How to find films given only Plateau border locations? There are two general aspects to this topic: a mathematical aspect (i.e. the mathematics of reconstructing of minimal surfaces) and a practical aspect (e.g. analysing/reconstructing tomographic data) The fundamental question: In principle can one reconstruct a foam given only the Plateau borders? Related questions: How to identify/find which borders and vertices form a cell? What are good algorithms to identify/find cells? How to determine bubble pressures? In fact, how easy it is to reconstruct films depends on whether borders are available as 1D curves only, or whether one can resolve the tricuspid cross-section of the borders (the shape of the tricuspid cross section would tells you the direction in which the films leave the borders -- and it is then just a matter of choosing the correct mean curvature for each film such that the directions match) Points mentioned in discussion: Uniqueness of solution? If I manage to reconstruct a foam based on the borders, is the reconstruction unique? If I dipped an `open foam skeleton' in soap solution -- would I reconstruct the original structure? Not necessarily, since films are now pinned to the existing skeleton and do not need to satisfy Plateau rules at the pinning points, i.e. the pinning points do not need to be force free, since the skeleton itself can resist imposed force. The structure that results from dipping the open foam skeleton in soap solution might have bubbles which differ in volume from the original bubbles (e.g. it would be theoretically possible to have all faces with zero mean curvature with the solid skeleton providing support) Reconstruction techniques have been developed by several groups (Paris, Dublin, Rennes, Aberystwyth) Iterative techniques are often used for reconstruction, e.g. in 2D, it is required to find 1 curvature per side that satisfies the various constraints: sum(Delta p) = 0 around a vertex (Delta p is pressure difference) sum(kappa l) + n pi/3 = 2 pi around a bubble (kappa is curvature, l is film length, n is number of edges) Delta p = kappa Procedure estimates l based on straight line length first, obtains bubble pressures and hence curvatures via the above mentioned constraints, and then reestimates the film lengths l for the curved films. Bubble areas can be obtained via area of inscribed polygons in the first instance, subsequently with corrections from the curved arcs. It is absolutely necessary to have curved arcs -- since flat faced bubbles are only a possibility for hexagonal bubbles (anything else would violate Plateau's rules) It is difficult to generalise in a direct fashion such a technique to 3D however -- as integral constraints on curvature apply to Gaussian curvature in 3D, whereas pressure difference is related to mean curvature Morevoer whereas in 2D connectivities are often known (and one is merely trying to determine edge curvatures and bubble areas), in 3D, cell connectivities are often not known a priori. Identifying cells in the first place is potentially more challenging than reconstructing bubbles given known cell topologies. (The 2D analogue of this would be to construct cells given only vertex locations -- not the edges connecting them) Cell identification algorithms (e.g. in 3D) tend to proceed by constructing faces made up of relatively few edges, and then choosing faces consisting of more and more edges. Vertices are excluded from the search list once they are attached to the required number of edges that Plateau's laws dictate. Such algorithms might fail to detect highly elongated bubbles properly (e.g. by trying to add a face which is not actually present). Such highly elongated bubbles could arise in some high capillary number applications e.g. rapid shear, emulsification, bubble breakage. Presence of solid obstacles within a flow can also produce elongated bubbles (even at low capillary number). Both Surface Evolver and Potts have been used in 3D reconstructions. Potts has some convenient computational features e.g. 1 pixel in simulation can be made to correspond to 1 pixel in experimental data, bubbles can be made to grow until they touch (giving you the topology); the numerical complexity of the problem grows only moderately with the number of bubbles; algorithms parallelize easily; wet as well as dry foams can be considered Gradient guided growth models have been used to grow bubbles based on experimental data sets. Each pixel in the data set obtains a score according to its distance from the nearest vertex. Local maxima of the score are used to identify bubble centres. Bubbles are grown out from these centres along the gradients of the score, until they touch. A Potts like model could also be used to correct/refine the results of the gradient guided growth (e.g. by adjusting cell boundary locations to find configurations with lower surface energy)