image to be filled with 0.
pix, Pixel, nf, nofits, NoFITS (default = 0)
If you set this to a true value, then FITS headers and
interpretation are ignored; the transformation is treated as being
in raw pixel coordinates.
j, J, just, justify, Justify (default = 0)
If you set this to 1, then output pixels are autoscaled to have unit
aspect ratio in the output coordinates. If you set it to a non-1
value, then it is the aspect ratio between the first dimension and
all subsequent dimensions -- or, for a 2-D transformation, the
scientific pixel aspect ratio. Values less than 1 shrink the scale
in the first dimension compared to the other dimensions; values
greater than 1 enlarge it compared to the other dimensions. (This
is the same sense as in the PGPLOTinterface.)
ir, irange, input_range, Input_Range
This is a way to modify the autoscaling. It specifies the range of
input scientific (not necessarily pixel) coordinates that you want
to be mapped to the output image. It can be either a nested array
ref or a piddle. The 0th dim (outside coordinate in the array ref)
is dimension index in the data; the 1st dim should have order 2.
For example, passing in either [[-1,2],[3,4]] or pdl([[-1,2],[3,4]])
limites the map to the quadrilateral in input space defined by the
four points (-1,3), (-1,4), (2,4), and (2,3).
As with plain autoscaling, the quadrilateral gets sparsely sampled
by the autoranger, so pathological transformations can give you
strange results.
This parameter is overridden by "orange", below.
or, orange, output_range, Output_Range
This sets the window of output space that is to be sampled onto the
output array. It works exactly like "irange", except that it
specifies a quadrilateral in output space. Since the output pixel
array is itself a quadrilateral, you get pretty much exactly what
you asked for.
This parameter overrides "irange", if both are specified. It forces
rectification of the output (so that scientific axes align with the
pixel grid).
r, rect, rectify
This option defaults TRUE and controls how autoscaling is performed.
If it is true or undefined, then autoscaling adjusts so that pixel
coordinates in the output image are proportional to individual
scientific coordinates. If it is false, then autoscaling
automatically applies the inverse of any input FITS transformation
*before* autoscaling the pixels. In the special case of linear
transformations, this preserves the rectangular shape of the
original pixel grid and makes output pixel coordinate proportional
to input coordinate.
m, method, Method
This option controls the interpolation method to be used.
Interpolation greatly affects both speed and quality of output. For
most cases the option is directly passed to interpND for
interpolation. Possible options, in order from fastest to slowest,
are:
· s, sample (default for ints)
Pixel values in the output plane are sampled from the closest
data value in the input plane. This is very fast but not very
accurate for either magnification or decimation (shrinking). It
is the default for templates of integer type.
· l, linear (default for floats)
Pixel values are linearly interpolated from the closest data
value in the input plane. This is reasonably fast but only
accurate for magnification. Decimation (shrinking) of the image
causes aliasing and loss of photometry as features fall between
the samples. It is the default for floating-point templates.
· c, cubic
Pixel values are interpolated using an N-cubic scheme from a
4-pixel N-cube around each coordinate value. As with linear
interpolation, this is only accurate for magnification.
· f, fft
Pixel values are interpolated using the term coefficients of the
Fourier transform of the original data. This is the most
appropriate technique for some kinds of data, but can yield
undesired "ringing" for expansion of normal images. Best suited
to studying images with repetitive or wavelike features.
· h, hanning
Pixel values are filtered through a spatially-variable filter
tuned to the computed Jacobian of the transformation, with
hanning-window (cosine) pixel rolloff in each dimension. This
prevents aliasing in the case where the image is distorted or
shrunk, but allows small amounts of aliasing at pixel edges
wherever the image is enlarged.
· g, gaussian, j, jacobian
Pixel values are filtered through a spatially-variable filter
tuned to the computed Jacobian of the transformation, with radial
Gaussian rolloff. This is the most accurate resampling method,
in the sense of introducing the fewest artifacts into a properly
sampled data set.
blur, Blur (default = 1.0)
This value scales the input-space footprint of each output pixel in
the gaussian and hanning methods. It's retained for historical
reasons. Larger values yield blurrier images; values significantly
smaller than unity cause aliasing.
sv, SV (default = 1.0)
This value lets you set the lower limit of the transformation's
singular values in the hanning and gaussian methods, limiting the
minimum radius of influence associated with each output pixel.
Large numbers yield smoother interpolation in magnified parts of the
image but don't affect reduced parts of the image.
big, Big (default = 0.2)
This is the largest allowable input spot size which may be mapped to
a single output pixel by the hanning and gaussian methods, in units
of the largest non-thread input dimension. (i.e. the default won't
let you reduce the original image to less than 5 pixels across).
This places a limit on how long the processing can take for
pathological transformations. Smaller numbers keep the code from
hanging for a long time; larger numbers provide for photometric
accuracy in more pathological cases. Numbers larer than 1.0 are
silly, because they allow the entire input array to be compressed
into a region smaller than a single pixel.
Wherever an output pixel would require averaging over an area that
is too big in input space, it instead gets NaN or the equivalent
(bad values are not yet supported).
phot, photometry, Photometry
This lets you set the style of photometric conversion to be used in
the hanning or gaussian methods. You may choose:
· 0, s, surf, surface, Surface (default)
(this is the default): surface brightness is preserved over the
transformation, so features maintain their original intensity.
This is what the sampling and interpolation methods do.
· 1, f, flux, Flux
Total flux is preserved over the transformation, so that the
brightness integral over image regions is preserved. Parts of
the image that are shrunk wind up brighter; parts that are
enlarged end up fainter.
VARIABLE FILTERING:
The 'hanning' and 'gaussian' methods of interpolation give
photometrically accurate resampling of the input data for arbitrary
transformations. At each pixel, the code generates a linear
approximation to the input transformation, and uses that linearization
to estimate the "footprint" of the output pixel in the input space.
The output value is a weighted average of the appropriate input spaces.
A caveat about these methods is that they assume the transformation is
continuous. Transformations that contain discontinuities will give
incorrect results near the discontinuity. In particular, the 180th
meridian isn't handled well in lat/lon mapping transformations (see
PDL::Transform::Cartography) -- pixels along the 180th meridian get the
average value of everything along the parallel occupied by the pixel.
This flaw is inherent in the assumptions that underly creating a
Jacobian matrix. Maybe someone will write code to work around it.
Maybe that someone is you.
map does not process bad values. It will set the bad-value flag of all
output piddles if the flag is set for any of the input piddles.
unmap
Signature: (data(); PDL::Transform a; template(); \%opt)
$out_image = $in_image->unmap($t,[<options>],[<template>]);
$out_image = $t->unmap($in_image,[<options>],[<template>]);
Map an image or N-D dataset using the inverse as a coordinate
transform.
This convenience function just inverts $t and calls map on the inverse;
everything works the same otherwise. For convenience, it is both a PDL
method and a PDL::Transform method.
t_inverse
$t2 = t_inverse($t);
$t2 = $t->inverse;
$t2 = $t ** -1;
$t2 = !$t;
Return the inverse of a PDL::Transform. This just reverses the
func/inv, idim/odim, itype/otype, and iunit/ounit pairs. Note that
sometimes you end up with a transform that cannot be applied or mapped,
because either the mathematical inverse doesn't exist or the inverse
func isn't implemented.
You can invert a transform by raising it to a negative power, or by
negating it with '!'.
The inverse transform remains connected to the main transform because
they both point to the original parameters hash. That turns out to be
useful.
t_compose
$f2 = t_compose($f, $g,[...]);
$f2 = $f->compose($g[,$h,$i,...]);
$f2 = $f x $g x ...;
Function composition: f(g(x)), f(g(h(x))), ...
You can also compose transforms using the overloaded matrix-
multiplication (nee repeat) operator 'x'.
This is accomplished by inserting a splicing code ref into the "func"
and "inv" slots. It combines multiple compositions into a single list
of transforms to be executed in order, fram last to first (in keeping
with standard mathematical notation). If one of the functions is
itself a composition, it is interpolated into the list rather than left
separate. Ultimately, linear transformations may also be combined
within the list.
No checking is done that the itype/otype and iunit/ounit fields are
compatible -- that may happen later, or you can implement it yourself
if you like.
t_wrap
$g1fg = $f->wrap($g);
$g1fg = t_wrap($f,$g);
Shift a transform into a different space by 'wrapping' it with a
second.
This is just a convenience function for two t_compose calls.
"$a->wrap($b)" is the same as "(!$b) x $a x $b": the resulting
transform first hits the data with $b, then with $a, then with the
inverse of $b.
For example, to shift the origin of rotation, do this:
$im = rfits('m51.fits');
$tf = t_fits($im);
$tr = t_linear({rot=>30});
$im1 = $tr->map($tr); # Rotate around pixel origin
$im2 = $tr->map($tr->wrap($tf)); # Rotate round FITS scientific origin
t_identity
my $xform = t_identity
my $xform = new PDL::Transform;
Generic constructor generates the identity transform.
This constructor really is trivial -- it is mainly used by the other
transform constructors. It takes no parameters and returns the
identity transform.
t_lookup
$f = t_lookup($lookup, {<options>});
Transform by lookup into an explicit table.
You specify an N+1-D PDL that is interpreted as an N-D lookup table of
column vectors (vector index comes last). The last dimension has order
equal to the output dimensionality of the transform.
For added flexibility in data space, You can specify pre-lookup linear
scaling and offset of the data. Of course you can specify the
interpolation method to be used. The linear scaling stuff is a little
primitive; if you want more, try composing the linear transform with
this one.
The prescribed values in the lookup table are treated as pixel-
centered: that is, if your input array has N elements per row then
valid data exist between the locations (-0.5) and (N-0.5) in lookup
pixel space, because the pixels (which are numbered from 0 to N-1) are
centered on their locations.
Lookup is done using interpND, so the boundary conditions and threading
behaviour follow from that.
The indexed-over dimensions come first in the table, followed by a
single dimension containing the column vector to be output for each set
of other dimensions -- ie to output 2-vectors from 2 input parameters,
each of which can range from 0 to 49, you want an index that has
dimension list (50,50,2). For the identity lookup table you could use
"cat(xvals(50,50),yvals(50,50))".
If you want to output a single value per input vector, you still need
that last index threading dimension -- if necessary, use "dummy(-1,1)".
The lookup index scaling is: out = lookup[ (scale * data) + offset ].
A simplistic table inversion routine is included. This means that you
can (for example) use the "map" method with "t_lookup" transformations.
But the table inversion is exceedingly slow, and not practical for
tables larger than about 100x100. The inversion table is calculated in
its entirety the first time it is needed, and then cached until the
object is destroyed.
Options are listed below; there are several synonyms for each.
s, scale, Scale
(default 1.0) Specifies the linear amount of scaling to be done
before lookup. You can feed in a scalar or an N-vector; other
values may cause trouble. If you want to save space in your table,
then specify smaller scale numbers.
o, offset, Offset
(default 0.0) Specifies the linear amount of offset before lookup.
This is only a scalar, because it is intended to let you switch to
corner-centered coordinates if you want to (just feed in o=-0.25).
b, bound, boundary, Boundary
Boundary condition to be fed to interpND
m, method, Method
Interpolation method to be fed to interpND
EXAMPLE
To scale logarithmically the Y axis of m51, try:
$a = float rfits('m51.fits');
$lookup = xvals(128,128) -> cat( 10**(yvals(128,128)/50) * 256/10**2.55 );
$t = t_lookup($lookup);
$b = $t->map($a);
To do the same thing but with a smaller lookup table, try:
$lookup = 16 * xvals(17,17)->cat(10**(yvals(17,17)/(100/16)) * 16/10**2.55);
$t = t_lookup($lookup,{scale=>1/16.0});
$b = $t->map($a);
(Notice that, although the lookup table coordinates are is divided by
16, it is a 17x17 -- so linear interpolation works right to the edge of
the original domain.)
NOTES
Inverses are not yet implemented -- the best way to do it might be by
judicious use of map() on the forward transformation.
the type/unit fields are ignored.
t_linear
$f = t_linear({options});
Linear (affine) transformations with optional offset
t_linear implements simple matrix multiplication with offset, also
known as the affine transformations.
You specify the linear transformation with pre-offset, a mixing matrix,
and a post-offset. That overspecifies the transformation, so you can
choose your favorite method to specify the transform you want. The
inverse transform is automagically generated, provided that it actually
exists (the transform matrix is invertible). Otherwise, the inverse
transform just croaks.
Extra dimensions in the input vector are ignored, so if you pass a 3xN
vector into a 3-D linear transformation, the final dimension is passed
through unchanged.
The options you can usefully pass in are:
s, scale, Scale
A scaling scalar (heh), vector, or matrix. If you specify a vector
it is treated as a diagonal matrix (for convenience). It gets left-
multiplied with the transformation matrix you specify (or the
identity), so that if you specify both a scale and a matrix the
scaling is done after the rotation or skewing or whatever.
r, rot, rota, rotation, Rotation
A rotation angle in degrees -- useful for 2-D and 3-D data only. If
you pass in a scalar, it specifies a rotation from the 0th axis
toward the 1st axis. If you pass in a 3-vector as either a PDL or
an array ref (as in "rot=>[3,4,5]"), then it is treated as a set of
Euler angles in three dimensions, and a rotation matrix is generated
that does the following, in order:
· Rotate by rot->(2) degrees from 0th to 1st axis
· Rotate by rot->(1) degrees from the 2nd to the 0th axis
· Rotate by rot->(0) degrees from the 1st to the 2nd axis
The rotation matrix is left-multiplied with the transformation
matrix you specify, so that if you specify both rotation and a
general matrix the rotation happens after the more general operation
-- though that is deprecated.
Of course, you can duplicate this functionality -- and get more
general -- by generating your own rotation matrix and feeding it in
with the "matrix" option.
m, matrix, Matrix
The transformation matrix. It does not even have to be square, if
you want to change the dimensionality of your input. If it is
invertible (note: must be square for that), then you automagically
get an inverse transform too.
pre, preoffset, offset, Offset
The vector to be added to the data before they get multiplied by the
matrix (equivalent of CRVAL in FITS, if you are converting from
scientific to pixel units).
post, postoffset, shift, Shift
The vector to be added to the data after it gets multiplied by the
matrix (equivalent of CRPIX-1 in FITS, if youre converting from
scientific to pixel units).
d, dim, dims, Dims
Most of the time it is obvious how many dimensions you want to deal
with: if you supply a matrix, it defines the transformation; if you
input offset vectors in the "pre" and "post" options, those define
the number of dimensions. But if you only supply scalars, there is
no way to tell and the default number of dimensions is 2. This
provides a way to do, e.g., 3-D scaling: just set
"{s="<scale-factor>, dims=>3}> and you are on your way.
NOTES
the type/unit fields are currently ignored by t_linear.
t_scale
$f = t_scale(<scale>)
Convenience interface to t_linear.
t_scale produces a tranform that scales around the origin by a fixed
amount. It acts exactly the same as "t_linear(Scale="\<scale\>)>.
t_offset
$f = t_offset(<shift>)
Convenience interface to t_linear.
t_offset produces a transform that shifts the origin to a new location.
It acts exactly the same as "t_linear(Pre="\<shift\>)>.
t_rot
$f = t_rot(<rotation-in-degrees>)
Convenience interface to t_linear.
t_rot produces a rotation transform in 2-D (scalar), 3-D (3-vector), or
N-D (matrix). It acts exactly the same as "t_linear(Rot="\<shift\>)>.
t_fits
$f = t_fits($fits,[option]);
FITS pixel-to-scientific transformation with inverse
You feed in a hash ref or a PDL with one of those as a header, and you
get back a transform that converts 0-originated, pixel-centered
coordinates into scientific coordinates via the transformation in the
FITS header. For most FITS headers, the transform is reversible, so
applying the inverse goes the other way. This is just a convenience
subclass of PDL::Transform::Linear, but with unit/type support using
the FITS header you supply.
For now, this transform is rather limited -- it really ought to accept
units differences and stuff like that, but they are just ignored for
now. Probably that would require putting units into the whole
transform framework.
This transform implements the linear transform part of the WCS FITS
standard outlined in Greisen & Calabata 2002 (A&A in press; find it at
"http://arxiv.org/abs/astro-ph/0207407").
As a special case, you can pass in the boolean option "ignore_rgb"
(default 0), and if you pass in a 3-D FITS header in which the last
dimension has exactly 3 elements, it will be ignored in the output
transformation. That turns out to be handy for handling rgb images.
t_code
$f = t_code(<func>,[<inv>],[options]);
Transform implementing arbitrary perl code.
This is a way of getting quick-and-dirty new transforms. You pass in
anonymous (or otherwise) code refs pointing to subroutines that
implement the forward and, optionally, inverse transforms. The
subroutines should accept a data PDL followed by a parameter hash ref,
and return the transformed data PDL. The parameter hash ref can be set
via the options, if you want to.
Options that are accepted are:
p,params
The parameter hash that will be passed back to your code (defaults
to the empty hash).
n,name
The name of the transform (defaults to "code").
i, idim (default 2)
The number of input dimensions (additional ones should be passed
through unchanged)
o, odim (default 2)
The number of output dimensions
itype
The type of the input dimensions, in an array ref (optional and
advisiory)
otype
The type of the output dimension, in an array ref (optional and
advisory)
iunit
The units that are expected for the input dimensions (optional and
advisory)
ounit
The units that are returned in the output (optional and advisory).
The code variables are executable perl code, either as a code ref or as
a string that will be eval'ed to produce code refs. If you pass in a
string, it gets eval'ed at call time to get a code ref. If it compiles
OK but does not return a code ref, then it gets re-evaluated with "sub
{ ... }" wrapped around it, to get a code ref.
Note that code callbacks like this can be used to do really weird
things and generate equally weird results -- caveat scriptor!
t_cylindrical
t_radial
$f = t_radial(<options>);
Convert Cartesian to radial/cylindrical coordinates. (2-D/3-D; with
inverse)
Converts 2-D Cartesian to radial (theta,r) coordinates. You can choose
direct or conformal conversion. Direct conversion preserves radial
distance from the origin; conformal conversion preserves local angles,
so that each small-enough part of the image only appears to be scaled
and rotated, not stretched. Conformal conversion puts the radius on a
logarithmic scale, so that scaling of the original image plane is
equivalent to a simple offset of the transformed image plane.
If you use three or more dimensions, the higher dimensions are ignored,
yielding a conversion from Cartesian to cylindrical coordinates, which
is why there are two aliases for the same transform. If you use higher
dimensionality than 2, you must manually specify the origin or you will
get dimension mismatch errors when you apply the transform.
Theta runs clockwise instead of the more usual counterclockwise; that
is to preserve the mirror sense of small structures.
OPTIONS:
d, direct, Direct
Generate (theta,r) coordinates out (this is the default);
incompatible with Conformal. Theta is in radians, and the radial
coordinate is in the units of distance in the input plane.
r0, c, conformal, Conformal
If defined, this floating-point value causes t_radial to generate
(theta, ln(r/r0)) coordinates out. Theta is in radians, and the
radial coordinate varies by 1 for each e-folding of the r0-scaled
distance from the input origin. The logarithmic scaling is useful
for viewing both large and small things at the same time, and for
keeping shapes of small things preserved in the image.
o, origin, Origin [default (0,0,0)]
This is the origin of the expansion. Pass in a PDL or an array ref.
u, unit, Unit [default 'radians']
This is the angular unit to be used for the azimuth.
EXAMPLES
These examples do transformations back into the same size image as they
started from; by suitable use of the "transform" option to unmap you
can send them to any size array you like.
Examine radial structure in M51: Here, we scale the output to stretch
2*pi radians out to the full image width in the horizontal direction,
and to stretch 1 radius out to a diameter in the vertical direction.
$a = rfits('m51.fits');
$ts = t_linear(s => [250/2.0/3.14159, 2]); # Scale to fill orig. image
$tu = t_radial(o => [130,130]); # Expand around galactic core
$b = $a->map($ts x $tu);
Examine radial structure in M51 (conformal): Here, we scale the output
to stretch 2*pi radians out to the full image width in the horizontal
direction, and scale the vertical direction by the exact same amount to
preserve conformality of the operation. Notice that each piece of the
image looks "natural" -- only scaled and not stretched.
$a = rfits('m51.fits')
$ts = t_linear(s=> 250/2.0/3.14159); # Note scalar (heh) scale.
$tu = t_radial(o=> [130,130], r0=>5); # 5 pix. radius -> bottom of image
$b = $ts->compose($tu)->unmap($a);
t_quadratic
$t = t_quadratic(<options>);
Quadratic scaling -- cylindrical pincushion (n-d; with inverse)
Quadratic scaling emulates pincushion in a cylindrical optical system:
separate quadratic scaling is applied to each axis. You can apply
separate distortion along any of the principal axes. If you want
different axes, use t_wrap and t_linear to rotate them to the correct
angle. The scaling options may be scalars or vectors; if they are
scalars then the expansion is isotropic.
The formula for the expansion is:
f(a) = ( <a> + <strength> * a^2/<L_0> ) / (abs(<strength>) + 1)
where <strength> is a scaling coefficient and <L_0> is a fundamental
length scale. Negative values of <strength> result in a pincushion
contraction.
Note that, because quadratic scaling does not have a strict inverse for
coordinate systems that cross the origin, we cheat slightly and use $x
* abs($x) rather than $x**2. This does the Right thing for pincushion
and barrel distortion, but means that t_quadratic does not behave
exactly like t_cubic with a null cubic strength coefficient.
OPTIONS
o,origin,Origin
The origin of the pincushion. (default is the, er, origin).
l,l0,length,Length,r0
The fundamental scale of the transformation -- the radius that
remains unchanged. (default=1)
s,str,strength,Strength
The relative strength of the pincushion. (default = 0.1)
honest (default=0)
Sets whether this is a true quadratic coordinate transform. The
more common form is pincushion or cylindrical distortion, which
switches branches of the square root at the origin (for symmetric
expansion). Setting honest to a non-false value forces true
quadratic behavior, which is not mirror-symmetric about the origin.
d, dim, dims, Dims
The number of dimensions to quadratically scale (default is the
dimensionality of your input vectors)
t_cubic
$t = t_cubic(<options>);
Cubic scaling - cubic pincushion (n-d; with inverse)
Cubic scaling is a generalization of t_quadratic to a purely cubic
expansion.
The formula for the expansion is:
f(a) = ( a' + st * a'^3/L_0^2 ) / (1 + abs(st)) + origin
where a'=(a-origin). That is a simple pincushion expansion/contraction
that is fixed at a distance of L_0 from the origin.
Because there is no quadratic term the result is always invertible with
one real root, and there is no mucking about with complex numbers or
multivalued solutions.
OPTIONS
o,origin,Origin
The origin of the pincushion. (default is the, er, origin).
l,l0,length,Length,r0
The fundamental scale of the transformation -- the radius that
remains unchanged. (default=1)
d, dim, dims, Dims
The number of dimensions to treat (default is the dimensionality of
your input vectors)
t_quartic
$t = t_quartic(<options>);
Quartic scaling -- cylindrical pincushion (n-d; with inverse)
Quartic scaling is a generalization of t_quadratic to a quartic
expansion. Only even powers of the input coordinates are retained, and
(as with t_quadratic) sign is preserved, making it an odd function
although a true quartic transformation would be an even function.
You can apply separate distortion along any of the principal axes. If
you want different axes, use t_wrap and t_linear to rotate them to the
correct angle. The scaling options may be scalars or vectors; if they
are scalars then the expansion is isotropic.
The formula for the expansion is:
f(a) = ( <a> + <strength> * a^2/<L_0> ) / (abs(<strength>) + 1)
where <strength> is a scaling coefficient and <L_0> is a fundamental
length scale. Negative values of <strength> result in a pincushion
contraction.
Note that, because quadratic scaling does not have a strict inverse for
coordinate systems that cross the origin, we cheat slightly and use $x
* abs($x) rather than $x**2. This does the Right thing for pincushion
and barrel distortion, but means that t_quadratic does not behave
exactly like t_cubic with a null cubic strength coefficient.
OPTIONS
o,origin,Origin
The origin of the pincushion. (default is the, er, origin).
l,l0,length,Length,r0
The fundamental scale of the transformation -- the radius that
remains unchanged. (default=1)
s,str,strength,Strength
The relative strength of the pincushion. (default = 0.1)
honest (default=0)
Sets whether this is a true quadratic coordinate transform. The
more common form is pincushion or cylindrical distortion, which
switches branches of the square root at the origin (for symmetric
expansion). Setting honest to a non-false value forces true
quadratic behavior, which is not mirror-symmetric about the origin.
d, dim, dims, Dims
The number of dimensions to quadratically scale (default is the
dimensionality of your input vectors)
t_spherical
$t = t_spherical(<options>);
Convert Cartesian to spherical coordinates. (3-D; with inverse)
Convert 3-D Cartesian to spherical (theta, phi, r) coordinates. Theta
is longitude, centered on 0, and phi is latitude, also centered on 0.
Unless you specify Euler angles, the pole points in the +Z direction
and the prime meridian is in the +X direction. The default is for
theta and phi to be in radians; you can select degrees if you want
them.
Just as the t_radial 2-D transform acts like a 3-D cylindrical
transform by ignoring third and higher dimensions, Spherical acts like
a hypercylindrical transform in four (or higher) dimensions. Also as
with t_radial, you must manually specify the origin if you want to use
more dimensions than 3.
To deal with latitude & longitude on the surface of a sphere (rather
than full 3-D coordinates), see t_unit_sphere.
OPTIONS:
o, origin, Origin [default (0,0,0)]
This is the Cartesian origin of the spherical expansion. Pass in a
PDL or an array ref.
e, euler, Euler [default (0,0,0)]
This is a 3-vector containing Euler angles to change the angle of
the pole and ordinate. The first two numbers are the (theta, phi)
angles of the pole in a (+Z,+X) spherical expansion, and the last is
the angle that the new prime meridian makes with the meridian of a
simply tilted sphere. This is implemented by composing the output
transform with a PDL::Transform::Linear object.
u, unit, Unit (default radians)
This option sets the angular unit to be used. Acceptable values are
"degrees","radians", or reasonable substrings thereof (e.g. "deg",
and "rad", but "d" and "r" are deprecated). Once genuine unit
processing comes online (a la Math::Units) any angular unit should
be OK.
t_projective
$t = t_projective(<options>);
Projective transformation
Projective transforms are simple quadratic, quasi-linear
transformations. They are the simplest transformation that can
continuously warp an image plane so that four arbitrarily chosen points
exactly map to four other arbitrarily chosen points. They have the
property that straight lines remain straight after transformation.
You can specify your projective transformation directly in homogeneous
coordinates, or (in 2 dimensions only) as a set of four unique points
that are mapped one to the other by the transformation.
Projective transforms are quasi-linear because they are most easily
described as a linear transformation in homogeneous coordinates (e.g.
(x',y',w) where w is a normalization factor: x = x'/w, etc.). In those
coordinates, an N-D projective transformation is represented as simple
multiplication of an N+1-vector by an N+1 x N+1 matrix, whose lower-
right corner value is 1.
If the bottom row of the matrix consists of all zeroes, then the
transformation reduces to a linear affine transformation (as in
t_linear).
If the bottom row of the matrix contains nonzero elements, then the
transformed x,y,z,etc. coordinates are related to the original
coordinates by a quadratic polynomial, because the normalization factor
'w' allows a second factor of x,y, and/or z to enter the equations.
OPTIONS:
m, mat, matrix, Matrix
If specified, this is the homogeneous-coordinate matrix to use. It
must be N+1 x N+1, for an N-dimensional transformation.
p, point, points, Points
If specified, this is the set of four points that should be mapped
one to the other. The homogeneous-coordinate matrix is calculated
from them. You should feed in a 2x2x4 PDL, where the 0 dimension
runs over coordinate, the 1 dimension runs between input and output,
and the 2 dimension runs over point. For example, specifying
p=> pdl([ [[0,1],[0,1]], [[5,9],[5,8]], [[9,4],[9,3]], [[0,0],[0,0]] ])
maps the origin and the point (0,1) to themselves, the point (5,9)
to (5,8), and the point (9,4) to (9,3).
This is similar to the behavior of fitwarp2d with a quadratic
polynomial.
AUTHOR
Copyright 2002, 2003 Craig DeForest. There is no warranty. You are
allowed to redistribute this software under certain conditions. For
details, see the file COPYING in the PDL distribution. If this file is
separated from the PDL distribution, the copyright notice should be
included in the file.
perl v5.18.1 2014-01-17 Transform(3)