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Introduction

Substantial success in non-destructive studying the crystal-lattice defects properties is deeply connected with applying the high-resolution X-ray diffractometry (XRD), the so-called reciprocal- space mapping (RSM) method [1-4] and relevant X-ray diffraction technique (XRD) technique [5- 9] (see references in [5-9] as well). Schematic setup of the XRDT is shown in Fig. 1. The key elements of the XRDT method are the precision rotation of the sample around the diffraction vector h (angle Φ in Fig. 1), as well as the noise filtering to the reference 2D imaging pattern data recorded via the CCD detector (D) with the acquisition system AS/PC (Fig. 1). The latter collect the 2D image pictures (IPs) information recorded in the reciprocal and real space, respectively, and it is utilized to decode the 2D IP data aiming the quantitative diagnostics of nanoscaled bulk materials. Using the direct methods to decode the experimental X-ray IPs data, recovering of elastic displacement fields around the crystal-lattice defects has been carried out in [5-9].

In practice, remaining in the modern issues frame of the materials science, one can state that plausible decoding the crystal-lattice-defects from the experimental X-ray IPs data depends on the two factors. The first is the mathematically correct solution of the inverse issue of the X-ray diffraction tomography. The second is obtaining the experimental data with good accuracy, if possible, excluding and or minimizing experimental measurement errors. In other words, it is important to use experimental data with an accuracy controlled. This means that for the experimental measurement it is necessary to achieve a level of noise that would allow reliable, at a level of significance not more than 10%, recognizing the elements of the image. Since these elements can have different sizes and shapes, it is difficult to give a general maximum limit for the noise level in all cases, we can only say that it should not exceed 10-20%. In [9], the inverse problem of the X- ray diffraction tomography has been discussed in the case of the simulated noisy 2D IPs data related to the Coulomb-types point defect in a crystal. For this, as the recovery criterion, the goodness-of- fit parameter (CP) strongly depending on the noise level of the 2D IPs data has been introduced.

In [10], authors have undertaken the endeavour to filtering the simulated noisy 2D IPs data for the Coulomb-types point defect in a crystal. It has been shown that the governed filter applied to the initial noise-levels of the order 3-10 % IPs data yields the noise levels reduction of the order of their values at least, and significantly decreases the goodness-of-fit CPs values of the optimization procedure.

Noteworthy is the fact, in [11-13] have been pointed out a technical capacity of filtering digital signals, and quantifying the signal and noise components to improve the signal-to-noise ratio. In [12] was shown how the noise reduction by the signals average can be achieved with acquisition system of the noise signal data. Recently [14] have shown that convolutional neural networks can be a powerful tool to remove noise from reconstructed 2D images, Concerning the noise 2D IPs, hereafter the term "filtering" does mean the procedure, which leads to an improvement in the signal- to-noise ratio independent of whether the filtering algorithms are used at the post-processing stage of collected images or statistical average by combining a set of the 2D IP frames.

The 2D IPs data have been numerically simulated in the case of the incident σ-polarized plane–wave X-radiation. As is known, the Gaussian noise satisfies to the probability density function (PDF)

where the dispersion r.m.s. is nothing else the noise level σ.There are some works [15-17], in which the powerful PC algorithms have been employed for modeling the noise 2D IPs data collected by the CCD area detectors in the XRD technique. All the above PC random-noise-generation algorithms are based on the inverse transform

Here Φ(y) is the PDF integral. Function Φ(y) alternates within the interval U(0, 1) with the r.m.s. Λ equal to unity; n is the current 2D IP-frame number, n = {1, 2, …, N}, N is the number of the 2D IP-frames; to be specific, the number N is large in a statistical sense.

Due to the inverse transform (2), the pseudorandom number set {yk} in some interval of U(−𝛥, 𝛥) can be generated according into some reasonable looking Gaussian function. The integer k is the pixel number of one 2D IP-frame, k = {1, 2, K}, K is the number of pixels in one 2D IP-frame. Under the Central Limit Theorem (CLT) of statistics, in the limit when N ≡ (𝑁 × 𝐾) →∞, the PDF 𝐺_𝑁 (𝑦_𝑘) tends to the Gaussian distribution G(y_k),

Gaussian noise, unlike Poisson’s noise, can already be considered to be an admixture in the reference data.

In the present study, for simplicity, without loss of generality, we will treat an issue of the 2D noise IPs data frames in the case of frame- and pixel-independent random noise.

Appropriately, one assumes that the noisy n-th IP frame is determined by the relationship

Before proceeding further, it is worth to notice that there are methods to generate the Gaussian pseudorandom numbers, which do not rely on inverse transform (2). For example, good statistical properties have the maximal entropy Wallace’s method [18] and the Wolfram Mathematica program package [19]. To generate the Gaussian pseudorandom numbers {y_k}, we implemented both the method based on the inverse transform (2) (see [15-17]) and the Wolfram Mathematica program package [19] when they fit our purposes. Following a general idea of reducing the signal-to-noise ratio due to the signal accumulation with the subsequent averaging [11-13], one considers the average of the noise {𝐼(𝑦_𝑘, 𝑛)} supposing that the regular IP-component 𝐼0(𝑘, 𝑛) does not depend on the frame number n. Then, by using equation (4) averaged over N images, we have

and according to a general statistics concept, the noise part in the averaged frame 〈{𝐼(𝑦𝑘, 𝑛)〉𝑁 decreases proportionally to

In this paper, we will learn how the Gaussian statistics works on an example of the XRDT noise 2D IPs of the crystal lattice Coulomb-types defect with the 3D displacement field function 𝑓_{𝐶𝑡𝑝𝑑}(𝐫 − 𝐫₀) under conditions of the X-ray diffraction tomography (XRDT), as described above. Special attention is paid for cumulative reducing of the noise distortion of the simulated XRDT 2D IPs onto the effective recovery of the 3D displacement field function 𝑓_{𝐶𝑡𝑝𝑑}(𝐫 − 𝐫₀) (cf. [9]).

Hereafter, without losing a generality, the denoising problem will be considered on the example of the Gaussian noisy image pictures that is commonly taken place in the case of the X-ray diffraction tomography (XRDT). In this paper, denoising of the reference XRDT data is discussed and analyzed in the case of the 2D noisy IPs of spherical inclusion incorporated within a single crystal Si(111);

the diffraction vector h = [220], an incident linear-polarized X-ray radiation with wavelength λ = 0.0709 nm, the Bragg angle θB = 10.65°, extinction length 𝛬 = 36.287 μm. The spherical inclusion is considered as the Coulomb-types point defect located at a point r0 with the 3D displacement field function

Hereafter, the sample thickness T is chosen to be equal to the X-ray extinction length 𝛬, each the simulated reference 2D IP frame of the square size = 𝛬2 contains of 3721 pixels. Accordingly, the linear pixel size is about 0.6 μm. Note that the linear spatial resolution of the X-ray synchrotron 2-20 keV hybrid pixel detectors used in the X-ray diffraction tomography is of the order of about 1μm. The goal of the study is to denoise the reference 2D Gaussian noise-type IPs data and to improve the quality of the digital reconstruction of the 3D function 𝑓_{𝐶𝑡𝑝𝑑}(𝐫 − 𝐫₀)digitally recovered.

In spite of the fact that the noise in the images obtained with CCD detectors has a Poisson distribution by nature, at high counting statistics the noise distribution is close to the Gaussian one, the case of which we will consider.

Random generation algorithm for the generating the Gaussian pseudorandom numbers

The computer algorithm code for generating the Gaussian pseudorandom numbers is employed according to the schemes, details of which may be found in the [15-19]. The results of the noise generation have been approved by applying a number of statistics criteria including the corresponding test-statistics values listed in Table 1 (cf. The R-Project for Statistical Computing. https://www.r-project.org and [20-27]). The p-value is widely used for testing the statistics hypothesis, specifically: to verify the null hypothesis (Statistics Central Limit Theorem – SCLT) that the pseudorandom numbers generated satisfy to the Gaussian distribution. If some model (e.g., the null hypothesis) has been chosen, one needs to establish some significance-level α of the test statistics as the cutoff one. In the statistics data analysis, the conventional cutoff value α is taken as α = 0.05. Should the p-value of the test-statistics is less than 0.05, it means that the pseudorandom number data are inconsistent with the null hypothesis and therefore, the null hypothesis has to be rejected. Typically, the null hypothesis is rejected if p < 0.05 and accepted for p > 0.05.

The generated Gaussian pseudorandom numbers has been successfully passed through Pearson’s test-statistics, see Table 2 [21]. Indeed, in all the cases the evaluated p-values exceed the level cutoff α =0.05. At the same time, it should be mentioned that the PC algorithm code elaborated is rather fast and takes the 1-2 sec to terminate the PC calculation procedure on the Intel Core i7-7820X processor.

Pearson's χ2 test-statistics approves a null hypothesis by declaring whether the frequency distribution of the generated pseudorandom numbers is in a good consistence with the Gaussian distribution. The well-known χK2-cumulative test-statistics function is determined as

where the GN(yk) is the generated frequency distribution due to the total event number N and the G(yk) is the pure Gaussian distribution function; (K-1) is the intervals number of the frequency distribution histogram (see Supplementary part for details).

Furthermore, the χ_K² -cumulative test-statistics may be used to evaluate the value p, which allows to formulate the “goodness-of-fit" criterion for generating the Gaussian pseudorandom numbers by the frame of stochastic processes. And for which, the freedom-degrees number is equal to the total events number N minus the reduction coefficient p = s+1, where s is the number used in fitting the desired frequency distribution, for the Gaussian distribution G(y_K) s=2.

Simulation and analysis of the noise-contamination-prone 2D IPs

Nowadays, as is shown in Fig.1, the XRDT technique allows one to collect and proceed the 2D IPs data {𝐼(𝑦𝑘, 𝑛)} in a relevant way that contain the {𝑦_𝑘|𝑛}-ensemble data. Indeed, each the nth frame 2D IP is nothing else the linear hybridizing of both the regular (none noise) and noise-contaminated components according to equation (5).

In terms of equation (5), the noise 2D IP frames have been calculated by using the Wolfram Mathematica program package [19] and the random number generator algorithm (RNGA, details of which are given in [15-17]), respectively. By using these algorithms for various string numbers of stochastic events, some distributions 𝐺_Ñ(𝑦_𝑘) of pseudorandom numbers {y_k} have been calculated for the integer ñ = 𝐾 × 𝑁, K=961 pixels per image, N is the number of the 2D IP-frames under consideration. The corresponding results are presented in Fig. 2. From Fig. 2, it follows that for N= 103 the generated distribution G_N(y_k) tends to the Gaussian distribution G(y_k) in a good agreement with the SCLT up to high accuracy.

To be specific, to highlight a goal of the present study, we are confined ourselves by consideration of the XRDT noise 2D IP-frames referred to the crystal-lattice Coulomb-types defect when rotation angle Φ = 0º. In the case of Φ = 0º, the numerically simulated 2D IP-frames for various noise levels σ are depicted in Figs. 3a-c, particularly: (a) σ=0, (b) σ=0.001, (c) σ =0.001, respectively. Accordingly, by using the Wolfram Mathematica program package [19] for generating the pseudorandom numbers{y_k}, the straightforward procedure to averaging the noise-contamination- prone 2D IP frames has been carried out. For comparison, the corresponding no averaged 2D IP- frame with K= 961 and N = unity and the averaged 2D IP-frame with K= 961 and N=961 have been simulated. The corresponding results are shown in Fig.4, 4a, 4b, the noise level σ=0.001.

Let us introduce into consideration the Erwartungwert noise gain (NoiseGain) determined as the ratio of average Mean to average Meanñ[yk]=

Accordingly, in our calculations 𝑀𝑒𝑎𝑛₉₆₁ = −0.0226414 and 𝑀𝑒𝑎𝑛_961×961 = −0.000911784 and thus, one obtains Noise Gain = 24.832 (remind that according to the SCLT the Erwartungwert noise values of Mean_Ñ[𝑦_𝑘] tends to zero for ñ→ ∞). In the case of the values N = 961 × 10n (n = 0, 1, 2), one applies the random number generator algorithm [15-17] (RNGA) algorithm package for generating the noise-contamination-prone 2D IP- frames. The corresponding Erwartungwert noise values of 𝑀𝑒𝑎𝑛_Ñ[𝑦_𝑘] are listed in Table 2, they allow one to state that the procedure of averaging some noise 2D IP-frames can be effective tool to achieve the statistical noise filtering of the initial XRDT 2D IPs. The noise-filtered 2D IPs data, which should be incorporated in the optimization recovery procedure allow to obtaining more stringent information about the nanoscaled crystal-lattice defects. Loosely speaking, the use of a large number of the noise 2D IP frames opens a way to enhance the precision of the XRDT technique.

To illustrate how equation (5) works in the case of the noise-contamination-prone 2D IPs, the calculated results of the 3D function 𝑓_{𝐶𝑡𝑝𝑑}(𝐫 − 𝐫₀) recovery are listed in Table 3. Having an aim to minimize the XRDT 2D IPs target function, all the calculations have been carried out by using the combined iterative quasi-Newton-Levenberg-Marquardt – Simulated Annealing algorithm (qNLMSA) (cf. [9]). All the calculation have been calculated for total grid crystal sizes along to the dimensionless coordinates (X, Y, Z)-coordinates are equal to (61, 61, 21); the voxel sizes along to (X, Y, Z)-coordinates are equal to: ΔX= π/30, ΔY= π/30 and ΔZ= π/20 in the units of Δ/π, the dimensionless thickness T of the sample Si(111) is chosen to be equal to π. From Table 3 it follows an assertion that optimized processing of the noise 2D IPs in terms of equations (5), (6) allows to decode the noise XRDT 2D IPs data up to the enhancing accuracy.

On the other hand, based on the calculation results listed in Table 3, one concludes that the averaging technique proposed is approved to work and improve a signal-to-noise ratio of the XRDT 2D IPs observed.

Discussion

A goal of the paper is to propose a statistical processing of the XRDT 2D IPs data to obtain quantitative information about the nanoscaled crystal-lattice defects in the case when a noise composes some part of the 2D IPs data observed.

Based on the simulated noise XRDT 2D IPs data, a statistical imperative noise-filtering procedure has been developed. It allows one to conclude that in particular, the statistical noise-averaging technique of the XRDT 2D IPs acquired secures retrieving of plausible information about small- size defects in a crystal with rather high accuracy. Both the RNGA algorithm and Wolfram Mathematica program have been employed for the computer simulating and processing of the noise XRDT 2D IPs. As can be seen from Table 3, the goodness-of-fit parameter can be significantly improved by noise-filtering procedure from high discrepancy values of more than 27% (obtained without noise-averaging technique) to values of less than 1% (it corresponds to an almost perfect reconstruction of the 3D displacement field function of Coulomb-type point defect). Based on the qNLMSA algorithm applied for solving the inverse XRDT issue, one has justified that the XRDT method may be a powerful tool for obtaining quantitative information about small-size crystal- lattice-defects, e.g., clusters, small dislocation loops etc.

As to a general solution of the inverse XRDT problem, a question of how the above statistical approach will work in a proper way for other kinds of crystal-lattice defects is still open for research. This is a good topic for future work. Staying on the study presented in the paper, we would like only to state that the elaborated statistical formalism works well and pushes ahead of the XRDT technique by accumulating the noise-filtered 2D-IPs data.

It is clear, the question of the easy-of-access XRDT 2D IPs rate is important. As is shown in the paper, statistical combining of the noise 2D IPs frames may effectively to improve the signal-to- noise ratio. In some cases, statistical processing of the 2D IP fragments may to reveal low-intensity details masked by a noise.

In present, the 2D IP-frame rate of the CCD detectors is quite high, and it can be altered within a wide range from 1 kHz to tens of MHz, only governed by reasons of the sufficient X-ray intensity per pixel [28]. Fast detectors would have reduced the time of the experiment, but in practice the flux of quanta after a highly absorbing sample turns out to be too small and the exposure time must be increased to achieve an acceptable relative level of noise in the image. Assuming the typical 2D IP-frame exposure time being about 10-20 sec [8], one can design a set of 100 frames under conditions when each of them contains, at least, 100 photons per the detector pixel in the 2D IP fragments needed.

Conclusion

In concluding, it is worth to mention that in the paper, the statistical filtering of the 2D IPs really relates to the experimental XRDT conditions in the two aspects as the measurement duration of the 2D IP frames from several milliseconds and so the Gaussian amplitude noise values. They correspond to the incident X-ray beam intensity of the order of 1014 photons/sec that is typical for most of the X-ray experimental facilities being in the modern synchrotrons of the 3rd - 4th generations.

All the above allows to make an assertion that the statistical noise filtering of the 2D IPs data proposed is a good way to improve a signal-to-noise ratio, at least, for investigating and analyzing the small-size defects by using the XRDT technique. Checking the conformity of the noise distribution in the frame combined from several frames against the normal law allows the experimenter to achieve a quality image by calculating the necessary number of individual measurements and conducting the necessary series of experiments.

Acknowledgement

Special gratitudes to Prof. V.E. Asadchikov, Dr. D.A. Zolotov for their permanent interest and fruitful discussions concerning the present study. This work was supported by the Ministry of Science and Higher Education of the Russian Federation within the State assignment FSRC «Crystallography and Photonics» RAS in part of developing methods of structural analysis using X rays and synchrotron radiation.

Supplementary part

Pearson's chi-squared test description

Pearson's chi-squared test (χ2) [S1] is a statistical test applied to sets of categorical data to evaluate how likely it is that any observed difference between the sets arose by chance. It is the most widely used of many chi-squared tests (e.g., Yates, likelihood ratio, portmanteau test in time series, etc.) – statistical procedures whose results are evaluated by reference to the chi-squared distribution. In contexts where it is important to improve a distinction between the test statistic and its distribution, names similar to Pearson χ-squared test or statistic are used.

It tests a null hypothesis stating that the frequency distribution of certain events observed in a sample is consistent with a particular theoretical distribution. The events considered must be mutually exclusive and have total probability 1. A common case for this is where the events each cover an outcome of a categorical variable.

Pearson's chi-squared test is used to assess three types of comparison: goodness of fit, homogeneity, and independence. A test of goodness of fit establishes whether an observed frequency distribution differs from a theoretical distribution. A test of homogeneity compares the distribution of counts for two or more groups using the same categorical variable A test of independence assesses whether observations consisting of measures on two variables, expressed in a contingency table, are independent of each other.

For all three tests, the computational procedure includes the following steps:

(1) Calculate the chi-square test statistic, which is the sum of squares of deviations between observed and theoretical frequencies, normalized to estimates of standard deviations.

(2) Determine the degrees of freedom, df, of that statistic.

(3) For a test of goodness-of-fit, df = Cats − Parms, where Cats is the number of observation details recognized by the model, and Parms is the number of parameters in the model adjusted to make the model best fit the observations: The number of details is reduced by the number of fitted parameters in the distribution.

(4) For a test of homogeneity, df = (Rows − 1)×(Cols − 1), where Rows corresponds to the number of details (i.e. rows in the associated contingency table), and Cols corresponds to the number of independent groups (i.e. columns in the associated contingency table).

(5) For a test of independence, df = (Rows − 1)×(Cols − 1), where Rows corresponds here to the number of details associated with one variable, and Cols corresponds to the number of details in the second variable.

(6) Select a desired level of confidence (significance level, p-value) for the result of the test.

(7) Compare (χ2) to the critical value from the chi-squared distribution with df degrees of freedom and the selected confidence level (one-sided, since the test is only in one direction, i.e. "is the test value greater than the critical value?"), which in many cases gives a good approximation of the distribution of (χ2).

(8) Sustain or reject the null hypothesis that the observed frequency distribution is the same as the theoretical distribution based on whether the test statistic exceeds the critical value of (χ2). If the test statistic exceeds the critical value of (χ2), the null hypothesis (H0 = there is no difference between the distributions) can be rejected, and the alternative hypothesis (H1 = there is a difference between the distributions) can be accepted, both with the selected level of confidence. If the test statistic falls below the threshold (χ2) value, then no clear conclusion can be reached, and the null hypothesis is sustained (we fail to reject the null hypothesis), though not necessarily accepted.

This method is an effective test of whether the theoretical observations from the constructed model correspond to the experimental data, since an incomplete or incorrect model leads to violations of the expected law of the distribution of the differences between them.

References

[S1] Pearson, Karl (1900) On the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling. Philosophical Magazine. Series 5: 50, 157–175.