|
PDL::Indexing - how to index piddles. |
PDL::Indexing - how to index piddles.
This manpage should serve as a first tutorial on the indexing and threading features of PDL.
This manpage is still in alpha development and not yet complete. ``Meta'' comments that point out deficiencies/omissions of this document will be surrounded by square brackets ([]), e.g. [ Hopefully I will be able to remove this paragraph at some time in the future ]. Furthermore, it is possible that there are errors in the code examples. Please report any errors to Christian Soeller (c.soeller@auckland.ac.nz)
Still to be done are (please bear with us and/or ask on the mailing list, see PDL::FAQ):
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.
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 manpage.)
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.
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 ;).
perldl> $im = sequence(5,5) perldl> 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] ]
perldl> 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 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:
perldl> $line = $im->slice(':,(2)')
perldl> $even = $im->slice(':,1:-1:2')
perldl> $area = $im->slice('3:4,3:1')
perldl> 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:
perldl> p $line [10 11 12 13 14]
Also note the difference in the ``Flow State'' of $area above
and below:
perldl> p $area perldl> help $area This variable is Double D [2,3] VC 0.00Kb
The following demonstrates that $im and $line really behave as you
would exspect from a pointer-like object (or in the dataflow picture:
the changes in $line's data are propagated back to $im):
perldl> $im++ perldl> p $line [11 12 13 14 15] perldl> $line += 2 perldl> 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 similiar 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:
n. The dimension of this index in the
resulting virtual pdl is 1. An example involving those first two index
formats:
perldl> $column = $im->slice('2,:')
perldl> $row = $im->slice(':,0')
perldl> p $column
[ [ 3] [ 8] [15] [18] [23] ]
perldl> p $row
[ [1 2 3 4 5] ]
perldl> help $column This variable is Double D [1,5] VC 0.00Kb
perldl> help $row This variable is Double D [5,1] VC 0.00Kb
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.
perldl> $line = $im->slice(':,(0)')
perldl> help $line
This variable is Double D [5] -C 0.00Kb
perldl> p $line [1 2 3 4 5]
Spot the difference to the previous example?
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 subimage composed of even lines.
perldl> $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 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.
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); ???? ]
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.
perldl> $line = $im->slice(':,(2)')
perldl> $line = zeroes(5);
perldl> $line++;
perldl> 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] ]
perldl> p $line [1 1 1 1 1]
But using .=
perldl> $line = $im->slice(':,(2)')
perldl> $line .= zeroes(5)
perldl> $line++
perldl> 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] ]
perldl> print $line [1 1 1 1 1]
Also, you can substitute
perldl> $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).
Related to the assignment feature is a little trap for the unwary: since perl currently does not allow subroutines to return lvalues the following shortcut of the above is flagged as a compile time error:
perldl> $im->slice(':,(2)') .= zeroes(5)->xvals->float
instead you have to say something like
perldl> ($pdl = $im->slice(':,(2)')) .= zeroes(5)->xvals->float
We hope that future versions of perl will allow the simpler syntax (i.e. allow subroutines to return lvalues). [Note: perl v5.6.0 does allow this, but it is an experimental feature. However, early reports suggest it works in simple situations]
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)')) .= $a->slice('-1:0,(1)')->sever;
which returns the same slice but severs the connection of the slice to its parent.
Having talked extensively about the slice function it should be noted that this is not the only PDL indexing function. There are additional indexing functions which are also useful (especially in the context of threading which we will talk about later). Here are a list and some examples how to use them.
dummy(X,x,Y) (0<=x<size_of_dummy_dim) just map to
the element with index (X,Y) of the parent pdl (where X and Y refer to
the group of indices before and after the location where the dummy
dimension was inserted.)
This example calculates the x coordinate of the centroid of an image (later we will learn that we didn't actually need the dummy dimension thanks to the magic of implicit threading; but using dummy dimensions the code would also work in a threadless world; though once you have worked with PDL threads you wouldn't want to live without them again).
# centroid ($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 pixelwise product with
# the repeated line of x-values
The rest is then summing the results of the pixelwise product together and normalising 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 greyscale 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 greyscale 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# 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# 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.
clumpperldl> $seqs = $stack->clump(2) perldl> 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.
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).
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).
[ 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.
[ 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
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.
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
greyscale. 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
greyvalue (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
greyscale). An approximation that works quite well is to compute the
grey intensity from each RGB triplet (r,g,b) as a weighted sum
greyvalue = 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 greyscale 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.
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 autocreated pdl (here 0D+2D=2D to yield a 2D
output of size (x,y)).
So the above command calls the core functioniality 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,$im(:,(x-1),(y-1)))
But this is done automatically by PDL without writing any explicit perl loops. We see that the command really creates an output pdl with the right dimensions and sets the elements indeed to the result of the computation for each pixel of the input image.
When even more pdls and extra dimensions are involved things get a bit more complicated. We will first give the general rules how the thread dimensions depend on the dimensions of input pdls enabling you to figure out the dimensionality of an autocreated output pdl (for any given set of input pdls and core dimensions of the PP-function in question). The general rules will most likely appear a bit confusing on first sight so that we'll set out to illustrate the usage with a set of further examples (which will hopefully also demonstrate that there are indeed many practical situations where threading comes in extremly handy).
Before we point out the other technical details of threading, please note this call for programming discipline when using threading:
In order to preserve human readability, PLEASE comment any nontrivial expression in your code involving threading. Most importantly, for any subroutine, include information at the beginning about what you expect the dimensions to represent (or ranges of dimensions).
As a warning, look at this undocumented function and try to guess what might be going on:
sub lookup {
my ($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,($o = null))
results in the semantic splitting of a's dimensions into:
core dimensions (2,4) and extra dimensions (7,1,3).
PDL->null before invocation) the number of dimensions of the created
pdl is equal to the sum of the number of core output dimensions +
number of loop dimensions. The size of the core output dimensions is
derived from the relevant dimension of input pdls (as specified in the
function definition) and the sizes of the other dimensions are equal
to the size of the loop dimension it is derived from. The
automatically created pdl will be physical (unless dataflow is in
operation).
In this context, note that you can run into the problem with
assignment to pdls containing greater-than-one dummy dimensions (see above).
Although your output pdl(s) didn't contain any dummy dimensions in the
first place they may end up with implicitly created dummy dimensions
according to R4.
As an example, suppose we have a (here unspecified) PP-function with the signature:
func((m,n),(m,n,o),(m),[o](m,o))
and you call it with 3 pdls a(5,3,10,11),
b(5,3,2,10,1,12), and c(5,1,11,12) as
func($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 greyscale 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 exspect 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 analyse 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(0,1),($ret=null));
and calculating all the sums of sub-slices over the third dimension is now almost too easy
# sums of pixles in time (assuming time is the third dim) sumover($stack->mv(0,2),($ret=null));
Finally, if you want to apply the operation to all elements (like max over
all elements or sum over all elements) regardless of the dimensions of
the pdl in question clump comes in handy. As an example look at the
definition of sum (as defined in Basic.pm):
sub sum {
PDL::Primitive::sumover($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
perldl> $im = zeroes(byte, 10,20) perldl> $line = exp(-rvals(10)**2/9) # threaded assignment perldl> $im .= $line # set every line of $im to $line perldl> $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 $im)
# you can say just dummy(0) since the rules of threading make it fit
perldl> index($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 explicitely state the size of the dummy
dimension that we created for $im; when we create it with size 1 (the
default) the rules of threading make it automatically fit to the
desired size (by rule R3, in our example the size would be 3 assuming
a palette of size (3,x)). Since situations like this do occur often in
practice this is actually why rule R3 has been introduced (the part
that makes dimensions of size 1 fit to the thread loop dim size). So
we can just say
perldl> index($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 ?
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 autocreation 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.
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 perldl> $mat = zeroes(4,3) perldl> $line = pdl (3.1416,2,-2) perldl> ($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 = $a->thread(2,1)
creates a new virtual pdl of dimension b(4,8) (which we call the
remaining dims) that also has 2 thread dimensions of size (2,7). For
the purposes of this document we write that symbolically as
b(4,8){2,7}. An important difference to the previous examples where
only implicit threading was used is the fact that the core dimensions
are matched against the remaining dimensions which are not
necessarily the first dimensions of the pdl. We will now specify how
the presence of thread dimensions changes the rules R1-R5 for
threadloops (which apply to the special case where none of the pdl
arguments has any thread dimensions).
The same restrictions apply with regard to implicit dummy dimensions (created by application of T4) as already mentioned in the section on implicit threading: if any of the output pdls has an (explicit or implicitly created) greater-than-one dummy dimension a runtime exception will be raised.
Let us demonstrate these rules at work in a generic case. Suppose we have a (here unspecified) PP-function with the signature:
func((m,n),(m),(),[o](m))
and you call it with 3 pdls a(5,3,10,11), b(3,5,10,1,12), c(10) and an
output pdl d(3,11,5,10,12) (which can here not be automatically
created) as
func($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))
Uhhmpf, this example was really not easy in terms of bookeeping. 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 .= $evecs; # initialise, no backpropagation
($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),"*"); # similiar thing by implicit threading with autocreated 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
perldl> $bb = zeroes(double, 2,3 );
perldl> minimum($vertices->thread(0)->clump->unthread(1),
$bb->slice('(0),:'));
perldl> maximum($vertices->thread(0)->clump->unthread(1),
$bb->slice('(1),:'));
Example 5:
# calculate a self-ratioed (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 divison
# implicit versus explicit
# alternatively calculate $aver with implicit threading and autocreation
sumover($stack->slice(":,:,0:$n")->mv(2,0),($aver=null));
$aver /= ($n+1);
#
In this paragraph we are going to illustrate when explicit threading is preferrable 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 exspected 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 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 mabe dataflow as well, see the PDL::Dataflow manpage). For further details check the PDL::PP manpage.
[ 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 manpage from here ]
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]())
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.
|
PDL::Indexing - how to index piddles. |