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Fabric Surface Roughness Evaluation Using Wavelet-Fractal Method Part I: Wrinkle, Smoothness and Seam Pucker TAE JIN KANG1, SOO CHANG KIM, IN HWAN SUL, JAE RYOUN YOUN,

AND

KWANSOO CHUNG

Intelligent Textile System Research Center, School of Materials Science and Engineering, Seoul National University, Seoul, Korea 151-744 ABSTRACT A wavelet-fractal method to calculate the fractal dimension is proposed to objectively evaluate the surface roughness of fabric wrinkle, smoothness appearance and seam pucker. The proposed method was validated using the fractal surfaces produced from the mathematical functions and compared with the box and cube counting methods. The more accurate three-dimensional mesh grid data points of wrinkle replicas, smoothness appearance replicas and seam pucker samples were obtained using a three-dimensional, noncontact scanning system. As a supplementary reference the standard roughness parameters, which differentiate the degree of fabric surface roughness, were also investigated. The results show that the fractal dimension measured by the wavelet-fractal method as well as the surface average mean curvature show the power to clearly discern the grades of wrinkle, smoothness appearance as well as seam pucker, and thus can evaluate the fabric surface roughness objectively and quantitatively

Fabric appearance properties such as wrinkle, smoothness appearance and seam pucker are important factors for quality control during manufacturing as well as aesthetic aspects for consumer choice. Therefore both fabric and garment manufacturers have made considerable efforts to control the fabric roughness and to establish a test method to quantify roughness. The accurate measurement of fabric surface roughness will contribute to determining the optimum processing conditions to improve the dimensional stability of fabric properties. The evaluation method of fabric surface properties has been based on subjectively comparing the specimen with either the standard replica [2, 3] or a photographic replica [1] by well-trained observers. A considerable amount of work has been done [4, 10, 14, 18] by many researchers to precisely evaluate fabric surface roughness. In our earlier work [6, 7], we extracted the fractal dimension to describe the degree of fabric surface roughness from three-dimensional (3-D) surface data using a laser scanning method or stereo vision technique. The first critical step in grading the fabric surface properties is obtaining the 3-D coordinates of the fabric surface precisely and calculating the fractal dimension of it. With the rapid development of 3-D surface measure1 To whom correspondence [email protected]

should

be

addressed:

e-mail:

ment technology, more accurate and finer 3-D surface data can be obtained. Using this system, it is expected that the fabric surface properties can be fully evaluated with greater accuracy. We adopted a 3-D non-contact scanning system to acquire 3-D data of the fabric surface of interest. The accurate calculation of the fractal dimension to represent the extent of the fabric surface roughness which, in our previous research, was evaluated by the application of an appropriate algorithm, remains a significant issue. Wavelet analysis has been a developing subject in recent years, especially in the field of surface metrology. Wang et al. [15, 16] and Xiong et al. [17] proposed a wavelet transform method as a means to calculate the fractal dimension of surface profiles. It is based on the fact that the wavelet transform method has the unique ability to characterize scale-invariant and space-invariant physical phenomena, which are relevant to the concepts of self-similarity and self-affinity, and form the essence of fractal geometry. These methods do not deal with 3-D surface coordinates by means of twodimensional (2-D) discrete wavelet transform (DWT) for one time but with 2-D profile data by means of a onedimensional DWT used repeatedly. In this study, to achieve objective and accurate evaluation of fabric surface roughness, a wavelet-fractal method using a 2-D profile was applied to the calculation

Textile Res. J. 75(11), 751–760 (2005) DOI: 10.1177/0040517505058855

© 2005 SAGE Publications

www.sagepublications.com

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of the fractal dimension of the fabric surface. In addition, the method to measure the fractal dimension from 3-D surface coordinates using a 2-D DWT (3DSWTM) was devised and validated by calculating the fractal dimensions of the fractal surfaces generated by three mathematical methods. The standard roughness parameters that are used frequently in practice were also introduced as a supplementary reference to rate the fabric surface roughness grade. The box-counting method, cube-counting method and DWT method using 2-D profile data (2DPWTM) were also compared with the 3DSWTM. Using a 3-D noncontact scanning system, we obtained the accurate 3-D coordinates of AATCC wrinkle replicas, smoothness appearance replicas and seam pucker samples that were made to simulate the fabric seam puckers with 3-D shapes. Then we evaluated the fractal dimension of surface using the method proposed herein from 3-D surface data. The results show that the waveletfractal method using much finer 3-D coordinates was more effective for the computation of the fractal dimension of the fabric surface and thus evaluated the degree of fabric surface roughness with greater accuracy.

Methods SYSTEM

AND

SAMPLES

The 3-D scanning system (Rexcan 460; Solutionix) was used to scan the rough fabric surface. It is equipped with a slit beam projector and sensor camera. When the specimen is laid on the table, slit beams of various widths are overlapped on it and the 3-D coordinates are measured from the differences of the beamed images. The number of scanned data points varies according to the area selected and the maximum number of data points was set to 1,450,000 points. We measured the topological surface shape of standard wrinkle replicas with five ratings, smoothness appearance replicas with six ratings and simulated seam pucker samples with five ratings. Unlike the fabric wrinkle and smoothness appearance, the seam pucker does not have a plastic standard replica but a photographic seam pucker replica. Therefore, double and single needle seam pucker samples of different grades from 1 to 5 that had been subjectively graded by human eye evaluation were used. Five specimens of seam puckers were prepared for each grade. The size of seam pucker samples was 300 mm ⫻ 80 mm. The number of scanned data points was about 180 000 for wrinkle standard replicas, 400 000 for smoothness appearance replicas and 100 000 for seam pucker samples. Computerized software to display the fabric surface topology from the points cloud

and to calculate the fractal dimension and surface roughness parameters was based on Matlab and C⫹⫹ Builder software. Figure 1 displays solid models of the wrinkle replicas, smoothness appearance replicas and seam pucker samples generated from 3-D scanned data points. In order to validate the precision and advantages of the wavelet-fractal method it should be compared with the existing box-counting method and cube-counting method [9] for well-known fractal surfaces with different theoretical fractal dimensions. Three-dimensional surface models can be constructed using the Weierstrass–Mandelbrot function [13, 15] and fractal Brownian motion generated with two different methods [12]; namely the midpoint displacement method and the interpolation method. A validated waveletfractal method then can be applied to calculate the fractal dimension of the fabric surface. When the surface is scanned by the 3-D scanning system, a large number of 3-D coordinates are obtained and a lot of time would be required to process all these data points. Moreover, it is difficult to apply the boxcounting method or the wavelet-fractal method using the 2-D profile of the points cloud. Therefore the values at the grid points of a uniformly spaced rectangular mesh superimposed over the points cloud were estimated from the nearest-neighbor interpolation. The construction of the mesh grid made the procedure faster and simpler. We constructed a 256 ⫻ 256 mesh grid (65 536 data points) for the purpose of computational efficiency and the application of the wavelet-fractal method which needs the number of data points to be equivalent to a power of 2 and a sufficient number of decomposition levels for a linear regression. The mesh grid data also have noise and this has to be filtered to reduce it. The noise might be attributed to the discontinuities of the mesh grid data and the very low wavelength (high frequency) range variation that has to be removed. The 2-D finite impulse response (FIR) filter was designed to smooth the mesh grid data and effectively eliminate the high frequency components. The filtered data are finally used to calculate the fractal dimension by using the wavelet-fractal method validated before as well as surface roughness parameters as a supplementary reference. Figure 2 illustrates the overall procedures for evaluating the fabric surface roughness using the wavelet-fractal method and surface roughness parameters. FILTERING The 2-D finite impulse response (FIR) filter is employed to smooth the mesh grid data having discontinuities and remove some noises caused by the variation of very high frequency range. The 2-D FIR filter is designed

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FIGURE 1. Solid models generated from scanned data points: (w1)–(w5) wrinkle replicas, (s1)–(s5) smoothness appearance replicas, (ssp1)–(ssp5) prepared single needle seam pucker samples, (dsp1)–(dsp5) prepared double needle seam pucker samples.

using the frequency sampling method, which contains the desired frequency response sampled at equally spaced points along the x and y frequency axes. The FIR filter can be implemented as a convolution of the form: ˆf 共 x,y兲 ⫽

冘 冘 h共i, j兲 f共 x ⫺ i,y ⫺ j兲

,

i 僆W j 僆W

where h(i, j) is the 2-D FIR filter, f(x, y) is the mesh grid data and W ⫽ {⫺M ⱕ i, j ⱕ M}. h(i, j), the frequency response of which is shown in Figure 3. The filter is a rotationally symmetric Gaussian low-pass filter of size 15 with standard deviation 3 and cutoff 0.6.

WAVELET TRANSFORM The discrete wavelet transform (WT) is defined as a decomposition of the original signal f filtered by a lowpass filter {hn} and a high-pass filter {gn}, with downsampling by a factor of two [5, 8]. In most practical applications, one performs the transform using the following recursive equations.

冘c ⫽ 冘d

c j,n ⫽

j⫺1,k

h k⫺2n

,

j⫺1,k

g k⫺2n

,

k 僆Z

d j,n

k 僆Z

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TEXTILE RESEARCH JOURNAL and three detailed parts, which represent the information of the horizontal, vertical and diagonal directions of the data. In the 2-D case, let the 2-D function be f(x, y) c 0,n1,n2 ⫽ f共n 1 T,n 2 T兲

,

where f(n1T, n2T)(n1 , n2 ⫽ 1,2 , . . . , M respectively) are discrete sampling data of a surface f(x, y), M is the number of data points and T is sampling distance. T is 1 since we use discrete mesh grid data. For 1 ⱕ j ⱕ J, c j,n1,n2 ⫽

冘 冘 冘 冘

c j⫺1,k1,k2 h k1⫺2n1 h k2⫺2n2

,

c j⫺1,k1,k2 h k1⫺2n1 g k2⫺2n2

,

c j⫺1,k1,k2 g k1⫺2n1 h k2⫺2n2

,

c j⫺1,k1,k2 g k1⫺2n1 g k2⫺2n2

,

k1,k2僆Z h d j,n ⫽ 1 ,n2

k1 ,k2僆Z v d j,n ⫽ 1 ,n2

k1 ,k2僆Z d d j,n ⫽ 1 ,n2

k1 ,k2僆Z

FIGURE 2. Overall procedures to evaluate the fabric surface roughness.

where cj,n and dj,n are approximating and detail coefficients at scale j, respectively. The 1-D multi-resolution wavelet decomposition can be easily extended to 2-D by introducing separable 2-D scaling and wavelet functions as the tensor products of their 1-D complements. The 2-D wavelet analysis operation produces one smooth part, which represents the coarse approximation of data,

where cj,n 1 ,n 2 are smooth components and dj,n 1 ,n 2 are detail components in direction corresponding to superscript at decomposition level j. h, v, and d mean horizontal, vertical, and diagonal directions, respectively. WAVELET FRACTAL METHOD Discrete Wavelet Transform Method Using 2-D Profile Data (2DPWTM) If ␺j,n (x) is a Daubechies wavelet base with a vanishing moment 2, and the continuous and bounded function

FIGURE 3. The frequency response of 2-D FIR filter of size 15 with standard deviation 3 and cutoff 0.6.

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f共 x兲 僆 C␣ (␣ is the exponent Ho¨lder continuous complex space), then the wavelet transform spectra {dj,n} have an inequality 兩d j,n兩 ⱕ E2 ⫺j共␣⫹0.5兲

scale, the linear regression with a slope ⫺ (␣ ⫹ 1) can be obtained using a least-square fit algorithm. Therefore, the fractal dimension of the surface is D ⫽ 3 ⫺ ␣ ⫽ 3 ⫹ 共slope ⫹ 1兲 ⫽ 4 ⫹ slope

,

where j ⫽ J, J ⫺ 1, . . . , 1; n ⫽ 1, 2, . . . , 2 (N is the power size of 2 of a discrete signal); 0 ⬍ ␣ ⬍ 1 is Lipschitz exponent; E is a constant. The fractal dimension of function f(x) is

.

N⫺j

D⫽2⫺␣

.

Let Emin ⱕ E, the first norms of {dj,n} have the equation d* j ⫽ E min2⫺j共 ␣ ⫹0.5兲

冘 兩d

,

2 N⫺j

j,n

where d*j ⫽

n⫽1

2N⫺j

兩 关15, 16兴 .

Practically, when d*j is plotted against j on a double logarithmic scale, the linear regression with a slope ⫺(␣ ⫹ 0.5) can be obtained using a least-square fit algorithm. Therefore the fractal dimension D of a 2-D profile is D ⫽ 2.5 ⫹ slope

.

Discrete Wavelet Transform Method using 3-D Surface Data (3DSWTM) So far, relatively little research has been reported on calculating the fractal dimension from 3-D surface data by using a 2-D DWT. The 3DSWTM is more computationally effective than the 2DPWTM. This is due to the fact that for 2DPWTM the wavelet transform is applied to all the 2-D profiles of the x- and y-directions and then the least square fit algorithm is repeatedly implemented. As the characteristics of the fabric surface cannot be completely explained using only the 2-D profiles of either direction, it is necessary to comprehensively consider all the data points of the surface to extract the parameters to describe the surface properties. Therefore the 3DSWTM based on the 2DPWTM introduced above and the variation method for the fractional Brownian motion images [11] is proposed.

冘 冘 兩d⬘

FRACTAL SURFACES To confirm the possibility of the 2DPWTM application to fabric surfaces and to validate the 3DSWTM, 3-D surface models were obtained using the Weierstrass– Mandelbrot function and two different types of fractional Brownian motion. The Weierstrass–Mandelbrot function (WMF) [13, 15] is continuous, self-affine and never smooth and so it can be used to simulate the deterministic rough surfaces, which exhibit a statistical resemblance to real surfaces. Fractional Brownian motion (FBM) [12] is also very important for the study of rough surfaces because its function is statistically self-affine. Two methods that deal with FBM, namely the midpoint displacement method (FBMmd) and the interpolated method (FBMi), were used here. The size of the generated surface data points was 256 ⫻ 256. Figure 4 shows the examples of theoretical fractal surfaces generated by the WMF, FBMmd, and FBMi as range images, in which gray scale equals elevation. SURFACE ROUGHNESS PARAMETERS The classical roughness parameters can be obtained from 3-D mesh data points with higher accuracy. These roughness parameters were used as a supplementary reference to help to estimate the degree of fabric surface roughness. The mean absolute deviation, standard deviation, mean height of peaks, and surface average mean curvature were measured in this study. The mean absolute deviation (MAD) is defined as MAD ⫽

d⬘j ⫽

n1⫽1 n2⫽1

dhj

2 N⫺j 䡠 2 N⫺j

SD ⫽

兩 ⫽ E *min2⫺j共 ␣ ⫹1兲

t ⫽ h,v,d

, dvj

where represents the horizontal information, the vertical information, d dj the diagonal information of a 3-D surface and E*min is a constant. Likewise, when dtj (t ⫽ h, v, d) is plotted against j on a double logarithmic

冘 冘 兩f i

i, j

⫺ ៮f 兩

,

j

where N 2 is the total number of mesh data points and ៮f is the average value of surface heights. The standard deviation value (SD) is computed by

2N⫺j 2N⫺j

j,n1,n2

1 N2

冑

1 N2

冘 冘 共f i

i, j

⫺ ៮f 兲 2

.

j

The mean height of peaks (MHP) is calculated as the average of the deviations above the reference value ៮f. MHP ⫽

1 N2

冘冘P i

j

i, j

,

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FIGURE 4. Range images for fractal surfaces generated with (a)–(d): WMF, (e)–(h): FBMmd and (i)–(l): FBMi : (a), (e), (i) D ⫽ 2.1; (b), (f), (j) D ⫽ 2.3; (c), (g), (k) D ⫽ 2.5; (d), (h), (l) D ⫽ 2.7.

where, Pi, j ⫽

再

fi, j ⫺ ៮f, if fi, j ⫺ ៮f ⬎ 0 0, otherwise

.

The surface average mean curvature (SAMC) is given by SAMC ⫽

1 N2

冘冘H i

2 i, j

,

j

where, H(i,j) is the mean curvature at each point of a surface.

Results and Discussion The 2DPWTM and 3DSWTM were confirmed and validated based on three types of fractal surfaces with different theoretical fractal dimensions and then compared with the box-counting method (BCM) and cubecounting method (CCM). For each surface generation model, 10 fractal surfaces with the fractal dimension 2.1, 2.3, 2.5, 2.7 and 2.9 were generated, respectively. The

results are the average values for 10 fractal surfaces with the same theoretical fractal dimensions. The wavelet transform is performed to the discrete data of the size 256 ⫻ 256, which are symmetrically extended in order to eliminate the boundary effect. The fractal dimensions of both x and y directions are averaged and added to 1 in the BCM and the 2DPWTM. In the 3DSWTM, the averages of absolute wavelet coefficients dhj and dvj , which represent the information of horizontal and vertical directions, respectively, are fitted to the decomposition levels by the least-square fit algorithm. Two fractal dimensions calculated using the horizontal and vertical information are averaged to obtain a single value, the so-called 3DSWTMhv. The information for the diagonal direction is excluded from calculating the fractal dimension to avoid overlapping parts with the horizontal and vertical ones. Figure 5 shows the calculated fractal dimensions to fractal surfaces produced with the WMF, FBMmd, and FBMi. The rougher surfaces correspond to the larger

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757 fractal dimensions. BCM and CCM have measured similar results with theoretical fractal dimension within specific narrow range, but showed a large deviation outside that range. In other words, BCM and CCM obtain higher fractal dimensions in the range of relatively small theoretical fractal dimensions but lower fractal dimensions in the range of relatively large theoretical fractal dimensions. The precision of BCM and CCM is not as good as the two wavelet-fractal methods, 2DPWTM and 3DSWTMhv, for fractal surfaces. Two wavelet-fractal methods show perfect linearity in a wide span of the fractal dimension. The results of the 3DSWTMhv are a little higher than those of the 2DPWTM, but the calculated errors are not very big. Therefore, it is expected that 2DPWTM and 3DSWTMhv are capable of evaluating the fabric surface roughness with higher accuracy than BCM and CCM. The fractal dimensions of wrinkle and smoothness appearance replicas were repeatedly measured five times for the same region of interest and averaged (Figures 6 and 7). Figure 8 shows the estimated fractal dimensions of single and double seam pucker samples for which grades were subjectively assigned in comparison with a photographic seam pucker replica. Table I lists the means and standard deviations from measuring the fractal dimensions with 2DPWTM and 3DSWTMhv. The deviation of the measurements for the same region of interest is of the order of 10⫺3 and it can denote the system reproducibility. The estimated fractal dimensions consistently decrease as the grade of the standard replicas or samples increase as expected. The fractal dimensions show higher value than the previous work [6, 7] since more data with greater precision are used to calculate the fractal dimension. As the degree of surface roughness

FIGURE 5. Application of various fractal dimension calculating methods to fractal surfaces produced with the (a) WMF, (b) FBMmd and (c) FBMi. FIGURE 6. Calculated fractal dimensions of AATCC wrinkle replicas.

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FIGURE 7. Calculated fractal dimensions of AATCC smoothness appearance replicas.

was relatively low for a fabric surface, the measured fractal dimensions by the BCM and CCM were generally higher than those of the wavelet-fractal methods. As for the results for generated fractal surfaces, the fractal dimensions estimated by the CCM and 3DSWTMhv were a little higher than those estimated by the BCM and 2DPWTM, respectively. Smoothness appearance replicas 3, 3.5, 4 require considerable effort to distinguish subtle differences by visual assessment and so observers may make different visual smoothness ratings even for the standard replicas, which fall into these categories. The BCM and CCM also cannot correctly differentiate smoothness grade 3 and 3.5 in terms of fractal dimension (Figure 7). But the fractal dimensions calculated by the 2DPWTM and 3DSWTMhv consistently decrease as the smoothness appearance grade increases. The results show that the wavelet fractal method is applicable to the evaluation of the extent of fabric surface roughness with added accuracy.

FIGURE 8. Calculated fractal dimensions of (a) single needle seam pucker samples and (b) double needle seam pucker samples.

TABLE I. Means (m) and standard deviations (␴) of measured fractal dimensions with 2DPWTM and 3DSWTMhv. Seam pucker samples Wrinkle replicas 2DPWTM

1 2 3 3.5 4 5

Smoothness replicas

3DSWTMhv

2DPWTM

Single needle

3DSWTMhv

2DPWTM

Double needle

3DSWTMhv

2DPWTM

3DSWTMhv

m

␴

m

␴

m

␴

m

␴

m

␴

m

␴

m

␴

m

␴

2.207 2.136 2.103 — 2.055 2.020

0.002 0.002 0.005 — 0.003 0.005

2.298 2.210 2.183 — 2.132 2.050

0.002 0.002 0.004 — 0.002 0.003

2.485 2.323 2.274 2.173 2.093 2.031

0.004 0.002 0.004 0.005 0.002 0.003

2.559 2.338 2.280 2.228 2.119 2.052

0.003 0.002 0.003 0.004 0.002 0.004

2.237 2.204 2.192 — 2.173 2.100

0.013 0.016 0.012 — 0.014 0.019

2.365 2.274 2.261 — 2.254 2.124

0.014 0.026 0.014 — 0.007 0.016

2.317 2.245 2.239 — 2.195 2.021

0.022 0.004 0.012 — 0.019 0.017

2.390 2.360 2.306 — 2.251 2.045

0.029 0.011 0.014 — 0.018 0.015

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FIGURE 9. Normalized surface roughness parameters for (a) AATCC wrinkle replicas, (b) AATCC smoothness appearance replicas, (c) prepared single needle seam pucker samples, (d) prepared double needle seam pucker samples.

The classical surface roughness parameters were normalized with a view to making its discriminatory power more visible as shown in Figure 9. The surface roughness parameters as a supplementary analytical tool also show the tendency to decrease as the wrinkle, smoothness appearance and seam pucker grade increase. Note that the MAD, SD and MPH values of smoothness appearance grade 2 are higher than those of grade 1. On the other hand, the SAMC values of smoothness appearance as well as wrinkle and seam pucker show a consistent and relatively linear decrease with an increase in grade. It is highly probable that the SAMC value is the most reliable supplementary parameter among surface roughness parameters for evaluating the fabric surface roughness. It stands to reason that the fractal dimension calculated by the wavelet-fractal method can be used to objectively judge the degree of fabric surface roughness, together with the SAMC value.

Conclusions In this study we have proposed the wavelet-fractal method to accurately calculate the fractal dimension as a descriptor of fabric surface roughness. The wavelet-fractal method was validated based on the fractal surfaces generated with the WMF, FBMmd, and FBMi. Its precision was higher than the box-counting method and cubecounting method. The more precise 3-D coordinates of wrinkle replicas, smoothness appearance replicas and seam pucker samples have been obtained using a 3-D

noncontact scanning system. We found that the fractal dimensions measured by the wavelet-fractal method have the ability to distinguish the differences of the degree of wrinkle, smoothness, and seam pucker with added accuracy. The SAMC value can be also used as a supplementary parameter for assessment of fabric surface roughness. ACKNOWLEDGMENT This work was supported by the SRC/ERC program of MOST/KOSEF (R11-2005-065).

Literature Cited 1. AATCC Test Methods 88B-1996, Smoothness of Seams in Fabrics after Repeated Home Laundering, AATCC Technical Manual, 75, 112–115 (2000). 2. AATCC Test Methods 124-1996, Appearance of Fabrics after Repeated Home Laundering, AATCC Technical Manual, 75, 205–208 (2000). 3. AATCC Test Methods 128-1999, Wrinkle Recovery of Fabrics: Appearance Method, AATCC Technical Manual, 75, 213–214 (2000). 4. Amirbayat, J., and Alagha, M. J., Objective Assessment of Wrinkle Recovery by Means of Laser Triangulation, J. Textile Inst. 87, 349 –355 (1996). 5. Daubechies, I., Orthonormal Bases of Compactly Supported Wavelets, Comm. Pure Appl. Math. 41, 909 –996 (1988). 6. Kang, T. J., Cho, D. H., and Kim, S. M., New Objective

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13. Russ., J. C., “Fractal Surfaces,” Plenum Press, New York, 1994. 14. Stylios, G., and Sotomi, J. O., Investigation of Seam Pucker in Lightweight Synthetic Fabrics as an Aesthetic Property: Parts I, II, J. Textile Inst. 84, 593– 610 (1993). 15. Wang, A. L., Yang, C. X., and Yuan, X. G., Evaluation of the Wavelet Transform Method for Machined Surface Topography Part I: Methodology Validation, Tribology Intl. 36, 517–526 (2003). 16. Wang, A.L., Yang, C.X., and Yuan, X.G., Evaluation of the Wavelet Transform Method for Machined Surface Topography Part II: Fractal Characteristic Analysis, Tribology Intl. 36, 527–535 (2003). 17. Xiong, F., Jiang, X. Q., Gao, Y., and Li, Z., Evaluation of Engineering Surfaces Using a Combined Fractal Modeling and Wavelet Analysis Method, Int. J. Mach. Tools Manufact. 41, 2187–2193 (2001). 18. Xu, B., and Reed, J. A., Instrumental Evaluation of Fabric Wrinkle Recovery, J. Textile Inst. 86, 129 –135 (1995).