A new shape factor measure for characterizing the cross-section of profiled fiber

Abstract

It is an important topic to characterize the cross-sectional shape of profiled fiber for quality control of textiles and performance design. In this paper, we propos e a new measure of the shape factor, which can be used to characterize the cross-sections of profiled fibers. Different from the existing indices of the shape factor which are based on the ratio of the radii of circumscribed and inscribed circles, the proposed measure uses the shape information residing in the boundary of profiled fiber cross-sections. It is defined as the coefficient of variation of the area sequence in the cross-sectional plane at each angular interval. The capacity of this measure is validated by various examples, which also indicates that the defined measure can distinguish the discrepancy degree between fiber cross-section and the area equivalent circle. It is robust and applicable to any arbitrary shape with solid cross-section, no matter whether it is convex or concave.

Keywords

Profiled fiber characterization fiber cross-section shape factor shape description

Introduction

Fiber cross-section analysis¹ is an important topic for understanding the properties of fiber and the downstream product made from it. The cross-sectional shape of the fiber is regarded as an important component of fiber surface geometry. Slight deviations from non-circularity of cross-sectional shape can lead to substantial differences in fiber processing behavior and in the performance characteristics of textile structures.²^–⁴ As for profiled fiber, quantitative characterization of fiber cross-sectional shape is an extraordinarily important part of component analysis and quality control.⁵ Itis indispensable when studying the influence mechanism of fiber cross-sectional non-circularity on the performance of end-use products.⁶^–¹²

At present, the fiber shape factor,¹³ which was defined as the degree of difference of fiber cross-section relative to the circular shape (sometimes refers to the degree of non-circularity), is commonly used as the characteristic parameter to describe the cross-sectional shape of profiled fiber, including the industry standard (FZ/T 50002-91) in identifying the cross-sectional shapes of synthetic fibers.¹³ More researches have proven the effects of cross-sectional shapes on many fiber properties and characteristics, such as luster,⁷ handle,⁸ wicking property,⁹ sound,¹⁰ bending modulus,¹¹ etc. Shape factor (non-circularity), like length and fineness, can be regarded as an index to describe the surface geometry of fiber. Therefore, the development of a suitable index to define the non-circularity degree in characterizing the cross-sectional shapes, mathematically, remains a meaningful topic of academic investigation.

Several indices of shape factor already exist. For example, the modification ratio defined as the ratio ofthe minimum distance to the boundary from the centroid, to the maximum distance to the boundary from the centroid;^1,14 the relative shape factor of the radius defined as the ratio of the subtraction of the radius of the inscribed circle from the radius of the circumscribed circle, to the radius of the circumscribed circle;^13,15 the ratio of the squared perimeter of the fiber cross-section to the area of the cross-section^1,14,16 (abbreviated as RSP in this paper, i.e., $RSP = \frac{P^{2}}{A}$ ); the ratio of the cross-sectional area to the area of the circumscribed circle, and the perimeter factor, which was defined as the ratio of the area of the fiber cross-section to the area of the circle with equal perimeter.¹⁷

However, it is quite possible for fibers with identical measured shape factors to have different non-circularity degrees.¹⁷ Furthermore, some indices have a strong correlation between each other, for example, RSP and the perimeter factor. The traditional characterization method based on the ratio of the radius of circumscribed and inscribed circles has shortcomings.¹³ For example, when the circumscribed and inscribed circles are not homocentric or the inscribed circles do not exist, it is impossible to find proper characteristic indices.

In this paper, we present a new measure of the shape factor. It is designed to characterize any arbitrary shape with solid cross-section and its value can directly and accurately depict the non-circularity degree. In our investigation, it is assumed that the discrepancy degree of profiled fiber cross-section relative to a circle is concretely equal to the discrepancy degree of fiber cross-section relative to the area equivalent circle whose center coincides with the centroid of the fiber object.

Our characterization system of fiber shape factor employs an automated fiber analysis system¹⁸ to automatically capture well-focused images of fiber cross-sections. Since previous reports had presented various techniques of fiber cross-sectional image capture¹⁹^–²¹ and image preprocessing,²²^–²⁴ we will focus mainly on the object description and the characterization of cross-sectional contour.

The rest of this paper is organized as follows. The derivation of the new circularity measure, CV_r₂, is introduced and several desirable properties of it are analyzed. Then several examples to demonstrate the capacity of CV_r₂ as a measure of fiber shape factor arediscussed.

New measure of fiber shape factor

Calculating parameters of equivalent circle area of fiber cross-sections

After image preprocessing and image segmentation, the fiber cross-sectional image (as shown in Figure 1a) can be converted to a binary image with the fiber and the background being separated completely (as shown in Figure 1b). The values of pixels in fiber objects are 1, the values of pixels in the background are 0. For each fiber object, we first determine the coordinates (x_o, y_o) of the centroid of the object:

(1)

where (x_i, y_i) are the pixels in the region of the fiber object. Np is the actual number of pixels in the region of the object. If we consider the area of a pixel as the measuring unit, the area S of the object can be obtained by counting the actual number of pixels in the region of the object, i.e. S = Np. The radius of the area equivalent circle, denoted by R, can be obtained accordingly. Figure 2 plots the area equivalent circle of a fiber cross-section object. The center of the circle coincides with the centroid of the object.

Figure 1 (a).

Original photomicrograph of profiled fibers with cruciform cross-sections; (b) the binary image.

Figure 2.

A fiber cross-section object and its area equivalent circle whose center coincides with the centroid of the object.

Generating the distance-versus-angle function of the fiber cross-section

Taking O as the origin, the pixel P₁ (the pixel with the minimum y-coordinate among the pixels with minimum x-coordinate) as the starting point, and the line OP₁ as the reference line, the distance D(θ_i) from O to each boundary pixel P₁ can be calculated in the clockwise direction, as shown in Figure 3a. This distance-versus-angle function D(θ) (0 ≤ θ ≤ 2π, i = 1, 2,…,N_b) is actually a function of the angle θ_i from the reference line OP₁ to the line OP_i specified in radians, as illustrated in Figure 3b. N_b is the total number of boundary pixels. In this way, we transform the two-dimensional image information to a one-dimensional function which contains useful information in characterizing the fiber shape factor quantitatively.

Figure 3.

Generating distance versus angle function D(θ); the plot of D(θ).

As shown in Figure 3b, D(θ) is valued discretely, i.e., D = [D(θ₁), D(θ₂),…, D(θ_Nb)], θ = [θ₁, θ₂,…, θ_Nb], θ₁ = 0, θ_Nb ≤ 2π. Let $\bar{D}$ be the mean value of D. The fluctuation of D against R fully reflects the fluctuation of the fiber boundary protruding from or hollowing out thecircumference of the area equivalent circle. Therefore, the discrete degree of D against R can be employed in investigating the fiber shape factor.

The coefficient of variation of a data set can effectively reflect its discrete degree relative to the mean. However, since $\bar{D} \neq R$ , as shown in Figure 3b, the discrete degree of D against R cannot be directly expressed by the coefficient of variation. We have to construct a new sequence on D whose relationship with R can be expressed by the coefficient of variation.

Equiangular sampling and derivation of a new measure CV_r2 of fiber shape factor

In fact, the function D(θ) satisfies a defined condition. The cross-sectional area of fiber equals to the area of the circle. To take advantage of this condition, we can divide its boundary into N segments according to the equiangular change Δθ. When Δθ is very small, each part of the cross-section can be regarded as a small fan, as illustrated in Figure 4a. Although theoretically smaller Δθ implies higher accuracy, according to the experimental experience, we take Δθ = 2π/180, thus N = 180. The radius of each small fan can be approximated by the mean value of D at each angular interval Δθ. Let r_i be the radius of the i-th fan, we get a 1-D sequence: r = [r₁, r₂,…, r_i,…, r_N], as illustrated in Figure4b. Hence the condition can be expressed as:

(2)

Deduction available is as follows:

(3)

With

\bar{r^{2}} = R^{2}

, also shown in Figure 5, the fluctuation degree of the sequence r_i² against R² can be expressed by the coefficient of variation

{CV}_{r^{2}}

that can be defined as a measure of fiber shape factor,

(4)

It ranges over the interval

[0, + \infty)

and its theoretical minimum value is 0, corresponding to the shape factor of a perfect circular cross-section. The bigger the value is, the greater the degree of difference is of fiber cross-section relative to the area equivalent circle. Obviously, the measure

{CV}_{r^{2}}

is invariant to translation, scaling, and rotation. It is suitable to be used as a measure of fiber shape factor.

Figure 4.

Dividing cross-sectional boundary into N parts according to equal change of polar angle; The sequence r obtained by calculating the radius of each small fan.

Figure 5.

The sequence r².

Boundary representation of concave cross-sections with deep depressions

The boundary representation described above may fail to apply when the shape of the cross-section is not convex or convex in a broad sense, since D(θ) should be a single-valued function. For concave cross-sections with deep depressions, if ray OP intersects with the boundary more than once, as shown in Figure 6, θ_i = θ_j = θ_k, D(θ_i) ≠ D(θ_j) ≠ D(θ_k), i ≠ j ≠ k, the aforementioned method cannot be used directly to obtain the value of ${CV}_{r^{2}}$ .

Figure 6.

Ray OP intersects with the boundary more than once when the cross-section has deep depressions.

In order to cope with this situation, we review the definition of the measure ${CV}_{r^{2}}$ :

(5)

where,

S_{fri} = \frac{1}{2} \cdot \frac{2 π}{N} \cdot r_{i}^{2}

S_{fRi} = \frac{1}{2} \cdot \frac{2 π}{N} \cdot R^{2}

. As indicated in equation (5), the meaning of

{CV}_{r^{2}}

is the coefficient of variation of the area sequence in the cross-sectional plane at each angular interval. Therefore, for those concave boundaries with deep depressions, like V-shaped, U-shaped, and C-shaped cross-sections,²⁵ we can get the sequence r² by calculating the sequence S_fr (as shown in Figure 7) or obtain

{CV}_{r^{2}}

by calculating

{CV}_{S_{fr}}

Figure 7.

The sequence S_fr; The sequence r².

Experiments illustrating CV_r2 behavior

15 types of basic shapes have been selected to test the capacity of the new measure ${CV}_{r^{2}}$ . The results are shown in Figure 8. The abscissas are the 15 shapes, the ordinates are values of ${CV}_{r^{2}}$ .

Figure 8.

Comparison of the ${CV}_{r^{2}}$ values of 15 basic shapes.

Examples in Figure 8 illustrate how the new measure acts. As for the ellipse, our subjective understanding is the ratio of the long axis and the short axis, namely, the bigger the ellipticity is, the greater the value of shape factor should be. The measured ${CV}_{r^{2}}$ shape factors of No. 1 to No. 5 satisfy this understanding.

If we approximate the circle with a regular polygon,the discrepancy degree of the equilateral triangle relative to the circle is the greatest, followed by the square, theequilateral pentagon, the equilateral hexagon, the equilateral heptagon, and the equilateral octagon. Theequilateral nonagon is very close to the circle. From the results of No. 5 to No. 11 and No. 13, it is clear that the values of ${CV}_{r^{2}}$ coincide with this tendency.

Among all quadrilaterals, the square should have the lowest shape factor. The discrepancy degree of the rectangle to the circle is bigger than that of the square. The discrepancy degree of the parallelogram to the circle is bigger than that of the rectangular. The flatter the parallelogram is, the bigger the discrepancy degree relative to the circle is. The measured ${CV}_{r^{2}}$ of No. 11, No. 12, No. 14 and No. 15 are all valued as expected.

To further verify our proposed parameter, we carry out an experiment with 24 types of common solid profiled fibers. To better illustrate the behavior of ${CV}_{r^{2}}$ , we also compute the values of γ (a derivation of RSP), which is defined as:

(6)

where P is the perimeter of the fiber cross-section, A is the area of the fiber cross-section.

Since the circle has the minimal perimeter for all shapes with the same area, those measures given by the ratio of perimeter to area, e.g.

RSP = \frac{P^{2}}{A}

and

FF = \frac{4 π A}{P^{2}}

(Form Factor²⁶), are the most basic measures of shape circularity or non-circularity. RSP is an common measure of fiber shape factor. FF is probably the most standard measure of shape circularity. Both of them are based on the ratio of perimeter to area. Hence we take

γ = \frac{P^{2}}{4 \cdot π \cdot A} - 1

to compare with

{CV}_{r^{2}}

. γ for all shape values in the range

[0, + \infty)

, which is the same as

{CV}_{r^{2}}

. Both γ and

{CV}_{r^{2}}

will be valued as 0 when the shape is a circle. The results are shown in Table 1.

Table 1.

${CV}_{r^{2}}$ and γ of 24 cross-sectional shapes

These cross-sectional shapes are ranked with respect to their measured

{CV}_{r^{2}}

shape factors in Table 1. Tofacilitate comparison, several results from the experiment are presented in groups according to their types, as shown in Table 2 to Table 5. It can be observed that the ranking obtained is in accordance with common perception, especially for trilobal shapes including No.6, No. 8, No. 11, No. 14, No. 19, and No. 23, as shown in Table 2; multi-leaf shapes No. 3, No. 4, No. 7, No. 11, and No. 15, as shown in Table 3. From the measured

{CV}_{r^{2}}

of No. 9, No. 12, No. 15, No. 22 and No. 24, it can be concluded that obvious oriented shapes have greater shape factors, i.e., the thinner/longer the shape is, the greater the shape factor is. These data prove that the index

{CV}_{r^{2}}

is capable of quantitatively characterizing fiber shape factors (non-circularity degrees).

Table 2.

Trilobal shape

Table 3.

Multi-leaf shapes

Table 4.

Slender shapes

Table 5.

Shapes with deep depressions

If the same shapes are ranked with respect to γ, a different ranking 1, 2, 9, 6, 13, 15, 7, 8, 16, 11, 4, 3, 22, 12, 20, 17, 14, 10, 18, 19, 24, 5, 21, 23 is obtained. Especially for multi-leaf shapes No. 3, No. 4, No. 7, No. 11, and No. 15, as shown in Table 3, the ranking becomes 15, 7, 11, 4, 3, which does not conform to the common perception when the shape has more leaves, its shape factor should be smaller since more leaves lead to the increase of the perimeter. Furthermore, γ is very sensitive to the change of the perimeter due to the presence of deep depressions in cross-sectional shapes (e.g., No. 5, No. 10, No. 21 or No. 23, as shown in Table 5). Such depressions lead to a perimeter increase which consequently leads to a highly measured γ. Therefore, using ${CV}_{r^{2}}$ to characterize shapes with deep depressions seems to be a rational choice.

Since we are dealing with digital images, the robustness of a computed value for the shape factor under different levels of image noises should be taken into account. Table 6 illustrates the robustness of

{CV}_{r^{2}}

and γ under different levels of noises. All four presented shapes have very similar measured

{CV}_{r^{2}}

even though the third and the fourth shapes have higher noise levels. However, γ can only cope with small levels of noise. Indeed, the last shape in Table 6 has approximately 3.9 times higher measured shape factor than the first shape. This indicates the computations of perimeters under noises may cause unwanted errors if taking measurement γ as the benchmark.

Table 6.

${CV}_{r^{2}}$ and γ of fiber cross-sections with different levels of noises

Conclusions

In this paper, we present a new measure ${CV}_{r^{2}}$ of profiled fiber shape factor by extracting the distance-versus-angle function of the fiber cross-section for both convex and concave shapes. The mathematical meaning of ${CV}_{r^{2}}$ is defined as the coefficient of variation of area sequence in the cross-sectional plane at each angular interval. It is invariant to translation, scaling, and rotation. The capacity of ${CV}_{r^{2}}$ as a measure of fiber shape factor is verified by basic shapes and practical examples. The experimental results show that ${CV}_{r^{2}}$ can distinguish the degree of discrepancy between fiber cross-section and its area equivalent circle. Compared with γ, ${CV}_{r^{2}}$ performs better in the case of cross-sectional shapes with deep depressions. Furthermore, ${CV}_{r^{2}}$ is robust with respect to noise.

Footnotes

Funding

This work was supported by the Natural Science Foundation of China (grant No. 60973072); the Fundamental Research Funds for the Central Universities (grant No. 2011D10102); and Foundation of Donghua University for Ph.D. candidates (grant No. BC201012).

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A new shape factor measure for characterizing the cross-section of profiled fiber

Abstract

Keywords

Introduction

New measure of fiber shape factor

Calculating parameters of equivalent circle area of fiber cross-sections

Generating the distance-versus-angle function of the fiber cross-section

Equiangular sampling and derivation of a new measure CVr2 of fiber shape factor

Boundary representation of concave cross-sections with deep depressions

Experiments illustrating CVr2 behavior

Conclusions

Footnotes

Funding

References

Equiangular sampling and derivation of a new measure CV_r2 of fiber shape factor

Experiments illustrating CV_r2 behavior