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¶
- •
- document Perl level threading
- •
- threadids
- •
- update and correct description of slice
- •
- new functions in slice.pd (affine, lag, splitdim)
- •
- 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 $res;
}
Would you agree that it might be difficult to figure out expected dimensions,
purpose of the routine, etc ? (If you want to find out what this piece of code
does, see below)
There are a couple of rules that allow you to figure out number and size of loop
dimensions (and if the size of your input pdls comply with the threading
rules). Dimensions of any pdl argument are broken down into two groups in the
following: Core dimensions (as defined by the PP-function, see
Appendix
B for a list of PDL primitives) and extra dimensions which comprises all
remaining dimensions of that pdl. For example calling a function
"func" with the signature "func((n,m),[o](n))" with a pdl
"a(2,4,7,1,3)" as "f($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¶
[ 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.