=head1 NAME

pdlindexing - how to index piddles.

=head1 DESCRIPTION

This manpage should serve as a first tutorial on the indexing and
threading features of L<PDL>.

As part of the current PDL alpha release it is also still in alpha
development and not yet complete. "Meta" comments that point out
deficiencies/omissions of this document will be surrounded by
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 (csoelle@sghms.ac.uk).

=head1 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 C<@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.

=head2 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 becomes the "child"
and the old one its "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.

[ This is probably not right any more;  maybe apart from the case of the
  yet to come vaffine trans (or some similar term)

A last definition in the terminology before we really get going is the
notion of "virtual" and "physical" pdls. A physical pdl doesn't have a
parent and can be thought of as being at the top of the hierarchy. It is
physical because it is guaranteed that there is a region in
memory that holds the data of this pdl (e.g. the image data we read
into $im) as opposed to a child that often only holds a "recipe" (a
transformation) specifying how to compute its data from the parent's
data. Distinction between the two cases can become important when you
want to interface data of a pdl to an external library function where
you often have to specify an array in memory that holds the data the
library function is going to work on.

]

=head2 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 ;).

 perldl> $im = rfits('image.fits')   # read in some stored data
 perldl> print $im                   # and show what we've got
 [
  [ 0  0  0  0  0]
  [ 1  2  3  4  5]
  [10 10 10 10 10]
  [ 0  2  0  2  0]
  [ 9  9  1  1  1]
 ]
 perldl> varhelp


[ here it might be appropriate to quickly talk about the C<varhelp>
function that provides information about pdls in the interactive
C<perldl> shell that comes with pdl: unfortunately this function is
not yet part of perldl but will be added in the near future to give
you information about the pdl objects that are currently in the main
symbol table ]

Now suppose we want to create a 1-D virtual 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> varhelp
 area    Float  (V)
 line    Float  (V)
 even    Float  (V)
 im      Float  (P)

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> print $line
 [10 10 10 10 10]

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> print $line
 [11 11 11 11 11]
 perldl> $line += 2
 perldl> print $im
 [
  [ 1  1  1  1  1]
  [ 2  3  4  5  6]
  [13 13 13 13 13]
  [ 1  3  1  3  1]
  [10 10  2  2  2]
 ]

Note how assignment operations on the child virtual pdls change the
parent physical pdl and vice versa. 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 L<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: 

=over 6
=item C<:>
Use the whole dimension
=item C<n>
Use only index C<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> varhelp


 perldl> print $column
 perldl> print $row
 [ Output will have to be added in a later patch to this doc ]

=item C<(n)>
Use only index C<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
get's important in assignments where left and right hand side have to
have appropriate dimensions.

 perldl> $line = $im->slice(':,(0)')
 perldl> varhelp
 [ Output will have to be added in a later patch to this doc ]
 perldl> print $line
 [ Output will have to be added in a later patch to this doc ]

Spot the difference to the previous example?

=item C<n1:n2> or C<n1:n2:n3>

Take the range of indices from C<n1> to C<n2> or (second form)
take the range of indices from C<n1> to C<n2> with step
C<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 C<n2> is smaller than C<n1> (in the example -1 is
equivalent to index 4) the elements in the virtual pdl are effectively
reverted with respect to its parent.
=item C<*[n]>
Add a dummy dimension. The size of this dimension will be 1 by default
or equal to C<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
C<dummy> function creates a new dummy dimension. So read on and check
its explanation below.


=item C<([n1:n2[:n3]]=i)>
[Not yet implemented ??????]
With an argument like this you make I<generalised diagonals>. The
I<diagonal> will be dimension no. C<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 C<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
=back

[ work in the new index function: $b = $a->index($c); ???? ]

=head2 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 I<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> print $im
 [ Output will have to be added in a later patch to this doc ]

But using '.='

 perldl> $line = $im->slice(':,(2)')
 perldl> $line .= zeroes(5)
 perldl> $line++
 perldl> print $im
 [ Output will have to be added in a later patch to this doc ]

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 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 I<not> do what you
wanted. But this may change in the future (perhaps making a copy of
the portion of the parent data first (internally) if lvalue and rvalue
pdl have a common parent).

=head2 Other functions that manipulate dimensions

Having talked extensively about the C<slice> function it should be
noted that this is not the only PDL indexing function. There
are some additional quite useful (we hope) indexing functions
(especially in the context of threading which we will talk about
later). Here are a list and some examples how to use them.

=over 4
=item *
C<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 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. :(

=item *
C<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; # 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->diag(0));  # sum all the diagonal elements


=item *
C<xchg> and C<mv>

C<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 C<xchg> will be used in the context of threading
but more about that later.

C<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

=item *
C<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

  perldl> $seqs = $stack->clump(2)
  perldl> varhelp
 [ Output will have to be added in a later patch to this doc ]


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.
=back

=head2 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 C<mv> function. I think
you get the idea (in spite of my convoluted explanations).


=head2 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 is flagged as a runtime error:

   $a = pdl [1,2,3];
   $b = $a->dummy(1,4);
   $b .= yvals(zeroes(3,4));

>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).

=head2 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.


=head2 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))


=head1 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.

=head2 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
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 tripel (r,g,b) as a weighted sum

    greyvalue = 77/256*r + 150/256*g + 29/256*b =
        inner(pdl[77,150,29]/256,pdl(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 C<inner> is a threading
aware function defined in the PDL::Primitive package)

 $w = pdl [77,150,29]/256;
 inner($w, $im, ($grey=null)); # this basically means $grey = inner($w,$im)

We have ended up with an (essentially) 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). See below for the funny way in which we call the
C<inner> function with the output variable as last argument. Well, we
still owe you an explanation how this 'magic' is achieved.

=head2 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 C<inner> function exspects two
one-dimensional (input) parameters from which it calculates a
zero-dimensional (output) parameter. We write that symbolically as
C<inner((n),(n),[o]())> and call it C<inner>'s I<core pattern>, 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
C<outer((n),(m),[o](n,m))>. The C<[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 I<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 (where we
call C<inner> using the autocreation syntax explained below)

  inner($w,$im,($grey = null));

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 I<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).

=head2 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, I<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 exspected
dimensions, purpose of the routine, etc ?
(If you want to find out what this piece of code does, see below)

=head2 How to figure out the loop dimensions

There are a couple of rules that allow you to figure out number and
size of loop dimensions (and if the size of your input pdls comply
with the threading rules). Dimensions of any pdl argument are broken
down into two groups in the following: Core dimensions (as defined by
the PP-function, see B<Appendix B> for a list of PDL primitives) and
extra dimensions which comprises all remaining dimensions of that
pdl. For example calling a function C<func> with the core pattern
C<func((n,m),[o](n))> with a pdl a(2,4,7,1,3) as C<f($a,($o = null))>
results in the semantic splitting of a's dimensions into:
core dimensions (2,4) and extra dimensions (7,1,3).

=over 6
=item 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.
=item R1
The number of (implicit) loop dimensions is equal to the maximal
number of extra dimensions taken over the set of pdl arguments.
=item 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.
=item 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.
=item 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.
=item R5
If output autocreation 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).
=back

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 I<R4>.

As an example, suppose we have a (here unspecified) PP-function with
the core pattern:

   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), 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 core pattern 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 I<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 C<sumover> operations show
a first example of how to make use of the dimension manipulating
commands. A quick look at sumover's core pattern 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 core pattern. 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 C<sumover>, C<prodover>, C<minimum> and
C<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 C<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 C<clump> comes in handy. As an example look at the
definition of C<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 .= pdl(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 C<index> function, core pattern
((n),(),[o]()) (see B<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 dimesnions 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 I<core functionality>
applied to another specified set of dimensions you use
the dimension manipulating commands to create a (or several)
I<virtual> pdl(s) so that from the point of view of the I<parent>
pdl(s) you get what you want (always having the core pattern of the
function in question and R1-R5 in mind!). Easy, isn't it ?

=head2 Output autocreation 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 [you might want to skip this
paragraph at first reading while you are only interested in the
general concepts but should read it before you start writing code]. As
you have probably already noted from the symbolic representation of
C<inner> and C<outer> they are really not invoked as you would
typically exspect from a proper function, namely

  $result = func($a,$b);

but instead with the output argument as specified in the symbolic
representation,

  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 C<PDL->null> that returns an essentially
"empty" pdl (for those interested in details it is...). 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 should say

  func($a,$b,($result=PDL->null));

Be warned that you can I<not> use output autocreation when using
explicit threading (for reasons explained in the following section on
B<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

  $result = null;  # create output pdl of appropriate size
                     only at first invocation
  for (0...$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
  }

Since there are also often occasions at which you you want to use the
normal "functional" invocation and do not care about the creation of a pdl
(or even want it to be created anyway) upcoming versions will support
a variant of any PP-function, for example in the case of C<inner> a
function C<f_inner>, defined to do something like
[ the f_ name is just a placeholder for what is ultimately adopted ]

   sub f_inner {
      my $tmp = null;   # request output autocreation
      inner(@_,$tmp);
      return $tmp;
   }

so that you can say

  $res = f_inner($a,$b);

as you would have exspected in the first place. But be aware of the
limitation on output creation when using B<explicit threading>. Most
likely the (yet to come) C<f_XXXX> functions will check for that first
before trying to invoke the C<XXXX> function.


=head2 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 I<explicit threading>.
Now, before we create the wrong impression: it is not
either I<implicit> or I<explicit>; the two flavours do mix. But more
about that later.

The two most used functions with explicit threading are C<thread>
and C<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);
      }
   }

C<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 C<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 I<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).

=over 9
=item T0
Core dimensions are matched against the first n I<remaining dimensions>
of the pdl argument (note the difference to R1). Any
further I<remaining dimensions> are I<extra dimensions> and are used
to determine the I<implicit loop dimensions>.
=item T1a
The number of I<implicit loop dimensions> is equal to the maximal
number of extra dimensions taken over the set of pdl arguments.
=item T1b
The number of I<explicit loop dimensions> is equal to the maximal
number of thread dimensions taken over the set of pdl arguments.
=item T1c
The total number of I<loop dimensions> is equal to the sum of
I<explicit loop dimensions> and I<implicit loop dimensions>. In the
thread loop, I<explicit loop dimensions> are threaded over first
followed by I<implicit loop dimensions>.
=item T2
The size of each of the I<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
I<explicit loop dimensions>) or extra dimension (for
I<implicit loop dimensions>).
=item T3
This rule applies to any I<explicit loop dimension> as well as any
I<implicit loop dimension>. For all pdls that have a given
I<thread/extra dimension> the size must be equal to the size of the
respective I<explicit/implicit loop dimension> or 1; otherwise you
raise a runtime exception. If the size of a I<thread/extra dimension>
of a pdl is one it is implicitly treated as a dummy
dimension of size equal to the I<explicit/implicit loop dimension>.
=item T4
If a pdl doesn't have a I<thread/extra dimension> that corresponds to
an I<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.
=item T4a
All pdls that do have I<thread dimensions> must have the same number of
thread dimensions.
=item T5
Output autocreation cannot be used if any of the pdl arguments has any
I<thread dimensions>. Otherwise R5 applies.
=back

The same restrictions apply with regard to implicit dummy dimensions
(created by application of C<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 core pattern:

   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 I<not> be automatically
created) as

   func($a->thread(1,3),$b->thread(0,3),$c,$d->thread(0,1))

>From the core pattern 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]
 # [ some of the examples in this document use already the (yet unimplemented)
 # zeroes( type, ... ) syntax ]
 # 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);
 #

=head2 Implicit versus explicit threading

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 I<greatly> reduce the syntactical complexity of PDL code (but keep
the admonition for documentation in mind). Once you
are comfortable with the I<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 ;).

=head1 PDL::PP

=head2 Concise description of 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
mabe dataflow as well, see L<Dataflow> manpage). For further details
check the L<PDL::PP> manpage.


=head1 Appendix A

=head2 Affine transformations - a special class of simple and powerful transformations

[ This is also something to be added in future releases. Do we already
have the general make_affine routine in PDL ? It is possible that we
will reference another appropriate manpage from here ]

=head1 Appendix B

=head2 Core patterns of standard PDL::PP compiled functions

A selection of core patterns 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 primitiv.pp

 # functions in primitiv.pp

  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]())

=head1 AUTHOR & COPYRIGHT

Copyright (C) 1997 Christian Soeller (csoelle@sghms.ac.uk) & Tuomas
J. Lukka (lukka@fas.harvard.edu).
Permission is granted for verbatim copying (and formatting) of this 
material as a part of PDL. Permission is explicitly not granted 
for distribution in book or any corresponding form. Email the authors
if you are unclear.