Estimating surface area by the isotropic fakir method from thick slices cut in an arbitrary direction

The proposed fakir method for estimating surface area is based on counting the intersections between the surface lying within a thick slice, and an isotropic spatial grid consisting of a combination of linear probes called fakir probes. An unbiased procedure using a directly randomized spatial grid rather than sections with randomized directions is presented. The method is applicable if perfectly registered serial sections of the surface are available in a thick slice while the direction of the slice can be arbitrary. The efficiency of the fakir method using different arrangements of orthogonal triplets of fakir probes is evaluated and it is shown that mutually shifted probes are superior to non-shifted ones. The application software for interactive counting of intersections between computer-generated fakir probes and the surface within the stack of digitized images is described and demonstrated by two examples: estimation of the surface area of individual tobacco cell chains using a confocal microscope, and estimation of the total area of exposed surface of mesophyll cells in a barley leaf using a wide-field transmission microscope.