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In addition to the area and perimeter percentages, for the three-dimensional reconstruction algorithms to be described below, it is also necessary to measure the autocorrelation functions for the individual phases and certain combinations of phases. For an M x N image, the autocorrelation function, S(x,y), takes the form:
Figure 2: Flowchart specifying segmentation algorithm for selecting pixel phase
values.
Figure 3: Final segmented two-dimensional image of Cement 116 issued by the
Cement and Concrete Reference Laboratory (NIST). Phases from
brightest to darkest are: C3A, gypsum, C4AF, C3S, C2S, and
porosity. Image is approximately 250 µm x 200 µm.
The user must input the size (M x N pixels) of the image and also a computational ``mask'' to be used in the correlation calculation. The mask is used so that simple logical and calculations may be employed in computing the correlation function and consists of the sum of 2 raised to the ID for each phase of interest. For example, to determine the autocorrelation for the two silicate phases (C3S and C2S), the mask would be (21 + 22) or 6. Similarly, the masks for C3S, C3A, and C4AF would be 2, 8, and 16, respectively. The user can also specify a scaling factor, n (1 or 2 for instance), indicating that only every nth pixel should be considered in the autocorrelation calculation, in case the resolution of the final processed SEM image is greater than the 1 µm/pixel normally employed in the cement hydration model.
Once S(x,y) is determined, it is converted to S(r) format, since for an
isotropic media the autocorrelation function should only be a function
of distance 
and not (x,y) [8]. For this
conversion, the program corrxy2r.c provided in Appendix A is used. It
accepts as input a file output by the program corrcalc and returns a file
containing a listing of S(r) vs. r. The following equation is utilized for the
conversion from Cartesian to polar coordinates:
)=S(r cos,r sin). For non-integer values of
(r cos) and (r sin), S(r, ) is obtained by bilinear
interpolation from the input values of S(x,y), which are only available for
integer values of x and y. The properties of the autocorrelation function are
such that S(0) provides the area fraction, Pa, of the phase(s) of interest
and as r 
, S(r) Pa2.
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