First Steps with PDL
Chapter 1 from the PDL Book
"Maybe there are a few civilizations out there that have decided to stay home, piddle around and send out some radio waves once in a while."
 Annette Foglino, Space: Is Anyone Out There? Most astronomers say yes, Life, 1 Jul 1989.
It can be very frustrating to read an introductory book which takes a long time teaching you the very basics of a topic, in a "Janet and John" style. While you wish to learn, you are anxious to see something a bit more exciting and interesting to see what the language can do.
Fortunately our task in this book on PDL is made very much easier by the highlevel of the language. We can take a tour through PDL, looking at the advanced features it offers without getting involved in complexity.
The aim of this section is to cover a breadth of PDL features rather than any in depth, to give the reader a flavour of what he or she can do using the language and a useful reference for getting started doing real work. Later sections will focus on looking at the features introduced here, in more depth.
Alright, let's do something
We'll assume PDL is correctly installed and set up on your computer system (see http://pdl.perl.org/ for details of obtaining and installing PDL).
For interactive use PDL comes with a program called perldl
. This allows
you to type raw PDL (and perl) commands and see the result right away. It
also allows command line recall and editing (via the arrow keys) on most
systems.
So we begin by running the perldl
program from the system
command line. On a Mac/UNIX/Linux system we would simply type perldl
in a terminal window. On a Windows system we would type perldl
in a command prompt window. If PDL is installed correctly this is
all that is required to bring up perldl
.
myhost% perldl perlDL shell v1.357 PDL comes with ABSOLUTELY NO WARRANTY. For details, see the file 'COPYING' in the PDL distribution. This is free software and you are welcome to redistribute it under certain conditions, see the same file for details. ReadLines, NiceSlice, MultiLines enabled Reading PDL/default.perldlrc... Found docs database /usr/lib/perl5/.../PDL/pdldoc.db Type 'help' for online help Type 'demo' for online demos Loaded PDL v2.006 (supports bad values) pdl>
We get a whole bunch of informational messages about what it is loading for
startup and the help system. Note; the startup is completely configurable,
an advanced user can completely customize which PDL modules
are loaded. We are left with the pdl
> prompt at which we can type commands. This kind
of interactive program is called a 'shell'. There is also pdl2
which is a newer version of the PDL shell with additional features.
It is still under development but completely usable.
Let's create something, and display it:
pdl> use PDL::Graphics::Simple pdl> imag (sin(rvals(200,200)+1))
The result should look like the image below  a two dimensional sin
function.
rvals
is a handy PDL function for creating an image whose pixel values are
the radial distance from the central pixel of the image. With these arguments
it creates a 200 by 200 'radial' image. (Try 'imag(rvals(200,200))
' and you
will see better what we mean!) sin()
is the mathematical sine function, this
already exists in perl but in the case of PDL is applied to all 40000 pixels at
once, a topic we will come back to. The imag()
function displays the image.
You will see the syntax of perl/PDL is algebraic  by which we mean it is very
similar to C and FORTRAN in how expressions are constructed. (In fact much
more like C than FORTRAN). It is interesting to reflect on how much C code
would be required to generate the same display, even given the existence of
some convenient graphics library.
That's all fine. But what if we wanted to achieve the same results in a standalone perl script? Well it is pretty simple:
use PDL; use PDL::Graphics::Simple; imag (sin(rvals(200,200)+1));
That's it. This is a complete perl/PDL program. One could run it by typing
perl filename
. (In fact there are many ways of running it, most systems
allows it to be setup so you can just type filename. See your local
Perl documentation  then the perlrun
manual page.)
Two comments:

The statements are all terminated by the '
;
' character. Perl is like C in this regard. When entering code at thepdl
command line the final ';
' may be omitted if you wish, note you can also use it to put multiple statements on one line. In our examples from now on we'll often omit thepdl
prompt for clarity. 
The directive
use PDL;
tells Perl to load the PDL module, which makes available all the standard PDL extensions. (Advanced users will be interested in knowing there are other ways of starting PDL which allows one to select which bits of it you want).
Whirling through the Whirlpool
Enough about the mechanics of using PDL, let's look at some real data! To work through these examples exactly you can download any needed input files from http://sourceforge.net/projects/pdl/files/PDL/PDL%20Book%20Example%20Data%20Set/ and we'll assume you are running any of these examples in the same directory as you have downloaded the input data files.
We'll be playing with an image of the famous spiral galaxy discovered by Charles Messier, known to astronomers as M51 and commonly as the Whirlpool Galaxy. This is a 'nearby' galaxy, a mere 25 million light years from Earth. The image file is stored in the 'FITS' format, a common astronomical format, which is one of the many formats standard PDL can read. (FITS stores more shades of gray than GIF or JPEG, but PDL can read these formats too).
pdl> $a = rfits("m51_raw.fits"); # m51_raw.fits is in current directory Reading IMAGE data... BITPIX = 32 size = 262144 pixels Reading 1048576 bytes BSCALE = && BZERO =
This looks pretty simple. As you can probably guess by now rfits
is the PDL
function to read a FITS file. This is stored in the perl variable $a
.
This is an important PDL concept: PDL stores its data arrays in simple perl
variables ($a, $x, $y, $MyData
, etc.). PDL data arrays are special arrays
which use a more efficient, compact storage than standard perl arrays (@a,
@x, ...
) and are much faster to access for numerical computations. To avoid
confusion it is convenient to introduce a special name for them, we call them
piddles (short for 'PDL variables') to distinguish them from ordinary Perl
'arrays', which are in fact really lists. We'll say more about this later.
Before we start seriously playing around with M51 it is worth noting that we can also say:
pdl> $a = rfits "m51_raw.fits";
Note we have now left off the brackets on the rfits
function. Perl is rather
simpler than C and allows one to omit the brackets on a function all together.
It assumes all the items in a list are function arguments and can be pretty
convenient. If you are calling more than one function it is however better to
use some brackets so the meaning is clear. For the rules on this 'list
operator' syntax see the Perl syntax documentation. From now on we'll mostly
use the list operator syntax for conciseness
Let's look at M51:
pdl> imag $a; pdl> imag $a,0,1000; # More contrast pdl> imag $a,0,300; # Even more contrast
A couple of bright spots can be seen, but where is the galaxy? It's the faint
blob in the middle: by default the display range is autoscaled linearly from
the faintest to the brightest pixel, and only the bright star slightly to the
bottom right of the center can be seen without contrast enhancement. We can
easily change that by specifying the black/white data values (Note: #
starts
a Perl comment and can be ignored  i.e. no need to type the stuff after it!):
You can see that imag
takes additional arguments to specify the display
range. In fact imag
takes quite a few arguments, many of them optional. By
typing 'help imag
' at the pdl
prompt we can find out all about the
function.
It is certainly a spiral galaxy with a few foreground stars thrown in for good measure. But what is that horrible stripey pattern running from bottom right to top left? That certainly is not part of the galaxy? Well no. What we have here is the uneven sensitivity of the detector used to record the image, a common artifact in digital imaging. We can correct for this using an image of a uniformly illuminated screen, what is commonly known as a 'flatfield'.
pdl> $flat = rfits "m51_flatfield.fits"; pdl> imag $flat;
This is shown in the next figure. Because the image is of a uniform field, the actual image reflects the detector sensitivity. To correct our M51 image, we merely have to divide the image by the flatfield:
pdl> $gal = $a / $flat; pdl> imag $gal,0,300; pdl> wfits $gal, 'fixed_gal.fits'; # Save our work as a FITS file
Well that's a lot better. But think what we have
just done. Both $a
and $flat
are images, with 512 pixels by
512 pixels. The divide operator '/
' has been applied over all
262144 data values in the piddles $a
and $flat
. And it was
pretty fast too  these are what are known as vectorized
operations. In PDL each of these is implemented by heavily optimized
C code, which is what makes PDL very efficient for procession of
large chunks of data. If you did the same operation using normal
perl arrays rather than piddles it would be about ten to twenty times slower
(and use ten times more memory). In fact we can
do whatever arithmetic operations we like on image piddles:
pdl> $funny = log(($gal/300)**2  $gal/100 + 4); pdl> imag $funny; # Surprise!
Or on 1D line piddles. On on 3D cubic piddles. In fact piddles can support an infinite number of dimensions (though your computers memory won't).
This the key to PDL: the ability to process large chunks of data at once.
Measuring the brightness of M51
How might we extract some useful scientific information out of this image? A simple quantity an astronomer might want to know is how the brightness of the the 'disk' of the galaxy (the outer region which contains the spiral arms) compares with the 'bulge' (the compact inner nucleus). Well let's find out the total sum of all the light in the image:
pdl> print sum($gal); 17916010
sum
just sums up all the data values in all the pixels in the
image  in this case the answer is 17916010. If the image is linear
(which it is) and if it was calibrated (i.e. we knew the relation
between data numbers and brightness units) we could work out the
total brightness. Let's turn it round  we know that M51 has
a luminosity of about 1E36 Watts, so we can work out what
one data value corresponds to in physical units:
pdl> p 10**36/sum($gal) 5.58159992096455e+28
This is also about 200 solar luminosities, (Note we have switched to using p
as a shorthand for print
 which only works in the pdl
and pdl2
shells)
which gives 4 billion solar luminosities for the whole galaxy.
OK we do not need PDL for this simple arithmetic, let's get back to
computations that involve the whole image.
How can we get the sum of a piece of an image, e.g. near the centre?
Well in PDL there is more than one way to do it (Perl aficionados call
this phenomenon TIMTOWTDI). In this case, because we really want
the brightness in a circular aperture, we'll use the rvals
function:
pdl> $r = rvals $gal; pdl> imag $r; ...
Remember rvals
? It replaces all the pixels in an image with its distance
from the centre. We can turn this into a mask with a simple
operation like:
pdl> $mask = $r<50; pdl> imag $mask; ...
The Perl less than operator is applied to all pixels in the image. You can see the result is an image which is 0 on the outskirts and 1 in the area of the nucleus. We can then simply use the mask image to isolate in a simple way the bulge and disk components (lower row) and it is then very easy to find the brightness of both pieces of the M51 galaxy:
pdl> $bulge = $mask * $gal pdl> imag $bulge,0,300 ... pdl> print sum $bulge; 3011125 pdl> $disk = $gal * (1$mask) pdl> imag $disk,0,300 ... pdl> print sum $disk 14904884
You can see that the disk is about 5 times brighter than the bulge in total, despite its more diffuse appearance. This is typical for spiral galaxies. We might ask a different question: how does the average surface brightness, the brightness per unit area on the sky, compare between bulge and disk? This is again quite straight forward:
pdl> print sum($bulge)/sum($mask); pdl> print sum($disk)/sum(1$mask);
We work out the area by simply summing up the 0,1 pixels in the mask image. The answer is the bulge has about 7 times the surface brightness than the disk  something we might have guessed from looking at the above figure, which tells astronomers its stellar density is much higher.
Of course PDL being so powerful, we could have figured this out in one line:
pdl> print ( avg($gal>where(rvals($gal)<50)) / avg($gal>where(rvals($gal)>=50)) ) 6.56590509414673
Twinkle, twinkle, little star
Let's look at something else, we'll zoom in on a small piece of the image:
pdl> $section = $gal(337:357,178:198); pdl> imag $section; # the bright star
Here we are introducing something new  we can see that PDL supports
extensions to the Perl syntax. We can say $var(a:b,c:d...)
to specify
multidimensional slices. In this case we have produced a subimage ranging
from pixel 337 to 357 along the first dimension, and 178 through 198 along the
second. Remember pdl data dimension indexes start from zero. We'll talk some
more about slicing and dicing later on. This subimage happens to contain
a bright star.
At this point you will probably be able to work out for yourself the amount of light coming from this star, compared to the whole galaxy. (Answer: about 2%) But let's look at something more involved: the radial profile of the star. Since stars are a long way away they are almost point sources, but our camera will blur them out into little disks, and for our analysis we might want an exact figure for this blurring.
We want to plot all the brightness of all the pixels in this section, against
the distance from the centre. (We've chosen the section to be conveniently
centered on the star, you could think if you want about how you might determine
the centroid automatically using the xvals
and yvals
functions). Well it
is simple enough to get the distance from the centre:
pdl> $r = rvals $section;
But to produce a onedimensional plot of one against the other we need to
reduce the 2D data arrays to one dimension. (i.e our 21 by 21 image section
becomes a 441 element vector). This can be done using the PDL clump
function, which 'clumps' together an arbitrary number of dimensions:
pdl> $rr = $r>clump(2); # Clump first two dimensions pdl> $sec = $section>clump(2); pdl> points $rr, $sec; # Radial plot
You should see a nice graph with points like those in the figure below showing the dropoff from the bright centre of the star. The blurring is usually measured by the 'Full Width Half Maximum' (FWHM)  or in plain terms how fat the profile is across when it drops by half. Looking at the plot it looks like this is about 23 pixels  pretty compact!
Well we don't just want a guess  let's fit the profile with a function.
These blurring functions are usually represented by the Gaussian
function. PDL comes with a whole variety of general purpose and
special purpose fitting functions which people have written for
their own purposes (and so will you we hope!). Fitting Gaussians
is something that happens rather a lot and there is surprisingly
enough a special function for this very purpose. (One could use
more general fitting packages like PDL::Fit::LM
or
PDL::Opt::Simplex
but that would require more care).
pdl> use PDL::Fit::Gaussian;
This loads in the module to do this. PDL, like Perl, is modular. We
don't load all the available modules by default just a convenient
subset. How can we find useful PDL functions and modules? Well
help
tells us more about what we already know, to find out
about what we don't know use apropos
:
pdl> apropos gaussian PDL::Fit::Gaussian ... Module: routines for fitting gaussians PDL::Gaussian Module: Gaussian distributions. fitgauss1d Fit 1D Gassian to data piddle fitgauss1dr Fit 1D Gassian to radial data piddle gefa Factor a matrix using Gaussian elimination. grandom Constructor which returns piddle of Gaussian random numbers ndtri The value for which the area under the Gaussian probability density function (integrated from minus infinity) is equal to the argument (cf erfi). Works inplace.
This tells us a whole lot about various functions and modules to do with
Gaussians. Note that we can abbreviate help
and apropos
with '?
' and '??
' when using the pdl
or pdl2
shells.
Let's fit a Gaussian:
pdl> use PDL::Fit::Gaussian; pdl> ($peak, $fwhm, $background) = fitgauss1dr($rr, $sec); pdl> p $peak, $fwhm, $background;
fitgauss1dr
is a function in the module the PDL::Fit::Gaussian manpage which fits
a Gaussian constrained to be radial (i.e. whose peak is at the origin).
You can see that, unlike C and FORTRAN, Perl functions can return
more than one result value. This is pretty convenient. You can see the
FWHM is more like 2.75 pixels. Let's generate a fitted curve with this
functional form.
pdl> $rrr = sequence(2000)/100; # Generate radial values 0,0.01,0,02..20 # Generate Gaussian with given FWHM
pdl> $fit = $peak * exp(2.772 * ($rrr/$fwhm)**2) + $background;
Note the use of a new function, sequence(N)
, which
generates a new piddle with N values ranging 0..(N1).
We are simply using this to generate the horizontal axis values
for the plot. Now let's overlay it on the previous plot.
pdl> hold; # This command stops new plots starting new pages pdl> line $rrr, $fit, {Colour=>2} ; # Line plot
The last line
command shows the PDL syntax for optional function
arguments. This is based on the Perl's built in hash syntax. We'll say
more about this later in the PDL::Book::PGPLOT manpage. The result should look a
lot like the figure above. Not too bad. We could perhaps do a bit
better by exactly centroiding the image but it will do for now.
Let's make a simulation of the 2D stellar image. This is equally easy:
pdl> $fit2d = $peak * exp(2.772 * ($r/$fwhm)**2); pdl> release; # Back to new page for new plots; pdl> imag $fit2d; ... pdl> wfits $fit2d, 'fake_star.fits'; # Save our work
But the figure below is a
boring. So far we have been using simple 2D graphics from the
PDL::Graphics::Simple
library. In fact PDL has more
than one graphics library (some see this as a flaw, some
as a feature!). Using the PDL::Graphics::TriD
library
which does OpenGL graphics we can look at our simulated
star in 3D (see the right hand panel);
pdl> use PDL::Graphics::TriD; # Load the 3D graphics module pdl> imag3d [$fit2d];
If you do this on your computer you should be able to look at the graphic from
different sides by simply dragging in the plot window with the mouse! You can
also zoom in and out with the right mouse button. Note that imag3d
has it's
a rather different syntax for processing it's arguments  for very good reasons
 we'll explore 3D graphics further in the PDL::Book::TriD manpage.
To continue: Select the TriD window and type q
Finally here's something interesting. Let's take our fake star and place it elsewhere on the galaxy image.
pdl> $newsection = $gal(50:70,70:90); pdl> $newsection += $fit2d; pdl> imag $gal,0,300;
We have a bright new star where none existed before! The Cstyle +=
increment operator is worth noting  it actually modifies the contents of
$newsection
inplace. And because $newsection
is a slice of $gal
the change also affects $gal
. This is an important property of slices  any
change to the slice affects the parent. This kind of parent/child
relationship is a powerful property of many PDL functions, not just slicing.
What's more in many cases it leads to memory efficiency, when this kind of
linear slice is stored we only store the start/stop/step and not a new copy of
the actual data.
Of course sometimes we DO want a new copy of the actual data, for example if we plan to do something evil to it. To do this we could use the alternative form:
pdl> $newsection = $newsection + $fit2d
Now a new version of $newsection
is created which has nothing to
do with the original $gal
. In fact there is more than one way to do
this as we will see in later chapters.
Just to amuse ourselves, lets write a short script to cover M51 with dozens of fake stars of random brightnesses:
use PDL; use PDL::Graphics::Simple; use PDL::NiceSlice; # must use in each program file
srand(42); # Set the random number seed $gal = rfits "fixed_gal.fits"; $star = rfits "fake_star.fits"; sub addstar { ($x,$y) = @_; $xx = $x+20; $yy = $y+20; # Note use of slice on the LHS! $gal($x:$xx,$y:$yy) += $star * rand(2); }
for (1..100) { $x1 = int(rand(470)+10); $y1 = int(rand(470)+10); addstar($x1,$y1); } imag $gal,0,1000;
This ought to give the casual reader some flavour of the Perl syntax  quite simple and quite like C except that the entities being manipulated here are entire arrays of data, not single numbers. The result is shown, for amusement, in the figure below and takes virtually no time to compute.
Getting Complex with M51
To conclude this frantic whirl through the possibilities of PDL, let's look at a moderately complex (sic) example. We'll take M51 and try to enhance it to reveal the largescale structure, and then subtract this to reveal smallscale structure.
Just to show off we'll use a method based on the Fourier transform  don't worry if you don't know much about these, all you need to know is that the Fourier transform turns the image into an 'inverse' image, with complex numbers, where each pixel represents the strength of wavelengths of different scales in the image. Let's do it:
pdl> use PDL::FFT; # Load Fast Fourier Transform package pdl> $gal = rfits "fixed_gal.fits";
Now $gal
contains real values, to do the Fourier transform it has to
have complex values. We create a variable $imag
to hold the imaginary
component and set to zero.(For reasons of efficiency complex numbers
are represented in PDL by separate real and imaginary arrays  more about this
in Chapter 2.)
pdl> $imag = $gal * 0; # Create imaginary component, equal to zero pdl> fftnd $gal, $imag; # Perform Fourier transform
fftnd
performs a Fast Fourier Transform, inplace, on arbitrarydimensioned data (i.e.
it is 'Ndimensional'). You can display $gal
after the FFT but you won't see
much. If at this point we ran ifftnd
to invert it we would get the original
$gal
back.
If we want to enhance the largescale structure we want to make a filter to only let through lowfrequencies:
pdl> $tmp = rvals($gal)<10; # Radiallysymmetric filter function pdl> use PDL::ImageND; # provides kernctr() pdl> $filter = kernctr $tmp, $tmp; # Shift origin to 0,0 pdl> imag $filter;
You can see from the image that $filter
is zero everywhere except near the origin (0,0)
(and the 3 reflected corners). As a result it only lets through lowfrequency wavelengths. So we multiply by the filter and FFT back to see the result (cmul
is complex multiplication):
pdl> ($gal2, $imag2) = cmul $gal, $imag, $filter, 0; pdl> ifftnd $gal2, $imag2; pdl> imag $gal2,0,300;
Well that looks quite a bit different! Just about all the highfrequency information has vanished. To see the highfrequency information we can just subtract our filtered image from the original to form the right hand image.
pdl> $orig = rfits "fixed_gal.fits"; pdl> imag $orig$gal2,0,100;
Roundoff
Well that is probably enough abuse of Messier 51. We have demonstrated the ease of simple and complex data processing with PDL and how PDL fits neatly in to the Perl syntax as well as extending it. You have come across basic arithmetical operations and a scattering of useful functions  and learned how to find more. You certainly ought now to have a good feel of what PDL is all about. In the next chapter we'll take a more comprehensive look at the basic parts of PDL that all keen PDL users should know.
Where to Read More
We hope this excerpt from the PDL Book has given you a taste of what PDL is capable of. Please download the full text to find out more.
As always your onestopshop for all things PDL is at http://pdl.perl.org.