# Texture Analysis is a Useful Tool to Assess the Complexity Profile of Microcirculatory Blood Flow
## Materials and Methods
### Numerical  
The raw PPG signals were imported to Matlab (Mathworks R2012, Natick, MA, USA) and smoothed with a moving average filter. The WT ([[http://noc.ac.uk/using-science/crosswavelet-wavelet-coherence](http://noc.ac.uk/using-science/crosswavelet-wavelet-coherence)]([http://noc.ac.uk/using-science/crosswavelet-wavelet-coherence](http://noc.ac.uk/using-science/crosswavelet-wavelet-coherence))) was then applied to the smoothed signals, which allowed the decomposition into their main frequency components. Wavelets are a family of functions constructed from translations and dilations of a single function called the \"mother wavelet\" \(\psi(t)\), collectively defined by:  
\[\psi_{a,b}(t)=\frac{1}{\sqrt{|a|}}\psi\Big{(}\frac{t-b}{a}\Big{)},\ a,b\in R,\ a\
eq 0 \tag{1}\]  
The parameter \(a\) is the scaling parameter or scale, measuring the degree of compression, while parameter \(b\) is the translation parameter, determining the time location of the wavelet. If \(|a|<1\), then the wavelet in the above equation is the compressed version of the mother wavelet and corresponds mainly to higher frequencies. If \(|a|>1\), then \(\psi_{a,b}(t)\) has a larger time-width than \(\psi(t)\) and corresponds to lower frequencies. Thus, wavelets have time-widths adapted to their frequencies. In this case, the central frequency of the wavelet was 6 Hz, with a discretization factor of 10 scales per octave. The detected components of the PPG signals occurred at the following frequency ranges: cardiac [1.6-0.7 Hz], respiratory [0.7-0.26 Hz], myogenic [0.26-0.1 Hz], neurogenic/sympathetic [0.1-0.045 Hz], endothelial NO-dependent (NOd) [0.045-0.015 Hz] and NO-independent (NOi) [0.015-0.007 Hz], which were in line with the ranges previously described for LDF [36]. From the WT, two main outputs were generated for each subject: (1) a scalogram and (2) a 3D periodogram. The WT scalogram (Figure 1) shows the time (here converted to sample number) evolution of the amplitude with a factor of \(2^{n}\) (here in color scale) for each signal scale. The approximate frequency of a component is obtained by dividing the central wavelet frequency by the scale of the component. From a visual inspection, the WT scalogram also presents an oscillatory pattern in pixel distribution on different scales, making it a suitable candidate for TA. The scalogram was then used as a grayscale image, where several regions of interest (ROI) were marked, each per component per phase of the protocol. Each ROI was then converted to a gray-level co-occurrence matrix (GLCM), a matrix where the number of rows and columns is equal to the number of gray levels (G) in the image, from which the TE was calculated as follows [37]:  
\[\text{Entropy}=-\sum_{\text{i}=0}^{\text{G-1}}\sum_{\text{j}=0}^{\text{G-1}} \text{P(i,j).\log[\text{P(i,j)}]} \tag{2}\]  
where \(P(i,j)\) denotes the probability of occurrence of a given element in the matrix. To correctly select each ROI, a control sine wave signal was generated with multiple components occurring at known frequencies (f1 = 3.2 Hz; f2 = 1.6 Hz; f3 = 0.7 Hz, f4 = 0.26 Hz; f5 = 0.1 Hz; f6 = 0.045 Hz; f7 = 0.015 Hz; f8 = 0.007 Hz), where each frequency is close to the ones that define the borders of the PPG components' detected frequency ranges. The sine wave signal was then deconstructed with the WT, and a reference scalogram was generated (Figure 2). From the 3D periodogram (Figure 3), a 2D time evolution of the amplitude of each component was constructed by averaging the amplitude at each time point for each spectral interval (Figure 4). Signals were acquired at a 100 Hz sampling rate, which translates to a total number of 180,000 samples for the entire 30 minute protocol. The wavelet period is a natural logarithmic representation of the wavelet scale, with the frequency in Hz exponentially related to this period. The time evolution was  
Figure 1: WT scalogram of the PPG signal from the control (top) and test (bottom) feet from a representative subject (20 y.o.).