Euler number vs Euler characteristic (problem about connectivity)

Hello community!

I have recently read some papers and found that Euler number is often used to express connectivity. The formula I see in this highly cited article( Vogel, H., & Roth, K. 2001. Quantitative morphology and network representation of soil pore structure. Advances in Water Resources, 24, 233–242 .)

is similar to one answers in a post said that:

If there are any similarities between the two? Euler number in this paper is equal to Euler characteristic in BoneJ? Can I think that C in the paper is equal to b1 in the answer, and H is equal to b2? Does b1 represent the connection(sorry, I don’t quite understand “handles” meaning)?

Because there is a Purify image process before BoneJ/Connectivity, should b0 be 1 in this case?

If Euler number is Euler characteristic, what is connectivity β1 = 1 - Δχ? According to the above answer formula, it is 1-(b0-b1+b2)? If b0 is 1, then it is b1-b2? If Euler characteristic can be used to express connectivity of objects, Why set this connectivity

This is very important for my work, hope to get your reply and check! :sob:

Thanks in advance
Mengying

Hi,

Yes Euler characteristic is sometimes called Euler number: https://en.wikipedia.org/wiki/Euler_characteristic. From the Wikipedia page you can see it has several definitions.

Yes, N - C + H is the same definition as Betti number. Cavities and handles are both kinds of holes in an object. A football has a cavity because it’s hollow, and a doughnut has a handle because there’s a hole through it.

Yes, after running Purify b0 = 1.

In BoneJ we calculate β1 = 1 - Δχ so that we see both the χ of a bone sample as a separate object, and it’s contribution to the χ of the whole bone. In practical terms Δχ corrects for any voxels that touch the edges of the image stack, because they affect topology. See https://www.sciencedirect.com/science/article/abs/pii/8756328293902456?via%3Dihub for more.

@mdoube Anything to add?

Best,
Richard

P.S. Considering this a forum of volunteers, it’s not considered polite to demand speedy answers on a weekend by posting the same question repeatedly

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Only thanks for taking the time to respond - I was waiting for normal office hours to start in HK before replying (and even then thought twice about it because I am on annual leave).

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Hi, @rimadoma
I’m very sorry to disturb your weekend. Thank you for reminding me of this. Because I didn’t notice it. Eh… Well, you know, this is a little different from ours. But I’m very supportive and I love this. Have a good weekend is a happy thing! So I’m really sorry to disturb your wonderful weekend, and I really thank you for being able to speak this matter out sincerely and let me understand. I’m also sorry to have disturbed other volunteers on the weekend before, especially @mdoube …… I will be careful about it in the future :heart:.

…Ok, I still have a few questions about connectivity. I hope you can help me to have a look if you have time. Thank you.

  1. Yes, Δχ can express the connection with the outside region. I want to use it to express my Euler number. My sample fits this situation. If I also would like to normalize Euler number with volume following the above paper, I figured it out like this (yellow marker column △ X / V).

    The results can be clearly divided into two parts, blue and orange. Blue has a significantly smaller value △ X / V, while orange has a larger value (Because they have a minus sign in front of them). By definition, Euler number decreases with increasing connectivity, i.e. blue has more connectivity than orange ones? However, this is contrary to my understanding. I have attached the data of porosity and pore number and will find that connectivity is closely related to the number of pores. Generally speaking, the increase of porosity will lead to the decrease of connectivity, but if we push it in this way, the blue porous objects have higher connectivity (△ X / V) than orange objects with few pores? I find this a little strange……I’m confused about with negative signs…

  2. How to look at the results of connectivity β1 = 1 - Δχ? That is, the greater the value, the greater the connectivity the sample has?

To sum up, I just wonder which index needs to choose to normalize the volume and how to judge the level of connectivity according to the value size. My mind is a little disorganized…

Thanks in advance!
Mengying

I’m starting to be out of my depth here, but the way I see it more porous samples would have a higher Δχ because when they connect to their neighbours they (may) add a lot of cavities to the whole object, and thus affect X more.

Hopefully you can access the Odgaard & Gundersen paper I linked, because it explains Euler correction (Δχ) a lot better than I can. To put it shortly Δχ is a way to get an estimate of the χ of a whole object e.g. trabecular bone by summing χ - Δχ of individual samples. The paper explains why naively summing just χ doesn’t work.

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Hi @rimadoma
Thank you for your time and advice. I’ll take a look at that paper. :relaxed: