Linearized deformations second take #488

adler-j · 2016-07-11T14:38:03Z

Second take at linearized deformations. I've cleaned up most of the implementation and updated the doc, should be ready for final review @chongchenmath, @kohr-h, @ozanoktem .

adler-j · 2016-07-11T14:45:23Z

Cleaned up commit history and implementation, also rebased to master. Should be reviewable.

chongchenmath · 2016-07-11T15:07:00Z

odl/deform/linearized.py

+
+        return PointwiseInner(self.domain, def_grad)
+
+    def __repr__(self):


What is this mean?

repr is short for representation and is intended to give a representation of the object. It is called when you write things like

>>> space = odl.rn(3) >>> space rn(3) # Here repr is called

and can also be explicitly invoked through the repr function

>>> space = odl.rn(3) >>> print(repr(space)) rn(3)

You can read some more in the official documentation.

Generally, you should be able to create the object from the repr string, but this is not always possible.

>>> np.array([1, 2, 3, 4, 5]) array([1, 2, 3, 4, 5]) # can recerate the array from this >>> np.eye(5) array([[ 1., 0., 0., 0., 0.], [ 0., 1., 0., 0., 0.], [ 0., 0., 1., 0., 0.], [ 0., 0., 0., 1., 0.], [ 0., 0., 0., 0., 1.]]) # can still recreate the array >>> np.eye(50) array([[ 1., 0., 0., ..., 0., 0., 0.], [ 0., 1., 0., ..., 0., 0., 0.], [ 0., 0., 1., ..., 0., 0., 0.], ..., [ 0., 0., 0., ..., 1., 0., 0.], [ 0., 0., 0., ..., 0., 1., 0.], [ 0., 0., 0., ..., 0., 0., 1.]]) # array too big, cannot recreate it

ODL generally works the same:

>>> odl.vector([1., 2, 3, 4, 5]) rn(5).element([1.0, 2.0, 3.0, 4.0, 5.0]) >>> odl.vector([1.] * 100) rn(100).element([1.0, 1.0, 1.0, ..., 1.0, 1.0, 1.0])

discr_space = odl.uniform_discr([-1, -1], [1, 1], (100, 100), interp='linear') template = odl.phantom.cuboid(discr_space, [-0.5, -0.5], [-0.25, 0]) fixed_templ_op = odl.deform.LinDeformFixedTempl(template)

Without explicit __repr__()

>>> fixed_templ_op LinDeformFixedTempl: ProductSpace(uniform_discr([-1.0, -1.0], [1.0, 1.0], [100, 100], interp=u'linear'), 2) -> uniform_discr([-1.0, -1.0], [1.0, 1.0], [100, 100], interp=u'linear')

With explicit __repr__()

>>> fixed_templ_op LinDeformFixedTempl(uniform_discr([-1.0, -1.0], [1.0, 1.0], [100, 100], interp=u'linear') .element([[0.0, 0.0, 0.0, ..., 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, ..., 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, ..., 0.0, 0.0, 0.0], ..., [0.0, 0.0, 0.0, ..., 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, ..., 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, ..., 0.0, 0.0, 0.0]]))

It seems the former is more clear. Maybe I am wrong.

Edit: @adler-j typeset python

While you are correct in some sense, it depends on your definition of clear. In python there are two "print" statements, __str__ and __repr__. __str__ is intended to give a short representation of the object so users know what is meant, while __repr__ should be used for debugging and be more precise.

Here I only implemented __repr__, mostly because I was lazy. We could want to implement __str__ as well however.

chongchenmath · 2016-07-12T07:59:20Z

odl/deform/linearized.py

+                                'instance if `domain` is None, got {!r}'
+                                ''.format(template))
+
+            domain = template.space.tangent_space


In differential geometry, one can attach to every point x of a differentiable manifold a tangent space. Here we attach a tangent_space to a space. It seems using space.tangent_space is not exact. Probably tangent bundle?

Since the space is flat they should be equivalent, but sure, the naming here is not perfect. We may need some more thought on it.

Yes. It's quite a geometric concept here. How about vector filed space, short for vf_space or more explicit name?

The source for the discussion regarding the notion of tangent space is that ODL lacks a notion of manifold (curved space). In ODL one has either a set or a vector space, the notion of a. manifold does not exist. We have already encountered the need to handle manifolds since data space in tomography is a non-Euclidian manifold. The operations required for points (=tomographic data) on such manifolds could however be rephrased as operations on the underlying coordinate space (products of intervals), ignoring the fact that a single coordinate space is insufficient as an atlas. Now, dealing with deformations requires us yet again to handle points (=vector fields) in curved spaces, i.e. manifolds. One option is to try to recast the operations to the underlying coordinate spaces, e.g. assuming a RKHS. Another is to introduce notions manifold in ODL with a tangent bundle. The latter seems to be quite time consuming though.

An option, and I think that's what @adler-j was trying to achieve, is to represent the space for the vector fields directly, without having the notion of manifolds. That's of course a bit limiting, as for the manifolds in acquisition geometries, but from my perspective it's a sufficient solution and can be handled quite smoothly with the means we have currently. I don't think we necessarily need RHKS to do that.

Well for the fixed template style deformations we need a proper domain for the deformations., and in the fixed deformation case we need to be able to give it as input. Basically we need a name for the space

ProductSpace(space.astype(space.real_dtype), space.ndim)

I picked tangent_space since it felt natural, but as @chongchenmath noted this may not be future proof since it is only correct for flat spaces.

@kohr-h was really on spot on what we try to achieve here. RKHS has nothing to do with this one and probably shouldn't need to be involved for now.

Are we talking about the set of deformations or the set of vector/velocity fields generating deformations? The set of deformations is not a vector space, it is a semi group and a Lie group at best. The set of vector/velocity fields generating deformations is however a vector space, which in some theoretical frameworks can be interpreted as the tangent space to the Lie group of deformations. RKHS was mentioned as an example of how this Lie group could be equipped coordinates. So, what are we trying to name here?

Are we talking about the set of deformations or the set of vector/velocity fields generating deformations?

We're talking about "the set of vector/velocity fields generating deformations".

As I suggested, one abbreviation of vector field space is probably a proper choice here.

Agree, something along those lines would be good.

adler-j · 2016-07-15T16:52:27Z

odl/deform/linearized.py

+    ----------
+    template : `DiscreteLpVector`
+        Template to be deformed by a displacement field.
+    displacement : `ProductSpace` element


ProductSpace of DiscreteLp element

adler-j · 2016-08-12T16:25:58Z

odl/deform/linearized.py

+                            'instance, got {!r}'.format(displacement[0]))
+
+        if templ_space is None:
+            templ_space = displacement[0].space


While equivalent, I feel it would be better to write displacement.space[0]

adler-j · 2016-08-12T16:32:52Z

This is starting to look very good, I only found some minor style issues. We should be able to merge soon.

kohr-h · 2016-08-15T06:44:50Z

odl/deform/linearized.py

+    ----------
+    template : `DiscreteLpVector`
+        Template to be deformed by a displacement field.
+    displacement : `ProductSpace` of `DiscreteLp` element


Should be same as below

kohr-h · 2016-08-15T08:19:38Z

I added an (approximate) inverse to the fixed-displacement operator since we're using it in the adjoint anyway. Fixed the comments.

kohr-h · 2016-08-16T11:57:21Z

Cleaned up the commit history a bit, will merge after tests have passed.

* Add linearized deformation operators * Add tests for the linearized deformations * Add examples of linerized deformations

Examples: - Better text, more informative comments and a more interesting displacement field in the deformation examples linear_deform.py: - Simplified imports - Improved docstrings, mostly adding missing documentation and some reformulations - Mathematical descriptions added to both operators - Removed optional `domain` parameter from the fixed-template operator since it complicates things and doesn't add much. - In the fixed displacement variant, it is needed to be able to support complex template spaces - here the support for implicitly discretizing a displacement field was removed for reasons of simplicity. Further, the parameter is now called `templ_space` since it is both domain and range. - Public properties for the fixed attributes `template` or `displacement` added, resp. lp_discr.py: - `_interp_by_axis` -> `__interp_by_axis` - Go back to old version of real space calculation in `DiscreteLp`. The current one arbitrarily added `self.real_space` to an instance. Tests: - Adapt to code changes

adler-j added status: review needed type: new feature labels Jul 11, 2016

adler-j force-pushed the linearized_deformations_simplified branch from ee2b210 to e22d8af Compare July 11, 2016 14:42

chongchenmath reviewed Jul 11, 2016
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This was referenced Jul 11, 2016

Linearized deformations #448

Closed

ENH: add deformation operators #465

Closed

chongchenmath reviewed Jul 12, 2016
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adler-j force-pushed the linearized_deformations_simplified branch 3 times, most recently from efa5d30 to eb3a60b Compare July 14, 2016 09:04

chongchenmath force-pushed the linearized_deformations_simplified branch from dc334e9 to cd5fd85 Compare July 15, 2016 10:52

adler-j reviewed Jul 15, 2016
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adler-j reviewed Aug 12, 2016
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kohr-h reviewed Aug 15, 2016
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kohr-h added merge? and removed status: review needed labels Aug 15, 2016

kohr-h force-pushed the linearized_deformations_simplified branch from c004719 to b5a5106 Compare August 16, 2016 11:56

chongchenmath and others added 13 commits August 16, 2016 14:12

ENH: implement linearized deformation

29b45df

* Add linearized deformation operators * Add tests for the linearized deformations * Add examples of linerized deformations

ENH: simplify linearized deformations implementation and test

1d612cb

TEST: update test of linearized deform to fixtures

f2a4f49

API: change DiscreteLp.interp to return 1d, add interp_by_axis

fffedac

ENH: add support for complex spaces in linearized deformations

06fc622

DOC: improve doc of linearized deformations

333e3d9

MAINT: Fix some minor things, mainly comments and formulations

9fb61bd

MAINT: remove obsolete import in __init__.py

39159da

BUG: fix vector_field_space in DiscreteLp after rebase

410db79

DOC: small improvements in doc of linearized deformations

c7b46d5

ENH: add inverse to fixed-displ. op and fix some formulations

fe764a6

REL: add linearized deformations to release notes

b8daded

kohr-h force-pushed the linearized_deformations_simplified branch from b5a5106 to b8daded Compare August 16, 2016 12:50

kohr-h merged commit e8b99b8 into master Aug 16, 2016

kohr-h deleted the linearized_deformations_simplified branch August 16, 2016 13:03

kohr-h mentioned this pull request Aug 18, 2016

Deformation operators #464

Closed

2 tasks

kohr-h removed the merge? label Sep 21, 2016

adler-j mentioned this pull request Oct 31, 2016

Vector field space of complex space #693

Closed

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Linearized deformations second take #488

Linearized deformations second take #488

adler-j commented Jul 11, 2016 •

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adler-j commented Aug 12, 2016

kohr-h Aug 15, 2016

kohr-h commented Aug 15, 2016

kohr-h commented Aug 16, 2016


		return PointwiseInner(self.domain, def_grad)

		def __repr__(self):

Linearized deformations second take #488

Linearized deformations second take #488

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adler-j commented Jul 11, 2016 •

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