Revisiting Sphere Unfolding Relationships for the Stereological Analysis of Segmented Digital Microstructure Images

Sphere unfolding relationships are revisited with a specific focus on the analysis of segmented digital images of microstructures. Since the features of such images are most easily quantified by counting pixels, the required equations are re-derived in terms of the histogram of areas (instead of diameters or radii) as inputs and it is shown that a substitution can be made that simplifies the calculation. A practical method is presented for utilizing negative number fraction bins (which sometimes arise from erroneous assumptions and/or insufficient numbers of observations) for the creation of error bars. The complete algorithm can be implemented in a spreadsheet. The derived unfolding equations are explored using both linear and logarithmic binning schemes, and the pros and cons of both binning schemes are illustrated using simulated data. The effects of the binning schemes on the stereological results are demonstrated and discussed with reference to their consequences for practical materials characterization situations, allowing for the suggestion of guidelines for proper application of this, and other, distribution-free stereological methods.