*1.1. Some Previous Results of Fractal Technique Application on Ceramics Samples*

Short Description of the Applied Technique for the Grain Cluster Shape Reconstruction An image from Figure 1 was imported into a pptx file and a well-definedgrid inserted on it. Equally spaced yellow points were marked on a portion of the contour and the following image was obtained (see Figure 2). Then, an ordered list of yellow points was scaled and registered in a file. After a few simulations on the program Fractal Real Finder [1,2], we find that with *p* = 10 and *L* = 2, a sufficiently good fit is obtained (*p<sup>L</sup>* = 100

points) (see the following plot in Figure 3).

**Figure 1.** A part of the microstructure morphology of BaTiO3 Ceramics sample.

**Figure 2.** Successful reconstruction of BaTiO3 sample structure by using fractal method.

**Figure 3.** Affine Fractal Regression of the reconstruction of BaTiO3 sample structure by using fractal method.

The statistical Kolmogorov-Smirnov test, as well as the respective plot, strongly confirms the good reconstruction of the original data. From the output of the program, the relevant fractal coefficients are 0.161007796 and 0.188832965.

With these estimated fractal coefficients, an upper estimative for the Hausdorff dimension (which will be denoted with D and explained more firmly in Section 2.1) was computed as the solution of the nonlinear equation:

$$8\left(\frac{1}{10}\right)^D + 0.161007796^D + 0.188832965^D = 1.\tag{1}$$

The calculated Hausdorff dimension is *D* = 1.06557. The result with fractal interpolation has fractal dimension estimated as *D* = 1.40792. Why these two methods give fractal dimensions so different? The difference between interpolation and regression is that the first one finds a function that matches all points, and the second finds a function that approximates the set of points, reducing the error of squared residuals. In practice, the fractal interpolation method introduces between every two points the fractal spikes. This provides a substantial over-estimative for the fractal dimension.
