Introduce Kendall's shape space

[mateuszbaran]

I've started working on Kendall's shape space (so this resolves #504 ). I'm not sure how much I'm going to cover here but at least I will make a small example of usage (maybe as a Pluto notebook if @kellertuer could help 😃 ).

[kellertuer]

Interesting!
I just decided with Brynjulf and a master student we supervise together to look into applications with Kendalls shape space to check how to maybe implement that in the future. So get this one as far as you like, she might continue from there.

[kellertuer]

Since I saw that often, we should also cover the PreShapeSpace.

And is maybe ShapeSpace too generic? There might be a lot of different shape spaces such that this might be more like a good common abstract thing?

[codecov]

Codecov Report

Merging #550 (597235b) into master (8fc0322) will increase coverage by 0.01%.
The diff coverage is 100.00%.

@@            Coverage Diff             @@
##           master     #550      +/-   ##
==========================================
+ Coverage   98.93%   98.94%   +0.01%     
==========================================
  Files         100      102       +2     
  Lines        9664     9767     +103     
==========================================
+ Hits         9561     9664     +103     
  Misses        103      103              

Impacted Files	Coverage Δ
src/Manifolds.jl	100.00% <ø> (ø)
src/differentiation/ode_callback.jl	100.00% <100.00%> (ø)
src/groups/rotation_action.jl	100.00% <100.00%> (ø)
src/manifolds/KendallsPreShapeSpace.jl	100.00% <100.00%> (ø)
src/manifolds/KendallsShapeSpace.jl	100.00% <100.00%> (ø)
src/manifolds/Sphere.jl	100.00% <100.00%> (ø)
src/tests/tests_general.jl	96.90% <100.00%> (ø)

📣 We’re building smart automated test selection to slash your CI/CD build times. Learn more

[mateuszbaran]

Interesting! I just decided with Brynjulf and a master student we supervise together to look into applications with Kendalls shape space to check how to maybe implement that in the future. So get this one as far as you like, she might continue from there.

Cool! I would be nice if she could make an example using Manifolds.jl 🙂 .

Since I saw that often, we should also cover the PreShapeSpace.

That's just a special case of ArraySphere so I was going to just write in documentation that users should use that when they need preshape space.

And is maybe ShapeSpace too generic? There might be a lot of different shape spaces such that this might be more like a good common abstract thing?

That's indeed a problem, there are many variants depending on the group that is acting (here are some examples: https://digitalcommons.wayne.edu/cgi/viewcontent.cgi?article=1417&context=oa_dissertations ). I doubt, though, that we need any abstraction except quotient manifolds for them. At least I'm not going to add any in this PR. Feel free to suggest a different name, I'm not particularly convinced ShapeSpace is the right choice but haven't come up with anything better, maybe something like KendallShapeSpace?

[kellertuer]

Introducing a new name or even type (that could have ArraySphare as Fallback with ExplicitDecorator) for the KendallsPreShapeSpace would be nice I think, also to have a few functions already on that level before “Quotienting out” rotations.

I would maybe go for KendallsShapeSpace since usually one has the genitive in the name, but yes tI think that would be better.

Concerning the master student – we will see. We just decided last week to go into that direction, so I will read a bit about that. I hope she will end up either contributing a few functions or a nice example/Tutorial. We will see.

[mateuszbaran]

Introducing a new name or even type (that could have ArraySphare as Fallback with ExplicitDecorator) for the KendallsPreShapeSpace would be nice I think, also to have a few functions already on that level before “Quotienting out” rotations.

OK, I think it would be easiest to make it a new KendallsPreShapeSpace <: AbstractSphere then.

I would maybe go for KendallsShapeSpace since usually one has the genitive in the name, but yes tI think that would be better.

That sounds fine.

[mateuszbaran]

Sorry for a small mistake, preshape space is actually a submanifold of ArraySphere, I've fixed it in the recent commit.

[kellertuer]

But then even better to have it as its own type as I proposed :)

[mateuszbaran]

Yes, exactly, it's definitely necessary 🙂 .

[mateuszbaran]

I've rewritten a part of https://geomstats.github.io/notebooks/14_real_world_applications__hand_poses_analysis_in_kendall_shape_space.html in Manifolds.jl 🙂 .

[kellertuer]

That sounds very interesting, I was a little ill a few days last week so I now am a little stressed to get back on (semester-)track. But I'll take a look as soon as I find time. Great that the PR directly also includes an example :)

I further hope the Quotient manifolds can show its potential here.

[mateuszbaran]

No problem, it will take me a few more days to finish this so you can focus on teaching.

Previous quotient manifold work really simplified the implementation here.

[kellertuer]

That was what I was hoping for, since we put quite some thought into the quotient manifold functions.

[kellertuer]

Here's a few comments.

[mateuszbaran]

I've looked at other features on shape spaces we could consider but they are complicated and not really essential so I'd prefer to merge this PR without them. #551 tracks those things.

[kellertuer]

Could we collect the others you found in there as well?

Besides that – I hope to find some time to review this PR soon then.

[mateuszbaran]

That issue already has the most important things. Geomstats also has a bunch of curvature derivatives I don't understand and I'm not sure what exactly they are for but that's something more general than Kendall shape spaces.

[kellertuer]

Here's a few comments.

In general it looks very nice, though I do not have an overview which other functions are available as well. I for example would love a tangent space basis (also the diagonalising one including curvature), if that exists. Maybe its even inherited from ArraySphere?

[mateuszbaran]

I for example would love a tangent space basis (also the diagonalising one including curvature), if that exists. Maybe its even inherited from ArraySphere?

For the pre-shape space basically everything can be worked out because it's just a fancy sphere. The only issue is that it can't reuse code for ArraySphere because of an additional constraint, which is which doesn't cause any issues with exp/log/PT but it does for bases. I'm just not sure how much work it would be.

For the shape space bases are much more complicated and I don't really want to work on it.