With Estimate Surface Area, the surface area of the structure is estimated with a method based on statistical measures.

The algorithm calculates an approximation of the surface area by statistical methods and not simply by adding up the voxel surfaces. Therefore, the surface of a sphere is approximated correctly. However, the estimation is slightly biased in case of highly anisotropic media. The method originates from statistical image analysis, where the determination of the four Minkowski measures (Volume, Surface Area, Integral of mean curvature, Integral of total curvature) from voxelized images is an essential task. To determine the surface area, the Crofton formula is used, which relates at first the 3D surface area to an integral over 2D boundary lengths of planar cross sections and then second these lengths to an integral over 1D rays. Based on this formula, an analysis of the intersection points of rays in all space directions with the structure allows determining the surface area.

The Estimate Surface Area options open when clicking the Edit… button. The Estimate Surface Area dialog includes the Surface Area Options tab and the Equations & References tab, which provides further information about the underlying algorithm from J. Ohser, F. Mücklich.

At the top of the dialog, enter the Result File Name. The result file is saved in the chosen project folder (File → Choose Project Folder in the menu bar).

Parallelization Settings

Surface Mode

Under Surface Mode choose the Materials or Material IDs for which the surface area should be calculated. The surface is defined as the boundary between the chosen quantity and the rest of the structure.

Pore Space computes the surface of the pore space, which is the union of Pore and all Fluid materials. It returns the same result as All Solid Materials, which computes the surface of all solid and porous materials combined to one material. Especially, All Solid Materials does not include the inner surfaces between different solid Materials or Material IDs. For your convenience, all Material IDs which denote pore space are shown at the bottom of the dialog.

If Chosen Material is selected, choose the desired Material from the pull-down menu that appears below. This will estimate the surface area between the chosen Material and any other Materials.

If Chosen Material ID is selected, choose the desired Material ID from the pull-down menu that appears below. This will estimate the surface area between the chosen Material ID and any other Material IDs.

You can also estimate the surface between only two selected Materials or Material IDs. Choose the surface mode Between Materials or Between Material IDs and select the two materials or Material IDs to be considered for the surface area.

Periodic

When selecting Periodic, the algorithm estimates the surface area assuming that the structure is periodic, so that the material that ends on one side of the volume reappears on the opposite side.

Observe the different results for the half sphere with radius 10µm touching the domain boundary. The surface area of a half sphere with 10µm radius is approximately 628 µm2 and the area of the circle from the cutting plane is approx. 314µm2.

GeoDict uses two methods to estimate the surface are, counting voxel surfaces (algorithm 1) and using an algorithm provided by J. Ohser, F. Mücklich (algorithm 2). Find out more about the algorithms in the Results section.

When not checking Periodic, the staggered grid causes half a voxel to be cut off at the domain boundary. Thus, the counted voxels are those from the sphere surface and the cutting plane at the domain boundary, while the algorithm by J. Ohser, F. Mücklich corrects it to the area of the curved sphere surface.

If Periodic is checked, the spherical surface of the cutting plane is counted twice in algorithm 1, as it is additionally reached from the opposite domain boundary. The algorithm by J. Ohser, F. Mücklich for the real surface area corrects this doubled counted surface area because the half voxels on each opposite domain side are connected.