Effect of the image resolution on the statistical descriptors of heterogeneous media

The characterization and reconstruction of heterogeneous materials, such as porous media and electrode materials, involve the application of image processing methods to data acquired by scanning electron microscopy or other microscopy techniques. Among them, binarization and decimation are critical in order to compute the correlation functions that characterize the microstructure of the above-mentioned materials. In this study, we present a theoretical analysis of the effects of the image-size reduction, due to the progressive and sequential decimation of the original image. Three different decimation procedures (random, bilinear, and bicubic) were implemented and their consequences on the discrete correlation functions (two-point, line-path, and pore-size distribution) and the coarseness (derived from the local volume fraction) are reported and analyzed. The chosen statistical descriptors (correlation functions and coarseness) are typically employed to characterize and reconstruct heterogeneous materials. A normalization for each of the correlation functions has been performed. When the loss of statistical information has not been significant for a decimated image, its normalized correlation function is forecast by the trend of the original image (reference function). In contrast, when the decimated image does not hold statistical evidence of the original one, the normalized correlation function diverts from the reference function. Moreover, the equally weighted sum of the average of the squared difference, between the discrete correlation functions of the decimated images and the reference functions, leads to a definition of an overall error. During the first stages of the gradual decimation, the error remains relatively small and independent of the decimation procedure. Above a threshold defined by the correlation length of the reference function, the error becomes a function of the number of decimation steps. At this stage, some statistical information is lost and the error becomes dependent on the decimation procedure. These results may help us to restrict the amount of information that one can afford to lose during a decimation process, in order to reduce the computational and memory cost, when one aims to diminish the time consumed by a characterization or reconstruction technique, yet maintaining the statistical quality of the digitized sample.

Figure 1

Statistically isotropic heterogenous materials: Impenetrable disks (ID, top left), partially overlapping disks (OD, top right), and coarse grained random heterogenous materials obtained with a Laplacian of Gaussian kernel (LoGK1, bottom left; LoGK3, bottom right), for two different values of the level-cut threshold. The corresponding surface fractions, as well as the parameters used for their conformation, are given in Tables 1 and 2.

Figure 2

Consecutive stages [from (a) $k=1$ to (h) $k=8]$ following a random decimation process, departing from the full-resolution image ($k=0$ with $N_0=M_0=4096$) of partially overlapping disks presented in Fig. 1.

Figure 3

Statistical descriptors along the random decimation process presented in Fig. 2. Top and bottom rows represent the descriptors for phases $j=0$ and $j=1$, respectively, whereas each column indicates the type of descriptor: (a) Surface fraction ${\varphi }_j$ and specific interface area per phase surface fraction $s/{\varphi }_j$ as functions of the $k\mathrm{th}$ decimation step, (b) normalized autocovariance function $F_{1,j}={\chi }_j$, (c) normalized line-path correlation function $F_{2,j}=L_j$, and (d) normalized pore-size distribution function $F_{3,j}=P_j$. The statistical descriptors in columns (b)–(d) are presented as functions of the distance $r$. Each color of the points in column (a) and the curves in columns (b)–(d) corresponds to a different $k\mathrm{th}$ decimation step, given by the horizontal position of the point in column (a).

Figure 4

Deviation of statistical descriptors, with respect to the full-resolution image, as a function of the $k\mathrm{th}$ step along the random decimation process, presented in Fig. 2. These deviations were obtained by applying Eq. (29). The subindices $\beta$ and $j$ correspond to the normalized statistical descriptor and phase, respectively, specified in Sec. 2b.

Figure 5

Ensemble average global error $\langle {\mathcal{E}}_k\rangle$, for each material described in Table 1, (a) impenetrable and (b) overlapping disks, and in Table 2, (c) LoGK 1, (d) LoGK 2, and (e) LoGK 3, as a function of the $k\mathrm{th}$ step along the three proposed decimation processes. The average and standard deviations (error bars) were obtained by applying Eqs. (31). The statistically sensitive decimation step $Z$ was computed with Eqs. (34), applied to the extrema, (—) $min\left({\ell }_{\beta ,j}\right)$ and ($\cdots$) $max\left({\ell }_{\beta ,j}\right)$, and the average, (- -) $\overline{{\ell }_{\beta ,j}}$, of the correlation lengths of the different descriptors $\beta$ and phases $j$.

Figure 6

Average statistical descriptors of the original images (with $k=0, M_0=N_0=4096$) of the $W=20$ different configurations of partially overlapping disks, with the same characteristics as the configuration presented in the top row, right column of Fig. 2. Each average statistical descriptor was computed with Eq. (32) and its corresponding correlation length is shown as a vertical dashed line. The subindices $\beta$ and $j$ correspond to the normalized statistical descriptor and phase, respectively, specified in Sec. 2b. For better visualization, only $1/10$ of the data points are shown.

Figure 7

Normalized coarseness (average and standard deviation of $W=20$ configurations) as a function of the $k\mathrm{th}$ step along the decimation process, for each material described in Tables 1 and 2. The curves were generated with the use of Eq. (33). Note that the curves corresponding to the LoGK configurations overlap. The $*$ symbol at each curve indicates the optimal decimation step $Z$ computed from its characteristic length scale $\ell =min\left({\ell }_{\beta ,j}\right)$ and Eqs. (34). The gray surface indicates the region for which $C_k^{*}>0.9$.

Figure 8

(a) Original image of a catalytic layer obtained by SEM. (b) Full-size binarized image. Decimated image after (c) two steps and (d) three steps of a bicubic process. (e)–(g) Statistical descriptors for the full-size and the decimated images. Top and bottom rows represent the descriptors for phases $j=0$ and $j=1$, respectively, whereas each column indicates the type of descriptor: (e) Normalized autocovariance function $F_{1,j,k}={\chi }_j$, (f) normalized line-path correlation function $F_{2,j,k}=L_j$, and (g) normalized pore-size distribution function $F_{3,j,k}=P_j$. In columns (e)–(g), solid circles (blue) correspond to the descriptors computed for the full-size binarized image, whereas open diamonds (yellow) and crosses (red) correspond to the descriptors computed for two-step and three-step decimated binarized images, respectively.

Figure 9

Quantitative information about the deviation from the full-resolution SEM image presented in Fig. 8. (a) Surface fraction ${\varphi }_j$ and specific interface area $s/{\varphi }_j$, (b) deviation of statistical descriptors $E_{\beta ,j,k}$ (for each descriptor $\beta$ and phase $j$, specified in Sec. 2b), and (c) global error as functions of the $k\mathrm{th}$ step of a bicubic decimation process.