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NAME
PDL::Indexing - Introduction to indexing and slicing ndarrays.
OVERVIEW
This man page should serve as a first tutorial on the indexing and broadcasting features
of PDL.
Like all vectorized languages, PDL automates looping over multi-dimensional data
structures ("ndarrays") using a variant of mathematical vector notation. The automatic
looping is called "broadcasting", similar to NumPy and Julia. PDL also automatically runs
broadcasting computation in parallel - see PDL::ParallelCPU.
A lot of the flexibility and power of PDL relies on the indexing and broadcasting features
of the Perl extension. Indexing allows access to the data of an ndarray in a very
flexible way. Broadcasting provides efficient vectorization of simple operations.
The values of an ndarray are stored compactly as typed values in a single block of memory,
not (as in a normal Perl list-of-lists) as individual Perl scalars.
In the sections that follow many "methods" are called out -- these are Perl operators that
apply to ndarrays. From the perldl (or pdl2) shell, you can find out more about each
method by typing "?" followed by the method name.
Dimension lists
A ndarray (PDL variable), in general, is an N-dimensional array where N can be 0 (for a
scalar), 1 (e.g. for a sound sample), or higher values for images and more complex
structures. Each dimension of the ndarray has a positive integer size. The "perl"
interpreter treats each ndarray as a special type of Perl scalar (a blessed Perl object,
actually -- but you don't have to know that to use them) that can be used anywhere you can
put a normal scalar.
You can access the dimensions of an ndarray as a Perl list and otherwise determine the
size of an ndarray with several methods. The important ones are:
nelem - the total number of elements in an ndarray
ndims - returns the number of dimensions in an ndarray
dims - returns the dimension list of an ndarray as a Perl list
dim - returns the size of a particular dimension of an ndarray
Indexing and Dataflow
PDL maintains a notion of "dataflow" between an ndarray and indexed subfields of that
ndarray. When you produce an indexed subfield or single element of a parent ndarray, the
child and parent remain attached until you manually disconnect them. This lets you
represent the same data different ways within your code -- for example, you can consider
an RGB image simultaneously as a collection of (R,G,B) values in a 3 x 1000 x 1000 image,
and as three separate 1000 x 1000 color planes stored in different variables. Modifying
any of the variables changes the underlying memory, and the changes are reflected in all
representations of the data.
There are two important methods that let you control dataflow connections between a child
and parent ndarray:
copy - forces an explicit copy of an ndarray
sever - breaks the dataflow connection between an ndarray and its parents (if any)
Broadcasting and Dimension Order
Most PDL operations act on the first few dimensions of their ndarray arguments. For
example, "sumover" sums all elements along the first dimension in the list (dimension 0).
If you feed in a three-dimensional ndarray, then the first dimension is considered the
"active" dimension and the later dimensions are "broadcast" dimensions because they are
simply looped over. There are several ways to transpose or re-order the dimension list of
an ndarray. Those techniques are very fast since they don't touch the underlying data,
only change the way that PDL accesses the data. The main dimension ordering functions
are:
mv - moves a particular dimension somewhere else in the dimension list
xchg - exchanges two dimensions in the dimension list, leaving the rest alone
reorder - allows wholesale mixing of the dimensions
clump - clumps together two or more small dimensions into one larger one
squeeze - eliminates any dimensions of size 1
Physical and Dummy Dimensions
• document Perl level broadcasting
• broadcastids
• update and correct description of slice
• new functions in slice.pd (affine, lag, splitdim)
• reworking of paragraph on explicit broadcasting
Indexing and broadcasting with PDL
A lot of the flexibility and power of PDL relies on the indexing and looping features of
the Perl extension. Indexing allows access to the data of an ndarray in a very flexible
way. Broadcasting provides efficient implicit looping functionality (since the loops are
implemented as optimized C code).
ndarrays are Perl objects that represent multidimensional arrays and operations on those.
In contrast to simple Perl @x style lists the array data is compactly stored in a single
block of memory thus taking up a lot less memory and enabling use of fast C code to
implement operations (e.g. addition, etc) on ndarrays.
ndarrays can have children
Central to many of the indexing capabilities of PDL are the relation of "parent" and
"child" between ndarrays. Many of the indexing commands create a new ndarray from an
existing ndarray. The new ndarray is the "child" and the old one is the "parent". The data
of the new ndarray is defined by a transformation that specifies how to generate (compute)
its data from the parent's data. The relation between the child ndarray and its parent are
often bidirectional, meaning that changes in the child's data are propagated back to the
parent. (Note: You see, we are aiming in our terminology already towards the new dataflow
features. The kind of dataflow that is used by the indexing commands (about which you will
learn in a minute) is always in operation, not only when you have explicitly switched on
dataflow in your ndarray by saying "$x->doflow". For further information about data flow
check the dataflow man page.)
Another way to interpret the ndarrays created by our indexing commands is to view them as
a kind of intelligent pointer that points back to some portion or all of its parent's
data. Therefore, it is not surprising that the parent's data (or a portion of it) changes
when manipulated through this "pointer". After these introductory remarks that hopefully
prepared you for what is coming (rather than confuse you too much) we are going to dive
right in and start with a description of the indexing commands and some typical examples
how they might be used in PDL programs. We will further illustrate the pointer/dataflow
analogies in the context of some of the examples later on.
There are two different implementations of this ``smart pointer'' relationship: the first
one, which is a little slower but works for any transformation is simply to do the
transformation forwards and backwards as necessary. The other is to consider the child
ndarray a ``virtual'' ndarray, which only stores a pointer to the parent and access
information so that routines which use the child ndarray actually directly access the data
in the parent. If the virtual ndarray is given to a routine which cannot use it, PDL
transparently physicalizes the virtual ndarray before letting the routine use it.
Currently (1.94_01) all transformations which are ``affine'', i.e. the indices of the data
item in the parent ndarray are determined by a linear transformation (+ constant) from the
indices of the child ndarray result in virtual ndarrays. All other indexing routines (e.g.
"->index(...)") result in physical ndarrays. All routines compiled by PP can accept
affine ndarrays (except those routines that pass pointers to external library functions).
Note that whether something is affine or not does not affect the semantics of what you do
in any way: both
$x->index(...) .= 5;
$x->slice(...) .= 5;
change the data in $x. The affinity does, however, have a significant impact on memory
usage and performance.
Slicing ndarrays
Probably the most important application of the concept of parent/child ndarrays is the
representation of rectangular slices of a physical ndarray by a virtual ndarray. Having
talked long enough about concepts let's get more specific. Suppose we are working with a
2D ndarray representing a 5x5 image (it's unusually small so that we can print it without
filling several screens full of digits).
pdl> $im = sequence(5,5)
pdl> p $im
[
[ 0 1 2 3 4]
[ 5 6 7 8 9]
[10 11 12 13 14]
[15 16 17 18 19]
[20 21 22 23 24]
]
pdl> help vars
PDL variables in package main::
Name Type Dimension Flow State Mem
----------------------------------------------------------------
$im Double D [5,5] P 0.20Kb
[ here it might be appropriate to quickly talk about the "help vars" command that provides
information about ndarrays in the interactive "perldl" or "pdl2" shell that comes with
PDL. ]
Now suppose we want to create a 1-D ndarray that just references one line of the image,
say line 2; or an ndarray that represents all even lines of the image (imagine we have to
deal with even and odd frames of an interlaced image due to some peculiar behaviour of our
frame grabber). As another frequent application of slices we might want to create an
ndarray that represents a rectangular region of the image with top and bottom reversed.
All these effects (and many more) can be easily achieved with the powerful slice function:
pdl> $line = $im->slice(':,(2)')
pdl> $even = $im->slice(':,1:-1:2')
pdl> $area = $im->slice('3:4,3:1')
pdl> help vars # or just PDL->vars
PDL variables in package main::
Name Type Dimension Flow State Mem
----------------------------------------------------------------
$even Double D [5,2] -C 0.00Kb
$im Double D [5,5] P 0.20Kb
$line Double D [5] -C 0.00Kb
$area Double D [2,3] -C 0.00Kb
All three "child" ndarrays are children of $im or in the other (largely equivalent)
interpretation pointers to data of $im. Operations on those virtual ndarrays access only
those portions of the data as specified by the argument to slice. So we can just print
line 2:
pdl> p $line
[10 11 12 13 14]
Also note the difference in the "Flow State" of $area above and below:
pdl> p $area
pdl> help $area
This variable is Double D [2,3] VC 0.00Kb
The following demonstrates that $im and $line really behave as you would expect from a
pointer-like object (or in the dataflow picture: the changes in $line's data are
propagated back to $im):
pdl> $im++
pdl> p $line
[11 12 13 14 15]
pdl> $line += 2
pdl> p $im
[
[ 1 2 3 4 5]
[ 6 7 8 9 10]
[13 14 15 16 17]
[16 17 18 19 20]
[21 22 23 24 25]
]
Note how assignment operations on the child virtual ndarrays change the parent physical
ndarray and vice versa (however, the basic "=" assignment doesn't, use ".=" to obtain that
effect. See below for the reasons). The virtual child ndarrays are something like "live
links" to the "original" parent ndarray. As previously said, they can be thought of to
work similar to a C-pointer. But in contrast to a C-pointer they carry a lot more
information. Firstly, they specify the structure of the data they represent (the
dimensionality of the new ndarray) and secondly, specify how to create this structure from
its parents data (the way this works is buried in the internals of PDL and not important
for you to know anyway (unless you want to hack the core in the future or would like to
become a PDL guru in general (for a definition of this strange creature see
PDL::Internals)).
The previous examples have demonstrated typical usage of the slice function. Since the
slicing functionality is so important here is an explanation of the syntax for the string
argument to slice:
$vpdl = $x->slice('ind0,ind1...')
where "ind0" specifies what to do with index No 0 of the ndarray $x, etc. Each element of
the comma separated list can have one of the following forms:
':' Use the whole dimension
'n' Use only index "n". The dimension of this index in the resulting virtual ndarray is
1. An example involving those first two index formats:
pdl> $column = $im->slice('2,:')
pdl> $row = $im->slice(':,0')
pdl> p $column
[
[ 3]
[ 8]
[15]
[18]
[23]
]
pdl> p $row
[
[1 2 3 4 5]
]
pdl> help $column
This variable is Double D [1,5] VC 0.00Kb
pdl> help $row
This variable is Double D [5,1] VC 0.00Kb
'(n)' Use only index "n". This dimension is removed from the resulting ndarray (relying on
the fact that a dimension of size 1 can always be removed). The distinction between
this case and the previous one becomes important in assignments where left and right
hand side have to have appropriate dimensions.
pdl> $line = $im->slice(':,(0)')
pdl> help $line
This variable is Double D [5] -C 0.00Kb
pdl> p $line
[1 2 3 4 5]
Spot the difference to the previous example?
'n1:n2' or 'n1:n2:n3'
Take the range of indices from "n1" to "n2" or (second form) take the range of
indices from "n1" to "n2" with step "n3". An example for the use of this format is
the previous definition of the sub-image composed of even lines.
pdl> $even = $im->slice(':,1:-1:2')
This example also demonstrates that negative indices work like they do for normal
Perl style arrays by counting backwards from the end of the dimension. If "n2" is
smaller than "n1" (in the example -1 is equivalent to index 4) the elements in the
virtual ndarray are effectively reverted with respect to its parent.
'*[n]'
Add a dummy dimension. The size of this dimension will be 1 by default or equal to
"n" if the optional numerical argument is given.
Now, this is really something a bit strange on first sight. What is a dummy
dimension? A dummy dimension inserts a dimension where there wasn't one before. How
is that done ? Well, in the case of the new dimension having size 1 it can be easily
explained by the way in which you can identify a vector (with "m" elements) with an
"(1,m)" or "(m,1)" matrix. The same holds obviously for higher dimensional objects.
More interesting is the case of a dummy dimensions of size greater than one (e.g.
"slice('*5,:')"). This works in the same way as a call to the dummy function creates
a new dummy dimension. So read on and check its explanation below.
'([n1:n2[:n3]]=i)'
[Not yet implemented ??????] With an argument like this you make generalised
diagonals. The diagonal will be dimension no. "i" of the new output ndarray and (if
optional part in brackets specified) will extend along the range of indices
specified of the respective parent ndarray's dimension. In general an argument like
this only makes sense if there are other arguments like this in the same call to
slice. The part in brackets is optional for this type of argument. All arguments of
this type that specify the same target dimension "i" have to relate to the same
number of indices in their parent dimension. The best way to explain it is probably
to give an example, here we make an ndarray that refers to the elements along the
space diagonal of its parent ndarray (a cube):
$cube = zeroes(5,5,5);
$sdiag = $cube->slice('(=0),(=0),(=0)');
The above command creates a virtual ndarray that represents the diagonal along the
parents' dimension no. 0, 1 and 2 and makes its dimension 0 (the only dimension) of
it. You use the extended syntax if the dimension sizes of the parent dimensions you
want to build the diagonal from have different sizes or you want to reverse the
sequence of elements in the diagonal, e.g.
$rect = zeroes(12,3,5,6,2);
$vpdl = $rect->slice('2:7,(0:1=1),(4),(5:4=1),(=1)');
So the elements of $vpdl will then be related to those of its parent in way we can
express as:
vpdl(i,j) = rect(i+2,j,4,5-j,j) 0<=i<5, 0<=j<2
[ work in the new index function: "$y = $x->index($c);" ???? ]
There are different kinds of assignments in PDL
The previous examples have already shown that virtual ndarrays can be used to operate on
or access portions of data of a parent ndarray. They can also be used as lvalues in
assignments (as the use of "++" in some of the examples above has already demonstrated).
For explicit assignments to the data represented by a virtual ndarray you have to use the
overloaded ".=" operator (which in this context we call propagated assignment). Why can't
you use the normal assignment operator "="?
Well, you definitely still can use the '=' operator but it wouldn't do what you want. This
is due to the fact that the '=' operator cannot be overloaded in the same way as other
assignment operators. If we tried to use '=' to try to assign data to a portion of a
physical ndarray through a virtual ndarray we wouldn't achieve the desired effect (instead
the variable representing the virtual ndarray (a reference to a blessed thingy) would
after the assignment just contain the reference to another blessed thingy which would
behave to future assignments as a "physical" copy of the original rvalue [this is actually
not yet clear and subject of discussions in the PDL developers mailing list]. In that
sense it would break the connection of the ndarray to the parent [ isn't this behaviour in
a sense the opposite of what happens in dataflow, where ".=" breaks the connection to the
parent? ].
E.g.
pdl> $line = $im->slice(':,(2)')
pdl> $line = zeroes(5);
pdl> $line++;
pdl> p $im
[
[ 1 2 3 4 5]
[ 6 7 8 9 10]
[13 14 15 16 17]
[16 17 18 19 20]
[21 22 23 24 25]
]
pdl> p $line
[1 1 1 1 1]
But using ".="
pdl> $line = $im->slice(':,(2)')
pdl> $line .= zeroes(5)
pdl> $line++
pdl> p $im
[
[ 1 2 3 4 5]
[ 6 7 8 9 10]
[ 1 1 1 1 1]
[16 17 18 19 20]
[21 22 23 24 25]
]
pdl> print $line
[1 1 1 1 1]
Also, you can substitute
pdl> $line .= 0;
for the assignment above (the zero is converted to a scalar ndarray, with no dimensions so
it can be assigned to any ndarray).
A nice feature in recent perl versions is lvalue subroutines (i.e., versions 5.6.x and
higher including all perls currently supported by PDL). That allows one to use the
slicing syntax on both sides of the assignment:
pdl> $im->slice(':,(2)') .= zeroes(5)->xvals->float
Related to the lvalue sub assignment feature is a little trap for the unwary: recent perls
introduced a "feature" which breaks PDL's use of lvalue subs for slice assignments when
running under the perl debugger, "perl -d". Under the debugger, the above usage gives an
error like: " Can't return a temporary from lvalue subroutine... " So you must use syntax
like this:
pdl> ($pdl = $im->slice(':,(2)')) .= zeroes(5)->xvals->float
which works both with and without the debugger but is arguably clumsy and awkward to read.
Note that there can be a problem with assignments like this when lvalue and rvalue
ndarrays refer to overlapping portions of data in the parent ndarray:
# revert the elements of the first line of $x
($tmp = $x->slice(':,(1)')) .= $x->slice('-1:0,(1)');
Currently, the parent data on the right side of the assignments is not copied before the
(internal) assignment loop proceeds. Therefore, the outcome of this assignment will depend
on the sequence in which elements are assigned and almost certainly not do what you
wanted. So the semantics are currently undefined for now and liable to change anytime. To
obtain the desired behaviour, use
($tmp = $x->slice(':,(1)')) .= $x->slice('-1:0,(1)')->copy;
which makes a physical copy of the slice or
($tmp = $x->slice(':,(1)')) .= $x->slice('-1:0,(1)')->sever;
which returns the same slice but severs the connection of the slice to its parent.
Other functions that manipulate dimensions
Having talked extensively about the slice function it should be noted that this is not the
only PDL indexing function. There are additional indexing functions which are also useful
(especially in the context of broadcasting which we will talk about later). Here are a
list and some examples how to use them.
"dummy"
inserts a dummy dimension of the size you specify (default 1) at the chosen location.
You can't wait to hear how that is achieved? Well, all elements with index "(X,x,Y)"
("0<=x<size_of_dummy_dim") just map to the element with index "(X,Y)" of the parent
ndarray (where "X" and "Y" refer to the group of indices before and after the location
where the dummy dimension was inserted.)
This example calculates the x coordinate of the centroid of an image (later we will
learn that we didn't actually need the dummy dimension thanks to the magic of implicit
broadcasting; but using dummy dimensions the code would also work in a broadcast-less
world; though once you have worked with PDL broadcasting you wouldn't want to live
without them again).
# centroid
($xd,$yd) = $im->dims;
$xc = sum($im*xvals(zeroes($xd))->dummy(1,$yd))/sum($im);
Let's explain how that works in a little more detail. First, the product:
$xvs = xvals(zeroes($xd));
print $xvs->dummy(1,$yd); # repeat the line $yd times
$prod = $im*xvs->dummy(1,$yd); # form the pixel-wise product with
# the repeated line of x-values
The rest is then summing the results of the pixel-wise product together and
normalizing with the sum of all pixel values in the original image thereby calculating
the x-coordinate of the "center of mass" of the image (interpreting pixel values as
local mass) which is known as the centroid of an image.
Next is a (from the point of view of memory consumption) very cheap conversion from
grey-scale to RGB, i.e. every pixel holds now a triple of values instead of a scalar.
The three values in the triple are, fortunately, all the same for a grey image, so
that our trick works well in that it maps all the three members of the triple to the
same source element:
# a cheap grey-scale to RGB conversion
$rgb = $grey->dummy(0,3)
Unfortunately this trick cannot be used to convert your old B/W photos to color ones
in the way you'd like. :(
Note that the memory usage of ndarrays with dummy dimensions is especially sensitive
to the internal representation. If the ndarray can be represented as a virtual affine
(``vaffine'') ndarray, only the control structures are stored. But if $y in
$x = zeroes(10000);
$y = $x->dummy(1,10000);
is made physical by some routine, you will find that the memory usage of your program
has suddenly grown by 100Mb.
"diagonal"
replaces two dimensions (which have to be of equal size) by one dimension that
references all the elements along the "diagonal" along those two dimensions. Here, we
have two examples which should appear familiar to anyone who has ever done some linear
algebra. Firstly, make a unity matrix:
# unity matrix
$e = zeroes(float, 3, 3); # make everything zero
($tmp = $e->diagonal(0,1)) .= 1; # set the elements along the diagonal to 1
print $e;
Or the other diagonal:
($tmp = $e->slice(':-1:0')->diagonal(0,1)) .= 2;
print $e;
(Did you notice how we used the slice function to revert the sequence of lines before
setting the diagonal of the new child, thereby setting the cross diagonal of the
parent ?) Or a mapping from the space of diagonal matrices to the field over which
the matrices are defined, the trace of a matrix:
# trace of a matrix
$trace = sum($mat->diagonal(0,1)); # sum all the diagonal elements
"xchg" and "mv"
xchg exchanges or "transposes" the two specified dimensions. A straightforward
example:
# transpose a matrix (without explicitly reshuffling data and
# making a copy)
$prod = $x x $x->xchg(0,1);
$prod should now be pretty close to the unity matrix if $x is an orthogonal matrix.
Often "xchg" will be used in the context of broadcasting but more about that later.
mv works in a similar fashion. It moves a dimension (specified by its number in the
parent) to a new position in the new child ndarray:
$y = $x->mv(4,0); # make the 5th dimension of $x the first in the
# new child $y
The difference between "xchg" and "mv" is that "xchg" only changes the position of two
dimensions with each other, whereas "mv" inserts the first dimension to the place of
second, moving the other dimensions around accordingly.
"clump"
collapses several dimensions into one. Its only argument specifies how many dimensions
of the source ndarray should be collapsed (starting from the first). An (admittedly
unrealistic) example is a 3D ndarray which holds data from a stack of image files that
you have just read in. However, the data from each image really represents a 1D time
series and has only been arranged that way because it was digitized with a frame
grabber. So to have it again as an array of time sequences you say
pdl> $seqs = $stack->clump(2)
pdl> help vars
PDL variables in package main::
Name Type Dimension Flow State Mem
----------------------------------------------------------------
$seqs Double D [8000,50] -C 0.00Kb
$stack Double D [100,80,50] P 3.05Mb
Unrealistic as it may seem, our confocal microscope software writes data (sometimes)
this way. But more often you use clump to achieve a certain effect when using implicit
or explicit broadcasting.
Calls to indexing functions can be chained
As you might have noticed in some of the examples above calls to the indexing functions
can be nicely chained since all of these functions return a newly created child object.
However, when doing extensive index manipulations in a chain be sure to keep track of what
you are doing, e.g.
$x->xchg(0,1)->mv(0,4)
moves the dimension 1 of $x to position 4 since when the second command is executed the
original dimension 1 has been moved to position 0 of the new child that calls the "mv"
function. I think you get the idea (in spite of my convoluted explanations).
Propagated assignments ('.=') and dummy dimensions
A subtlety related to indexing is the assignment to ndarrays containing dummy dimensions
of size greater than 1. These assignments (using ".=") are forbidden since several
elements of the lvalue ndarray point to the same element of the parent. As a consequence
the value of those parent elements are potentially ambiguous and would depend on the
sequence in which the implementation makes the assignments to elements. Therefore, an
assignment like this:
$x = pdl [1,2,3];
$y = $x->dummy(1,4);
$y .= yvals(zeroes(3,4));
can produce unexpected results and the results are explicitly undefined by PDL because
when PDL gets parallel computing features, the current result may well change.
From the point of view of dataflow the introduction of greater-size-than-one dummy
dimensions is regarded as an irreversible transformation (similar to the terminology in
thermodynamics) which precludes backward propagation of assignment to a parent (which you
had explicitly requested using the ".=" assignment). A similar problem to watch out for
occurs in the context of broadcasting where sometimes dummy dimensions are created
implicitly during the broadcast loop (see below).
Reasons for the parent/child (or "pointer") concept
[ this will have to wait a bit ]
XXXXX being memory efficient
XXXXX in the context of broadcasting
XXXXX very flexible and powerful way of accessing portions of ndarray data
(in much more general way than sec, etc allow)
XXXXX efficient implementation
XXXXX difference to section/at, etc.
How to make things physical again
[ XXXXX fill in later when everything has settled a bit more ]
** When needed (xsub routine interfacing C lib function)
** How achieved (->physical)
** How to test (isphysical (explain how it works currently))
** ->copy and ->sever
Broadcasting
In the previous paragraph on indexing we have already mentioned the term occasionally but
now its really time to talk explicitly about "broadcasting" with ndarrays: within the
framework of PDL it could probably be loosely defined as an implicit looping facility. It
is implicit because you don't specify anything like enclosing for-loops but rather the
loops are automatically (or 'magically') generated by PDL based on the dimensions of the
ndarrays involved. This should give you a first idea why the index/dimension manipulating
functions you have met in the previous paragraphs are especially important and useful in
the context of broadcasting. The other ingredient for broadcasting (apart from the
ndarrays involved) is a function that is broadcasting aware (generally, these are PDL::PP
compiled functions) and that the ndarrays are "broadcast" over. So much about the
terminology and now let's try to shed some light on what it all means.
Implicit broadcasting - a first example
There are two slightly different variants of broadcasting. We start with what we call
"implicit broadcasting". Let's pick a practical example that involves looping of a
function over many elements of a ndarray. Suppose we have an RGB image that we want to
convert to grey-scale. The RGB image is represented by a 3-dim ndarray "im(3,x,y)" where
the first dimension contains the three color components of each pixel and "x" and "y" are
width and height of the image, respectively. Next we need to specify how to convert a
color-triple at a given pixel into a grey-value (to be a realistic example it should
represent the relative intensity with which our color insensitive eye cells would detect
that color to achieve what we would call a natural conversion from color to grey-scale).
An approximation that works quite well is to compute the grey intensity from each RGB
triplet (r,g,b) as a weighted sum
grey-value = 77/256*r + 150/256*g + 29/256*b =
inner([77,150,29]/256, [r,g,b])
where the last form indicates that we can write this as an inner product of the 3-vector
comprising the weights for red, green and blue components with the 3-vector containing the
color components. Traditionally, we might have written a function like the following to
process the whole image:
my @dims=$im->dims;
# here normally check that first dim has correct size (3), etc
$grey=zeroes(@dims[1,2]); # make the ndarray for the resulting grey image
$w = pdl [77,150,29] / 256; # the vector of weights
for ($j=0;$j<dims[2];$j++) {
for ($i=0;$i<dims[1];$i++) {
# compute the pixel value
$tmp = inner($w,$im->slice(':,(i),(j)'));
set($grey,$i,$j,$tmp); # and set it in the grey-scale image
}
}
Now we write the same using broadcasting (noting that "inner" is a broadcasting aware
function defined in the PDL::Primitive package)
$grey = inner($im,pdl([77,150,29]/256));
We have ended up with a one-liner that automatically creates the ndarray $grey with the
right number and size of dimensions and performs the loops automatically (these loops are
implemented as fast C code in the internals of PDL). Well, we still owe you an
explanation how this 'magic' is achieved.
How does the example work ?
The first thing to note is that every function that is broadcasting aware (these are
without exception functions compiled from concise descriptions by PDL::PP, later just
called PP-functions) expects a defined (minimum) number of dimensions (we call them core
dimensions) from each of its ndarray arguments. The inner function expects two one-
dimensional (input) parameters from which it calculates a zero-dimensional (output)
parameter. We write that symbolically as "inner((n),(n),[o]())" and call it "inner"'s
signature, where n represents the size of that dimension. n being equal in the first and
second parameter means that those dimensions have to be of equal size in any call. As a
different example take the outer product which takes two 1D vectors to generate a 2D
matrix, symbolically written as "outer((n),(m),[o](n,m))". The "[o]" in both examples
indicates that this (here third) argument is an output argument. In the latter example the
dimensions of first and second argument don't have to agree but you see how they determine
the size of the two dimensions of the output ndarray.
Here is the point when broadcasting finally enters the game. If you call PP-functions with
ndarrays that have more than the required core dimensions the first dimensions of the
ndarray arguments are used as the core dimensions and the additional extra dimensions are
broadcast over. Let us demonstrate this first with our example above
$grey = inner($im,$w); # w is the weight vector from above
In this case $w is 1D and so supplied just the core dimension, $im is 3D, more
specifically "(3,x,y)". The first dimension (of size 3) is the required core dimension
that matches (as required by inner) the first (and only) dimension of $w. The second
dimension is the first broadcast dimension (of size "x") and the third is here the second
broadcast dimension (of size "y"). The output ndarray is automatically created (as
requested by setting $grey to "null" prior to invocation). The output dimensions are
obtained by appending the loop dimensions (here "(x,y)") to the core output dimensions
(here 0D) to yield the final dimensions of the auto-created ndarray (here "0D+2D=2D" to
yield a 2D output of size "(x,y)").
So the above command calls the core functionality that computes the inner product of two
1D vectors "x*y" times with $w and all 1D slices of the form "(':,(i),(j)')" of $im and
sets the respective elements of the output ndarray "$grey(i,j)" to the result of each
computation. We could write that symbolically as
$grey(0,0) = f($w,$im(:,(0),(0)))
$grey(1,0) = f($w,$im(:,(1),(0)))
.
.
.
$grey(x-2,y-1) = f($w,$im(:,(x-2),(y-1)))
$grey(x-1,y-1) = f($w,$im(:,(x-1),(y-1)))
But this is done automatically by PDL without writing any explicit Perl loops. We see
that the command really creates an output ndarray with the right dimensions and sets the
elements indeed to the result of the computation for each pixel of the input image.
When even more ndarrays and extra dimensions are involved things get a bit more
complicated. We will first give the general rules how the broadcast dimensions depend on
the dimensions of input ndarrays enabling you to figure out the dimensionality of an auto-
created output ndarray (for any given set of input ndarrays and core dimensions of the PP-
function in question). The general rules will most likely appear a bit confusing on first
sight so that we'll set out to illustrate the usage with a set of further examples (which
will hopefully also demonstrate that there are indeed many practical situations where
broadcasting comes in extremely handy).
A call for coding discipline
Before we point out the other technical details of broadcasting, please note this call for
programming discipline when using broadcasting:
In order to preserve human readability, PLEASE comment any nontrivial expression in your
code involving broadcasting. Most importantly, for any subroutine, include information at
the beginning about what you expect the dimensions to represent (or ranges of dimensions).
As a warning, look at this undocumented function and try to guess what might be going on:
sub lookup {
my ($im,$palette) = @_;
my $res;
index($palette->xchg(0,1),
$im->long->dummy(0,($palette->dim)[0]),
($res=null));
return $res;
}
Would you agree that it might be difficult to figure out expected dimensions, purpose of
the routine, etc ? (If you want to find out what this piece of code does, see below)
How to figure out the loop dimensions
There are a couple of rules that allow you to figure out number and size of loop
dimensions (and if the size of your input ndarrays comply with the broadcasting rules).
Dimensions of any ndarray argument are broken down into two groups in the following: Core
dimensions (as defined by the PP-function, see Appendix B for a list of PDL primitives)
and extra dimensions which comprises all remaining dimensions of that ndarray. For example
calling a function "func" with the signature "func((n,m),[o](n))" with an ndarray
"$x(2,4,7,1,3)" as "f($x,($o = null))" results in the semantic splitting of x's dimensions
into: core dimensions "(2,4)" and extra dimensions "(7,1,3)".
R0 Core dimensions are identified with the first N dimensions of the respective ndarray
argument (and are required). Any further dimensions are extra dimensions and used to
determine the loop dimensions.
R1 The number of (implicit) loop dimensions is equal to the maximal number of extra
dimensions taken over the set of ndarray arguments.
R2 The size of each of the loop dimensions is derived from the size of the respective
dimensions of the ndarray arguments. The size of a loop dimension is given by the
maximal size found in any of the ndarrays having this extra dimension.
R3 For all ndarrays that have a given extra dimension the size must be equal to the
size of the loop dimension (as determined by the previous rule) or 1; otherwise you
raise a runtime exception. If the size of the extra dimension in an ndarray is one
it is implicitly treated as a dummy dimension of size equal to that loop dim size
when performing the broadcast loop.
R4 If an ndarray doesn't have a loop dimension, in the broadcast loop this ndarray is
treated as if having a dummy dimension of size equal to the size of that loop
dimension.
R5 If output auto-creation is used (by setting the relevant ndarray to "PDL->null"
before invocation) the number of dimensions of the created ndarray is equal to the
sum of the number of core output dimensions + number of loop dimensions. The size of
the core output dimensions is derived from the relevant dimension of input ndarrays
(as specified in the function definition) and the sizes of the other dimensions are
equal to the size of the loop dimension it is derived from. The automatically
created ndarray will be physical (unless dataflow is in operation).
In this context, note that you can run into the problem with assignment to ndarrays
containing greater-than-one dummy dimensions (see above). Although your output ndarray(s)
didn't contain any dummy dimensions in the first place they may end up with implicitly
created dummy dimensions according to R4.
As an example, suppose we have a (here unspecified) PP-function with the signature:
func((m,n),(m,n,o),(m),[o](m,o))
and you call it with 3 ndarrays "$x(5,3,10,11)", "$y(5,3,2,10,1,12)", and "$z(5,1,11,12)"
as
func($x,$y,$z,($d=null))
then the number of loop dimensions is 3 (by "R0+R1" from $y and $z) with sizes
"(10,11,12)" (by R2); the two output core dimensions are "(5,2)" (from the signature of
func) resulting in a 5-dimensional output ndarray $c of size "(5,2,10,11,12)" (see R5) and
(the automatically created) $d is derived from "($x,$y,$z)" in a way that can be expressed
in pdl pseudo-code as
$d(:,:,i,j,k) .= func($x(:,:,i,j),$y(:,:,:,i,0,k),$z(:,0,j,k))
with 0<=i<10, 0<=j<=11, 0<=k<12
If we analyze the color to grey-scale conversion again with these rules in mind we note
another great advantage of implicit broadcasting. We can call the conversion with an
ndarray representing a pixel (im(3)), a line of rgb pixels ("im(3,x)"), a proper color
image ("im(3,x,y)") or a whole stack of RGB images ("im(3,x,y,z)"). As long as $im is of
the form "(3,...)" the automatically created output ndarray will contain the right number
of dimensions and contain the intensity data as we expect it since the loops have been
implicitly performed thanks to implicit broadcasting. You can easily convince yourself
that calling with a color pixel $grey is 0D, with a line it turns out 1D grey(x), with an
image we get "grey(x,y)" and finally we get a converted image stack "grey(x,y,z)".
Let's fill these general rules with some more life by going through a couple of further
examples. The reader may try to figure out equivalent formulations with explicit for-
looping and compare the flexibility of those routines using implicit broadcasting to the
explicit formulation. Furthermore, especially when using several broadcast dimensions it
is a useful exercise to check the relative speed by doing some benchmark tests (which we
still have to do).
First in the row is a slightly reworked centroid example, now coded with broadcasting in
mind.
# broadcast mult to calculate centroid coords, works for stacks as well
$xc = sumover(($im*xvals(($im->dims)[0]))->clump(2)) /
sumover($im->clump(2));
Let's analyze what's going on step by step. First the product:
$prod = $im*xvals(zeroes(($im->dims)[0]))
This will actually work for $im being one, two, three, and higher dimensional. If $im is
one-dimensional it's just an ordinary product (in the sense that every element of $im is
multiplied with the respective element of "xvals(...)"), if $im has more dimensions
further broadcasting is done by adding appropriate dummy dimensions to "xvals(...)"
according to R4. More importantly, the two sumover operations show a first example of how
to make use of the dimension manipulating commands. A quick look at sumover's signature
will remind you that it will only "gobble up" the first dimension of a given input
ndarray. But what if we want to really compute the sum over all elements of the first two
dimensions? Well, nothing keeps us from passing a virtual ndarray into sumover which in
this case is formed by clumping the first two dimensions of the "parent ndarray" into one.
From the point of view of the parent ndarray the sum is now computed over the first two
dimensions, just as we wanted, though sumover has just done the job as specified by its
signature. Got it ?
Another little finesse of writing the code like that: we intentionally used
"sumover($pdl->clump(2))" instead of "sum($pdl)" so that we can either pass just an image
"(x,y)" or a stack of images "(x,y,t)" into this routine and get either just one
x-coordinate or a vector of x-coordinates (of size t) in return.
Another set of common operations are what one could call "projection operations". These
operations take a N-D ndarray as input and return a (N-1)-D "projected" ndarray. These
operations are often performed with functions like sumover, prodover, minimum and maximum.
Using again images as examples we might want to calculate the maximum pixel value for each
line of an image or image stack. We know how to do that
# maxima of lines (as function of line number and time)
maximum($stack,($ret=null));
But what if you want to calculate maxima per column when implicit broadcasting always
applies the core functionality to the first dimension and broadcasts over all others? How
can we achieve that instead the core functionality is applied to the second dimension and
broadcasting is done over the others. Can you guess it? Yes, we make a virtual ndarray
that has the second dimension of the "parent ndarray" as its first dimension using the
"mv" command.
# maxima of columns (as function of column number and time)
maximum($stack->mv(1,0),($ret=null));
and calculating all the sums of sub-slices over the third dimension is now almost too easy
# sums of pixels in time (assuming time is the third dim)
sumover($stack->mv(2,0),($ret=null));
Finally, if you want to apply the operation to all elements (like max over all elements or
sum over all elements) regardless of the dimensions of the ndarray in question "clump"
comes in handy. As an example look at a definition of "sum" (summarised from
Basic/Ufunc/ufunc.pd):
sub sum {
PDL::Ufunc::sumover($name->clump(-1),($tmp=null));
return $tmp; # return a 0D ndarray
}
We have already mentioned that all basic operations support broadcasting and assignment is
no exception. So here are a couple of broadcasted assignments
pdl> $im = zeroes(byte, 10,20)
pdl> $line = exp(-rvals(10)**2/9)
# broadcasted assignment
pdl> $im .= $line # set every line of $im to $line
pdl> $im2 .= 5 # set every element of $im2 to 5
By now you probably see how it works and what it does, don't you?
To finish the examples in this paragraph here is a function to create an RGB image from
what is called a palette image. The palette image consists of two parts: an image of
indices into a color lookup table and the color lookup table itself. [ describe how it
works ] We are going to use a PP-function we haven't encoutered yet in the previous
examples. It is the aptly named index function, signature "((n),(),[o]())" (see Appendix
B) with the core functionality that "index(pdl (0,2,4,5),2,($ret=null))" will return the
element with index 2 of the first input ndarray. In this case, $ret will contain the value
4. So here is the example:
# a broadcasted index lookup to generate an RGB, or RGBA or YMCK image
# from a palette image (represented by a lookup table $palette and
# an color-index image $im)
# you can say just dummy(0) since the rules of broadcasting make it fit
pdl> index($palette->xchg(0,1),
$im->long->dummy(0,($palette->dim)[0]),
($res=null));
Let's go through it and explain the steps involved. Assuming we are dealing with an RGB
lookup-table $palette is of size "(3,x)". First we exchange the dimensions of the palette
so that looping is done over the first dimension of $palette (of size 3 that represent r,
g, and b components). Now looking at $im, we add a dummy dimension of size equal to the
length of the number of components (in the case we are discussing here we could have just
used the number 3 since we have 3 color components). We can use a dummy dimension since
for red, green and blue color components we use the same index from the original image,
e.g. assuming a certain pixel of $im had the value 4 then the lookup should produce the
triple
[palette(0,4),palette(1,4),palette(2,4)]
for the new red, green and blue components of the output image. Hopefully by now you have
some sort of idea what the above piece of code is supposed to do (it is often actually
quite complicated to describe in detail how a piece of broadcasting code works; just go
ahead and experiment a bit to get a better feeling for it).
If you have read the broadcasting rules carefully, then you might have noticed that we
didn't have to explicitly state the size of the dummy dimension that we created for $im;
when we create it with size 1 (the default) the rules of broadcasting make it
automatically fit to the desired size (by rule R3, in our example the size would be 3
assuming a palette of size "(3,x)"). Since situations like this do occur often in practice
this is actually why rule R3 has been introduced (the part that makes dimensions of size 1
fit to the broadcast loop dim size). So we can just say
pdl> index($palette->xchg(0,1),$im->long->dummy(0),($res=null));
Again, you can convince yourself that this routine will create the right output if called
with a pixel ($im is 0D), a line ($im is 1D), an image ($im is 2D), ..., an RGB lookup
table (palette is "(3,x)") and RGBA lookup table (palette is "(4,x)", see e.g. OpenGL).
This flexibility is achieved by the rules of broadcasting which are made to do the right
thing in most situations.
To wrap it all up once again, the general idea is as follows. If you want to achieve
looping over certain dimensions and have the core functionality applied to another
specified set of dimensions you use the dimension manipulating commands to create a (or
several) virtual ndarray(s) so that from the point of view of the parent ndarray(s) you
get what you want (always having the signature of the function in question and R1-R5 in
mind!). Easy, isn't it ?
Output auto-creation and PP-function calling conventions
At this point we have to divert to some technical detail that has to do with the general
calling conventions of PP-functions and the automatic creation of output arguments.
Basically, there are two ways of invoking PDL routines, namely
$result = func($x,$y);
and
func($x,$y,$result);
If you are only using implicit broadcasting then the output variable can be automatically
created by PDL. You flag that to the PP-function by setting the output argument to a
special kind of ndarray that is returned from a call to the function "PDL->null" that
returns an essentially "empty" ndarray (for those interested in details there is a flag in
the C pdl structure for this). The dimensions of the created ndarray are determined by the
rules of implicit broadcasting: the first dimensions are the core output dimensions to
which the broadcasting dimensions are appended (which are in turn determined by the
dimensions of the input ndarrays as described above). So you can say
func($x,$y,($result=PDL->null));
or
$result = func($x,$y)
which are exactly equivalent.
Be warned that you can not use output auto-creation when using explicit broadcasting (for
reasons explained in the following section on explicit broadcasting, the second variant of
broadcasting).
In "tight" loops you probably want to avoid the implicit creation of a temporary ndarray
in each step of the loop that comes along with the "functional" style but rather say
# create output ndarray of appropriate size only at first invocation
$result = null;
for (0...$n) {
func($x,$y,$result); # in all but the first invocation $result
func2($y); # is defined and has the right size to
# take the output provided $y's dims don't change
twiddle($result,$x); # do something from $result to $x for iteration
}
The take-home message of this section once more: be aware of the limitation on output
creation when using explicit broadcasting.
Explicit broadcasting
Having so far only talked about the first flavour of broadcasting it is now about time to
introduce the second variant. Instead of shuffling around dimensions all the time and
relying on the rules of implicit broadcasting to get it all right you sometimes might want
to specify in a more explicit way how to perform the broadcast loop. It is probably not
too surprising that this variant of the game is called explicit broadcasting. Now, before
we create the wrong impression: it is not either implicit or explicit; the two flavours do
mix. But more about that later.
The two most used functions with explicit broadcasting are broadcast and unbroadcast. We
start with an example that illustrates typical usage of the former:
[ # ** this is the worst possible example to start with ]
# but can be used to show that $mat += $line is different from
# $mat->broadcast(0) += $line
# explicit broadcasting to add a vector to each column of a matrix
pdl> $mat = zeroes(4,3)
pdl> $line = pdl (3.1416,2,-2)
pdl> ($tmp = $mat->broadcast(0)) += $line
In this example, "$mat->broadcast(0)" tells PDL that you want the second dimension of this
ndarray to be broadcast over first leading to a broadcast loop that can be expressed as
for (j=0; j<3; j++) {
for (i=0; i<4; i++) {
mat(i,j) += src(j);
}
}
"broadcast" takes a list of numbers as arguments which explicitly specify which dimensions
to broadcast over first. With the introduction of explicit broadcasting the dimensions of
an ndarray are conceptually split into three different groups the latter two of which we
have already encountered: broadcast dimensions, core dimensions and extra dimensions.
Conceptually, it is best to think of those dimensions of an ndarray that have been
specified in a call to "broadcast" as being taken away from the set of normal dimensions
and put on a separate stack. So assuming we have an ndarray "x(4,7,2,8)" saying
$y = $x->broadcast(2,1)
creates a new virtual ndarray of dimension "y(4,8)" (which we call the remaining dims)
that also has 2 broadcast dimensions of size "(2,7)". For the purposes of this document we
write that symbolically as "y(4,8){2,7}". An important difference to the previous examples
where only implicit broadcasting was used is the fact that the core dimensions are matched
against the remaining dimensions which are not necessarily the first dimensions of the
ndarray. We will now specify how the presence of broadcast dimensions changes the rules
R1-R5 for broadcast loops (which apply to the special case where none of the ndarray
arguments has any broadcast dimensions).
T0 Core dimensions are matched against the first n remaining dimensions of the ndarray
argument (note the difference to R1). Any further remaining dimensions are extra
dimensions and are used to determine the implicit loop dimensions.
T1a The number of implicit loop dimensions is equal to the maximal number of extra
dimensions taken over the set of ndarray arguments.
T1b The number of explicit loop dimensions is equal to the maximal number of broadcast
dimensions taken over the set of ndarray arguments.
T1c The total number of loop dimensions is equal to the sum of explicit loop dimensions
and implicit loop dimensions. In the broadcast loop, explicit loop dimensions are
broadcasted over first followed by implicit loop dimensions.
T2 The size of each of the loop dimensions is derived from the size of the respective
dimensions of the ndarray arguments. It is given by the maximal size found in any
ndarrays having this broadcast dimension (for explicit loop dimensions) or extra
dimension (for implicit loop dimensions).
T3 This rule applies to any explicit loop dimension as well as any implicit loop
dimension. For all ndarrays that have a given broadcast/extra dimension the size must
be equal to the size of the respective explicit/implicit loop dimension or 1;
otherwise you raise a runtime exception. If the size of a broadcast/extra dimension of
an ndarray is one it is implicitly treated as a dummy dimension of size equal to the
explicit/implicit loop dimension.
T4 If an ndarray doesn't have a broadcast/extra dimension that corresponds to an
explicit/implicit loop dimension, in the broadcast loop this ndarray is treated as if
having a dummy dimension of size equal to the size of that loop dimension.
T4a All ndarrays that do have broadcast dimensions must have the same number of broadcast
dimensions.
T5 Output auto-creation cannot be used if any of the ndarray arguments has any broadcast
dimensions. Otherwise R5 applies.
The same restrictions apply with regard to implicit dummy dimensions (created by
application of T4) as already mentioned in the section on implicit broadcasting: if any of
the output ndarrays has an (explicit or implicitly created) greater-than-one dummy
dimension a runtime exception will be raised.
Let us demonstrate these rules at work in a generic case. Suppose we have a (here
unspecified) PP-function with the signature:
func((m,n),(m),(),[o](m))
and you call it with 3 ndarrays "a(5,3,10,11)", "b(3,5,10,1,12)", "c(10)" and an output
ndarray "d(3,11,5,10,12)" (which can here not be automatically created) as
func($x->broadcast(1,3),$y->broadcast(0,3),$c,$d->broadcast(0,1))
From the signature of func and the above call the ndarrays split into the following groups
of core, extra and broadcast dimensions (written in the form "pdl(core dims){broadcast
dims}[extra dims]"):
a(5,10){3,11}[] b(5){3,1}[10,12] c(){}[10] d(5){3,11}[10,12]
With this to help us along (it is in general helpful to write the arguments down like this
when you start playing with broadcasting and want to keep track of what is going on) we
further deduce that the number of explicit loop dimensions is 2 (by T1b from $a and $b)
with sizes "(3,11)" (by T2); 2 implicit loop dimensions (by T1a from $b and $d) of size
"(10,12)" (by T2) and the elements of are computed from the input ndarrays in a way that
can be expressed in pdl pseudo-code as
for (l=0;l<12;l++)
for (k=0;k<10;k++)
for (j=0;j<11;j++) effect of treating it as dummy dim (index j)
for (i=0;i<3;i++) |
d(i,j,:,k,l) = func(a(:,i,:,j),b(i,:,k,0,l),c(k))
Ugh, this example was really not easy in terms of bookkeeping. It serves mostly as an
example how to figure out what's going on when you encounter a complicated looking
expression. But now it is really time to show that broadcasting is useful by giving some
more of our so called "practical" examples.
[ The following examples will need some additional explanations in the future. For the
moment please try to live with the comments in the code fragments. ]
Example 1:
*** inverse of matrix represented by eigvecs and eigvals
** given a symmetrical matrix M = A^T x diag(lambda_i) x A
** => inverse M^-1 = A^T x diag(1/lambda_i) x A
** first $tmp = diag(1/lambda_i)*A
** then A^T * $tmp by broadcasted inner product
# index handling so that matrices print correct under pdl
$inv .= $evecs*0; # just copy to get appropriately sized output
$tmp .= $evecs; # initialise, no back-propagation
($tmp2 = $tmp->broadcast(0)) /= $evals; # broadcasted division
# and now a matrix multiplication in disguise
PDL::Primitive::inner($evecs->xchg(0,1)->broadcast(-1,1),
$tmp->broadcast(0,-1),
$inv->broadcast(0,1));
# alternative for matrix mult using implicit broadcasting,
# first xchg only for transpose
PDL::Primitive::inner($evecs->xchg(0,1)->dummy(1),
$tmp->xchg(0,1)->dummy(2),
($inv=null));
Example 2:
# outer product by broadcasted multiplication
# stress that we need to do it with explicit call to my_biop1
# when using explicit broadcasting
$res=zeroes(($x->dims)[0],($y->dims)[0]);
my_biop1($x->broadcast(0,-1),$y->broadcast(-1,0),$res->(0,1),"*");
# similar thing by implicit broadcasting with auto-created ndarray
$res = $x->dummy(1) * $y->dummy(0);
Example 3:
# different use of broadcast and unbroadcast to shuffle a number of
# dimensions in one go without lots of calls to ->xchg and ->mv
# use broadcast/unbroadcast to shuffle dimensions around
# just try it out and compare the child ndarray with its parent
$trans = $x->broadcast(4,1,0,3,2)->unbroadcast;
Example 4:
# calculate a couple of bounding boxes
# $bb will hold BB as [xmin,xmax],[ymin,ymax],[zmin,zmax]
# we use again broadcast and unbroadcast to shuffle dimensions around
pdl> $bb = zeroes(double, 2,3 );
pdl> minimum($vertices->broadcast(0)->clump->unbroadcast(1), $bb->slice('(0),:'));
pdl> maximum($vertices->broadcast(0)->clump->unbroadcast(1), $bb->slice('(1),:'));
Example 5:
# calculate a self-rationed (i.e. self normalized) sequence of images
# uses explicit broadcasting and an implicitly broadcasted division
$stack = read_image_stack();
# calculate the average (per pixel average) of the first $n+1 images
$aver = zeroes([stack->dims]->[0,1]); # make the output ndarray
sumover($stack->slice(":,:,0:$n")->broadcast(0,1),$aver);
$aver /= ($n+1);
$stack /= $aver; # normalize the stack by doing a broadcasted division
# implicit versus explicit
# alternatively calculate $aver with implicit broadcasting and auto-creation
sumover($stack->slice(":,:,0:$n")->mv(2,0),($aver=null));
$aver /= ($n+1);
#
Implicit versus explicit broadcasting
In this paragraph we are going to illustrate when explicit broadcasting is preferable over
implicit broadcasting and vice versa. But then again, this is probably not the best way of
putting the case since you already know: the two flavours do mix. So, it's more about how
to get the best of both worlds and, anyway, in the best of Perl traditions: TIMTOWTDI !
[ Sorry, this still has to be filled in in a later release; either refer to above examples
or choose some new ones ]
Finally, this may be a good place to justify all the technical detail we have been going
on about for a couple of pages: why broadcasting ?
Well, code that uses broadcasting should be (considerably) faster than code that uses
explicit for-loops (or similar Perl constructs) to achieve the same functionality.
Especially on supercomputers (with vector computing facilities/parallel processing) PDL
broadcasting is implemented in a way that takes advantage of the additional facilities of
these machines. Furthermore, it is a conceptually simple construct (though technical
details might get involved at times) and can greatly reduce the syntactical complexity of
PDL code (but keep the admonition for documentation in mind). Once you are comfortable
with the broadcasting way of thinking (and coding) it shouldn't be too difficult to
understand code that somebody else has written than (provided they gave you an idea what
expected input dimensions are, etc.). As a general tip to increase the performance of your
code: if you have to introduce a loop into your code try to reformulate the problem so
that you can use broadcasting to perform the loop (as with anything there are exceptions
to this rule of thumb; but the authors of this document tend to think that these are rare
cases ;).
PDL::PP
An easy way to define functions that are aware of indexing and broadcasting (and the universe
and everything)
PDL:PP is part of the PDL distribution. It is used to generate functions that are aware of
indexing and broadcasting rules from very concise descriptions. It can be useful for you
if you want to write your own functions or if you want to interface functions from an
external library so that they support indexing and broadcasting (and maybe dataflow as
well, see PDL::Dataflow). For further details check PDL::PP.
Appendix A
Affine transformations - a special class of simple and powerful transformations
[ This is also something to be added in future releases. Do we already have the general
make_affine routine in PDL ? It is possible that we will reference another appropriate man
page from here ]
Appendix B
signatures of standard PDL::PP compiled functions
A selection of signatures of PDL primitives to show how many dimensions PP compiled
functions gobble up (and therefore you can figure out what will be broadcasted over). Most
of those functions are the basic ones defined in "primitive.pd"
# functions in primitive.pd
#
sumover ((n),[o]())
prodover ((n),[o]())
axisvalues ((n)) inplace
inner ((n),(n),[o]())
outer ((n),(m),[o](n,m))
innerwt ((n),(n),(n),[o]())
inner2 ((m),(m,n),(n),[o]())
inner2t ((j,n),(n,m),(m,k),[o]())
index (1D,0D,[o])
minimum (1D,[o])
maximum (1D,[o])
wstat ((n),(n),(),[o],())
assgn ((),())
# basic operations
binary operations ((),(),[o]())
unary operations ((),[o]())
AUTHOR & COPYRIGHT
Copyright (C) 1997 Christian Soeller (c.soeller@auckland.ac.nz) & Tuomas J. Lukka
(lukka@fas.harvard.edu). All rights reserved. Although destined for release as a man page
with the standard PDL distribution, it is not public domain. Permission is granted to
freely distribute verbatim copies of this document provided that no modifications outside
of formatting be made, and that this notice remain intact. You are permitted and
encouraged to use its code and derivatives thereof in your own source code for fun or for
profit as you see fit.