A λ = 533 nm, 50 mW output power green DPSS laser was used as the light source. A 40× microscope objective combined with a f = 50 mm biconvex lens were used to expand and collimate the laser beam. An Alfa a 200 CCD camera with a 3872 × 2592 pixels resolution and 6.095 × 7.253 mm pixel size was used for capturing and recording of speckle patterns. A roughness comparator (Microfinish, Figure 1) was used as the sample to be illuminated by the laser to obtain optical profiles and speckle patterns corresponding to the comparator sections characterized by different roughness ranges: 0.1 - 0.2 µm (lapped), 0.4 - 0.8 µm (rectified) and 1.6 - 12.7 µm (profiled). The green laser beam was directed to a point in the central part of each section to be analyzed on the comparator (Figure 2(a)), where a mechanical translational system allowed the longitudinal displacement through the sections in order to obtain the corresponding optical point scanning. The generated scattered light at a θ = 45˚ reflection angle, containing information about surface point-to-point irregularities and variations, was

collected by means of a biconvex lens (f = 200 mm) and directed to a power photodetector coupled to a data acquisition unit, which was also coupled to a PC in order to record it for latter analysis. For speckle optical patterns obtaining (Figure 2(b)), the green laser beam was amplified to a ~10 mm diameter with a 40X microscope objective combined with a f = 50 mm biconvex lens. A ~78.53 mm² area of the comparator was illuminated with the laser beam considering that its corresponding reflected speckle pattern, recorded with a CCD Alfa a 200 camera, contains the variations and irregularities existing on this surface area.

An application software was used for digital data processing of each recorded speckle pattern. Intensity profiles (1 × 1616 pixel vectors) corresponding to ~78.53 mm² illuminated cross-section were obtained; the resulting vector represents the image average intensity profile which also contains information about surface variations and irregularities of the considered surface area.

The point-to-point and surface speckle optical profiles were used to perform a qualitative analysis of the considered surface section of the comparator by means of RPs; for a quantitative analysis the RQA technique were also used. The Visual Recurrence Analysis application software (VRA Ver. 4.9) was used for RPs building [26] ; here an one-dimensional time series is expanded to a high-dimensional space where underlying system dynamics is generated inside an artificial phase-space called embedding space [27] , and where trajectories in the rebuild space are topologically equivalent to the original space [28] . For such a reason, determination of time- delay (τ) is necessary to perform a correct rebuild of the process dynamics, same as a proper embedding dimension (m). The mutual information function is used for optimal delay time value determining and also for phase-space rebuilding [29] ; if optimal time-delay value is too short, the coordinates used for rebuild could be not enough independent for carrying new information about the system trajectory through the phase-space. Moreover, if optimal time-delay is too long, coordinates may be random with respect to each other. For embedding dimension value selection (m), the false nearest neighbor method is used [30] . The m value determines the number of components of the states-system rebuild vector where for noiseless time series, using embedding dimension high values causes no problems. Oh the other hand, for time series containing noise, this is amplified and the quality of prediction is deteriorated, so it is necessary to fix a proper m value, in such a way that the system dynamics is optimally resolved.

In this work, the Commandline Recurrence Plots application software was used for complex structures quantification [31] , where only these RQA following parameters were considered: Recurrence Rate (%REC), Determinism (%DET) and Laminarity (%LAM). Recurrence Rate is the percentage of recurrence points in the phase-space, excluding main diagonal points:

(1).

Implicit periodic processes are characterized by higher %REC values. Determinism is the percentage of recurrence points forming parallel segments on the main diagonal, which length reaches or exceeds minimum length threshold; %DET allows to tell between dispersed recurrence points and those points organized in diagonal patterns, and contains information about duration of a stable interaction as longer is an interaction, as higher is the %DET value:

where l is diagonal line length parallel to the identity line; P(l) is the frequency distribution of diagonal lines and R_ij is the recurrence of a state. Laminarity is the percentage of recurrence points included in linear segments vertical to the diagonal; %LAM measures chaotic transitions and is related to the quantity of laminar phases in the system (intermittence):

(3).

The Kolmogorov Entropy is a measure of information loss index (o gain) along an attractor of certain data set. Numerically speaking, K may be estimated as the Rényi Entropy, being the Information Theory of Shannon Entropy a particular case. On the other hand, the product of the Shannon Entropy and the Boltzmann constant is the Thermodynamic Entropy [8] . The Kolmogorov Entropy is defined as follows [8] [32] :

where {X = x_i} is the random variable and; is the joint probability than x (t = Dt) trajectory is in the i₁, x (t = 2Dt) in i₂, and x (t = NDt) in i_N. Here, application software RRCHAOS [33] have been applied for the maximum-likelihood estimation of Kolmogorov Entropy (KML), proposed by Schouten [34] ; KML is usually expressed in bits/seconds, where a finite positive value (0 < KML < ∞) means that underlying time series/data is chaotic, a value equal to zero (KML = 0) represents an ordered system (regular, cyclic or constant), and an infinite value (KML → ∞) infers that the system is an stochastic one (random) [8] .

The Approximate Entropy (ApEn) quantifies the regularity and complexity in real time series, where time series with higher ApEn values may present considerable irregularity fluctuations (more complex), while lower ApEn values would indicate more regular time series (less complex). Given its simplicity, ApEn has been applied in many fields of science and engineering including psychology, geophysics and financial systems. A detailed description of the method may be found on specialized literature [8] - [11] . The Approximate Entropy estimated for a finite value N is defined by:

For calculation of ApEna Matlab code implementation by Danny Kaplan was used here [35] [36] .

In Figure 3, data series of optical profiles and speckle patterns are shown, corresponding to different conventional machineries which surface roughness vary between R_a = 0.2 - 12.7 mm.

For qualitative (RPs) and quantitative (RQA) optical and speckle profiles analysis, a mutual information function analysis was performed to obtain the optimal time-delay value (τ) and consequently, by using the false neighbor method, it was used to obtain the embedding dimension value (m) where maximum dimension value was set on 10 and then, from obtained τ and computed m, RPs were generated with VRA software [26] . The corresponding RQA analysis was performed using Commandline Recurrence Plots software [31] considering τ and m values; the average phase-space diameter was also computed to obtain the optimal Threshold Radius (ε) [36] [37] . A set of the parameters described above is shown in Table 1.

The analysis consisted in generating RPs from the optical and speckle patterns of different machining processes for different surface roughness degrees. In Figure 4, corresponding RPs generated without threshold value ε are displayed.

The information about the dynamics of a time series/data is usually obtained from the density of points and line structures in a RPs [25] ; a recursive homogeneous plot without a structure typically indicates an autonomous or stationary process such as white noise, unlike oscillating systems which present periodic recurrent or diagonally orientated structures (diagonal lines or discernible patterns). The presence of vertical or horizontal lines indicates a presence of intermittence or laminarity, whereas sudden changes on the process or the dynamics as same as extreme events are characterized by white areas or bands [7] .

As it can be observed on Fig. 4, surface roughness R_a offers an initial visual inspection for identification of changes on dynamics of the structure, reflected on RPs textures which are defined as small scale patterns and revealed by the color and/or texture type (generated by the machining process) in the RP. The optical profiles (Figure 3(a), Figure 3(c), Figure 3(e), Figure 3(g), Figure 3(i), Figure 3(k), Figure 3(m)) and their corresponding RPs (Figure 4(a), Figure 4(c), Figure 4(e), Figure 4(g), Figure 4(i), Figure 4(k), Figure 4(m)) present certain discernible patterns or diagonally oriented structures as surface roughness increases; this agrees with surface texture of each process shown on Table 1, which might implies that sharp cutting forces produced during machining process generate very regular signals, and therefore a better machinability. On the other hand, speckle profiles (Figure 3(b), Figure 3(d), Figure 3(f), Figure 3(h), Figure 3(j), Figure 3(l), Figure 3(n)) only present vertical and horizontal lines as roughness degree increases, which indicate no presence of oscillations with some periodicity (i.e. no correlation among them), and also implying a stochastic behavior. This last may be related to the effect of surface roughness on the speckle pattern formation (Table 1), where this generates a random intensity distribution, as the result of multiple interference of light scattered from each irregularity on the rough surface.

Although RPs offer an initial visual comparison of surface roughness, it is necessary to evaluate it in a quantitative way. In order to perform Recurrence Quantification Analysis (RQA), the m, τ, and ε parameters (indicated in Table 1) are necessary, as also to consider the Euclidean distance as a parameter to determine the distance between new vectors; in our case a minimum diagonal of 2 was considered. In Figure 5(a) the Recurrence Rate (%REC) was calculated for optical and speckle profiles as a function of surface roughness R_a; the average %REC for the profiles was 47.69% y 60.23%, respectively, which implies that analyzed surface textures were generated by processes with certain degree of periodicity.

The calculated Determinism (%DET) is shown in Figure 5(b), where high %DET values can be adverted for different R_a degrees. The average %DET was 99.756% for optical profiles and 91.534% for speckle ones; this is reflected on surface texture nature of machining processes (see Table 1 images) as %DET reaches higher values when machining produces signals with more stable (regular) correlation and interaction degrees, due to cutting forces presented in the process. Unlike the above, if cutting forces during machining process were unstable (irregular), it may generate signals with low %DET values, and therefore, a low machinability.

In Figure 5(c), the calculated Laminarity (%LAM) is presented; as %LAM may be defined as an alternation of states or phases in dynamical systems, in our particular case this could be interpreted as the alternation of regular (stable) and irregular (unstable) surface textures, due to cutting forces. The average value %LAM for optical and speckle profiles was 99.856% and 95.006%, respectively.

Figure 6 represents the maximum likelihood Kolmogorov Entropy (KML) calculated from optical and speckle series. For the optical ones, KML exhibit a lower rate of information creation for every lapping, rectified and profiled cases, where an average value KML = 0.02 bits/seg was obtained; this value is characteristic for regular oscillating (certain periodicity degree) and therefore less complex series, which is reflected on the optical profiles shown in Figure 3. For the case of speckle profiles, a KML = 0.676 bits/seg (R_a = 0.2 µm) correspond to the zone I (lapping process); at zones II (rectified process) and III (profiled process), increments with maxima values at KML = 1.237 bits/seg (R_a = 0.8 µm), KML = 1.043 bits/seg (R_a = 1.6 µm) and KML = 1.13 bits/seg (R_a = 3.2 µm) were obtained. Such increments may be associated with a high sensitivity to external conditions or perturbations as the alternation of regular and irregular texture surfaces, for example, presented on speckle pattern intensities.

Regarding Approximate Entropy (ApEn) calculation shown in Figure 7, for optical profiles the obtained ApEn = 0.07717 low average value evidences the presence of regular oscillations, while for speckle profiles the calculated ApEn = 1.89, higher than obtained for optical profiles, may be associated to different irregularities between the processes as surface roughness increases, which reflected on the speckle pattern intensities. An advantage of ApEn with respect to KML is its computational effectiveness as same as its algorithm, which may produce acceptable results from certain relatively small number of observations.