Multiscale multifractal detrended-fluctuation analysis of two-dimensional surfaces

Two-dimensional (2D) multifractal detrended fluctuation analysis (MF-DFA) has been used to study monofractality and multifractality on 2D surfaces, but when it is used to calculate the generalized Hurst exponent in a fixed time scale, the presence of crossovers can bias the outcome. To solve this problem, multiscale multifractal analysis (MMA) was recent employed in a one-dimensional case. MMA produces a Hurst surface $h(q,s)$ that provides a spectrum of local scaling exponents at different scale ranges such that the positions of the crossovers can be located. We apply this MMA method to a 2D surface and identify factors that influence the results. We generate several synthesized surfaces and find that crossovers are consistently present, which means that their fractal properties differ at different scales. We apply MMA to the surfaces, and the results allow us to observe these differences and accurately estimate the generalized Hurst exponents. We then study eight natural texture images and two real-world images and find (i) that the moving window length (WL) and the slide length (SL) are the key parameters in the MMA method, that the WL more strongly influences the Hurst surface than the SL, and that the combination of $\mathrm{WL}=4$ and $\mathrm{SL}=4$ is optimal for a 2D image; (ii) that the robustness of $h(2,s)$ to four common noises is high at large scales but variable at small scales; and (iii) that the long-term correlations in the images weaken as the intensity of Gaussian noise and salt and pepper noise is increased. Our findings greatly improve the performance of the MMA method on 2D surfaces.

Figure 1

Two samples of synthetic surfaces merged by 2D multifractal signals and uniformly distributed random noises.

Figure 2

Multifractal structures of the merged surfaces. The main panel is the $F_q\left(s\right)$ versus $s$ in double-log plot. The vertical lines show two examples of the fitting windows for the small scales $s\in [6,24]$ and the large scales $s\in [30,120]$. Inset is the $h\left(q\right)$ curves calculated for the small (red dash line) and for the large scales (green solid line).

Figure 3

Hurst surface $h(q,s)$ dependence calculated for the surfaces A1 and A2 with WL = 4 and SL = 4.

Figure 4

2D synthetic fractal surfaces generated by fBM model and fGn model with known Hurst exponents.

Figure 5

Estimated multifractal long range correlation scaling exponents for the fBm surfaces B1 and B2 with given Hurst indices $H$. Error bars indicate standard deviation calculated from 20 independent realizations of the corresponding processes.

Figure 6

Estimated multifractal long range correlation scaling exponents for the fGn surfaces B3 and B4 with given Hurst indices $H$. Notation as in Fig. 5.

Figure 7

Multifractal structure of the synthetic multifractal surface generated p model by with parameter $p$ = [0.1, 0.2, 0.3, 0.4]. (a) is the fluctuation function $F_q\left(s\right)$ versus scale $s$ calculated using 2D MF-DFA with kinds of $q$. Inset: corresponding $h\left(q\right)$ dependence. (b) is the Hurst surface for the synthetic surface in (a).

Figure 8

Eight natural textures in Brodatz album.

Figure 9

Hurst surface $h(q,s)$ dependence calculated for the texture image C1 with different parameter combinations {WL, SL}. The red points correspond to $h(2,s)$.

Figure 10

Two images in the real world.

Figure 11

Hurst surface $h(q,s)$ dependence calculated for the image D1 added on different noises. The red points corresponds to $h(2,s)$. The presented results have been averaged over 20 realizations of the test series.

Figure 12

The $h(2,s$) calculated for the images D1 and D2 added on different noises. The presented results have been averaged over 20 realizations of the test series.

Figure 13

Average distance error of $h(q,s)$ over all the scales calculated for the kinds of noised images. The presented results have been averaged over 20 realizations of the test series.

Figure 14

Hurst surface $h(q,s)$ dependence calculated for the image D1 added on Gaussian noises with different PSNR. The red points corresponds to $h(2,s)$. The presented results have been averaged over 20 realizations of the test series.

Figure 15

Hurst surface $h(q,s)$ dependence calculated for the image D1 added on salt and pepper noises with different PSNR. The red points corresponds to $h(2,s)$. The presented results have been averaged over 20 realizations of the test series.

Figure 16

Average distance error of $h(q,s)$ over all the scales calculated for the noised image of D1 with different PSNR. The presented results have been averaged over 20 realizations of the test series.

Figure 17

Relative error of the Hurst exponent $h(2,s)$ calculated for the noised images of D1 and D2 with different PSNR. The upper one denotes the image D1; the bottom one denotes the image D2. The presented results have been averaged over 20 realizations of the test series.