Err about composing linear transformations... If I find the seperate
transformation matrices required, isnt it enough that I just multiply them
to find the composite transformation matrix? Say three seperately applied
rotation matrices do what I want, isnt the composed matrix the multplication
of the three?
On Wed, Apr 30, 2008 at 6:09 PM, Roban Kramer <[EMAIL PROTECTED]>
wrote:
> A while ago I wrote code to mathematically compose linear
> transformations. I make no guarantee of its correctness or robustness,
> but it might be useful to someone:
>
> =head2 t_compose_linear
>
> Mathematically compose transformations and combine the C<pre> and
> C<post> translations into a single C<post> translation. The
> C<t_compose> function just does the transformations in order, while
> this function creates a single mathematical transformation.
>
> =cut
>
> sub t_compose_linear{
> my (@t_in) = @_;
>
> # set up the initial output matrix and post translation
> my $out_matrix = identity($t_in[-1]->{'params'}->{'matrix'});
> my $out_post = zeroes($out_matrix->dim(-1));
>
> foreach my $in (reverse @t_in) {
> # check to make sure we have the right type of transformation
> unless(UNIVERSAL::isa($in,'PDL::Transform::Linear')) {
> Carp::cluck( "PDL::Transform::t_inverse_linear: ".
> "got a transform that is not linear.\n"
> );
> return undef;
> }
> unless(defined $in->{params}->{inverse}) {
> Carp::cluck( "PDL::Transform::t_inverse_linear: ".
> "got a transform with no inverse.\n"
> );
> return undef;
> }
>
> my $in_matrix = $in->{'params'}->{'matrix'};
>
> # get the post and pre translations of the input xform
> my $in_post = topdl($in->{'params'}->{'post'});
> my $in_pre = topdl($in->{'params'}->{'pre'});
> $in_post = $in_post->copy();
> $in_pre = $in_pre->copy();
>
> # if they're single element piddles, make them vectors
> $in_post = $in_post * ones($in_matrix->dim(1))
> if ($in_post->nelem() < $in_matrix->dim(1));
> $in_pre = $in_pre * ones($in_matrix->dim(1))
> if ($in_pre->nelem() < $in_matrix->dim(1));
>
> # now combine the pre and post translations into a single post
> $in_post = $in_post + ($in_pre x $in_matrix);
>
> # and convert that into the combined post translation
> $out_post = $in_post + matmult($out_post,$in_matrix);
> $out_matrix = $out_matrix x $in_matrix;
> }
>
>
> On Wed, Apr 30, 2008 at 8:52 AM, Craig DeForest
> <[EMAIL PROTECTED]> wrote:
> >
> > That would be the part in NOTES where it says
> >
> >
> > Composition works OK but should probably be done in a more
> sophisticated
> > way so that, for example, linear transformations are combined at the
> > matrix level instead of just strung together pixel-to-pixel.
> >
> > For 100-10,000 coordinates you shouldn't worry. That will fit entirely
> in
> > the CPU cache for most modern machines, so the performance hit isn't bad.
> > When I say build your own rotation matrix, I mean as a PDL using
> elementwise
> > calculation or matrix multiplication.
> >
> > The comment in the notes, and the point I made, is that if you have three
> > linear operators to string together, it is much faster (54
> multiplications)
> > to multiply your three 3x3 matrices together, than to apply each matrix
> in
> > order with the data (27 times 3N multiplications). But with only (say)
> 1000
> > points, that means each operation will take under 9000 multiplications,
> or
> > about 30-100 microseconds if you're using a recent machine and the data
> are
> > in CPU cache.
> >
> > My suggestion: try it using the Transform composition; if it is too slow,
> > you can make it faster by applying your composition Transform to the
> > identity matrix, and then pass the resulting matrix into t_linear to make
> a
> > single transform.
> >
> > Cheers,
> > Craig
> >
> > On Apr 30, 2008, at 6:31 AM, Sina Türeli wrote:
> >
> > Okay this part seems important. Where in the documentation does it write
> > that? So if to rotate around an arbitrary axis, I compose three rotation
> > matrices T^-1.R.T<v>, where T takes the arbitrary axis to x axis, R does
> the
> > rotation around x axis and T^-1 maps the arbitrary axis back to its
> original
> > direction how much of an inefficiency are we talking about. This is
> likely
> > to operate on a data set of anywhere between 100 to 10000 coordinates.
> These
> > T and R operators are all t_linear rotation operator in PDL. When saying
> > build your own rotation matrix do you mean from scratch and by just using
> > multplication and inverse operations defined in PDL and not any t_linear
> > opeartions...
> >
> > Thanks alot for your help
> >
> > On Tue, Apr 29, 2008 at 11:23 PM, Craig DeForest <
> [EMAIL PROTECTED]>
> > wrote:
> >
> > >
> > >
> > > You can compose transformations to get to the axis you want, but as you
> > will have seen in the documentation it is inefficient because the code
> just
> > strings the transformations together. If your data are big then you will
> > want instead to build your own rotation matrix (or extract the one in the
> > transform) to minimize the number of passes. You can still use Transform
> to
> > encapsulate the operations, which is good in case you later want to
> > generalize. t_linear will accept a matrix if you want.
> > >
> > >
> > >
> > >
> > > On Apr 29, 2008, at 1:26 PM, "Sina Türeli" <[EMAIL PROTECTED]>
> wrote:
> > >
> > >
> > >
> > >
> > > Thanks for the answers. One more question, is there any build in
> function
> > for rotationa around an arbitrary axis of the object? If there isnt I am
> > planning to first rotate all the object so that the arbitrary axis
> concides
> > with say x axis, rotate the object around the x axis and apply the
> inverse
> > of the first transformation to put the arbirtrary axis back in its place.
> > But somehow this seems computationally really inefficient. I am might
> also
> > think of a way to transform rotations around an arbitrary axis to their
> > correspoding transformation angles around x,y,z axis that also is I
> assume
> > possible...
> > >
> > >
> > > On Tue, Apr 29, 2008 at 7:37 PM, Sina Türeli <[EMAIL PROTECTED]>
> wrote:
> > >
> > > >
> > > >
> > > > Ok, for a certain program I am writing (protein folding), I need to
> be
> > able perform rotations. I was first planning to do it manually by
> defining
> > rotation matrices and change of basis matrices etc but I think pdl might
> > save me time. However I am not sure how to use its use PDL::Transform to
> do
> > so. Here is a piece of code that I was using to experiment with pdl use
> PDL;
> > > > use PDL::Transform;
> > > >
> > > > @a = [[1,0,0],[0,1,0],[0,0,1]];
> > > >
> > > > $c= pdl @a;
> > > >
> > > > $e = t_rot(45,45,45);
> > > >
> > > > $c = $e * $c
> > > >
> > > > print $c;
> > > >
> > > > I was hoping this would rotate my 1,1,1 vector in all directions by
> 45
> > degrees but it gives the error. "Hash given as a pdl - but not {PDL}
> key!".
> > I am not able to understand what this error is for? Also I have seen no
> > tutorial where these rotationa matrices are explained so I would
> appreciate
> > any help, thanks.
> > > >
> > > > --
> > > > "Vectors have never been of the slightest use to any creature.
> > Quaternions came from Hamilton after his really good work had been done;
> and
> > though beautifully ingenious, have been an unmixed evil to those who have
> > touched them in any way, including Maxwell." - Lord Kelvin
> > > >
> > > >
> > > >
> > > >
> > >
> > >
> > >
> > > --
> > > "Vectors have never been of the slightest use to any creature.
> Quaternions
> > came from Hamilton after his really good work had been done; and though
> > beautifully ingenious, have been an unmixed evil to those who have
> touched
> > them in any way, including Maxwell." - Lord Kelvin
> > >
> > > _______________________________________________
> > >
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> > >
> >
> >
> >
> > --
> > "Vectors have never been of the slightest use to any creature.
> Quaternions
> > came from Hamilton after his really good work had been done; and though
> > beautifully ingenious, have been an unmixed evil to those who have
> touched
> > them in any way, including Maxwell." - Lord Kelvin
> >
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> >
> >
>
--
"Vectors have never been of the slightest use to any creature. Quaternions
came from Hamilton after his really good work had been done; and though
beautifully ingenious, have been an unmixed evil to those who have touched
them in any way, including Maxwell." - Lord Kelvin
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