DragonFly On-Line Manual Pages
INDEXING(1) User Contributed Perl Documentation INDEXING(1)
NAME
PDL::Indexing - Introduction to indexing and slicing piddles.
OVERVIEW
This man page should serve as a first tutorial on the indexing and
threading features of PDL.
Like all vectorized languages, PDL automates looping over arrays using
a variant of mathematical vector notation. The automatic looping is
called "threading", in part because ultimately PDL will implement
parallel processing to speed up the loops.
A lot of the flexibility and power of PDL relies on the indexing and
threading features of the Perl extension. Indexing allows access to
the data of a piddle in a very flexible way. Threading provides
efficient vectorization of simple operations.
The values of a piddle 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 PDLs. From the perldl (or pdl2) shell,
you can find out more about each method by typing "?" followed by the
method name.
Dimension lists
A piddle (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 piddle
has a positive integer size. The "perl" interpreter treats each piddle
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 a piddle as a Perl list and otherwise
determine the size of a piddle with several methods. The important
ones are:
nelem - the total number of elements in a PDL
ndims - returns the number of dimensions in a PDL
dims - returns the dimension list of a PDL as a Perl list
dim - returns the size of a particular dimension of a PDL
Indexing and Dataflow
PDL maintains a notion of "dataflow" between a piddle and indexed
subfields of that piddle. When you produce an indexed subfield or
single element of a parent piddle, 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 PDL:
copy - forces an explicit copy of a PDL
sever - breaks the dataflow connection between a PDL and its parents
(if any)
Threading and Dimension Order
Most PDL operations act on the first few dimensions of their piddle
arguments. For example, "sumover" sums all elements along the first
dimension in the list (dimension 0). If you feed in a three-
dimensional piddle, then the first dimension is considered the "active"
dimension and the later dimensions are "thread" dimensions because they
are simply looped over. There are several ways to transpose or re-
order the dimension list of a PDL. 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
o document Perl level threading
o threadids
o update and correct description of slice
o new functions in slice.pd (affine, lag, splitdim)
o reworking of paragraph on explicit threading
Indexing and threading 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 a pdl object in a very flexible way. Threading provides
efficient implicit looping functionality (since the loops are
implemented as optimized C code).
Pdl objects (later often called "pdls") 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 pdls.
pdls can have children
Central to many of the indexing capabilities of PDL are the relation of
"parent" and "child" between pdls. Many of the indexing commands create
a new pdl from an existing pdl. The new pdl is the "child" and the old
one is the "parent". The data of the new pdl is defined by a
transformation that specifies how to generate (compute) its data from
the parent's data. The relation between the child pdl 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 pdl by saying "$a->doflow". For
further information about data flow check the dataflow man page.)
Another way to interpret the pdls 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 piddle a
``virtual'' piddle, which only stores a pointer to the parent and
access information so that routines which use the child piddle actually
directly access the data in the parent. If the virtual piddle is given
to a routine which cannot use it, PDL transparently physicalizes the
virtual piddle 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 piddle are determined by a
linear transformation (+ constant) from the indices of the child piddle
result in virtual piddles. All other indexing routines (e.g.
"->index(...)") result in physical piddles. All routines compiled by
PP can accept affine piddles (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
$a->index(...) .= 5;
$a->slice(...) .= 5;
change the data in $a. The affinity does, however, have a significant
impact on memory usage and performance.
Slicing pdls
Probably the most important application of the concept of parent/child
pdls is the representation of rectangular slices of a physical pdl by a
virtual pdl. Having talked long enough about concepts let's get more
specific. Suppose we are working with a 2D pdl representing a 5x5 image
(its 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 pdls in the interactive
"perldl" or "pdl2" shell that comes with PDL. ]
Now suppose we want to create a 1-D pdl that just references one line
of the image, say line 2; or a pdl 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 a pdl
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" pdls are children of $im or in the other (largely
equivalent) interpretation pointers to data of $im. Operations on
those virtual pdls 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 pdls change the
parent physical pdl and vice versa (however, the basic "=" assignment
doesn't, use ".=" to obtain that effect. See below for the reasons).
The virtual child pdls are something like "live links" to the
"original" parent pdl. 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 pdl) 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 = $a->slice('ind0,ind1...')
where "ind0" specifies what to do with index No 0 of the pdl $a, 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 pdl 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
pdl (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
pdl 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 pdl and (if optional part in brackets specified)
will extend along the range of indices specified of the
respective parent pdl'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 a pdl that refers to
the elements along the space diagonal of its parent pdl (a cube):
$cube = zeroes(5,5,5);
$sdiag = $cube->slice('(=0),(=0),(=0)');
The above command creates a virtual pdl 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 $vpdl will then be related to those of its
parent in way we can express as:
vpdl(i,j) = rect(i+2,j,4,5-j,j) 0<=i<5, 0<=j<2
[ work in the new index function: "$b = $a->index($c);" ???? ]
There are different kinds of assignments in PDL
The previous examples have already shown that virtual pdls can be used
to operate on or access portions of data of a parent pdl. 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 pdl 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 pdl through
a virtual pdl we wouldn't achieve the desired effect (instead the
variable representing the virtual pdl (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 pdl 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 piddle,
with no dimensions so it can be assigned to any piddle).
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 pdls refer to overlapping portions of data in the parent
pdl:
# revert the elements of the first line of $a
($tmp = $a->slice(':,(1)')) .= $a->slice('-1:0,(1)');
Currently, the parent data on the right side of the assignments is not
copied before the (internal) assignment loop proceeds. Therefore, the
outcome of this assignment will depend on the sequence in which
elements are assigned and almost certainly not do what you wanted. So
the semantics are currently undefined for now and liable to change
anytime. To obtain the desired behaviour, use
($tmp = $a->slice(':,(1)')) .= $a->slice('-1:0,(1)')->copy;
which makes a physical copy of the slice or
($tmp = $a->slice(':,(1)')) .= $a->slice('-1:0,(1)')->sever;
which returns the same slice but severs the connection of the slice to
its parent.
Other functions that manipulate dimensions
Having talked extensively about the slice function it should be noted
that this is not the only PDL indexing function. There are additional
indexing functions which are also useful (especially in the context of
threading which we will talk about later). Here are a list and some
examples how to use them.
"dummy"
inserts a dummy dimension of the size you specify (default 1) at
the chosen location. You can't wait to hear how that is achieved?
Well, all elements with index "(X,x,Y)" ("0<=x<size_of_dummy_dim")
just map to the element with index "(X,Y)" of the parent pdl (where
"X" and "Y" refer to the group of indices before and after the
location where the dummy dimension was inserted.)
This example calculates the x coordinate of the centroid of an
image (later we will learn that we didn't actually need the dummy
dimension thanks to the magic of implicit threading; but using
dummy dimensions the code would also work in a thread-less world;
though once you have worked with PDL threads you wouldn't want to
live without them again).
# centroid
($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 piddles with dummy dimensions is
especially sensitive to the internal representation. If the piddle
can be represented as a virtual affine (``vaffine'') piddle, only
the control structures are stored. But if $b in
$a = zeroes(10000);
$b = $a->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 = $a x $a->xchg(0,1);
$prod should now be pretty close to the unity matrix if $a is an
orthogonal matrix. Often "xchg" will be used in the context of
threading 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 pdl:
$b = $a->mv(4,0); # make the 5th dimension of $a the first in the
# new child $b
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 pdl should be collapsed (starting
from the first). An (admittedly unrealistic) example is a 3D pdl
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 threading.
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.
$a->xchg(0,1)->mv(0,4)
moves the dimension 1 of $a 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 sublety related to indexing is the assignment to pdls containing
dummy dimensions of size greater than 1. These assignments (using ".=")
are forbidden since several elements of the lvalue pdl 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:
$a = pdl [1,2,3];
$b = $a->dummy(1,4);
$b .= 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 threading where sometimes dummy dimensions
are created implicitly during the thread 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 threading
XXXXX very flexible and powerful way of accessing portions of pdl 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
Threading
In the previous paragraph on indexing we have already mentioned the
term occasionally but now its really time to talk explicitly about
"threading" with pdls. The term threading has many different meanings
in different fields of computing. 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 pdls 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 threading. The other ingredient for threading (apart
from the pdls involved) is a function that is threading aware
(generally, these are PDL::PP compiled functions) and that the pdls are
"threaded" over. So much about the terminology and now let's try to
shed some light on what it all means.
Implicit threading - a first example
There are two slightly different variants of threading. We start with
what we call "implicit threading". Let's pick a practical example that
involves looping of a function over many elements of a pdl. Suppose we
have an RGB image that we want to convert to grey-scale. The RGB image
is represented by a 3-dim pdl "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 pdl 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 threading (noting that "inner" is a
threading 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 pdl
$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 threading 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 pdl 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
pdl.
Here is the point when threading finally enters the game. If you call
PP-functions with pdls that have more than the required core dimensions
the first dimensions of the pdl arguments are used as the core
dimensions and the additional extra dimensions are threaded 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 thread
dimension (of size "x") and the third is here the second thread
dimension (of size "y"). The output pdl 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 pdl (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 $im and sets the respective elements of
the output pdl "$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,$im(:,(x-1),(y-1)))
But this is done automatically by PDL without writing any explicit Perl
loops. We see that the command really creates an output pdl with the
right dimensions and sets the elements indeed to the result of the
computation for each pixel of the input image.
When even more pdls and extra dimensions are involved things get a bit
more complicated. We will first give the general rules how the thread
dimensions depend on the dimensions of input pdls enabling you to
figure out the dimensionality of an auto-created output pdl (for any
given set of input pdls and core dimensions of the PP-function in
question). The general rules will most likely appear a bit confusing on
first sight so that we'll set out to illustrate the usage with a set of
further examples (which will hopefully also demonstrate that there are
indeed many practical situations where threading comes in extremely
handy).
A call for coding discipline
Before we point out the other technical details of threading, please
note this call for programming discipline when using threading:
In order to preserve human readability, PLEASE comment any nontrivial
expression in your code involving threading. Most importantly, for any
subroutine, include information at the beginning about what you expect
the dimensions to represent (or ranges of dimensions).
As a warning, look at this undocumented function and try to guess what
might be going on:
sub lookup {
my ($im,$palette) = @_;
my $res;
index($palette->xchg(0,1),
$im->long->dummy(0,($palette->dim)[0]),
($res=null));
return $res;
}
Would you agree that it might be difficult to figure out expected
dimensions, purpose of the routine, etc ? (If you want to find out
what this piece of code does, see below)
How to figure out the loop dimensions
There are a couple of rules that allow you to figure out number and
size of loop dimensions (and if the size of your input pdls comply with
the threading rules). Dimensions of any pdl argument are broken down
into two groups in the following: Core dimensions (as defined by the
PP-function, see Appendix B for a list of PDL primitives) and extra
dimensions which comprises all remaining dimensions of that pdl. For
example calling a function "func" with the signature
"func((n,m),[o](n))" with a pdl "a(2,4,7,1,3)" as "f($a,($o = null))"
results in the semantic splitting of a'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 pdl 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 pdl arguments.
R2 The size of each of the loop dimensions is derived from the size
of the respective dimensions of the pdl arguments. The size of a
loop dimension is given by the maximal size found in any of the
pdls having this extra dimension.
R3 For all pdls 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 a pdl is one it is implicitly
treated as a dummy dimension of size equal to that loop dim size
when performing the thread loop.
R4 If a pdl doesn't have a loop dimension, in the thread loop this
pdl 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 pdl to
"PDL->null" before invocation) the number of dimensions of the
created pdl 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
pdls (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 pdl will be
physical (unless dataflow is in operation).
In this context, note that you can run into the problem with assignment
to pdls containing greater-than-one dummy dimensions (see above).
Although your output pdl(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 pdls "a(5,3,10,11)", "b(5,3,2,10,1,12)", and
"c(5,1,11,12)" as
func($a,$b,$c,($d=null))
then the number of loop dimensions is 3 (by "R0+R1" from $b and $c)
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 pdl $c of size "(5,2,10,11,12)" (see R5) and (the automatically
created) $d is derived from "($a,$b,$c)" in a way that can be expressed
in pdl pseudo-code as
$d(:,:,i,j,k) .= func($a(:,:,i,j),$b(:,:,:,i,0,k),$c(:,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 threading. We can
call the conversion with a pdl 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 pdl 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
threading. 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 threading to the explicit formulation.
Furthermore, especially when using several thread 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 threading in mind.
# threaded 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 $im has more dimensions further
threading is done by adding appropriate dummy dimensions to
"xvals(...)" according to R4. More importantly, the two sumover
operations show a first example of how to make use of the dimension
manipulating commands. A quick look at sumover's signature will remind
you that it will only "gobble up" the first dimension of a given input
pdl. But what if we want to really compute the sum over all elements of
the first two dimensions? Well, nothing keeps us from passing a virtual
pdl into sumover which in this case is formed by clumping the first two
dimensions of the "parent pdl" into one. From the point of view of the
parent pdl the sum is now computed over the first two dimensions, just
as we wanted, though sumover has just done the job as specified by its
signature. Got it ?
Another little finesse of writing the code like that: we intentionally
used "sumover($pdl->clump(2))" instead of "sum($pdl)" so that we can
either pass just an image "(x,y)" or a stack of images "(x,y,t)" into
this routine and get either just one x-coordiante or a vector of
x-coordinates (of size t) in return.
Another set of common operations are what one could call "projection
operations". These operations take a N-D pdl as input and return a
(N-1)-D "projected" pdl. These operations are often performed with
functions like sumover, prodover, minimum and maximum. Using again
images as examples we might want to calculate the maximum pixel value
for each line of an image or image stack. We know how to do that
# maxima of lines (as function of line number and time)
maximum($stack,($ret=null));
But what if you want to calculate maxima per column when implicit
threading always applies the core functionality to the first dimension
and threads over all others? How can we achieve that instead the core
functionality is applied to the second dimension and threading is done
over the others. Can you guess it? Yes, we make a virtual pdl that has
the second dimension of the "parent pdl" as its first dimension using
the "mv" command.
# maxima of columns (as function of column number and time)
maximum($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),($ret=null));
Finally, if you want to apply the operation to all elements (like max
over all elements or sum over all elements) regardless of the
dimensions of the pdl in question "clump" comes in handy. As an example
look at the definition of "sum" (as defined in "Ufunc.pm"):
sub sum {
PDL::Ufunc::sumover($name->clump(-1),($tmp=null));
return $tmp->at(); # return a Perl number, not a 0D pdl
}
We have already mentioned that all basic operations support threading
and assignment is no exception. So here are a couple of threaded
assignments
pdl> $im = zeroes(byte, 10,20)
pdl> $line = exp(-rvals(10)**2/9)
# threaded 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 pdl. In this case, $ret will contain the value 4.
So here is the example:
# a threaded 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 $im)
# you can say just dummy(0) since the rules of threading make it fit
pdl> index($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 threading code works; just go ahead
and experiment a bit to get a better feeling for it).
If you have read the threading 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 $im; when we create it with size 1 (the
default) the rules of threading make it automatically fit to the
desired size (by rule R3, in our example the size would be 3 assuming a
palette of size "(3,x)"). Since situations like this do occur often in
practice this is actually why rule R3 has been introduced (the part
that makes dimensions of size 1 fit to the thread loop dim size). So we
can just say
pdl> index($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 threading 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
pdl(s) so that from the point of view of the parent pdl(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
$result = func($a,$b);
and
func($a,$b,$result);
If you are only using implicit threading 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 pdl that is returned
from a call to the function "PDL->null" that returns an essentially
"empty" pdl (for those interested in details there is a flag in the C
pdl structure for this). The dimensions of the created pdl are
determined by the rules of implicit threading: the first dimensions are
the core output dimensions to which the threading dimensions are
appended (which are in turn determined by the dimensions of the input
pdls as described above). So you can say
func($a,$b,($result=PDL->null));
or
$result = func($a,$b)
which are exactly equivalent.
Be warned that you can not use output auto-creation when using explicit
threading (for reasons explained in the following section on explicit
threading, the second variant of threading).
In "tight" loops you probably want to avoid the implicit creation of a
temporary pdl in each step of the loop that comes along with the
"functional" style but rather say
# create output pdl of appropriate size only at first invocation
$result = null;
for (0...$n) {
func($a,$b,$result); # in all but the first invocation $result
func2($b); # is defined and has the right size to
# take the output provided $b's dims don't change
twiddle($result,$a); # do something from $result to $a for iteration
}
The take-home message of this section once more: be aware of the
limitation on output creation when using explicit threading.
Explicit threading
Having so far only talked about the first flavour of threading 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
threading to get it all right you sometimes might want to specify in a
more explicit way how to perform the thread loop. It is probably not
too surprising that this variant of the game is called explicit
threading. 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 threading are thread and
unthread. 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->thread(0) += $line
# explicit threading 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->thread(0)) += $line
In this example, "$mat->thread(0)" tells PDL that you want the second
dimension of this pdl to be threaded over first leading to a thread
loop that can be expressed as
for (j=0; j<3; j++) {
for (i=0; i<4; i++) {
mat(i,j) += src(j);
}
}
"thread" takes a list of numbers as arguments which explicitly specify
which dimensions to thread over first. With the introduction of
explicit threading the dimensions of a pdl are conceptually split into
three different groups the latter two of which we have already
encountered: thread dimensions, core dimensions and extra dimensions.
Conceptually, it is best to think of those dimensions of a pdl that
have been specified in a call to "thread" as being taken away from the
set of normal dimensions and put on a separate stack. So assuming we
have a pdl "a(4,7,2,8)" saying
$b = $a->thread(2,1)
creates a new virtual pdl of dimension "b(4,8)" (which we call the
remaining dims) that also has 2 thread dimensions of size "(2,7)". For
the purposes of this document we write that symbolically as
"b(4,8){2,7}". An important difference to the previous examples where
only implicit threading was used is the fact that the core dimensions
are matched against the remaining dimensions which are not necessarily
the first dimensions of the pdl. We will now specify how the presence
of thread dimensions changes the rules R1-R5 for thread loops (which
apply to the special case where none of the pdl arguments has any
thread dimensions).
T0 Core dimensions are matched against the first n remaining
dimensions of the pdl 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 pdl arguments.
T1b The number of explicit loop dimensions is equal to the maximal
number of thread dimensions taken over the set of pdl arguments.
T1c The total number of loop dimensions is equal to the sum of explicit
loop dimensions and implicit loop dimensions. In the thread loop,
explicit loop dimensions are threaded 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 pdl arguments. It is given by the
maximal size found in any pdls having this thread 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 pdls that have a given
thread/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 thread/extra dimension
of a pdl is one it is implicitly treated as a dummy dimension of
size equal to the explicit/implicit loop dimension.
T4 If a pdl doesn't have a thread/extra dimension that corresponds to
an explicit/implicit loop dimension, in the thread loop this pdl is
treated as if having a dummy dimension of size equal to the size of
that loop dimension.
T4a All pdls that do have thread dimensions must have the same number
of thread dimensions.
T5 Output auto-creation cannot be used if any of the pdl arguments has
any thread 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 threading: if any of the output pdls 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 pdls "a(5,3,10,11)", "b(3,5,10,1,12)", "c(10)"
and an output pdl "d(3,11,5,10,12)" (which can here not be
automatically created) as
func($a->thread(1,3),$b->thread(0,3),$c,$d->thread(0,1))
From the signature of func and the above call the pdls split into the
following groups of core, extra and thread dimensions (written in the
form "pdl(core dims){thread 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 threading 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
pdls 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 threading 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 threaded 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->thread(0)) /= $evals; # threaded division
# and now a matrix multiplication in disguise
PDL::Primitive::inner($evecs->xchg(0,1)->thread(-1,1),
$tmp->thread(0,-1),
$inv->thread(0,1));
# alternative for matrix mult using implicit threading,
# first xchg only for transpose
PDL::Primitive::inner($evecs->xchg(0,1)->dummy(1),
$tmp->xchg(0,1)->dummy(2),
($inv=null));
Example 2:
# outer product by threaded multiplication
# stress that we need to do it with explicit call to my_biop1
# when using explicit threading
$res=zeroes(($a->dims)[0],($b->dims)[0]);
my_biop1($a->thread(0,-1),$b->thread(-1,0),$res->(0,1),"*");
# similar thing by implicit threading with auto-created pdl
$res = $a->dummy(1) * $b->dummy(0);
Example 3:
# different use of thread and unthread to shuffle a number of
# dimensions in one go without lots of calls to ->xchg and ->mv
# use thread/unthread to shuffle dimensions around
# just try it out and compare the child pdl with its parent
$trans = $a->thread(4,1,0,3,2)->unthread;
Example 4:
# calculate a couple of bounding boxes
# $bb will hold BB as [xmin,xmax],[ymin,ymax],[zmin,zmax]
# we use again thread and unthread to shuffle dimensions around
pdl> $bb = zeroes(double, 2,3 );
pdl> minimum($vertices->thread(0)->clump->unthread(1), $bb->slice('(0),:'));
pdl> maximum($vertices->thread(0)->clump->unthread(1), $bb->slice('(1),:'));
Example 5:
# calculate a self-rationed (i.e. self normalized) sequence of images
# uses explicit threading and an implicitly threaded division
$stack = read_image_stack();
# calculate the average (per pixel average) of the first $n+1 images
$aver = zeroes([stack->dims]->[0,1]); # make the output pdl
sumover($stack->slice(":,:,0:$n")->thread(0,1),$aver);
$aver /= ($n+1);
$stack /= $aver; # normalize the stack by doing a threaded division
# implicit versus explicit
# alternatively calculate $aver with implicit threading and auto-creation
sumover($stack->slice(":,:,0:$n")->mv(2,0),($aver=null));
$aver /= ($n+1);
#
Implicit versus explicit threading
In this paragraph we are going to illustrate when explicit threading is
preferable over implicit threading 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 threading ?
Well, code that uses threading 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 threading will be
implemented in a way that takes advantage of the additional facilities
of these machines. Furthermore, it is a conceptually simply 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 threading
way of thinking (and coding) it shouldn't be too difficult to
understand code that somebody else has written than (provided he 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
threading 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 threading
(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 threading 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 threading (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 threaded 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.
perl v5.20.2 2015-05-24 INDEXING(1)