**9. Analysis in spatial domains**

A basic statistical feature of *R(d)* is autocorrelation between distances. Autocorrelation depends on the lag *h* (i.e. selected distances between places of force evaluation). The main characteristics of autocorrelation is covariance function *C(h)*

$$\mathbf{C}(h) = \text{cov}(R(d), R(d+h)) = E(\left(R(d) - E(R(d)) \left(R(d+h) - E(R(d))\right)\right) \tag{50}$$

and autocorrelation function ACF(h) defined as normalized version of C(h).

$$ACF(h) = \frac{\text{cov}\{R(0) \ R(h)\}}{v} = \frac{c(h)}{c(0)}\tag{31}$$

*ACF* is one of main characteristics for detection of short and long-range dependencies in dynamic series. It could be used for preliminary inspection of data.

Spectral density function is therefore generally useful for evaluation of hidden periodicities. The statistical geometry of an isotropic random Gaussian surface could be expressed in the terms of the moment of power spectral function called spectral moments (Zhang

> *H <sup>k</sup> m gd <sup>k</sup>*

The *mo* is equal to the variance oh heights and *m2* is equal to the variance of slopes between

integration of the spectrogram. These frequency bound can be converted to the wavelength

The roughness *Rq* = SD (standard deviation) is simply *Rq m* <sup>0</sup> and the density of summits

A basic statistical feature of *R(d)* is autocorrelation between distances. Autocorrelation depends on the lag *h* (i.e. selected distances between places of force evaluation). The main

> cov( (0) ( )) ( ) ( ) (0) *R Rh ch ACF h*

*ACF* is one of main characteristics for detection of short and long-range dependencies in

*<sup>m</sup> DS m*

4 <sup>2</sup> 6 3

*Ch Rd Rd h E Rd ERd Rd h ERd* ( ) cov( ( ), ( )) (( ( ) ( ( )) ( ( ) ( ( )))) (30)

 

 

. For rough SHV from fig. 4, the selected spectral moments have the following

(28)

(29)

*v c* (31)

*<sup>L</sup>* are high and low frequency bounds of

and the short wavelength limit is

 

*H* and

Gopalakrishnan, 1996).

*l*

*L L* 2 / 

is defined as

values:

bound frequencies. The frequencies

 zero moment = 0.0006234. first moment = 0.2755. second moment = 0.0865. fourth moment = 6.387e-006.

 spectral variance = 0.010637. spectral skewness = -0.0006368. spectral kurtoisis = 0.000342.

**9. Analysis in spatial domains** 

() *<sup>L</sup>*

limits. The long wavelength limit is *lH H* 2 /

Corresponding spectral statistical characteristics are:

characteristics of autocorrelation is covariance function *C(h)*

and autocorrelation function ACF(h) defined as normalized version of C(h).

dynamic series. It could be used for preliminary inspection of data.

The computation of sample autocorrelation directly from definition for large data is tedious. The spectral density is the Fourier transform of covariance function *C(h)*

$$\mathbf{g}(o) = \frac{1}{2\pi} \int\_0^\infty \mathbf{C}(t) \, \exp(-i \; o \; t) dt \tag{32}$$

The ACF is inverse Fourier transform of spectral density.

$$\text{ACF}(h) = \bigcap\_{\alpha=0}^{\infty} \text{S}(\alpha) \text{ exp}(i \text{ o } h) d o \tag{33}$$

These relations show that characteristics in the space and frequency domain are interchangeable.

For rough SHV from fig. 4 is *ACF* till lag 320 component in the fig. 31.

Fig. 31. ACF for twill fabric

The dashed lines in fig. 31 are the approximate 95% confidence limits of the autocorrelation function of an IID process of the same length. Sample autocorrelations lying outside the 95% confidence intervals of an IID process are marked by black circles. The slow decrease of ACF for large lags indicates long-range correlation, which may be due to non-stationarity and/or dynamic non-linearity.

In spatial statistics variogram is more frequent (Kulatilake et. al, 1998) which is defined as one half variance of differences (*R*(d) - *R*(d+h))

$$I\,\,^r(h) = 0.5\,\,D\,[R(d) - R(d+h)]\,\,\,\,\tag{34}$$

The variogram is relatively simpler to calculate and assumes a weaker model of statistical stationarity, than the power spectrum. Several estimators have been suggested for the variogram. The traditional estimator is

Woven Fabrics Surface Quantification 149

The surface moments play central role in description of surfaces topography. The parameters m2,0 and m0,2, which are the 2nd surface spectral moments, denote the variance of slope in two vertical directions along cross direction and machine direction. The parameter m1,1 represents the association-variance of slope in these two directions. These parameters are generally dependent of the selected coordinate system. From the known

moments m2,0, m1,1 and m0,2 by using of linear regression. The maxima and minima of eqn.

*<sup>p</sup>* called principal direction given by relation

2

As one measure of anisotropy the so-called long-crestedness 1/g has been proposed

*m*

For an isotropic surface is *g =* 1 and for degenerated one-dimensional spectrum *g* = 0. The

2\* \*

For *AN* = 0 surface perfectly is isotropic and for *AN* = 1 surface is anisotropic. Lower *AN* 

For investigation of surface roughness anisotropy the twill fabric (see fig. 3) and Krull fabric were selected. The *R(d)* traces have been obtained by means of KES apparatus in the

The surface moments m2,0, m0,2 and m1,1 were computed from eqn. (37) by using of linear least squares regression. Estimated surface moments and AN anisotropy measure are given

The proposed technique is capable to estimate roughness characteristics of anisotropic surfaces

*m m m*

> 2min 2max

2,0 0,2 1,1 2,0 0,2

*mm m*

*m m* 

p

*g*

better criterion of anisotropy has been proposed in the form (Thomas et. al, 1999)

1

Kawabata SMD of individual profiles of twill fabric at chosen directions

typical for textile structures. Beside the anisotropy measure *AN* the direction

m2max will be probably necessary for deeper description of textiles surface roughness.

*AN*

characteristic indicates low degree of anisotropy.

polar graph in fig. 32 and for Krull fabric in fig. 33.

tan

2 2,0 1,1 0,2 *mm m m* ( ) cos 2 cos sin sin

 

*<sup>i</sup>* for selected set of directions

values of 2( ) *m*

(37) are

These occur in the angle

following directions:

in the table 1.

(Longuet-Higgins, 1957), where

2 2

2 max 2min 2,0 0,2 2,0 0,2 1,1 ( , ) 0,5 \* [( ) {( ) 4 }] *m m mm mm m* (40)

1,1

2,0 0,2

 

*<sup>i</sup>* i = 1,..n it is possible to estimate the surface

2 2

(41)

*<sup>m</sup>* (42)

(43)

*<sup>í</sup>* is plotted as

*<sup>p</sup>* and values of

2

*<sup>í</sup>* 0o (weft direction), 30o, 45o, 60o and 90o (warp direction). The

(39)

 

$$\text{G(}h\text{)} = \frac{1}{2M\text{(}h\text{)}} \sum\_{j=1}^{M(h\text{)}} \left(\text{R}(d\_j) - \text{R}(d\_{j+h})\right)^2 \tag{35}$$

where *M*(h) is the number of pairs of observations separated by lag *h*. Problems of bias in this estimate when the stationarity hypothesis becomes locally invalid have led to the proposal of more robust estimators. It can be summarized that simple statistical characteristics are able to identify the periodicities in data but the reconstruction of "clean" dependence is more complicated.
