sub-level set filtration for cubical complexes

[catanzaromj]

I work with cubical complexes and would like to filter by their gray scale values. Is this currently implemented in giotto?

Specifically, suppose I have an image, exactly analogous to the mnist handwritten digits example already included in giotto. I would like to filter by the gray scale values of the image. In practice, this is a sublevel set filtration on the 'underlying' cubical complex given by the pixels in the image.

An even better work flow might be not to start with the full image, but first apply some threshold binarizer. Then on the binarized image, filter by the original grey scale values. Is there an easy way to do this in giotto already?

Thanks

[ulupo]

Hi @catanzaromj! Thanks for the questions!

Specifically, suppose I have an image, exactly analogous to the mnist handwritten digits example already included in giotto. I would like to filter by the gray scale values of the image. In practice, this is a sublevel set filtration on the 'underlying' cubical complex given by the pixels in the image.

At the moment, I'm not sure why CubicalPersistence wouldn't do exactly what you are asking. I may be misreading your question, though.

Concerning general tools for images, the complete selection is here. We always welcome contributions, though :)

[catanzaromj]

Hi @ulupo. Thank you for the quick response.

I think I discovered one of my errors.

import sklearn

X = sklearn.datasets.load_digits()
digit = X['data'][0]

from gtda.homology import CubicalPersistence
cub_pers = CubicalPersistence(homology_dimensions=[0,1])

Up to this point, everything is fine. However, calling cub_pers.fit_transform(digit) versus cub_pers.fit_transform([digit]) produces two different results, neither of which is an error. I would have thought the former would have thrown an error as it does for VietorisRipsPersistence, but maybe this is too subtle to catch with the usual shape verifier.

Is the second idea I mentioned implemented already? This is natural because you're constructing a filtered complex on the shape/object you potentially care about rathern than the underlying grid of all pixels. In particular, filtering the underlying square by intensity produces a persistence diagram where the infinite persistence points are always just a single class in degree 0 and none in higher dimensions. However, if the underlying cubical complex has topology of its own (like a binarized figure 8 or zero), then the infinite classes reflect the topology of the shape. I have found this second approach to be very useful in medical imaging applications. I'm happy to try and contribute this if its not present already.

[ulupo]

but maybe this is too subtle to catch with the usual shape verifier

Ah yes, I see what you mean! But I always found this to be a feature, not a bug. This way, a 2D array input (instead of 3D array or collection of 2D arrays) can be interpreted as a collection of 1-dimensional cubical complexes. One can do e.g. the sub-level set filtration of a collection of time series in this way!

Concerning your second point, I think it would effectively be implemented by setting the pixel values in the points which would be binarized to 0 as np.inf, since this means they never appear in the filtration. What do you think?

[catanzaromj]

Do you accept PRs with respect to documentation or examples? Are there any examples of applying cubical persistence 'directly' instead of first using a filtration in the documentation? I'd be happy to help out given how much use I've gotten out of giotto-tda

[ulupo]

Absolutely we do accept PRs! CubicalPersistence is only used in the MNIST notebook at the moment, so all non-trivial examples would be there. Happy to discuss a contribution.

[catanzaromj]

What would be the best way to proceed? Should I create some examples in a PR?

[ulupo]

It's important for all parties to minimize time waste so perhaps it might help if you could first create an issue briefly describing the proposed addition (don't worry about being complete), so that we can agree on it before moving forward with a PR.

[catanzaromj]

Great, thanks!