Using chaotic dynamics to characterize the complexity of rough surfaces

In this work, we use the basic ingredients of chaotic dynamics (stretching and folding of phase space points) for the characterization of the complexity of microscopy images of rough surfaces. The key idea is to use an image as the initial condition of a chaotic discrete dynamical system, such as the Arnold cat map, and track its transformations during the first iterations of the map. Since the basic effects of the Arnold map are the stretching and folding of image texture, the application of the map leads to an enhancement of the high frequency content of images along with an increase of discontinuities in pixel intensities. We exploit these effects to quantify the complexity of $S$ type (lying between homogeneity and randomness) of the image texture since the first (enhancement of high frequencies) can be used to quantify the distance of texture from randomness and noise and the second (the proliferation of discontinuities) the distance from order and homogeneity. The method is validated in synthetic images which are generated from computer generated surfaces with controlled correlation length and fractal dimension and it is applied in real images of nanostructured surfaces obtained from a scanning electron microscope with very interesting results.

Figure 1

Examples of scanning electron microscope (SEM) images depicting rough surfaces with complex morphologies.

Figure 2

Schematic of the relationship of type-$S$ and type-$R$ complexity measures with the transition from order to full randomness. From left to right we present examples of different surfaces ranging from order (symmetry or homogeneity) (left) to randomness (right) and their relation to the above mentioned measures. We can observe that although type-$R$ complexity grows monotonically with randomness, type-$S$ measures exhibit a maximum for the surfaces that lie between order and randomness.

Figure 3

The first iterations of the Arnold cat map which shows the gradual transformation of the image to a wrinkled pattern and then to a fully noisy image along with the last iterations just before the return to the original initial image after 168 iterations.

Figure 4

The first iterations of Arnold cat map of an original image ($800\times 800$ pixels) until the fifth iteration when the stripes start to become incredibly thin just before noise takes over in the next iteration.

Figure 5

Schematic illustration of our methodology: (a) Profile perpendicular to the stripes; (b) 100 profiles of the same length perpendicular to the stripes; (c) An example (of one) of the aforementioned profiles; (d) the average FFT of the profiles with the corresponding linear fit in our region of interest.

Figure 6

Typical examples of synthetic images generated from the Fourier filtering technique using various values of correlation length $\xi$ and roughness exponent $\alpha$. From left to right we show surfaces with ascending correlation length (=10, 50, 100 pixels) and three different roughness exponents ($\alpha =1$ top row, $\alpha =0.5$ middle row, and $\alpha =0.2$ bottom row). One can notice the transition from uncorrelated randomness at small $\xi$ to smooth textures at large $\xi$ as well as the enhancement of high frequency fluctuations when $\alpha$ shifts to smaller values.

Figure 7

Top-down SEM images of aluminum etched in hydrochloric acid for different durations: (a) 5 min in HCl, magnitude $\times 2000$, SEI sensor; (b) 7 min in HCl, magnitude $\times 2000$, LEI sensor; (c) 8 min HCl magnitude $\times 2000$, LEI sensor; (d) 15 min in HCl, magnitude $\times 2000$, LEI sensor.

Figure 8

(a) Examples of typical Fourier spectra $F_n\left(k\right)$ of the first five iterations of Arnold cat map when the initial image is generated from a rough surface with $\xi =50$ pixels and $\alpha =1$. One can observe the gradual flattening of Fourier spectra with map iterations demonstrating the randomization of image with map iterations. (b) Examples of $F_1\left(k\right)$ for initial images with five different correlation lengths (10, 30, 50, 70, 100 pixels) where one can notice the proximity of smaller correlation lengths to randomness.

Figure 9

The transition from order or homogeneity (left) to noise or randomness (right). In the top row we present the greyscale images of our surfaces while in the bottom row we show the $F_n\left(k\right)$ in pixel ($y$ axis) versus $k$ in $\mathrm{pixe}{\mathrm{l}}^{-1}$ ($x$ axis) for the first five iterations of Arnold cat map. (a) 2D sinusoidal surface with full order; (b) surface with correlation length $\xi =100$ pixels and roughness exponent $\alpha =1$; (c) surface with correlation length $\xi =50$ pixels and roughness exponent $\alpha =1$; (d) surface with correlation length $\xi =5$ pixels and roughness exponent $\alpha =1$; (e) fully noisy surface where Gaussian noise dominates.

Figure 10

Diagram shows the value of $S{D}_{\mathrm{b}}$ ($y$ axis) for the surfaces shown in Fig. 9 going from order ($I_{\mathrm{FO}}$) to noise ($I_{\mathrm{FR}}$), with $I_{\mathrm{FO}}$ being the 2D sinusoidal surface and $I_{\mathrm{SA}}$ being the surfaces with correlation lengths $\xi =100$, 50, and 5 pixels and roughness exponent $\alpha =1$ respectively. $I_{\mathrm{FR}}$ stands for the fully noisy surface.

Figure 11

(a) Plot of $S{D}_{\mathrm{b}}$ ($y$ axis) versus roughness exponent ($x$ axis) for three different correlation lengths (=10, 50, 80). (b) Contour plot of the $S{D}_{\mathrm{b}}$ versus correlation length ($x$ axis) and roughness exponent ($y$ axis).

Figure 12

$S{D}_{\mathrm{b}}$ versus etching time for the experimental images described in Sec. 4b. One can notice the maximum that complexity exhibits at 8 min in the middle of etching process.