Estimation of fiber system orientation for nonwoven and nanofibrous layers: local approach based on image analysis

Abstract

Analysis of textile materials often includes measurement of structural anisotropy or directional orientation of textile object systems. To that purpose, the real-world objects are replaced by their images, which are analyzed, and the results of this analysis are used for decisions about the product(s). Study of the image data allows one to understand the image contents and to perform quantitative and qualitative description of objects of interest. This paper deals in particular with the problem of estimating the main orientation of fiber systems. Firstly, we present a concise survey of the methods suitable for estimating orientation of fiber systems stemming from the image analysis. The methods we consider are based on the two-dimensional discrete Fourier transform combined with the method of moments. Secondly, we suggest abandoning the currently used global, that is, all-at-once, analysis of the whole image, which typically leads to just one estimate of the characteristic of interest, and advise replacing it with a “local analysis”. This means splitting the image into many small, non-overlapping pieces, and estimating the characteristic of interest for each piece separately and independently of the others. As a result we obtain many estimates of the characteristic of interest, one for each sub-window of the original image, and – instead of averaging them to get just one value – we suggest analyzing the distribution of the estimates obtained for the respective sub-images. The proposed approach seems especially appealing when analyzing nonwoven textiles and nanofibrous layers, which may often exhibit quite a large anisotropy of the characteristic of interest.

Keywords

Fiber system digital image Fourier analysis covariance matrix analysis moments of image nanofibers layers histogram kernel density estimator

Fibrous textile materials are used almost everywhere around us. They have various applications, for example fabric for clothing, materials for technical purposes (automotive, composites, geotextiles, building industry, filtering, etc.) or materials for special purposes (textiles for medicine, structures for scaffolds and tissue engineering, etc.). It is well known that the actual properties of all textile materials substantially depend on the properties of individual fibers, together with the arrangement and/or structure that they form.

Arrangement and/or directional orientation of individual fibers greatly influences mechanical properties of linear and planar textiles. Analogously, orientation of fibers in fibrous porous materials influences properties such as permeability, absorbance of liquids, etc. Many papers confirm an intuitive feeling that fiber orientation considerably influences properties of the final textile products. For examples, see Tahir and Tafreshi,¹ Maoa and Russella² and Murugan and Ramakrishna.³ The first paper is devoted to the modeling of in-plane and through-plane fiber orientations on a fibrous medium's transverse permeability. The second one demonstrates that fiber orientation is a major factor influencing the anisotropy of permeability. Finally, in the third paper its authors show that the orientation of the fibers is one of the most important features of a perfect tissue scaffold made of nanofibers.

From this standpoint, measurement of the directional orientation of a fibrous structure is an important part of quantitative measurements, both in textile metrology and practice. Here we must not neglect to mention an especially influencing series of papers by Pourdeyhimi and Davis⁴ and Pourdehyhimi et al.⁵ devoted to measuring fiber orientations. Another important branch of research in the domain of fiber orientation of fibrous materials is its mathematical modeling. For recent stimulating papers, see, for example, Murugan and Ramakrishna,³ Neckář and Das⁶ and Neckář et al.⁷

Recall that although the first automated image analysis appeared in 1990, most of the measurements are currently performed either manually or with the use of specialized software, where evaluation of object orientation is not performed automatically but is affected by subjective human decisions. Therefore, the interest in fully automatic measurement is growing in an attempt to decrease costs of manufacturing; see, for example, Tsai and Huang^8,9 or Kang et al.¹⁰

Processing image data allows us to understand image content and to perform quantitative and qualitative descriptions of objects of interest. These objects are either randomly placed or they follow a certain directional placement. In textiles, typical objects can be linear textiles, fibers, threads, cross-sections of fibers, etc., and the systems containing these objects can, in general, be planar textiles, webs, fiber layers, woven fabrics, knitted fabrics, nonwoven textiles, nanofibrous layers, cross-sections of layers or projections of three-dimensional (3D) structures to two dimensions, etc.

In this paper we concentrate especially on estimation of fiber orientation of nanofibrous and nonwoven layers. One of the main tools utilized in this paper is discrete Fourier transform, which has been used for image analysis for a very long time, and which has often been applied both in textile research and industrial applications. As noticed by Wood¹¹ and Ravandi and Toriumi,¹² Fourier transform also allows pattern characterization and measuring the changes in pattern definition and estimation of directionality and density of yarns on fabric surfaces. The second main tool used is analysis of the sample image covariance matrix. Relevant literature and some basic results are discussed in more detail in the following section.

This paper is organized as follows. In the following section, four methods suitable for estimation of the fibers' orientation, that is, rose of directions, spectral approach, moment approach, and a combination of the spectral and moment approaches, are described. As already mentioned, among the main tools are two-dimensional discrete Fourier transform (2DDFT) and eigenvalue decomposition of the sample image covariance matrix. The first of the novelties of this paper is in the Combining spectral and moment approaches in estimation of fiber orientation section, where a combination of spectral and moment methods is suggested in an effort to increase advantages of both approaches. Further, the third section concentrates on the main new idea of this paper, which consists of abandoning the currently used “global approach”, in which it is typical to characterize the fiber orientation in the entire image using just one number (index), and replacing it with a “local approach”. This new approach means splitting the image into many smaller, non-overlapping sub-windows, estimating the fibers' orientation in these sub-windows separately and providing the user with an estimate of the distribution of the fiber orientation in these sub-windows. A big advantage of this suggested approach is that it allows us to characterize the fiber orientation in materials that do not exhibit isotropic structure, for which the characterization by one number (index) can be totally misleading. Examples of its application to both real and simulated data are presented. We would like to emphasize that the proposed algorithms were tested especially on images of nanofibrous and nonwoven layers.

Estimation of fiber orientation

In this section we describe four methods that have been shown to be useful for the estimation of fiber orientation. Firstly, in the Estimation of structural anisotropy of planar systems section, we concentrate on estimation of the structural anisotropy of planar systems using the rose of directions, rose of intersections, and angular density. Secondly, in the Spectral approach and estimation of fiber orientation section, we turn to estimation of the fiber orientation via spectral approach. The main tool here is the 2DDFT. Thirdly, in the Moments of the image function and estimation of fiber orientation section, we show that sample moments of the image file and the corresponding sample covariance matrix contain a lot of information that can be used for the estimation of the fiber orientation. Finally, in the Combining spectral and moment approaches in estimation of fiber orientation section we show how the spectral- and moment-based approaches can be mutually combined.

Estimation of structural anisotropy of planar systems

The quantitative characteristics of structural anisotropy in planar systems are of practical interest in the domain of estimating the fiber orientation. Among them, the rose of directions R(D) and the rose of intersections R(I), the polar plot of the mean density of intersections of the fiber system with the line of given orientation, seem to be most popular. Therefore, angular density f(α) of lengths of threads or fibers, that is, density of lengths of threads or fibers oriented in an angular segment α ± α/2, is one of the most typical characteristics of planar anisotropy. The polar plot of f(α) is called the rose of directions. The estimate of R(D) is accessible by means of stereological lattice testing systems or by using an image analyzer, which may not be available to everybody. Therefore, a simple experimental graphical method for estimation of f(α) is worthy of our interest. Let us concisely summarize the basic idea behind one of the proposals; for details see Rataj and Saxl.¹³

This method uses the set of angles α₁,…,α_n situated at the top of a fiber system that is monitored for construction of the rose of intersections R(I). The rose of directions as an estimate of function f(α) is obtained graphically from the rose of intersections R(I) through the construction of the so-called Steiner compact. As Rataj and Saxl^13,14 pointed out, R(I) is unfortunately an indirect characteristic of the anisotropy and serious difficulties may be encountered when calculating R(D) from R(I), because an estimate of the second derivative of R(I) is needed. The maximal number of angles n that is affordable in practice is, according to the experience of the authors of this paper, at most n = 18. Otherwise the method becomes non-precise and time consuming.

Results of the method of the rose of directions have been included in this paper to offer an opportunity to compare our newly suggested approach with a totally different one, which is used as a standard in many places.

Spectral approach and estimation of fiber orientation

Techniques of image analysis based on the spectral approach are especially suitable for describing textural images; images of planar textiles can be considered good examples. We prefer to use the 2DDFT, which transforms images from the spatial domain to the frequency domain. The key point is that dominating directions (gradients of image function) in the directional textures correspond to large magnitudes of frequency components distributed along the straight lines in the Fourier spectrum.⁸ The Fourier transform is rotation-dependent. Indeed, rotating the original image by an angle will rotate the corresponding frequency plane by the same angle. Moreover, the transform of horizontal lines in the spatial domain appears as vertical lines in the Fourier domain, that is, a line in the spatial image and its transformation are orthogonal to each other. Due to this relationship, the Fourier transform is useful for describing regular (periodic) patterns in woven fabric images,^{10–12,15,16} where information about weft or warp sets of yarns is concentrated in the Fourier spectrum in the vertical or horizontal direction, respectively. In contrast, purely random textures, for example random noise, cause the frequency components in the power spectrum to be approximately isotropic and possess a nearly circular shape.⁹

Fourier spectrum and the properties mentioned above can be useful for description of structural anisotropy or directional orientation of textile object systems. There are a several articles dealing with the measurement of fiber orientation distribution with the aid of Fourier transform. Measurement of fiber orientation in nonwovens is presented by Pourdeyhimi and Davis,⁴ Pourdehyhimi et al.⁵ and Jeddi et al.,¹⁷ where orientation is estimated by scanning the Fourier spectrum radially with annulus of a specific width at a radius from the center. Fiber orientations in simulated images of nonwovens are analyzed by Zhang et al.¹⁸

Estimation of directional orientation in fiber-objected systems with the aid of the Fourier transform can be found in various other engineering areas, for example surface characterization in mechanical engineering and metallography,^19,20 estimation of cell and fiber orientation distribution in biology,^21,22 determination of fiber orientation in composites,^23,24 distribution of fiber orientation on paper surface,²⁵ etc.

Let us take a closer look at this method. Let f(x,y) be the gray level at pixel coordinates (x,y). Let the size of a spatial domain image be M × N. The corresponding 2DDFT (for details see, e.g., Gonzales et al.,²⁶ p.108) is given by

F (u, v) = \sum_{x = 0}^{M - 1} \sum_{y = 0}^{N - 1} f (x, y) e^{- j 2 π (\frac{ux}{M} + \frac{vy}{N})},

where

u = 0, 1, \dots, M - 1

and

v = 0, 1, \dots, N - 1

are frequency variables and

x = 0, 1, \dots, M - 1

y = 0, 1, \dots, N - 1

are spatial variables. If f(x,y) is a real function, its Fourier transform is a complex function. For the subsequent visual analysis it is suitable to calculate spectrum |F(u,v)| and display it as an image. Recall that the power spectrum is defined as a squared modulus of the spectrum, that is, P(u,v) = |F(u,v)|². For a visual representation of F(u,v) scaled to eight-bit gray levels, the spectrum is usually converted using the log transformation (for details see, e.g., Gonzales et al.,²⁶ p.114):

Q (u, v) = \log (1 + | F (u, v) |) .

Examples of textile fiber systems in the form of grayscale images and the corresponding spectra are displayed in Figures 1(a) and 1(b). Figure 1(a1) represents a system of simulated fibers with a preferred direction, and an image size of 800 × 800 pixels. Figure 1(a2) represents a system of real viscose fibers, with an image size of 500 × 500 pixels. Figure 1(a3) represents a nanofibrous layer, with an image size of 500 × 500 pixels.

Figure 1.

(a) Grayscale images of textile fibrous systems. (b) Corresponding power spectra.

Readers interested in the fast graphical representation of the directional arrangements of textile objects, which is based on 2DDFT, are referred to by Tunak and Linka,²⁷ where anisotropy estimation is performed summing all frequency components in the directional vector of a certain angle α through the whole range of angles. As an estimate of the rose of directions, the sums are plotted onto the polar diagram. The main advantages of this method include its speed and the possibility of monitoring with a very small angular step (equal to one degree).

Figure 2.

(a) Brodatz texture D15. (b) Rose of direction constructed by means of Steiner compact according to Rataj and Saxl.¹³ (c) Estimation of directional orientation according to Tunak and Linka.²⁷ (d) Estimate of directional orientation using image moments.

Figure 3.

(a) Image of polyvinylidene fluoride nanofibers (1000 × 1000 pixels). (b) Estimates of orientation according to Tunak and Linka.²⁷ (c) and (d) Gray-level map of orientation (sub-window sizes 40 × 40 and 20 × 20 pixels). (e) and (f) Directional vectors in original image. (g) and (h) Density histogram and kernel estimates of the density of orientations.

Figure 4.

(a) Image of polycaprolactone nanofibers (1000 × 1000 pixels). (b) Estimates of orientation according to Tunak and Linka.²⁷ (c) Gray-level map of orientation (sub-window size 30 × 30 pixels). (d) Directional vectors in original image. (e) Density histogram and kernel estimates of the density of orientations.

Moments of the image function and estimation of fiber orientation

As mentioned earlier, directions of significant frequency components corresponding to the dominating directions in the spatial domain can be found in the power spectrum. Therefore, it is appealing to transform the power spectrum to a binary image in which only significant frequencies remain, and then to analyze this binary image. In practice it means that we can concentrate on cluster(s) of white pixels as object(s) or region(s) of interest, which considerably simplifies the task.

It is well known that an important description of an image file is provided in its moments. For a two-dimensional (2D) image function f(x,y) the moment of order (p,q) is defined (for details see, e.g., Gonzales et al.,²⁶ p.470) as

m_{pq} = \sum_{x} \sum_{y} x^{p} y^{q} f (x, y), p, q = 0, 1, 2, \dots,

and the central moments of order (p, q) are defined as

μ_{pq} = \sum_{x} \sum_{y} (x - \bar{x})^{p} (y - \bar{y})^{q} f (x, y) .

Recall that moments of the image function are interesting characteristics of objects in the image, that is:

$m_{00} / (M \times N)$ in a binary image is a proportion of white points in the image;

ratios of first-order moments m₁₀ and m₀₁ with the moment of zero order, that is, $\bar{x} = m_{10} / m_{00} and \bar{y} = m_{01} / m_{00}$ , determine the center of gravity of objects or regions of interest.

Another useful piece of information about the image can be found using the corresponding data covariance function:

cov f (x, y) = (\frac{μ_{20}}{μ_{00}} \frac{μ_{11}}{μ_{00}} \frac{μ_{11}}{μ_{00}} \frac{μ_{02}}{μ_{00}}) .

Indeed, the eigenvectors of cov f(x,y) represent the major/minor axis of the data ellipse and are orthogonal to each other. Eigenvalues $λ_{1}$ and $λ_{2}$ of cov f(x,y) are of the form

λ_{i} = \frac{μ_{20} + μ_{02}}{2} \pm \frac{\sqrt{4 μ_{11}^{2} + (μ_{20} - μ_{02})^{2}}}{2}, i = 1, 2 .

The corresponding eigenvectors are the half-lengths of the major and minor axes. Moreover, the eigenvector corresponding to the largest eigenvalue represents a slope of the major eccentricity of an ellipse matrix, and the angle θ between the x-axis and direction of the major axis of the data ellipse is

θ = \frac{1}{2} \arctan (\frac{2 μ_{11}}{μ_{20} - μ_{02}}),

while the eccentricity is given by

L = \sqrt{1 - λ_{2} / λ_{1}} .

For an example see Figure 8, where Figure 8(a) represents the image plane.

Combining spectral and moment approaches in estimation of fiber orientation

It feels appealing to combine the spectral approach and method of moments as follows.

At first we transform the power spectrum to a binary image via thresholding, so that only significant frequencies remain. Global thresholding was used for binarization of the image, with the threshold value being set to 0.5 of the maximum of the logarithm of the power spectra $Q (u, v)$ . 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 corresponding 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 (in red in Figure 8(b)) are computed only for the region of interest. It is obvious that its orientation reflects the predominant directions of objects in the spatial domain.

Examples

This section illustrates the considered approaches on examples of both real and simulated data.

Example 2.1

As the first example, we present in Figure 9(a) a 500 × 500 pixel grayscale image of viscose fibers. In Figure 9(b) the corresponding power spectrum rotated by 90° is shown, and in Figure 9(c) a binary image of its most significant frequencies is displayed.

Example 2.2

As the second example we present the situation well known from the literature, which has already been used by many other authors for a comparison between methods. Graphical representations of the results of all methods described above can be found in Figures 2(a)–(d). The results of all approaches are coherent and the correspondence in the preferred directions is evident.

Figure 2(a) represents a Brodatz texture D15 (for details see Brodatz texture²⁸).

Figure 2(b) shows the rose of directions evaluated by the simple graphical experimental method as described by Rataj and Saxl.¹³

Figure 2(c) shows a polar diagram as an estimator of the rose of directions according to Tunak and Linka;²⁷ the main direction is marked by a small red circle.

A red-color ellipse with the lengths of major and minor axes and its orientation can be seen in Figure 2(d).

Estimation of the fiber system orientation for nonwoven textiles and nanofibrous layers: a local approach

The methods described in the second section are usually used for estimating the orientation of fibers in the whole image. However, it turns out that for textile fibrous layers, such as nonwoven textiles or nanofibrous layers, which often bear the stamps of non-homogeneity and anisotropy, more detailed analysis would be more suitable. The reason is that it is difficult, and often even impossible, to characterize the whole image by just one number that would not necessarily describe the reality satisfactorily.

This led us to an idea of abandoning the currently used “global approach”, for which it is typical to use just one number as an estimate of the characteristic of interest, which is difficult to interpret, especially for nonwoven and nanofibrous layers. Instead, it seems appealing to replace this “global point of view” with the “local one”. By the local point of view we mean splitting the image into many small, non-overlapping pieces (sub-windows) covering it, and estimating the characteristic of interest for each piece independently from the others. As a result we obtain many estimates of the characteristic of interest, one for each sub-window of the original image, and instead of averaging them in order to get just one value we analyze and further discuss distribution of these estimates.

Formally, as an estimator of the fiber orientation in every sub-window, any of the approaches described in the second section may be used. We suggest using the method described in the Combining spectral and moment approaches in estimation of fiber orientation section, that is, a combination of spectral and moment approaches. The threshold value used should be half of the maximum of the logarithm of power spectrum calculated in the given sub-window. On the other hand, use of the rose of directions for small-sized sub-windows does not seem appropriate; see discussions in Rataj and Saxl.^13,14

Detailed description of the method

Let us describe our idea in more detail. At first, we divide the image area into K smaller, non-overlapping sub-windows of a certain size and estimate the direction arrangement for each sub-window separately. Instead of one value characterizing the whole image, we thus obtain K estimates of the parameter of interest, for example, fiber orientation, etc. We denote them T₁,…,T_K and estimate the distribution of T_i values by either histogram or kernel estimator. Recall that procedures for calculation of both histograms and kernel estimators are routinely available in all software packages and/or image analyzers. For details about these estimators, see, for example, Scott.²⁹ It is evident that the more evenly dispersed the values of T_i are, the more uniform the considered characteristic is.

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,²⁹ 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 500 × 500 pixels into sub-windows of 30 × 30 pixels, it leads to slightly fewer than 300 sub-windows, which is adequate to produce good-quality estimates of the desired characteristic. The larger the image size is, the better estimates we will receive.

On the other hand, the size of the sub-window substantially influences the quality of fiber orientation estimates. 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. This phenomenon is illustrated in Figures 10(d)–(f). We are convinced that the optimal choice depends, among other circumstances, especially on the fiber thickness and curvature. In the case of nanofibrous layers we analyzed, the curvature is rather small. Unfortunately, we are not aware of any theoretical results that would help us to solve the problem of how to choose an optimal sub-window size, so for now we leave it an open problem. In the simulated example the thickness was set to one pixel, while in experimental examples the typical thickness is, according to our measurements, between one and three pixels, sometimes larger. In subsection Examples we present the results for the sub-window size being 20–30 times the mean thickness of the fibers, which yielded good results.

Experimental design: polymers, processing parameters, and image capturing used

Three kinds of polymers were used for the presented experiments, all being dissolved in suitable solvent to prepare polymer solutions. Details of the experiments follow.

The first polymer used was water soluble polyvinylalcohol (PVA) 16 wt%, produced by Chemicke Zavody Novaky, Slovakia, diluted to the concentration 12 wt%. Experiments were conducted with NanospiderTM; the distance between the roller and the collector was 120 mm. The voltage used was 45 kV, the duration of the experiment 15 minutes, ambient humidity 40%. Selected results are shown in Figures 1(a3), 5, and 6.

The second polymer used was polycaprolactone (PCL) by Sigma Aldrich. It is biodegradable polyester with a low melting point and molecular weight Mw = 45.000. The final concentration was 16 wt% and chloroform and acetone (9:1) were used as solvents. The distance between the syringe and the collector was 15 cm (Figures 4 and 7).

The third polymer used was polyvinylidene fluoride (PVDF), a highly non-reactive and pure thermoplastic fluoropolymer, prepared from Solef 1006 by Solvay Solexis with the melt flow index MFI = 120 g/min. For the experiment 16 wt% solution was prepared using dimethylacetamide (DMAc) as a solvent. Results are presented in Figure 3.

A needleless electrospinning method called Nanospider was used for the PVA polymer. A feeding pump with hypodermic syringe and conductive needle was used for PVDF and PCL polymers.

Images were captured by a Phenom FEI Scanning Electron Microscope and stored as image matrices in an eight-bit gray-level range. This device is capable of magnifying a specimen from 20 times up to 20,000 times with an image resolution up to 2048×2048 pixels. Matlab software and built-in functions of the Image Processing Toolbox were used for realization of the proposed algorithm; for details see Matlab.³⁰

Examples

To get a taste of the idea, take a look at the results of the proposed approach applied to both simulated and real data. Parameters of tested images are summarized in Table 1.

Table 1.

Parameters of tested images

Figure no.	Size of image	Material
1(a1), 11(a)	800 × 800 pix	Simulated system of fibers; length, gray level, and position of fibers was randomly generated from the uniform distribution. Orientation of fibers was randomly generated from the uniform distribution in the interval [0, 45°].
1(a2), 9(a)	500 × 500 pix	Viscose fibers
1(a3)	PVA nanofibers
2(a)	640 × 640 pix	Brodatz texture (D15)²⁸
3(a)	1000 × 1000 pix	PVDF nanofibers
8(a), 11(a)	1000 × 1000 pix	PCL nanofibers
9(a), 10(a)	1000 × 1000 pix	PVA nanofibers

PVA: polyvinylalcohol; PVDF: polyvinylidene fluoride; PCL: polycaprolactone.

Example 3.1

First we analyzed a simulated fiber system. Data of the size 800 × 800 pixels are shown in Figure 11(a). It represents a system of 500 linear fibers randomly oriented in the interval [0°, 45°]. The figure was divided into sub-windows of 20 × 20 pixels. Preferred orientation of objects for every sub-window is represented by the directional vector displayed in red (where the ratio of major-to-minor axis length is greater than 2). Moreover, orientation in degrees is displayed in the gray-level scale. Results of estimating the orientation in the form of a gray-level map can be seen in Figure 11(b). Figure 11(c) displays the directional vector in the original image. Distribution of estimates of the orientations in the respective sub-windows in the form of density histogram and kernel density estimates are shown in Figure 11(d). We can see that the distribution of orientation of fibers is in very good agreement with the input parameter used for generating the image.

Example 3.2

This example corresponds to the real data comprising oriented viscose fibers. Image size in this case was 500 × 500 pixels and can be seen in Figure 10(a). We divided the original image into sub-windows of 20 × 20 pixels. Analogous to the previous example, results of the fiber orientation estimates can be found in Figures 10(b)–(d).

Algorithm 1. Calculation of the estimate of fiber system orientation

Setup phase

Prepare functions calculating kernel density estimator and histogram from given data (notice that Matlab offers for that purpose functions KSDENSITY and HIST).

Read data image $f (x, y), x = 0, 1, \dots, M - 1, y = 0, 1, \dots, N - 1 .$

Set sub-window size $k \times l$ (usually $k = l$ ).

Split data into $L = [\frac{M}{k}] \times [\frac{N}{l}]$ non-overlapping sub-windows $W_{1}, \dots, W_{L} .$ (note: $[x]$ means round toward minus infinity)

Main loop (analysis of individual sub-windows W_i)

for $i = 1 : L$ do

Calculate Fourier transform $F_{i} = F_{i} (u, v), u = 0, 1, \dots, k - 1, v = 0, 1, \dots, l - 1,$ of W_i

Calculate log transform $Q_{i} (u, v) = log (1 + | F_{i} (u, v) |)$

Calculate threshold $t_{i} = 0.5 \times max {Q_{i} (u, v)}$

Prepare new binary image file (in frequency domain) $B_{i} = B_{i} (u, v), u = 0, 1, \dots, k - 1, v = 0, 1, \dots, l - 1,$ using thresholding, i.e.

B_{i} (u, v) = {0 if Q_{i} (u, v) < t_{i} 1 if Q_{i} (u, v) \geq t_{i}

Calculate moments of B_i

m_{pq} = \sum_{u} \sum_{v} u^{p} v^{q} B_{i} (u, v) p, q = 0, 1, 2

Calculate central moments of B_i

μ_{pq} = \sum_{u} \sum_{v} (u - \bar{u})^{p} (v - \bar{v})^{q} B_{i} (u, v)

Calculate corresponding covariance function

cov (B_{i}) = (\frac{μ_{20}}{μ_{00}} \frac{μ_{11}}{μ_{00}} \frac{μ_{11}}{μ_{00}} \frac{μ_{02}}{μ_{00}})

Calculate eigenvectors of $cov (B_{i})$ , i.e.

λ_{j} = \frac{μ_{20} + μ_{02}}{2} \pm \frac{\sqrt{4 μ_{11}^{2} + (μ_{20} - μ_{02})^{2}}}{2} j = 1, 2

Calculate orientation (eccentricity) of B_i

θ_{i} = \frac{1}{2} arctan (\frac{2 μ_{11}}{μ_{20} - μ_{02}})

and store it

end for

Estimating distribution of directions

Apply kernel density estimator and histogram estimator to the orientations $θ_{1}$ ,…, $θ_{L}$ to estimate their distribution (for given image data $f (x, y))$ .

Output

Distribution of directions of orientation of individual sub-windows of the global image.

Example 3.3

Results of the analysis of other real data can be found in Figure 3. Figure 3(a) represents an image of the PVDF nanofibers produced via electro-spinning. The size of the image is 1000 × 1000 pixels. In Figures 3(c) and (d), gray-level maps of the orientation are presented for the sub-windows at 40 × 40 pixels and 20 × 20 pixels, respectively, together with the corresponding directional vectors. Figures 3(e) and (f) display directional vectors in the original image and Figures 3(g) and (h) represent distribution of orientation estimates in the form of density histograms and kernel density estimators for the sub-window at sizes of 40 × 40 pixels and 20 × 20 pixels, respectively. As can be seen, regions marked by a middle gray level represent regions without preferred orientations. A polar diagram of an orientation estimate constructed according to Tunak and Linka²⁷ is presented in Figure 3(b). Bimodal density histograms are in accordance with the polar diagram. Moreover, we can see from Figures 3(c) and (d) that the placement of the local maxima (peaks) is practically the same for both sizes of the sub-window. On the other hand, a smaller sub-window size provides smoother results.

Example 3.4

Finally, Figures 4 –7 represent several typical situations one can encounter in practice when analyzing images of nanofiber layers produced via electro-spinning. Experiments are described in the Experimental design: polymers, processing parameters, and image capturing used section. Let us comment on the obtained results in more detail.

Figure 4 shows a situation in which one part of the image exhibits a strong orientation of fibers, while the other part corresponds to a more-or-less anisotropic case. Both histogram and kernel estimates clearly indicate it.

Figure 5 presents another interesting situation in which seemingly no direction of the fibers appears to dominate the image, at least optically. It reflects the desire of most manufacturers and users, who usually ask for maximum homogeneity in the product. The obtained results, however, show that direction of fibers from the left to the right slightly prevails: compare Figure 5(e).

Figure 6 displays another typical situation when two directions of fibers with different slopes dominate the image. The two dominating orientations are relatively close to each other and the larger sub-window size allows better discrimination between them.

Finally, Figure 7 corresponds to a situation with very strong orientation of fibers in only one direction, that is, from the left to the right. We can clearly see that both histogram and kernel estimates describe the situation very well. Notice that we would get the same results when rotating the original image. Of course, they would be shifted by the angle of rotation.

Figure 5.

(a) Image of polyvinylalcohol nanofibers (1000 × 1000 pixels). (b) Estimates of orientation according to Tunak and Linka.²⁷ (c) Gray-level map of orientation (sub-window size 30 × 30 pixels). (d) Directional vectors in original image. (e) Density histogram and kernel estimates of the density of orientations.

Figure 6.

Figure 7.

Figure 8.

(a) Ellipse. (b) Region of interest.

Figure 9.

(a) Image of viscose fibers (500 × 500 pixels). (b) Power spectrum of the image. (c) Estimate of directional orientation using image moments.

Figure 10.

(a) Real fiber system of viscose fibers (500 × 500 pixels). (b) Gray-level map of orientation (sub-window size 20 × 20 pixels). (c) Directional vectors in original image. (d) Density histogram and kernel density estimates for sub-window size 10 × 10 pixels, (e) sub-window size 20 × 20 pixels, and (f) sub-window size 30 × 30 pixels.

Figure 11.

(a) Simulated fiber system (800 × 800 pixels). (b) Gray-level map of orientations (sub-window size 20 × 20 pixels). (c) Directional vectors in original image. (d) Histogram and kernel density estimate of the fiber orientation. (Color online only).

Moreover, the obtained results clearly demonstrate that the local idea, that is, “divide-and-conquer”, is a very efficient tool, especially when we are trying to describe the orientation of fibers in non-homogeneous cases when one, two, or a few directions of fibers prevail in the image.

Our approach is schematically described in Algorithm 1. We present only the pseudocode, which can be used in any programming language. Therefore, all technicalities taking into account memory constraints of a given computer, programming language constructions, etc., which may be of key importance for an effective implementation, are missing. Parallelization is not considered.

Table 2 represents computational speed needed for the analysis of examples presented in the paper. We can see that the CPU time needed for the analysis of one sub-window is practically the same regardless of whether its size is 10 × 10 or 40 × 40. Recall that all calculations were carried out using Matlab on a PC computer equipped with Intel Xeon W3530 @ 2.80 GHz, 4.00 GB RAM, and MS Windows 7 Pro 64-bit.

Table 2.

CPU time needed for the analysis of examples presented in the paper. Different sub-window sizes are considered

Figure no.	Size of the image	Sub-window size	Overall time needed for analysis of the whole image [s]	Number of sub-windows	Time needed for analysis of one sub-window [ms]
5	800 × 800	20 × 20	26.8	1600	16.75
6	500 × 500	10 × 10	40.3	2500	16.12
20 × 20	10.4	625	16.64
30 × 30	4.5	256	17.58
7	1000 × 1000	20 × 20	41.0	2500	16.40
40 × 40	11.1	625	17.76
8	1000 × 1000	30 × 30	18.7	1089	17.17
9	1000 × 1000	30 × 30	19.1	1089	17.54
10	1000 × 1000	30 × 30	19.0	1089	17.45
11	1000 × 1000	30 × 30	18.9	1089	17.36

Unfortunately, a decrease in the sub-window size dramatically (in a quadratic way) increases the number of sub-windows and, consequently, the overall time needed for the analysis of the whole image. Nevertheless, there exists at least a partial remedy to this problem, namely, parallelization. We would like to point out that one of the greatest advantages of our method is the fact that it can easily be parallelized using, for example, the so-called CUDA approach. This means that the main processor (CPU) is responsible just for splitting the analyzed image into smaller sub-windows (sub-images) and distributing them to a set of graphical processing units (GPUs) for further analysis. Analysis of each sub-image is identical and follows the procedure given in Algorithm 1. Finally, the results from individual GPUs are collected by the main CPU (“put together”), and further analyzed. We are convinced that the parallelization of calculations is a way of applying our method in real practice for very large dataset as in the quality control of nonwoven materials. For details on CUDA see, for example, CUDA Parallel Computing Platform.³¹

Conclusions

In this paper we concentrated on estimation of fiber orientation of nanofibrous and nonwoven layers. Let us summarize two most important novelties proposed in this paper. Firstly, it turned out that combination of the discrete Fourier transform and results of analysis of the sample image covariance matrix is effective for this purpose. Secondly, the most important point consists of abandoning the currently used “global approach” that characterizes an entire image using just one number (index) and replaces that with the distribution of the fiber orientation in sub-windows of the original image, the so-called “local approach”. Among the main advantages of this approach are the possibility of describing more precisely fiber orientation of the materials (layers) that exhibit a large anisotropy of the characteristic of interest, for which the characterization by one number (index) as used typically nowadays can be totally misleading.

We devoted our effort to the practically important case of nanofibrous and nonwoven materials, concentrating on the task of monitoring the structure orientation in such systems. However, the results indicate that the suggested method might be used for finding other quantitative characteristics of textile materials as well.

It also appears that the proposed method can be effectively used for estimating the directional orientation of fibrous textile materials from the point of view of their homogeneity, eventual defects, random violation of regularity of the structure, etc.

We think that the methods of the second and third sections are suitable for implementation in software for analysis of fiber orientation, and expect that ongoing research will bring further improvements.

Footnotes

Funding

This work was supported by the Czech Science Foundation (grant no. 201/09/0755) and TAO/TA, Technology Agency of the Czech Republic (project no. TA03010609).

Acknowledgments

The authors are grateful to all reviewers and editor-in-chief for careful reading and detailed comments and suggestions, which greatly improved the manuscript. Support from the IAP research Network P7/06 is also gratefully acknowledged.

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