# Statistical Analysis of Nanofiber Mat AFM Images by Gray-Scale-Resolved Hurst Exponent Distributions

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

## 3. Results

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- False-color maps of gray-scale-resolved Hurst exponent distributions enable distinguishing between different fiber distributions, especially between images with many and with only few fibers.
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- Polynomial background subtraction levels out the upper channel numbers for which a signal occurs and should thus be performed, especially in order to avoid ignoring hidden features (cf. Figure 9a,b).
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- Aligning rows and deleting horizontal errors leaves the images and thus, the false-color maps nearly unaltered.
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- The effect of sharpening once differs from sample to sample and must be investigated further in the next study.
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- Sharpening twice modifies not only the image in an unnaturally looking way, but also the false-color maps of the gray-scale-resolved Hurst exponent distributions and should thus be avoided.
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- What could not be performed yet, but has to be tested in the near future, is a comparison with a highly accurate AFM with increased quality of the original images, as reported in [36].
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- Generally, the meaning of the “noise” in the range of Hurst exponents values below approx. 0.25 needs further investigation to find out whether this range should be ignored since it is mostly based on errors in the images, or whether it can oppositely be used as part of the “fingerprint” of such AFM images. To explain the threshold value of 0.25, we can refer to the composition of two random processes in analogy to [37]. In our model the first random process would concern the formation of tiny features in textiles (or their maps) and the second one would be the random walk itself. Further referring to [25], such Hurst exponent values (being the inverse of the fractal dimension of the random walk therein) would be far below the outcome of the random walk on any percolation model in 2D.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Original atomic force microscopy (AFM) images (

**a**–

**d**), taken on different positions of an electrospun nanofiber mat, as explained in detail in the text.

**Figure 2.**Post-treatment steps, performed on sample a: (

**a**) polynomial background subtraction; (

**b**) aligning rows; (

**c**) deleting horizontal errors; (

**d**) sharpening once.

**Figure 3.**Gray-scale distribution (excerpt) of Figure 2a: (

**a**) channels 32–39; (

**b**) channels 104–111; (

**c**) channels 112–119; (

**d**) channels 208–215. Comparing with Figure 2a, one can conclude that (

**a**) is connected with deepest features that are darkest, while (

**d**) is correlated with the top features that are brightest; hence, the gray scale reflects the 3D structure of the image.

**Figure 4.**Example of the random walks evaluation on AFM gray-scale-resolved images: (

**a**) linear relation between ln ($\u27e8\mathsf{\Delta}{r}^{2}\u27e9$ and ln(n); (

**b**) Hurst exponent distribution.

**Figure 5.**(

**a**) Gray-scale-resolved Hurst exponent distribution of sample a in the original version, with the color-code giving the counts of the included histograms; (

**b**) classical distribution for channel 20; classical distribution for channel 21 (

**c**).

**Figure 6.**Gray-scale-resolved Hurst exponent distribution of sample a: (

**a**) original (identical to Figure 5a); (

**b**) after polynomial background subtraction; (

**c**) after aligning rows; (

**d**) after deleting horizontal errors; (

**e**) after sharpening; (

**f**) after sharpening twice.

**Figure 7.**Gray-scale-resolved Hurst exponent distribution of sample b: (

**a**) original; (

**b**) after polynomial background subtraction; (

**c**) after aligning rows; (

**d**) after deleting horizontal errors; (

**e**) after sharpening; (

**f**) after sharpening twice.

**Figure 8.**Gray-scale-resolved Hurst exponent distribution of sample c: (

**a**) original; (

**b**) after polynomial background subtraction; (

**c**) after aligning rows; (

**d**) after deleting horizontal errors; (

**e**) after sharpening; (

**f**) after sharpening twice.

**Figure 9.**Gray-scale-resolved Hurst exponent distribution of sample c: (

**a**) original; (

**b**) after polynomial background subtraction; (

**c**) after aligning rows; (

**d**) after deleting horizontal errors; (

**e**) after sharpening; (

**f**) after sharpening twice.

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**MDPI and ACS Style**

Blachowicz, T.; Domino, K.; Koruszowic, M.; Grzybowski, J.; Böhm, T.; Ehrmann, A.
Statistical Analysis of Nanofiber Mat AFM Images by Gray-Scale-Resolved Hurst Exponent Distributions. *Appl. Sci.* **2021**, *11*, 2436.
https://doi.org/10.3390/app11052436

**AMA Style**

Blachowicz T, Domino K, Koruszowic M, Grzybowski J, Böhm T, Ehrmann A.
Statistical Analysis of Nanofiber Mat AFM Images by Gray-Scale-Resolved Hurst Exponent Distributions. *Applied Sciences*. 2021; 11(5):2436.
https://doi.org/10.3390/app11052436

**Chicago/Turabian Style**

Blachowicz, Tomasz, Krzysztof Domino, Michał Koruszowic, Jacek Grzybowski, Tobias Böhm, and Andrea Ehrmann.
2021. "Statistical Analysis of Nanofiber Mat AFM Images by Gray-Scale-Resolved Hurst Exponent Distributions" *Applied Sciences* 11, no. 5: 2436.
https://doi.org/10.3390/app11052436