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 high-level 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.
We'll assume PDL is correctly installed and set up on your computer system (see https://pdl.perl.org/ for details of obtaining and installing PDL).
Many of the graphics examples in this chapter use a simplified interface to whatever PDL Graphics library you have installed. However, this simplified interface is not included with the normal PDL distribution. To get it, install PDL::Graphics::Simple from CPAN. This should do it (perhaps prefixed with 'sudo'):
myhost% cpan PDL::Graphics::SimpleFor 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.
Figure of a two dimensional sin function.
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 the pdl 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 the pdl 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).
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 https://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; 
Figure of the raw image m51_raw.fits shown with progressively greater contrast using the imag command.
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!):
pdl> imag $a,0,1000; # More contrast 
pdl> imag $a,0,300; # Even more contrastYou 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:
Figure: The 'flatfield' image showing the detector sensitivity of the raw data.
pdl> $gal = $a / $flat;
pdl> imag $gal,0,300;
pdl> wfits $gal, 'fixed_gal.fits'; # Save our work as a FITS fileWell 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:
Figure: The M51 image corrected for the flatfield.
pdl> $funny = log(($gal/300)**2 - $gal/100 + 4);
pdl> imag $funny; # Surprise!Or on 1-D line piddles. On on 3-D cubic piddles. In fact piddles can support an infinite number of dimensions (though your computers memory won't).
This the key to PDL: the ability to process large chunks of data at once.
How might we extract some useful scientific information out of this image? A simple quantity an astronomer might want to know is how the brightness of the the 'disk' of the galaxy (the outer region which contains the spiral arms) compares with the 'bulge' (the compact inner nucleus). Well let's find out the total sum of all the light in the image:
pdl> print sum($gal);
17916010sum 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+28This 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;
... 
Figure: Using rvals to generate a mask image to isolate the galaxy bulge and disk. Top row: radial gradient image $r, and radial gradient masked with less than operator $r < 50. Bottom row: Bulge and disk of the galaxy.
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 $disk
14904884You can see that the disk is about 5 times brighter than the bulge in total, despite its more diffuse appearance. This is typical for spiral galaxies. We might ask a different question: how does the average surface brightness, the brightness per unit area on the sky, compare between bulge and disk? This is again quite straight forward:
pdl> print sum($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)) )
6.56590509414673Let'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 starHere 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 sub-image 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 sub-image 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 one-dimensional 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 plotYou should see a nice graph with points like those in the figure below showing the drop-off 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 2-3 pixels - pretty compact!
Figure: Radial light profile of the bright star with fitted curve.
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 PDL::Fit::Gaussian 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..(N-1). 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 plotThe 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 PDL::Book::PGPLOT. 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 $fit2d, 'fake_star.fits'; # Save our workBut the figure below is a boring. So far we have been using simple 2D graphics from the PDL::Graphics::Simple library. In fact PDL has more than one graphics library (some see this as a flaw, some as a feature!). Using the PDL::Graphics::TriD library which does OpenGL graphics we can look at our simulated star in 3D (see the right hand panel);
Figure: Two different views of the 2D simulated Point Spread Function.
pdl> use PDL::Graphics::TriD; # Load the 3D graphics module
pdl> imag3d [$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 PDL::Book::TriD.
To continue: Select the TriD window and type qFinally 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 C-style += increment operator is worth noting - it actually modifies the contents of $newsection in-place. 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 + $fit2dNow 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.
Figure: M51 covered in fake stars.
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 large-scale structure, and then subtract this to reveal small-scale 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; see PDL::Complex.)
pdl> $imag = $gal * 0; # Create imaginary component, equal to zero
pdl> fftnd $gal, $imag; # Perform Fourier transformfftnd performs a Fast Fourier Transform, in-place, on arbitrary-dimensioned data (i.e. it is 'N-dimensional'). 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 large-scale structure we want to make a filter to only let through low-frequencies:
pdl> $tmp = rvals($gal)<10; # Radially-symmetric 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 $filter is zero everywhere except near the origin (0,0) (and the 3 reflected corners). As a result it only lets through low-frequency wavelengths. So we multiply by the filter and FFT back to see the result (cmul is complex multiplication):
pdl> ($gal2, $imag2) = cmul $gal, $imag, $filter, 0;
pdl> ifftnd $gal2, $imag2;
pdl> imag $gal2,0,300; 
Figure: Fourier filtered smoothed image and contrast enhanced image with the smoothed image subtracted.
Well that looks quite a bit different! Just about all the high-frequency information has vanished. To see the high-frequency 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;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.
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 one-stop-shop for all things PDL is at https://pdl.perl.org.