Provided by: pdl_2.085-1ubuntu1_amd64 
NAME
PDL::Transform - Coordinate transforms, image warping, and N-D functions
SYNOPSIS
use PDL::Transform;
my $t = PDL::Transform::<type>->new(<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")
DESCRIPTION
PDL::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
broadcasted 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 = PDL::Transform::Radial->new; # 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;
$x = rfits('m51.fits'); # Substitute path if necessary!
$ts = t_linear(Scale=>3); # Scaling transform
$w = pgwin(xs);
$w->imag($x);
## Grow m51 by a factor of 3; origin is at lower left.
$y = $ts->map($x,{pix=>1}); # pix option uses direct pixel coord system
$w->imag($y);
## Shrink m51 by a factor of 3; origin still at lower left.
$c = $ts->unmap($x, {pix=>1});
$w->imag($c);
## Grow m51 by a factor of 3; origin is at scientific origin.
$d = $ts->map($x,$x->hdr); # FITS hdr template prevents autoscaling
$w->imag($d);
## Shrink m51 by a factor of 3; origin is still at sci. origin.
$e = $ts->unmap($x,$x->hdr);
$w->imag($e);
## A no-op: shrink m51 by a factor of 3, then autoscale back to size
$f = $ts->map($x); # 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 broadcast 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 ndarray 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.
MODULE INTERFACE
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 an ndarray to be interpreted as a collection of
N-vectors (with index in the 0th dimension). The output is a similar ndarray.
For convenience this is both a PDL method and a PDL::Transform method.
map
Signature: (k0(); pdl *in; pdl *out; pdl *map; SV *boundary; SV *method;
long big; double blur; double sv_min; char flux; SV *bv)
match
$y = $x->match($c); # Match $c's header and size
$y = $x->match([100,200]); # Rescale to 100x200 pixels
$y = $x->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
$y = $x->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 $x to the new size and maintains any rotation or skew in its
scientific-to-pixel coordinate transform.
map
$y = $x->map($xform,[<template>],[\%opt]); # Distort $x with transform $xform
$y = $x->map(t_identity,[$pdl],[\%opt]); # rescale $x 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 PGPLOT
interface.)
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 an ndarray. 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. This method uses a
lookup table to speed up calculation of the Gaussian weighting.
• G
This method works exactly like 'g' (above), except that the Gaussian values are computed explicitly
for every sample -- which is considerably slower.
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. The parameter has slightly different meanings for Gaussian and
Hanning interpolation. For Hanning interpolation, numbers smaller than unity control the sharpness of
the border at the edge of each pixel (so that blur=>0 is equivalent to sampling for non-decimating
transforms). For Gaussian interpolation, the blur factor parameter controls the width parameter of
the Gaussian around the center of each pixel.
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 = 0.5)
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-broadcast 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 bad value.
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.
BAD VALUES:
map() supports bad values in the data array. Bad values in the input array are propagated to the output
array. The 'g' and 'h' methods will do some smoothing over bad values: if more than 1/3 of the weighted
input-array footprint of a given output pixel is bad, then the output pixel gets marked bad.
map does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is
set for any of the input ndarrays.
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,$g);
Shift a transform into a different space by 'wrapping' it with a second.
This is just a convenience function for two "t_compose" calls. "$x->wrap($y)" is the same as "(!$y) x $x
x $y": the resulting transform first hits the data with $y, then with $x, then with the inverse of $y.
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 = PDL::Transform->new;
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 broadcasting 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 broadcasting
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:
$x = float rfits('m51.fits');
$lookup = xvals(128,128) -> cat( 10**(yvals(128,128)/50) * 256/10**2.55 );
$t = t_lookup($lookup);
$y = $t->map($x);
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});
$y = $t->map($x);
(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 transform 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_cylindrical" is an alias for "t_radial"
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.
$x = 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
$y = $x->map($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.
$x = 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
$y = $ts->compose($tu)->unmap($x);
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 v5.38.2 2024-04-10 Transform(3pm)