Monitoring homogeneity of textile fiber orientation

Abstract

Replacement of human control by robotic control is one of the key issues in the textile industry, because it is generally believed that it can decrease the costs of producers on the one hand and increase the satisfaction of clients on the other side. Therefore, this paper deals with the proposal of monitoring homogeneity in fibrous systems when the attention is focused on keeping prescribed directional orientation of fibers. The first part concentrates on the estimation of the orientation of fibers in fibrous systems when the information comes from scanned images of the material. Emphasis is on the use of 2D Fourier transform and a local approach based on idea “conquer and rule”. Such a local approach appears crucial for a proposal how to detect eventual defects, random violation of structural regularity, and such like, when statistical testing of fit is used as a tool. To illustrate the method, it was necessary to prepare data with a prescribed form and orientation of fibers. To that purpose we developed algorithms enabling us to simulate images imitating typical textile fibrous systems with prescribed properties, both containing and not containing defects. Linear and highly nonlinear fibers can be generated. The paper is of a methodological nature, and most of conclusions are based on the analysis of simulated data. Experimental verification of the suggested method is in progress.

Keywords

fiber orientation 2D Fourier transform goodness-of-fit test quality measurement structure properties

Fibrous materials surround us almost everywhere. They are not only the basic material for clothing, but they have numerous applications ranging from nanofibrous layers, composites, geotextiles in the building industry, filtering, and so on, to materials for special purposes in medicine such as structures for scaffolds and tissue engineering, among others. It is well known that the actual properties of all fibrous materials substantially depend on the properties of individual fibers, together with the arrangement and/or structure they form. The arrangement and/or directional orientation of individual fibers will greatly influence the mechanical properties of linear and planar materials. Analogously, the orientation of fibers in fibrous porous materials influences properties such as permeability, absorbency of liquids, and so on. Therefore, the determination of directional orientation of any fibrous structure is an important part of quantitative measurements, not only in textile metrology and practice.^1–7

To be able to analyze structural anisotropy and/or directional orientation, real world objects are nowadays replaced by their images and/or scans, which are afterwards examined in detail. Results of these analyses are then used for decision(s) about the product’s quality, homogeneity, etc. Study of the image data, moreover, allows us to better understand the image contents and to formulate quantitative and qualitative descriptions of objects we are interested in. Examples of certain textile fiber systems with both random orientation of fibers and with a predominant orientation of fibers are displayed in Figure 1(a)–(c).

Figure 1.

Examples of real textile fiber systems. Image of: (a) viscose fibers, (b) randomly distributed nanofibers, (c) nanofibers with specific orientation.

In this paper we particularly deal with the problem of estimating the main orientation of fibers in fibrous systems. To this end, several methods were proposed, based on image analysis. These methods are concisely summarized is the next section, and then one possible approach is described enabling the simulation of fibrous systems with described properties. Simulated images are used for evaluating the properties of the algorithm suggested for monitoring quality for the case when the prescribed homogeneity of directional orientation of fibers is required. The section after that describes how to use the suggested methods and algorithms for quality monitoring of fibrous layers when homogeneity of the fiber orientation is an issue. The final section contains selected conclusions.

Estimation of the orientation of fibers in fibrous systems

Recall that any digital image can be modeled by a two-dimensional function f(x,y), where (x,y) are plane coordinates and the amplitude f at any point (x,y) is either color intensity or gray level. Processing of image data can be divided into several fundamental steps:

image acquisition;

image enhancement;

image segmentation;

image representation and description.

More details about all these steps can be found in, for example, the monograph by Gonzales.⁸ The main focus of this paper is on the last two steps; more precisely, on the representation and description of the fiber system orientation.

Tunák and Linka⁹ proposed a method enabling estimation of fiber orientation in gray level images, which is based on a spectral approach and uses 2D discrete Fourier transformation, which transforms images from the spatial to the frequency domain. Different variants of this technique have been used in image processing for a long time. They are useful for pattern characterization and measuring changes in the studied patterns.^10–19 In our case we specifically focus on the directionality of yarns in fabric surfaces, but our approach can be also used for studying periodicity or density of yarns in fabrics and such like. The key point is that dominating directions (gradients of image function) in the spatial domain correspond to the large magnitudes of frequency components distributed along the straight lines in the Fourier spectrum. The basic idea of our method stems from transformation of a power spectrum to a binary image via thresholding, so that only significant frequencies remain. For binarization of the image we use global thresholding with the threshold value set to half of the maximum of the power spectrum, or rather its logarithm. The directional orientation of significant frequencies in the frequency domain, rotated by 90°, corresponds to the directional orientation of objects in the spatial domain. In the resulting binary image we consider a cluster of white pixels as a region of interest and analyze it further. Orientation and length of major and minor axes of the “covering ellipse”, which have the same normalized second central moment as the region of interest, are computed. For illustration, see Figure 2(a)–(c). It is obvious that orientation of the covering ellipse reflects the predominant directions of the objects in the spatial domain. For a more detailed description of this algorithm, see Tunák et al.²⁰

Figure 2.

(a) Image of viscose fibers (500 × 500 pixels). (b) Power spectrum rotated by 90°. (c) Estimate of directional orientation using image moments; white pixels were obtained by thresholding.

If applied to the whole image, the procedure described above can be used for the estimation of the fiber orientation in it, and considered as a global approach to the problem. However, it turns out that for many fibrous or nanofibrous layers a more detailed analysis is needed, which calls for an analysis from a local point of view. Therefore, we suggested dividing the image area into K small, non-overlapping pieces called sub-windows, and estimating the characteristic of interest (being the orientation of fibers in our case) for each such piece separately and independently from the other sub-windows, and to further analyze the obtained results.²⁰ As a result we obtain many estimates of the characteristic of interest, say T₁, … , T_K, one for each sub window of the original image. Instead of averaging them in order to get just one value, we estimate the distribution of T_i values either by a histogram or kernel density estimator.

To illustrate the method, the local type analysis was performed for a fibrous system shown in Figure 1. Selected results are presented in Figure 3. We start description of the results with Figure 3(a1), which represents a real system of viscose fibers. The image was divided into sub-windows of 30 × 30 pixels in size, and the fiber orientation was estimated individually for each sub window. The results can be seen in Figure 3(a2), where the preferred orientation of fibers for every sub window is represented by the directional vector displayed in red (provided the ratio of major-to-minor axis length is greater than 2). Moreover, orientation in degrees is displayed on the gray-level scale. Distributions of estimates of the fiber orientations in the respective sub-windows, expressed in the forms of a density histogram and kernel density estimate, are shown in Figure 3(a3).

Figure 3.

Examples of analyses of images presented in Figure 1. (a1)–(c1) Initial images. (a2)–(c2) Gray level map of estimated orientation: (a2) sub-window size 30 × 30 pixels, (b2) and (c2) sub-window size 20 × 20 pixels. (a3)–(c3) Density histogram and kernel density estimates of orientation of fibers.

Figures 3(b1)–(b3) present a typical situation when there is no prevailing orientation of fibers in the image, at least not optically visible. Finally, Figures 3(c1)–(c3) display another standard situation when two directions of fibers with different slopes dominate the image. For more examples of analysis performed on both simulated and real fiber systems see Tunák et al.²⁰ In all cases we can see a very good agreement between the estimated distribution of orientation of fibers and the input image.

The size of the whole image and splitting it into small sub-windows considerably influence the quality of the histogram and/or quality of the kernel density estimator for the density of fiber orientation. As shown and discussed by Scott,²¹ sample sizes of 100–200 observations are reasonable for obtaining good-quality estimates of regular densities if the integrated mean square error is used as the quality measure. Notice that if we split an image of 1000 × 1000 pixels into sub-windows of 40 × 40 pixels, it leads to slightly fewer than 625 sub-windows, which is adequate to produce good quality estimates of the desired characteristic.

As mentioned in a previous article,²⁰ too-small sub-window sizes are not able to capture information about the fiber orientation, while too-large sub-window sizes do not bring new information. We are convinced, and our calculations confirm it, that the optimal choice depends, among other circumstances, especially on the fiber thickness and curvature. In the simulated examples the thickness was set between one and three pixels, sometimes larger. In our examples below we present the results for the sub-window size being 20–30 times the mean thickness of the fibers, which yielded good results.

Simulation of fiber systems

For testing and evaluation of the algorithm proposed for quality monitoring of fibrous system with respect to the homogeneity of directional orientation of fibers it is also necessary to prepare images of a simulated fibrous system with the predetermined orientation of fibers. We can imagine fibers in a digital image as a mixture of compact objects where length is much larger than width. Of course, the simplest objects to be considered here are linear ones; however, as will be shown later, we will simulate also nonlinear objects, which correspond much more to the reality.

For the purpose of simulating a simple fiber system we start by representing a fiber as a rectangular object, where one dimension of the rectangle is much larger than the other dimension, see Figure 4. It is worth noticing that using random generation of scale, location, orientation and gray level of n such objects, one can obtain images similar to the images of a real fiber system.

Figure 4.

(a) Rectangular (square) object. Transformation by (b) scaling ( $t_{11}$ = 4, $t_{22}$ = 0.5), (c) rotation (α = π/4), (d) translation ( $t_{31}$ = 1, $t_{32}$ = 1.5).

For coordinate transformations we use geometric mappings in the form

(X, Y) = T (X_{0}, Y_{0}),

(1)

where (X₀, Y₀) are the original coordinates, (X, Y) are the transformed coordinates, and T is a transformation matrix. Commonly used transformations are affine transformations of the form

[X Y 1] = [X_{0} Y_{0} 1] T = [X_{0} Y_{0} 1] [t_{11} t_{12} 0 t_{21} t_{22} 0 t_{31} t_{32} 1] [\cos α \sin α 0 - \sin α \cos α 0001]

(2)

With the aid of equation (2) we can scale, rotate, translate and shear a set of coordinates depending on the elements of matrix T; for details see Gonzalez et al.⁸ Figure 4 shows examples of the identity, scaling, rotation, and overall translation. Shear transformation is missing because we do not need it for our purposes. Evidently, we can combine individual affine transformations and create transformation matrix T of size 3 × 3, which is the product of individual transformation matrices.

A simulated fiber system can be generated using a combination of individual elementary transformations. More precisely:

the size of an individual fiber is generated by random scaling in both x and y direction using parameters $t_{11}$ and $t_{22}$ ;

orientation is described by random angle of rotation via parameter α;

the location of the fiber is set with the help of translation in both x and y direction using parameters $t_{31}$ and $t_{32}$ .

In subsequent examples the parameters of the transformation matrices were randomly generated from the uniform distribution on the interval (a, b). (X ∼ U(a, b) means that random variable X has uniform distribution on interval (a, b); density in this interval is equal to 1/(b − a) and is equal to zero otherwise.)

In addition, gray levels of the respective objects were also generated randomly. The resulting fiber system is generated as a set of n objects and the results are stored as a digital image matrix.

Examples of generated fibrous systems are presented in Figure 5(a1)–(a3), representing images of the size 1000 × 1000 pixels. Magnified crops of these images of the size 200 × 200 pixels are presented below the global image. The parameters of these simulations were fixed as follows:

$t_{11}$ ∼ U(20, 60), $t_{22}$ ∼ U(0.5, 1), $t_{31}$ , $t_{32}$ ∼ U(0, 1000);

the objects are randomly orientated with α ∼ U(0, 2π);

the gray levels gl ∼ U(0.5, 1), where 0 represents black and 1 white;

the number n of objects (i.e. fibers) is : (a1) n = 10 000, (a2) n = 15 000, (a3) n = 20 000.

Figure 5.

Examples of generated fibrous systems images of 1000 × 1000 pixels in size and magnified crop of images of size 200 × 200 below. Linear objects were simulated.

The scale values of the objects are generated using t₁₁ ∼ U(20, 60) and t₂₂ ∼ U(0.5, 1). The objects have random locations t₃₁, t₃₂ ∼ U(0, 1000), the number of objects n = 15 000. The gray levels were generated using gl ∼ U (0.5, 1). The objects are randomly oriented with the angle (b1) α ∼ U(0, π/4), (b2) α ∼ U(0, π/2), (b3) α ∼ U(0, 2π−π/4).

In a more complicated case, we will consider simulated fiber as a sine wave. We generated coordinates x in the interval (−π, π) with the discrete step of 0.5, that is 13 coordinates. y coordinates are calculated as y₁ = sin(ax), where parameter a represent period of function and affects the shape or “crimp” of generated fiber. Parameter a is generated randomly with a ∼ U(−1.5, 1.5). Then function y₂ = y₁ + b is calculated and randomly generated parameter b ∼ U(0.1, 0.2) affect the “thickness” of generated fiber. Four examples of random generation of a set of coordinates in sine wave form can be seen in Figure 6.

Figure 6.

Examples of random shape of fiber in a form of sine wave. (a) a = –1.4246, b = 0.1421; (b) a = –0.0360, b = 0.1498; (c) a = –0.4489, b = 0.1478; (d) a = 0.5536, b = 0.1474.

As in the case of the rectangular object mentioned above, with the aid of equation (2) we can scale, rotate, translate and shear a set of coordinates depending on the elements of matrix T. A simulated fiber system then can be generated using a combination of respective elementary transformations.

Figure 7 (a1)–(a3) represent examples of a generated fibrous system of size 1000 × 1000 pixels (magnified crops of these images of size 200 × 200 pixels are presented below). The parameters of the simulation were fixed as follows: t₁₁ ∼ U(10, 20), t₂₂ ∼ U(0, 10), t₃₁ t₃₂ ∼ U(0, 1000); the objects are randomly orientated with α ∼ U(0, 2π); the gray levels gl ∼ U(0.5, 1), where 0 represents black and 1 white; the number of fibers is (a1) n = 5000, (a2) n = 7500, (a3) n = 10,000.

Figure 7.

Examples of generated fibrous systems images of 1000 × 1000 pixels in size and magnified crop of images of size 200 × 200 below.

In Figure 7 (b1)–(b3) randomly generated systems are presented, with a magnified crop of the images below the global images. The scale values of the objects are generated using t₁₁ ∼ U(10, 20) and t₂₂ ∼ U(0, 10). The objects have random locations t₃₁, t₃₂ ∼ U(0, 1000), the number of objects is n = 7500. The gray levels were generated using gl ∼ U (0.5, 1). The objects are randomly oriented with the angle (b1) α ∼ U(0, π/4), (b2) α ∼ U(0, π/2), (b3) α ∼ U(0, 2π−π/4).

Monitoring homogeneity of orientation of fibers

It is well known that statistical methods can be used for monitoring non-homogeneities in arrangement of a textile material. Recall that univariate control charts can be used as a tool for monitoring the weaving density in textile industry.²² Megahed et al.^23,24 provide an excellent overview of the use of control charts based on image data for various industrial applications. In the case of monitoring several quality variables, multivariate control charts can be used as described in Lauro et al.²⁵ and Tunák and Linka.²⁶ Another approach was used by Linka and Volf ²⁷ who, on the basis of the textural characteristics of the second order, studied the possibility of detecting non-homogeneity in nonwoven materials using methods of classification and regression trees.

Moreover, it is evident that the above described method can be effectively used for estimation of the directional orientation of fibers in fibrous materials, because “any” deviation of the obtained estimate of distribution of directional orientation from the expected (desired) one indicates a failure of the regular structure and can thus reflect possible defects, random violations of the regularity of the structure, and so on, which should be detected.

Therefore, in this paper we are looking at the problem of monitoring homogeneity of produced material from a different point of view than described in the papers mentioned above. The main difference is that instead of the generally assumed global approach we suggest combining local analysis of images describing produced material as described earlier, with classical χ² goodness-of-fit testing. The basic idea is to estimate distribution of fiber orientation in the image first and then to compare it with some “standard”. For the standard, which serves as a null hypothesis, we can take either a theoretical distribution or a distribution specified by the producers’ and/or consumers’ requirements and agreements. In the case of a random orientation of fibers, being a typical requirement for nonwoven textile materials, the standard corresponds to the uniform distribution. For other materials, such as, for example, slivers, another distribution describing required prevailing directional orientation of fibers must be specified. It is evident that the standard can also be obtained as an estimate of the distribution of fiber orientation based on a sample of the monitored material when both the producer and the consumer are satisfied with the output of the production.

Recall that the χ² goodness-of-fit test is a well-known statistical procedure intended for testing the null hypothesis that the observations form a sample from a specified distribution against the alternative that the hypothesis is not true, which is performed by grouping data into the non-overlapping bins, calculating both observed and expected counts in the respective bins, and computing the statistic

χ^{2} = \sum_{i = 1}^{r} \frac{(O_{i} - E_{i}) 2}{E_{i}}

(3)

where O_i denotes the observed counts, E_i the expected counts, and r is the number of bins (classes).

Under quite general conditions this statistic follows asymptotically the χ² distribution with r-1 degrees of freedom. This allows not only deciding about the null hypothesis, but also calculating corresponding p-values. Recall that when the p-value is smaller than the predetermined significance level, α, which is in technical practice usually set to 0.05, the test rejects the null hypothesis. For details about χ² testing of fitness and its applications in industry see, for example, Nagla,²⁸ Montgomery,²⁹ or Greenwood and Nikulin.³⁰

Figure 8 shows datas and results of monitoring process when a sliding-window technique for detection and localization of change in orientation of fibers in a fibrous system is used. An image of the size 2000 × 4000 pixels displays a simulated fibrous system with the random orientation of objects, which in the middle contains specifically oriented fibers with a strong directional orientation α ∈ (0, π/6). We calculated a χ² goodness-of-fit test against the uniform distribution for a sliding window (inspection area of the size 1000 × 1000 pixels), which is systematically moved over the whole image area either with or without overlapping. Overlapping was set to the half of the size of the monitoring window. Each sliding window was then split into 625 non-overlapping sub-windows of the size 40 × 40, and for each sub-window directional orientation of fibers estimated as described earlier. We set the number of bins r = 18 that is, each bin covers the angle π/18 = 10°.

Figure 8.

(a) Image of the size 2000 × 4000 pixels representing simulated fiber system with random orientation of objects. In the middle the orientation of fibers is from interval (0, π/6), and corresponds to the failure of regular structure. (b) Inspection windows of the size 1000 × 1000 pixels without overlapping. (c) Inspection windows of the size 1000 × 1000 pixels with overlapping.

Figure 9 show results of the individual analysis of orientation of fibers represented in Figure 8 according to the methods suggested in Tunák et al.²⁰ for sub-windows with index of inspected area 1, 3, and 11 in Table 2, where Figure 9(a1–c1) display original image, Figure 9(a2–c2) represent orientation in form of directional vectors displayed in red and Figure 9(a3–c3) presents histograms and kernel density estimates of the fiber orientations.

Figure 9.

(a1, b1, c1) Original images, selected sliding windows from Figure 8(a). (a2, b2, c2) Gray level map of orientation (sub window size 40 × 40). (a3), (b3), (c3) Density histogram and kernel density estimation of orientation.

Significance level α was set to 0.05. If the p-value is smaller than selected significance level, the inspected area is considered as the area with a distribution of orientation different from the uniform one. Results of the chi-squared goodness-of-fit testing are shown in Tables 1 and 2, where r (row) and c (column) represent coordinates of the left top corner of the inspected area either without or with overlapping.

Table 1.

Table of results with row and column indices of the top left corner of the monitoring window without overlapping and corresponding p-values for data presented in Figure 8(b)

Index	1	2	3	4	5	6	7	8
r	1	1	1	1	1001	1001	1001	1001
c	1	1001	2001	3001	1	1001	2001	3001
p-value	0.4	0	0	0.78	0.98	0	0	0.92

Table 2.

Table of results with row and column indices of the top left corner of the monitoring window with overlapping and corresponding p-value for data presented in Figure 8(c)

Index	1	2	3	4	5	6	7	8	9	10	11
r	1	1	1	1	1	1	1	501	501	501	501
c	1	501	1001	1501	2001	2501	3001	1	501	1001	1501
p-value	0.4	0.24	0	0	0	0.53	0.78	0.72	0.11	0	0

Index	12	13	14	15	16	17	18	19	20	21
r	501	501	501	1001	1001	1001	1001	1001	1001	1001
c	2001	2501	3001	1	501	1001	1501	2001	2501	3001
p-value	0	0.42	0.4	0.98	0.99	0	0	0	0.76	0.92

Corresponding p-values plotted against the index of inspected area (both without and with overlapping) can be seen in Figure 10(a), (b). Four, respectively eight, windows located in the middle of the image are indicated as areas with the changed orientation (p-values smaller than 0.05). White squares in Figure 8(b), (c) represent areas with defects.

Figure 10.

Plot of p-values vs inspected area (a) with overlapping, (b) without overlapping.

On the other hand, Figure 11 presents a situation with random orientation of linear fibers. There are no defective areas in the image and the results of testing “confirm” that. Finally, Figures 12 and 13 represent datas and results of a monitoring process for detection and localization of a change in orientation of linear fibers in the simulated fibrous system when fibers in the middle of the image are oriented with α ∈ (0, π/3), α ∈ (0, π/2), respectively. As can be seen from Figures 12(b) and 13(b), the proposed monitoring process is effective even in the case when directional non-homogeneity covers only a quarter of the inspected window area.

Figure 11.

(a) Image of the size 2000 × 4000 pixels displaying a simulated fibrous system with the random orientation of objects. (b) Plot of p-values vs index of the inspected area of the size 1000 × 1000, overlapping of windows is applied.

Figure 12.

(a) Image of the size 2000 × 4000 pixels displaying a simulated fibrous system with the random orientation of linear objects. In the middle the orientation of fibers is from interval (0, π/3). (b) Plot of p-values vs index of inspected area of the size 1000 × 1000, overlapping of windows is applied.

Figure 13.

(a) Image of the size 2000 × 4000 pixels displaying a simulated fibrous system with the random orientation of objects. In the middle the orientation of fibers is from interval (0, π/2). (b) Plot of p-values vs index of inspected area of the size 1000 × 1000, overlapping of windows is applied.

Figures 14 –16 represent datas and results of a monitoring process for detection and localization of a change in orientation of nonlinear fibers—that is, generated as a sine waves. Fibers in the middle of the image are oriented with α ∈ (0, π/6), α ∈ (0, π/3), α ∈ (0, π/2), respectively.

Figure 14.

(a) Image of the size 2000 × 4000 pixels displaying simulated fibrous system with non linear random orientation of objects. In the middle the orientation of fibers is from interval (0, π/6), and corresponds to the failure of regular structure. (b) Inspection windows of the size 1000 × 1000 pixels without overlapping. (c) Plot of p-values vs index of inspected area of the size 1000 × 1000, overlapping used.

Figure 15.

(a) Image of the size 2000 × 4000 pixels displaying simulated fibrous system with non linear random orientation of objects. In the middle the orientation of fibers is from interval (0, π/3), and corresponds to the failure of regular structure. (b) Inspection windows of the size 1000 × 1000 pixels without overlapping. (c) Plot of p-values vs index of inspected area of the size 1000 × 1000, overlapping used.

Figure 16.

(a) Image of the size 2000 × 4000 pixels displaying simulated fibrous system with non linear random orientation of objects. In the middle the orientation of fibers is from interval (0, π/2), and corresponds to the failure of regular structure. (b) Inspection windows of the size 1000 × 1000 pixels without overlapping. (c) Plot of p-values vs index of inspected area of the size 1000 × 1000, overlapping used.

Conclusions

In this contribution we focused on estimation of the orientation of fibrous systems based on image analysis and presented several possible solutions. Suggested methods can be effectively used for estimation of the directional orientation of fibrous materials with respect to their homogeneity, random violation of regularity of this structure, occurrence of possible defects, and so on.

Procedure for monitoring homogeneity of orientation of fibers was tested only on simulated samples with predefined (theoretical) distribution of the orientation of the objects (simulated fibers). For materials where we required homogeneity of orientation of fibers in all directions, it corresponds to uniform distribution in interval (0, 2π). For other types of materials a highly directional fiber arrangement may be prescribed, followed by different type of distribution. In the case of theoretical distribution where the orientation is unknown, we can estimate the distribution from a representative sample (“learning sample”), which we regard as a standard corresponding to the required quality.

The proposed algorithm can be used both for estimation of orientation and, eventually, for monitoring the homogeneity of the orientation for the images of fibrous systems in nano, micro, and macro scale. It also appears to be promising for real applications, and testing and verifying for different types of real textile fibrous materials is in progress.

Footnotes

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The work of JA was partially supported by the Czech Science Foundation under Grant Number P403/15/09663S. Support from the BELSPO IAP P7/06 StUDyS network is also prominently acknowledged.

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