Efficient 3D erosion dilation analysis by sub-pixel EDT

The micro-structure of materials or natural substrates contributes significantly to macroscopic properties like mechanical strength or filtration performance. The intrinsic volumes or their densities are versatile geometric characteristics of micro-structures and can be estimated efficiently from 3D binary images whose foreground represents the structure of interest. A recent algorithm generalizes this approach to gray value images. That is, the intrinsic volumes are derived for each possible global gray value threshold in the image in a single pass through the image. Here, it is combined with a sub-pixel precise Euclidean distance transform enabling efficient so-called erosion dilation analysis. That means, the densities of the intrinsic volumes are simultaneously computed not only for the structure itself but also for all erosions and dilations by spherical structuring elements of this structure. That way, the algorithm reveals additional information on the local size of critical features of the structure or its complement. The algorithm's power is demonstrated by means of computed tomography image data of rigid foams and stacks of scanning electron microscopies of nano-porous membrane layers sliced by a focused ion beam.