How to calculate tortuosity of the phases in a material from its cross section 2D image?

I have cross section image of material consistent of two solid phase and pore phase, how to calculate tortuosity of phases in this image ? and any references related to this will be much appreciated. Please see the attached image.

@iarganda can you please look into this, or tag somebody who knows about it ?

Hello @sam1990,

Regarding tortuosity, you can try to skeletonize your phases and then study the ratio between the Euclidean distance and the actual distance of each skeleton branch.


@iarganda Hi Ignacio, thank you for your response, can you please provide with literature reference where they have used such methodology to calculate tortuosity?

Have you checked the wikipedia page for tortuosity? There you are some references.

as far as I know, it’s quite common a definition for tortuosity (see Wikipedia). But with this image only, I’m afraid you won’t get representative results: you have too few particles, and almost all of them touch the edges. I hope you have larger images.

@iarganda yes I have checked and most of them do it for 3-D image or if they are doing it for 2D then they usually use LBM or random walk simulations, I am not exactly aware about them. I was looking for simple image analysis methods as you suggested by analyze skeleton.

yes @Nicolas , I do have a large scale image its just 20% fraction of image. But I didn’t find any paper where they used such methodology on 2-D image to give tortuosity for a material. Different researchers used simulation studies for 3D or 2D to get tortuosity value.

Have a look at this one:

The geodesic length is a concept close to the skeleton, so I would take that for a reason justifying the use of the skeleton.
And also:

In “Background”, they give a lot of references.


Thanks @Nicolas I will have a look at these.


just to mention that MorphoLibJ also provides some facilities for measuring the tortuosity of particles. It uses a defiinitioin of tortuosity based on the ratio of geodesic distance over max Feret diameter.


Thank you @dlegland, I forgot this.
Just for my understanding: Are the max Feret diameter endings the same as the geodesic diameter ones?

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not necessarily, and in general, not. A simple example is given by the letter “C”. The extremities of the maximum Feret diameter will be given by the top and the bottom points of the curve (assuming the letter elongated in the vertical direction). The extremities given by geodesic diameter will correspond to the two extremities at the right of the curve.

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I see. So, one must be careful when this definition is used, as it goes somehow against the intuitive definition of tortuosity, i.e. the length of the long path against the length of the short path, both having the same starting and ending points (e.g. in )

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There are usually two cases:

  • particle analysis -> tortuosity of each particle can be defined from the Geodesic diameter. Several definitions may be found however (see e.g.
  • microstructure / continuous phase analysis -> tortuosity is usually defined as the ratio of the geodesic distance between two image planes over the size of the image (c.f. the article from Peyrega and Jeulin).

Maybe a last comment: geodesic paths often comes very close from the phase boundary. One possibility is also to compute phase skeleton first, then computing geodesic distances on the binary image of the skeleton.