Quantitative analysis of fiber composite

microstructure

by: Jan Gajdošík

Thesis advisor: Doc. Ing. Michal Šejnoha Ph.D.

Ing. Jan Zeman Ph.D.

CTU in Prague

2004

2

Table of contents

QUANTITATIVE ANALYSIS OF FIBER COMPOSITE MICROSTRUCTURE 1

TABLE OF CONTENTS 2

ABSTRACT 4

ABSTRAKT 4

1. INTRODUCTION TO PROBLEM OF MICROSTRUCTURE ANALYSIS 5

2. DESCRIPTION OF MICROSTRUCTURE 7

2.1. Quantification of microstructure morphology 7

2.2. Basic concepts and hypotheses 8 2.2.1. Concept of an ensemble 9 2.2.2. Ergodic hypothesis 9 2.2.3. Statistical homogenity 10 2.2.4. Statistical isotropy 10

2.3. Microstructure description 11 2.3.1. n-point probability functions 11 2.3.2. Lineal path function 13

2.4. Numerical evaluation of microstructural statistics 15 2.4.1. n-point probability functions 15 2.4.2. Lineal path function 17

3. TAKING AND MODIFYING IMAGES 19

3.1. Application of image analysis in composite material research 19

3.2. C/E composites description 21

3.3. Tresholding 22

3.4. Image analysis of graphite fiber laminate 23 3.4.1. Preparation of samples 23

3

3.4.2. Taking images with camera 23 3.4.3. Taking images with scanner 24 3.4.4. Modification of input using program PHOTOSHOP 25 3.4.5. Modification using program LUCIA G 26

3.5. Conclusion of image analysis 29

4. STATISTICAL DESCRIPTION WITH VARIOUS BOUNDARY

CONDITIONS 30

4.1. Boundary conditions 30

4.2. Evaluation of the two-point probability function using Fast Fourier Transform method 32 4.2.1. Graphs of measures 36

4.3. Evaluation of the two-point probability function using the Monte Carlo Method 40 4.3.1. Description of used algorithm 40 4.3.2. Periodic boundary conditions 42 4.3.3. Plain boundary conditions 44 4.3.4. Mirror boundary conditions 46

4.4. Comparison of periodic boundary conditions with others 47

4.5. Conclusions 50

BIBLIOGRAPHY 52

5. APPENDIX: DESIGN OF OBJECT ORIENTED CODE SERVING FOR

STATISTICAL RESEARCH OF MICROSTRUCTURE 56

5.1. Introduction 56

5.2. General structure 57

5.3. The two-point probability function 58

5.4. Output possibilities 58

4

Abstract

This thesis is concerned with the description of the process of retrieving color images

from real composite systems together with their transformation into binary images. The fiber

composites were used as a real composite, because of its future perspective use. The main part

is dedicated to the comparison of various boundary conditions and processes for obtaining the

two-point probability function with various boundary conditions. The dominant purpose of

this thesis is to determine the differences in three boundary conditions. In particular, the plain

(no condition), mirror and periodic boundary conditions are considered. The possible

computational methods are taken in consideration, too. The speed of evaluation is one of the

most important issues and as such is emphasized.

Abstrakt

V p�edkládané diplomové práci se v�nuji popisu postupu získávání snímk� skute�ného

kompozitu a následného p�evodu snímk� na idealizované bitmapy. Pro tento ú�el byl použit

kompozitní materiál jednosm�rn� vyztužený vlákny a to s ohledem na jeho velmi perspektivní

využití ve stavebnictví. Hlavní �ást mé práce se zabývá srovnáváním r�zných okrajových

podmínek použitých p�i výpo�tu dvoubodové charakteristické funkce a také popisu postupu

získávání dvoubodové pravd�podobnostní funkce s r�znými okrajovými podmínkami.

D�ležitý úkol je ur�it rozdíly mezi r�znými okrajovými podmínkami. Prosté (žádné),

zrcadlové a periodické okrajové podmínky jsou vzaty v úvahu. Možné výpo�etní metody jsou

zmín�ny a popsány. Rychlost výpo�tu je jedním z nejd�ležit�jších faktor� a proto je

zd�razn�na.

5

1. Introduction to problem of microstructure analysis

The goal of this work is an extension of the knowledge in the field of evaluation and

modeling of a microstructure, so-called microanalysis. The microstructure generally consists

of two or more phases with fundamentally different properties. The microstructure description

has already been an objective of many works (see later references). Scientists can usually use

only a part of the section taken through the sample of a given microstructure and then

evaluate the measurable properties of a material (volume fractions, two and three point

probability functions, lineal path function etc.). This process is repeated on many sections in

order to determine average values or other characteristics.

One of the basic tasks is to decide which approach is the most suitable one for

examination of sections close to the sample boundary. In the neighborhood of edges the

images are inaccurate or rather incomplete (“cut” fibers of composite). The method of

investigation in this area that is rather unsure, which leads to further problems. For example,

the two point probability function is defined as a probability of finding two specified points in

specified phases. It is important to determine what to do in the case when one of the selected

points is found outside of the image. Such a pair can be either disposed or the distribution of

microstructure outside of the image can be predicted in some way. It is also very important to

determine the best size of the examined microstructure. In other words to decide, whether it is

comparable to compute two point probability functions of a large ensemble of small samples

and then averaged over the number of samples or rather use a big sample.

The very important factor of a reliable description of the real microstructure is the

preparation of an image representing such a microstructure. Note that samples of

microstructure are usually taken using microstructure sampling and conversion into electronic

form. This is described in the first part of this thesis.

It is worth mentioning that real microstructures usually fall into the category of

random composites. Literature offers many statistical descriptors of random composite

material. One of the most suitable descriptors is n-point probability function Sn defined in

[47], which gives the probability of simultaneous finding of n points randomly thrown into

the composite medium, which are in the same phase. Unfortunately, the determination of

higher order descriptors is computationally very demanding.

The first work dedicated to obtaining the function Sn and others for random

microstructure was presented in Corson’s work [6]. Although the suggested approach was

6

very slow and demanding (it was based on manual evaluation of photographs of samples of

real microstructures) the specified principle (evaluation of Sn values in lattice and subsequent

averaging) was used and expanded in following works. In [3] Corson’s technique was

automatized by image processing technology and finally [41] a simple simulation method for

setting Sn function suitable even for nondigitalized samples was proposed.

The thesis is separated into four main sections. The first one (Description of

microstructure) is devoted to the description of geometrical properties of the microstructure.

The basic descriptors and correlation between them are described in this section. The second

part is concerned with the description of microstructure images processing. The technique for

obtaining quality images of real woven composite (graphite fibers in epoxy matrix) and

following conversion into binary images is also presented. The third section (Statistic

descriptions with various boundary conditions) provides statistical processing of obtained

images. The different boundary conditions are described therein. The conclusion follows.

Appendix (Design of object oriented code serving for statistical research of microstructure) is

dedicated to the description of the developed program.

7

2. Description of microstructure

2.1. Quantification of microstructure morphology

Traditional micromechanical analysis of composite media with disordered

microstructure is typically based on very limited microstructural information such as volume

fraction of individual phases1. However, when an additional knowledge of the real

microstructure is available, the estimates of local fields can be improved by treating random

composites (see, e.g., [21], [22], [23], [45], [46], [50], [51] and references herein). Such a

modeling framework is considered throughout this text.

This opening chapter outlines evaluation of various statistical descriptors, which arise

in the analysis of binary microstructures with random arrangement of individual phases. With

regard to specific applications discussed in the following chapters (analysis of the graphite

fiber tow embedded in the polymer matrix), the background introduced in this chapter is quite

general and can be applied to any two-phase random heterogeneous medium of arbitrary

phase geometry2.

Section 2.2 reviews basic concepts and hypotheses associated with quantification of

microstructure morphology. Individual statistical descriptors used in the present work are

introduced in Section 2.3. The methods of their numerical evaluation are presented in

Section 2.4.

1 Namely, stiffness and compliance averaging method [14], [37], [48], dilute approximation [8],

selfconsistent method [4], [15], differential scheme [34] and Mori-Tanaka method [1], [33] fall into this

category. 2 In the case of particulate composites, a variety of specialized microstructural descriptors can be used

for microstructure characterization, see, e.g, [36], [38], [42], [45], [46], [49] and improved estimates of local

fields [30], [35], [45]. Moreover, we refer a more theoretically oriented reader to [42] for mathematically

rigorous discussion related to subjects of this chapter.

8

2.2. Basic concepts and hypotheses

Motivation. To introduce the subject, imagine a collection of a large number of micrographs

describing the geometry of a two phase fibrous composite. An example of such a micrograph

is displayed in Figure 2.1. Figure 2.1a represents a portion of a graphite–fiber tow containing

approximately twelve thousand fibers. A random cut consisting of about three hundred fibers

is shown in Figure 2.1b. Although having a large number of fibers, one can hardly assume

that such a representative sample can completely describe the morphology of the whole

composite. Simply taking similar micrographs from other parts of the fiber tow indicates

visual difference in the microstructure from sample to sample. At this point, we should

perhaps ask ourselves whether there is a reliable approach in modeling of composite

materials, which permits in some way incorporating elements of real microstructure into the

analysis. The answer is affirmative once we recognize the random nature of geometrical

arrangements of phases and treat random composites – it means that the particular

microstructure of a given part of a fiber tow yields only one possible arrangement of phases.

Therefore, instead of determining the exact value of some quantity at a given point (which is

sample dependent), attention is given to its expected or averaged or macroscopic value, which

incorporates information from all samples taken from a material.

(a) (b)

Figure 2.1: A real micrograph of a transverse plane section of the fiber tow

9

2.2.1. Concept of an ensemble

To reflect a random character of a heterogeneous medium it is convenient to introduce

the concept of an ensemble – a set of a large number of systems which are different in their

microscopic details but they are entirely identical within a point of view of macroscopic scale

(see, e.g., [1], [13], [21], [22], [25], [51]). Random nature of such material systems further

suggests that individual members of the ensemble, to be statistically representative of the

composite, should be sufficiently large compared to the microscopic length scale (e.g., fiber

diameter).

To begin, consider a sample space S defined here as a collection of material samples

similar to one of Figure 2.1b. Formation of S then opens a way to provide an estimate for

effective or expected value of some quantity, say stress or strain field, through the process of

its averaging over all systems in the ensemble. To proceed, identify individual members of

this space by � and define p(�) as the probability density of � in S (see [19], [21], [23], [51]

for further reference). Then, the ensemble average of function F(x, �) at a point x is provided

by

( ) ( ) ( ) .d,, �=S

pFF αααα xx (2.1)

Following the above definition would lead to experimental determination of the

ensemble average of function F(x, �) for a given point x through the cumbersome procedure

of manufacturing a large number of samples (which form the ensemble space S), measuring

F(x, �) for every sample and then its averaging for all samples. Therefore, it appears

meaningful to introduce certain hypotheses regarding the ensemble average, which

substantially simplify this task.

2.2.2. Ergodic hypothesis

This hypothesis demands all states available to an ensemble of the systems to be

available to every member of the system in the ensemble as well [1], [21], [25], [40], [42].

Once this hypothesis is adopted, spatial or volume average of function F(x,�) given by

10

( ) ( ) ,d,1

, � +=V

FV

F yyxx αα (2.2)

which is independent of and identical to the ensemble average for |V| � �, i.e.,

( ) ( ),,, αα xx FF = (2.3)

for all � ∈ S. This hypothesis allows an examination of only one arbitrary member of the

sample space, provided that the sample is “sufficiently large”. A possible way to fulfill this

condition is to assume a periodic composite described by a unit cell Y. Then [40],

( ) ( ) ,d,1

d,1

lim �� +=+∞→ YVV

FY

FV

yyxyyx αα (2.4)

so for the ergodic periodic composite medium, the ensemble average of F(x,�) is equal to the

volume average taken over the unit cell.

2.2.3. Statistical homogenity

Suppose that function F depends on n vectors x1, . . . , xn. If the material is statistically

homogeneous the ensemble average of F is invariant with respect to translation [1], [42], [47],

so the relation

( ) ( ),,...,,..., 11 yxyxxx −−= nn FF (2.5)

holds for an arbitrary value of y. The most common choice is to set y = x1, so

( ) ( ) ( ),,...,,...,,0,..., 1121121 nnn FFF xxxxxxxx =−−= (2.6)

where xij = xj − xi.

2.2.4. Statistical isotropy

Further simplification arises when assuming the material to be statistically isotropic

[1], [42], [47]. In such a case, the ensemble average is not only independent of the position of

the coordinate system origin but also of the coordinate system rotation. Under this hypothesis,

the ensemble average depends on the absolute value of vectors x12, . . . , x1n only:

( ) ( ),,..., 112 ijn rFF =xx (2.7)

where rij = xij, i = 1, . . . , n, j = (i + 1), . . . , n.

11

2.3. Microstructure description

A number of statistical descriptors are available to characterize the microstructure of a

two-phase random medium. This section describes two specific sets of descriptors which

proved to be useful in the present work. First, a set of general n-point probability functions,

applicable to an arbitrary two-phase composite, is introduced. A different statistical function

deserves attention when phase connectivity information is to be captured in more detail. The

lineal path function is discussed as an example of such a descriptor.

Both types of functions are introduced through a fundamental random function

relevant to the microstructure configuration. Then, statistical moments of this function are

identified as descriptors of the microstructure morphology. Finally, similarities and

differences between individual types of statistical descriptors are discussed.

2.3.1. n-point probability functions

Fundamental function and statistical moments. Consider an ensemble of a two-phase

random medium. To provide a general statistical description of such a systems it proves useful

to characterize each member of the ensemble by a stochastic function – characteristic function

�r(x,�), which is equal to one when point x lies in the phase r of the sample and equals to

zero otherwise [1], [42], [47],

( )��

��� ∈

=��

�

� ,if,1,otherwise,0,

ααχ r

r

Dxx (2.8)

where Dr(�) denotes the domain occupied by the r-th phase. Except where noted, composites

consisting of clearly distinguishable continuous matrix phase are considered. Therefore, r =

m, f is further assumed to take values m for the matrix phase while symbol f is reserved for the

second phase. For such a system the characteristic functions �f (x,�) and �m(x,�) are related

by

( ) ( ) 1,, =+ αχαχ xx fm (2.9)

Following [18, 238, 258, 286], we write the ensemble average of the product of

characteristic functions

( ) ( ) ( ),,,,..., 11,..., 11αχαχ nrrnrr nn

S xxxx �= (2.10)

12

where function Sr1,...,rn referred to as the general n-point probability gives the probability of

finding n points x1, . . . , xn randomly thrown into a medium located in the phases r1, . . . , rn.

Functions of the first and second order. Hereafter, we limit our attention to functions of the

order of one and two, since higher-order functions are quite difficult to determine in practice3.

Therefore, description of a random medium will be provided by the one-point probability

function Sr(x)

( ) ( ),,αχ xx rrS = (2.11)

which simply gives the probability of finding the phase r at x and by the two-point probability

function Srs(x1, x2)

( ) ( ) ( ),,,, 2121 αχαχ xxxx srrsS = (2.12)

which denotes the probability of finding simultaneously the phase r at x1 and the phase s at x2.

In general, evaluation of these characteristics may prove to be prohibitively difficult.

Fortunately, a simple method of attack can be adopted when accepting an assumption

regarding the material as statistically homogeneous, so that (compare with Eq. (2.5))

( ) ,rr SS =x (2.13)

( ) ( )., 2121 xxxx −= rsrs SS (2.14)

Further simplification arises when assuming the medium to be statistically isotropic.

Then Srs(x1,x2) reduces to (see also Eq. (2.7))

( ) ( ).2121 xxxx −=− rsrs SS (2.15)

Finally, making an ergodic assumption allows a substitution of the one-point

correlation function by its volume average, i.e., volume concentration or volume fraction of

the r-th phase cr,

.rr cS = (2.16)

Limiting values. In addition, the two-point probability function Srs incorporates the one-point

probability function Sr for certain values of its arguments such that

( ) ( ),,:for 12121 xxxxx rrsrs SS δ== (2.17)

( ) ( ) ( ),,lim:for 21212121

xxxxxxxx srrs SSS =∞→−

→∞− (2.18)

where symbol �rs stands for Kronecker’s delta. Relation (2.17) states that the probability of

finding two different phases at a single point is equal to 0 (see also Eq. (2.9)) or is given by

3 Note, however, that relatively efficient procedures for approximation of higher-order probability

functions for ergodic and statistically isotropic media were recently proposed in [45] and [46].

13

the one-point probability function if phases are identical. Equation (2.18) manifests that for

large distances points x1 and x2 are statistically independent. This relation is often denoted as

the no-long range orders hypothesis (see e.g. [30], [50]).

Finally, according to Eq. (2.9), we may determine one and two-point probability

functions for all phases provided that these functions are given for one arbitrary phase. For

one-point probability function of statistically homogeneous and ergodic medium, this relation

assumes a trivial form

.1 fm cc −= (2.19)

Relations for the two-point probability functions of statistically uniform and ergodic

medium are summarized in Table 2.14.

Known Function

Smm(x) Smf(x) Sff(x)

Smm(x) Smm(x) cm - Smf(x) cm - cf + Sff(x)

Smf(x) cm - Smm(x) Smf(x) cf - Sff(x)

Sff(x) cf - cm + Smm(x) cf - Smf(x) Sff(x)

Table 2.1: Relations among two-point probability functions

2.3.2. Lineal path function

As already noted in the previous section, the determination of probability functions of

order higher than two, encounters serious difficulties, both analytical and numerical5.

However, the importance of these functions for the characterization of morphology and

overall properties of heterogeneous materials is substantial (see, e.g., [8], [31], [32], [45] and

references therein). To overcome this difficulty, one can study low-order microstructural

descriptors based on a more complex fundamental function which contains more detailed

information about phase connectedness and hence certain information about long-range

orders. The lineal path function [29] described in this section is a representative of such

indicators.

4 Note that, by definition (2.12) and assumption of statistical homogeneity, Srs(x) = Ssr(x). 5 See, e.g., [3], [6], [8] for discussion of procedures for determination of third-order probability

functions for statistically isotropic ergodic media.

14

Fundamental function and statistical moments. To maintain a formal similarity with the

discussion of the n-point probability functions, we introduce a random function �r(x1, x2, �)

as

( )��

��� ⊂

=��

�

� ,,if,1,otherwise,0,, 21

21

ααλ r

r

Dxxxx (2.20)

i.e., a function which equals to 1 when the segment x1x2 is contained in the phase r for the

sample and zero otherwise. The lineal path function6, denoting the probability that the x1x2

segment lies in the phase r, then follows directly from the ensemble averaging of this function

[29]

( ) ( ).,,, 2121 αλ xxxx rrL = (2.21)

Under the assumptions of statistical homogeneity and isotropy, the function simplifies

equivalently to relations (2.5) and (2.7)

( ) ( ),, 2121 xxxx −= rr LL (2.22)

( ) ( ).2121 xxxx −=− rr LL (2.23)

Limiting values. Obviously, if the points x1 and x2 coincide, the lineal path function is

nothing else but the one-point probability function; for points x1 and x2 that are far apart the

lineal path function vanishes,

( ) ( ),,:for 12121 xxxxx rr SL == (2.24)

( ) ,0,lim:for 212121

=∞→−∞→−

xxxxxx rL (2.25)

The substantial difference between the lineal path function and n-point probability

function is that the functions related to different phases cannot be, in general, uniquely

determined by relations similar to Table 2.1. This is just another confirmation of the fact that

this function contains additional information which needn’t be captured by low-order

probability functions7.

6 The lineal path function can be related to the lineal contact distribution function Hr

l(u) introduced in

[42]. Indeed, the lineal contact distribution function for a line l starting at the origin and the r-th phase is defined

by relation Hrl(u) = 1 − P({Dr ∩ ul} = ∅)/(1 − cr). Then, e.g., for r = m, we get Hf

l(u) = 1 − Lm(ul)/(1 − cf ) and

finally Lm(ul) = cm(1 − Hfl). See also [152, 247].

7 For various deterministic periodic microstructures with smooth boundaries between phases, however,

the extensive numerical studies reported in [39] led the authors to the conjecture that the two-point probability

functions are sufficient to uniquely reconstruct the given microstructure (up to the translation and possible

inversion of the image).

15

2.4. Numerical evaluation of microstructural statistics

Numerical evaluation of microstructural statistics introduced in the previous sections

proceeds as follows. We begin with the n-point probability functions assuming an ergodic

medium. An approach suitable for digitized media is explored. Then, a procedure for the

determination of lineal path function is proposed. Both descriptors are evaluated for a selected

representative of theoretical microstructural models8.

2.4.1. n-point probability functions

To determine Sr1,...,rn we recall that the general n-point probability gives the probability

of finding n points x1, . . . , xn randomly thrown into a medium located in the phases r1, . . . ,

rn. Among all functions, the one–point and two–point probability functions deserve a special

attention. In view of Table 2.1 we further consider only the matrix probability functions.

To follow the above definition, the one–point matrix probability function Sm gives the

chance of finding a randomly placed point located in the matrix phase. To determine this

quantity, a simple Monte-Carlo like simulation can be utilized – we throw randomly a point

into the microstructure and count successful “hits“ into the matrix phase. Then, the value of

function Sm can be estimated as

,'

nn

Sm ≈ (2.26)

where n’ is the number of successful hits and n denotes the total number of throws. Entirely

similar procedure can be employed to determine values of Smm(x)9.

8 An interested reader may consult the overview article [45] or the books [42], [46] for exhaustive and

systematic discussion of various microstructural models. 9 For statistically isotropic microstructure, Smith and Torquato [41] proposed more efficient procedure for the

determination of Smm(||x12||). Instead of tossing a line corresponding to x into a medium, a sampling template is

used for the determination of two-point probability function. See also [49] for comparison of this method with

approaches presented hereafter.

16

Figure 2.2: Idealized binary image of Figure 2.1b. Bitmap

resolution is 976 × 716 pixels

Another, more attractive, approach is available when the real microstructure is

replaced by its binary image. A binary version of Figure 2.1b is shown in Figure 2.2. Such a

digitized micrograph can be imagined as a discretization of the characteristic function

�r(x,�), usually presented in terms of a W × H bitmap. Denoting the value of �r for the pixel

located in the i-th row and j-th column as �r(i, j) allows writing the first two moments of

function �r for an ergodic and statistically homogeneous medium in the form10

( ),,1 1

0

1

0��

−

=

−

=

=W

i

H

jrr ji

WHS χ (2.27)

( ) ( )( ) ( ) ( ),,,1

,1 1

��−

=

−

=

++−−

=M

m

N

n

i

ii

j

jjsr

nNmMrs njmiji

jjiinmS χχ (2.28)

where im = max(0,−m), iM = min(W,W−m) and jn = max(0,−n), jN = min(H,H−n). Observe that

to compute function Sr requires O(WH) operations, while O((WH)2) operations are needed for

function Srs. This might be computationally demanding, particularly for a large micrograph,

and does not seem to bring any advantages over simulation techniques.

10 Throughout the text, the C-language type of array indexing is consistently used, i.e., we denote the first

element of an array � as �0 and the last element of the array as �L−1, where L is the array length.

17

The required number of operations, however, can be reduced when writing the two–

point probability function Srs for a periodic ergodic medium as a correlation of functions �r

and �s, recall Eq. (2.4),

( ) ( ) ( ) .d1 yyxyx += � sY rrs Y

S χχ (2.29)

Then the Fourier transform of Srs is provided by [53]

( ) ( ) ( ),~~1~��� srrs Y

S χχ= (2.30)

where � now stands for the complex conjugate. Taking advantage of the periodicity of

function �r one may implement the Discrete Fourier Transform (DFT) [5] when evaluating

Eq. (2.30). To shed a light on this subject we first write the discrete version of Eq. (2.29) in

the form

( ) ( ) ( ) ( )( ),%,%,1

,1

0

1

0��

−

=

−

=

++=W

i

H

jsrrs HnjWmiji

WHnmS χχ (2.31)

where symbol “%” stands for modulo. The above equation, usually termed the cyclic

correlation [5], readily implies periodicity of function Srs. Note that the correlation property of

DFT holds for cyclic correlation. Referring to Eq. (2.30) it is given by the following relation

( ){ } ( ){ } ( ){ }.,DFT,DFT,DFT nmnmnmS srrs χχ= (2.32)

The inverse DFT denoted as IDFT then serves to derive function Srs at the final set of

discrete points as [3]

( ) ( ){ } ( ){ }{ }.,DFT,DFTIDFT1

, nmnmWH

nmS srrs χχ= (2.33)

This method is very economical and its accuracy depends only on the selected

resolution of the digitized medium. Usually, the Fast Fourier Transform, which needs only

O(WH log(WH) + WH) operations, is called to carry out the numerical computation11.

2.4.2. Lineal path function

Following the definition of the lineal path function as a probability of finding a

segment randomly thrown into a medium contained in a given phase, Eq. (2.21), an

11 The public-domain package FFTW version 2.1.3 [10] was used for the evaluation of (2.33).

18

elementary Monte Carlo-based procedure can be again used for its evaluation, i.e., we

randomly throw segments into a medium and count the cases when the segment meets the

given condition. Computationally more intensive approach, however, can be employed

following the idea of sampling template introduced in [41].

To that end, we form a sampling template with dimensions TW × TH pixels. Then, we

draw a set of segments from the center of a template to the points on the template boundary

separated by given discrete steps �W and �H. If the DDA algorithm (see, e.g., [16]) is used for

a construction of segments, the template can be rapidly assembled using only integer

operations. Moreover, this algorithm can be effectively combined with the bitmap

representation of the microstructure. Once a template is formed, the values of the lineal path

function for a given direction starts with placing the template center at a given point found,

say, in phase r and then marking the pixel at which the segment corresponding to the selected

direction meets the other phase, say s12. Then, counters corresponding to pixels of a given

segment which are closer to the center than the marked pixel are increased by one while

remaining counters are left unchanged. The value of the lineal path function can be then

obtained either by stochastic sampling (randomly throwing template center into a medium) or

deterministic sampling (template center is successively placed in all pixels of a bitmap) and

averaging the obtained results. Moreover, the latter method allows us to actually use only a

half of the sampling template, provided that the analyzed microstructure is statistically

homogeneous. Note that even though this procedure basically needs only integer-based

operations, it is still substantially slower than the FFT-based approach. Hence, a relatively

sparse sampling template is unavoidable if one wishes to keep the efficiency of this procedure

comparable to the determination of the two-point probability function.

12 In particular, the matrix phase is checked when Lf function is determined while the fiber phase represents

the “stop condition” for the Lm function.

19

3. Taking and modifying images

3.1. Application of image analysis in composite material research1

A rapid development of image analysis device enabling us to obtain digital record of

image information has occurred in the last decade [54]. Computer based image acquisition,

processing and analysis system allows scientists, engineers, researchers, etc., capturing video

information into the digital world of computers, where images can be viewed, enhanced,

analyzed, measured, annotated, archived, transmitted and more [17].

Image data acquisition is processed by various image input devices according to the

type of scanned specimen:

• The scanners are mainly used for square stationary patterns (prints, slides, negatives).

• Individual images of micro- and macro-objects are scanned by digital photocamera

either directly or by the camera connected to microscope.

• Sequence of images is recorded by the digital CCD camera (usually connected to

other optical equipment).

Imaging system consists of the following components in addition to the computer

(Figure 3.1):

• Image input device

• Frame grabber, or image capture computer add-in board, and any associated interface

cables.

• Software to acquire, analyze, measure, process and store images.

1 This section is based o work[44]

20

Figure 3.1: Imaging system diagram [17]

Computer added image processing may consist of:

• Detection of optical system, conversion and digitizing influence on the distortion of

image data and its subsequent correction

• Image adjustment before own analysis (e.g. decomposition into color components,

conversion to other color spaces, filtration)

• Basic image data analysis (e.g. measurement of lengths and areas, determination of

color component histograms in individual color spaces)

• Complex image data analysis (e.g. harmonic, wavelet, fractal)

The development of image analysis device has qualified its use in material

engineering. The opportunity of direct structure viewing increases the chances of difficult

structure investigation (e.g. various composite structures). The outcome of material image

analysis is basic information of image structure and its attributes, which are used in the proper

description of recorded objects or processes.

21

3.2. C/E composites description

Carbon epoxy (C/E) composites are materials which possess similar structure of the

reinforcement and matrix built on carbon base. In this thesis, structural research of plain

weave reinforced C/E composites is further described. These materials are prepared by

molding together a number of plies, each ply consisting of preimpregnated woven carbon-

fibre fabric (Figure 3.2).

Figure 3.2: a) Plain weave fabric b) Plain weave reinforced composite

Plain weave reinforced composite structure has been the topic of many works arisen in

last few years. These works have been oriented on the description of woven reinforcement,

relation between the structure and mechanical properties of a composite, influence of carbon

tows deformation caused by woven technology on effectiveness of the composite

reinforcement [7], [18], [20], [52].

Many researchers employ themselves in the creation of geometrical models of woven

reinforcement and their application in the prediction of composite properties. These models

have been usually based on idealized description of woven structure [24], [27], [28]. In the

current thesis, however, we restrict our attention to analysis of a cross-section of a tow, which

is unidirectional fibrous composite.

Expansion of image analysis and its use in composite research have allowed detailed

examination of real woven reinforcement structure and specification of its geometry

description. Also other structural characteristics (e.g. shape and distribution of individual

structural components, its size and volume fraction, etc.) can be measured and evaluated

directly [26].

22

However, image analysis of composite structures requires careful composite specimen

preparation and skilled worker for image acquisition and analysis to obtain valuable data set

for effective description of a real structure.

3.3. Tresholding

In this part, one of the terms used in the description of deriving binary images from the

real microstructure is explained.

The thresholding [43] is defined as „an image operation which produces a binary

image from a gray scale image“ [12]. Furthermore, the thresholding can produce a binary one

on the output digital image whenever a pixel value on the input digital image is below a

specified threshold level. A binary zero is produced otherwise. Although the composite

material is not always composed of two phases, it is considered as a two-phase composite for

simplicity. Nonetheless, the definition can be easily extended by increasing the number of

thresholding values.

The threshold values may be determined interactively by the operator. While the video

display affords direct comparison of the thresholded image with the working image, the

software enables the operator to modify or fix the thresholded images. Since it is assumed that

the phases of the composite could be distinct in the original image, the thresholding is done by

referring to the histogram on the video screen. If the original image has enough resolution and

little noise, then the histogram can provide most of the information needed to choose the

threshold value required for generating an approximated geometry.

Furthermore, if the volume fraction of every phase in the two dimensional image is

given, the software may provide a function to calculate the ratio between pixel values, by

which the threshold value is determined. On the other hand, if there is some noise, e.g., due to

bad resolution of the captured image, it has to be eliminated manually by the drawing function

in the software so that the desired geometry can be obtained. This additional operation is

called adjusting. Once a threshold value has been chosen, it is easy to convert the selected

image into a binary image. If the obtained image does not seem to be a satisfactory

representation of the original one, new threshold values may be chosen until the satisfaction

with the binary image is achieved.

23

3.4. Image analysis of graphite fiber laminate2

The laminated composite material in the shape of tube is used in the present study. In

particular, the fiber tow with the assumption of unidirectional fibers embedded in the matrix is

considered.

In this study LUCIA G software is used. It is an image editing software, enabling the

processing of color images – like, e.g., ADOBE PHOTOSHOP, and supporting conversion

into binary image. Binary image is a color image reduced just to two colors (black and white),

which is suitable when dealing with two-phase composites. Approach of converting images

into binary images is very efficient for the present material pattern. The program is designed

for scientific work – it allows programming simple macros leading to a great improvement of

the work efficiency.

3.4.1. Preparation of samples

A part of the laminated tube was put into the form of blunted cone and bonded by

dentacryle. The blunted cone was very steep and almost cylindrical. After solidification of

dentacryle a piece of material perpendicular to the longitudinal axis of the tube was cut. Both

surfaces perpendicular to the longitudinal axis of the tube were parallel. The sample was

grinded with metallographic grinder with decreasing abrasivity with 280, 400, 600 particles

per mm2. The perfectly plane parallel surfaces were achieved this way, which was necessary

for the quality of taken images. At the end, the sample was polished with diamond substance

on felt pad. The whole procedure was repeated many times to obtain images of the desired

quality.

3.4.2. Taking images with camera

Images of low quality were gained with the use of video camera. However, it was

impossible to convert these images with the help of the program LUCIA G. Brightness of

2 This section is based on work [11]

24

images was uneven, that was the reason, why it was not possible to use the function threshold

or to increase contrast. For illustration, we present a preview (Figure 3.3) of these images.

Each image was gained as an average of two hundred images made using microscope. It was

tried to get useable output from these inputs, but it was just proved, that the only way to arrive

at suitable results is a complicated increase of intensity, which could not often be done

sufficiently. The subsequent tresholding was again unsatisfactory, owing to the low contrast

so that the final manual drawing of missing parts of fibers with the use of original image was

necessary.

In overall it took about 3 hours to obtain one binary output. The output was

comparable with outputs gained with improved technique, but lasted too long. That is the

main reason, why for getting useable images we switched to scanner with higher resolution

and better illumination.

Figure 3.3: Input image made by camera

3.4.3. Taking images with scanner

Taking images by scanner was done using the microscope with lenses 12,5 with lateral

output. First, a few color images were taken, in order to determine the effectiveness of this

technique. An example is shown in Figure 3.4. This image served as a starting point in the

process of getting final simplified diagrams. Nevertheless, even this technique was too long

25

and inefficient in order to get larger sets of images. Therefore, the direct modification in input

of scanner images was used.

Figure 3.4: Color output made by scanner3

3.4.4. Modification of input using program PHOTOSHOP

First, the sample was scanned with full color, but it was found impossible to

determine, using only visual determination, whether the image had sufficient quality. Balance

of brightness is important factor of quality of the image, but it is hard to decide, whether the

image is balanced enough for subsequent modification.

That was the reason, why the first part of the image modification in the analytical

graphical program LUCIA G (increasing of contrast) was changed and left to scanning of

sample. During scanning the contrast was adjusted (increased) and saved in the grayscale.

Note that because of increased contrast the images seem to be monochromatic, but they are

not. As suggested by previous works, images with about 100 fibers are sufficient for the

subsequent work. The resolution of scanner was 4 400x3 600 pixels and the saved images had

3 The dark color stripe in the left part of the image is caused by defect of the scanner. The

example shown here is just for illustration, images without defects were used for modification.

Unluckily the undefected ones were lost by computer collapse.

26

resolution 1 500x1 500 pixels. The image was cut out from the large scanned image. Owing to

the sizes of images it was theoretically possible to receive up to four independent and non

overlapping images from one prescan. In reality, it was almost impossible to get all four

samples, especially because of unbalanced exposure. Thanks to increased contract it was,

despite of absence of color in images, possible to determine, whether the image was reliable

or not. If edges were very sharp and evidently artificially made it was obvious that the sample

was in bad quality. Such images were automatically disposed from the set of inputs.

Figure 3.5: Input grayscale image made by scanner.

3.4.5. Modification using program LUCIA G

There is only passive version of LUCIA G available on our faculty. It allows just

modification of images, but not grabbing or direct editing camera or scanner input.

The aim of modification in program LUCIA G was to obtain binary (monochromatic)

images. In these images the fiber should be idealized by circles with the equal radii. The

centers of these circles are placed in gravity centre of original imperfect fibers. The total area

of imperfect fibers equals to the total area of idealized fibers. The number of fibers is

identical, too. These conditions fulfill expectation of the same material fraction in real and

idealized image.

27

Input images (Figure 3.5) were modified in program LUCIA G. The program creates

binary image based on tresholding of input color images. The tresholding works as setting of

range of color scales or intensity and saturation (depends, whether the RGB or HIS

representation of image is used), for which the output binary image should be black (or

white). More detailed explanation of this procedure can be found in section 3.3. The function

could be used only providing the quality (balanced) illuminated image was

sufficientpresented. In case of gradual edges it was better to first increase the contrast. This

was fulfilled by input modification from the scanner in program PHOTOSHOP.

Figure 3.6: Binary image created in LUCIA G by tresholding.

The binary image in Figure 3.6 was not good enough for the analysis. Thanks to

existing scratches on the sample surface the image was locally incomplete – parts of fibers

were missing (Figure 3.6). It was necessary to complete the shape of fibers to the expected

shape. Original shape was usually evident from fragments of a single fiber. To automatize this

step the function convex envelope was used. It was accepted that the shape of fibers is circular

or oval, but nevertheless still non concave.

It was necessary to erase too small “cut” pieces of fibers at edges of the image. If the

major part of shape was preserved at the edge, it was completed to expected original shape

(based on similarity with nearby fibers).

28

Next, the fibers were separated from each other. The separation was done with the use

of function automatic separation of objects in LUCIA G. The function is very efficient,

provided that sufficient setup of number of steps and size of neighborhood is done, but just

with the use of powerful computer.

Figure 3.7: Binary image after first step of modification in LUCIA G.

The area of fibers, number of objects and positions of their gravity centers of the

image in Figure 3.7 was measured. After that the binary image with the same area, number of

objects and positions of their gravity centers was generated. The only difference was that in

the idealized image all objects were circles with the same radius. Whole technique was done

with the help of macro in LUCIA G. There is a tool in the program, which allows getting

history of used commands and setting of various functions. This tool was used for creating

macro in programming language (sort of C++). Resulting image, which is suitable for

statistical simulations, is shown in Figure 3.8.

29

Figure 3.8: Example of final output.

3.5. Conclusion of image analysis

This section presented the most effective approach for obtaining binary images of real

composite material. This approach was tested and used for retrieval of c. 25 images of real

microstructure of the composite material. The next chapter discusses the main objective of

this work; process of statistical description of the microstructure.

30

4. Statistical description with various boundary conditions

4.1. Boundary conditions

There are many possibilities how to statistically describe microstructure. Some of

them are summarized in chapter 2 (Description of microstructure). In this thesis the attention

is focused on the two-point probability function. This function is quite simple and easy to

compute when compared with other descriptors. The two point probability function can be

determined applying various boundary conditions. The literature offers the three main

possibilities. Recall that the two point probability function can be obtained by throwing a

needle of certain length in the image and counting how many times the ends fall in the

selected color. The question is what to do, when one end of the needle falls out of the image.

The first possibility is to expect that the image of microstructure is periodically

repeated in each direction. This condition is the simplest one, as it allows using very efficient

computational method described below. This first possibility is shown in Figure 4.1.

Figure 4.1: Example of binary image of microstructure with periodical boundary conditions.

The second possibility is to expect that the microstructure is mirrored. This approach

is more time consuming, but is presented for the sake of comparison with other results. In

case of unidirectional fibers it is probably not very efficient, but research of microstructure

consists of many others fields of research, where technique can be very useful(Figure 4.2)

31

Figure 4.2: Example of binary image of microstructure with mirror boundary conditions.

The third possibility, here termed as plain boundary conditions, is to disregard the

throws, where at least one of the ends of the needle falls out of the image. This approach is

probably the most accurate one, but significantly reduces the amount of obtained information.

It is evident that long needles can be thrown just a few times to be totally contained in the

image. If the needle with length of the diagonal of the rectangle is used, it can be thrown just

one time. That is the reason, why the data of two point probability function have to be reduced

of data with low reliability. (The needle is too long and therefore can be thrown just few

times.) (Figure 4.3)

Figure 4.3: Example of binary image with no mirroring or repeating.

32

Based on the previous work it was possible to use 25 images, but only 18 images were

eventually selected for the analysis. The visual evaluation consists of elimination samples

which seem to be unuseful. If about one fifth of the image was fiber free it was clear, that the

image experienced some error.

4.2. Evaluation of the two-point probability function using Fast

Fourier Transform method

The fast Fourier transform method already explained in section 2.4.1 can be used only

with the periodic boundary conditions. This method is much faster than the classical Monte-

Carlo method and was used to as a basis to compare effectiveness of both methods.

After selecting suitable inputs the two point probability function was calculated. In the

first step the edge of the input image was cut to obtain a square bitmap. The reason was that

the edge could be influenced by inaccuracy when taking images. That means that the largest

image had size of 1 148 x 1 148 pixels. In the next step the size was reduced by 70 pixels in

both directions to 1 078 x 1 078 pixels. The smaller image was selected 10 times from random

positions of the largest image. The step 70 pixels was chosen, because the average size of

fibers in images is c. 70 pixels. In each subsequent step the size was reduced by 70 pixels in

both directions and the random selection was increased 10 times compared to the previous

case. Sizes of binary images are summarized in Table 4.1.

33

Height of bitmap Width of bitmap Size of bitmap Number of bitmaps

1 148 1 148 1 317 904 1

1 078 1 078 1 162 084 10

1 008 1 008 1 016 064 20

938 938 879 844 30

868 868 753 424 40

798 798 636 804 50

728 728 529 984 60

658 658 432 964 70

588 588 345 744 80

518 518 268 324 90

448 448 200 704 100

378 378 142 884 110

308 308 94 864 120

238 238 56 644 130

168 168 28 224 140

98 98 9 604 150

Table 4.1: Sizes of binary images used in FFTM. Units in table are pixels.

After preparation and sorting of bitmaps the two point probability function was

calculated for each bitmap. The results of this calculation were stored in the matrix of the

same dimensions as the bitmap. Next step was to create the average result for each size. The

average was calculated not only on the whole set, but also on partial sets constructed from 2,

3, 4, … , Number of bitmaps of the specified size bitmaps. The value of average two-point

probability function at the position ij was obtained as average value of independently

computed two-point probability functions at the position ij.

The last step was to determine a suitable measure of a difference among the two-point

probability functions. As “an exact” two point probability function the function calculated on

the largest image was chosen. To calculate the measure M, the following formula was used

��−

=

−

=

−=

1

0

1

0

..1 W

i

H

j ij

ijij

S

PS

HWM

. (4.1)

In equation (4.1) W means the number of used points in the direction of width of the bitmap,

H the number of used points in the direction of height of the bitmap. If every point of bitmap

34

is used for evaluation of the measure W means the width of bitmap and H is the height of

bitmap. Sij is the value of the two point probability function of an image taken in the point

with coordinates i and j. Pij is the value of the two point probability function of the largest

bitmap. This enables comparison of measures taken from bitmaps with variable sizes.

Another measure MM was also considered and compared to measure M. The used

formula follows

��−

=

−

=

−=

1

0

1

0 ),(..

.1 W

i

H

j ij

ijij

jidistS

PS

HWM (4.2)

in which

��

��

� =∧=

+= 00,1

otherwise,22),( ji

jijidist (4.3)

The variable dist should guarantee that the values of the two point probability function, which

are more distant from the origin, have smaller significance than those which are closer. Quite

surprisingly, this correction results in rather negligible difference. Both kinds of measures are

compared in Table 4.2.

35

Number in the first column is the number of the two point probability functions from which the

average two point probability function was computed. The dimensions of the bitmap size over

which the averaging was carried out, are 1 008 x 1 008 pixels

Number Measure M Measure MM M/MM abs((M/MM)i-(M/MM)avg)

1 0,0000141996 4,640461E-11 305994,60 523,93

2 0,0000144160 4,708017E-11 306201,10 317,43

3 0,0000150730 4,915416E-11 306648,10 129,58

4 0,0000151216 4,931404E-11 306639,04 120,51

5 0,0000153525 5,005355E-11 306721,30 202,77

6 0,0000152362 4,970160E-11 306553,11 34,58

7 0,0000152719 4,981294E-11 306585,20 66,67

8 0,0000153377 5,001717E-11 306648,50 129,97

9 0,0000153923 5,019365E-11 306658,11 139,58

10 0,0000154202 5,028348E-11 306665,33 146,80

11 0,0000153869 5,018206E-11 306620,93 102,40

12 0,0000153068 4,992734E-11 306580,52 61,99

13 0,0000152420 4,972088E-11 306550,69 32,16

14 0,0000151973 4,958246E-11 306504,76 13,77

15 0,0000151477 4,942139E-11 306501,29 17,23

16 0,0000151423 4,940891E-11 306469,82 48,70

17 0,0000151201 4,934129E-11 306439,09 79,44

18 0,0000151799 4,953054E-11 306474,55 43,98

19 0,0000151807 4,953722E-11 306451,19 67,34

20 0,0000151436 4,941407E-11 306463,32 55,21

Average M/MM: Max difference:

Max difference/Average = 0,17% 306518,53 523,93

Table 4.2:Example of comparison of measure M and MM.

Because the measure M and measure MM were the same, just multiplied by almost a

constant value, in the subsequent research, only the measure M was used as it is more simple

to compute.

36

4.2.1. Graphs of measures

This part presents an overview of the measures M evaluated from the computed two

point probability functions. It is evident that bigger bitmaps allow getting results with better

reliability. Quite surprisingly, even for lower resolutions there is no common value, where all

measures stabilize. The measure of the two point probability function oscillates around

different value.

The resolution displayed above each graph represents the area over which the measure

M was evaluated. The values at horizontal axis are the numbers of used bitmaps for evaluation

of the average two point probability function. The accuracy does not gain zero value at

numbers of repeats 11, 21, 31, … . The line connecting the value of measure with zero is

displayed just for the lucidity of graphs. The described example follows.

938 ×××× 938pixels

37

1078 ×××× 1078pixels 1008 ×××× 1008pixels

1,05E-05

1,06E-05

1,07E-05

1,08E-05

1,09E-05

1,10E-05

1,11E-05

1,12E-05

1 2 3 4 5 6 7 8 9 10

Number of repeats

Mea

sure

0,00E+00

2,00E-06

4,00E-06

6,00E-06

8,00E-06

1,00E-05

1,20E-05

1,40E-05

1,60E-05

1,80E-05

1 3 5 7 9 11 13 15 17 19

Number of repeats

Mea

sure

938 ×××× 938pixels 868 ×××× 868pixels

0,00E+00

2,00E-06

4,00E-06

6,00E-06

8,00E-06

1,00E-05

1,20E-05

1,40E-05

1,60E-05

1,80E-05

2,00E-05

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

Number of repeats

Mea

sure

0,00E+00

5,00E-06

1,00E-05

1,50E-05

2,00E-05

2,50E-05

1 4 7 10 13 16 19 22 25 28 31 34 37 40

Number of repeats

Mea

sure

38

798 ×××× 798pixels 728 ×××× 728pixels

0,00E+00

5,00E-06

1,00E-05

1,50E-05

2,00E-05

2,50E-05

3,00E-05

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49

Number of repeats

Mea

sure

0,00E+00

5,00E-06

1,00E-05

1,50E-05

2,00E-05

2,50E-05

3,00E-05

3,50E-05

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57

Number of repeats

Mea

sure

658 ×××× 658pixels 588 ×××× 588pixels

0,00E+00

5,00E-06

1,00E-05

1,50E-05

2,00E-05

2,50E-05

3,00E-05

3,50E-05

4,00E-05

4,50E-05

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69

Number of repeats

Mea

sure

0,00E+00

1,00E-05

2,00E-05

3,00E-05

4,00E-05

5,00E-05

6,00E-05

1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76

Number of repeats

Mea

sure

39

518 ×××× 518pixels 448 ×××× 448pixels

0,00E+00

1,00E-05

2,00E-05

3,00E-05

4,00E-05

5,00E-05

6,00E-05

7,00E-05

8,00E-05

1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86

Number of repeats

Mea

sure

0,00E+00

2,00E-05

4,00E-05

6,00E-05

8,00E-05

1,00E-04

1,20E-04

1 9 17 25 33 41 49 57 65 73 81 89 97

Number of repeats

Mea

sure

378 ×××× 378pixels 308 ×××× 308pixels

0,00E+00

5,00E-05

1,00E-04

1,50E-04

2,00E-04

2,50E-04

1 10 19 28 37 46 55 64 73 82 91 100 109

Number of repeats

Mea

sure

0,00E+00

5,00E-05

1,00E-04

1,50E-04

2,00E-04

2,50E-04

3,00E-04

3,50E-04

1 10 19 28 37 46 55 64 73 82 91 100 109 118

Number of repeats

Mea

sure

40

238 ×××× 238pixels 168 ×××× 168pixels

0,00E+00

1,00E-04

2,00E-04

3,00E-04

4,00E-04

5,00E-04

6,00E-04

7,00E-04

1 11 21 31 41 51 61 71 81 91 101 111 121

Number of repeats

Mea

sure

0,00E+00

2,00E-04

4,00E-04

6,00E-04

8,00E-04

1,00E-03

1,20E-03

1 12 23 34 45 56 67 78 89 100 111 122 133

Number of repeats

Mea

sure

98 ×××× 98pixels

0,00E+00

1,00E-03

2,00E-03

3,00E-03

4,00E-03

5,00E-03

6,00E-03

7,00E-03

1 12 23 34 45 56 67 78 89 100 111 122 133 144

Number of repeats

Mea

sure

4.3. Evaluation of the two-point probability function using the

Monte Carlo Method

4.3.1. Description of used algorithm

Due to computational demands of the Monte-Carlo method the simplifying conditions

had to be applied to the evaluation. The detail description of the used algorithm is presented in

this section. Two possibilities of simplification were considered. First, the value of the two

41

point probability function could be computed for all available vectors of function, but not all

data contained by the graph are used for the function evaluation. In other words the needles of

every available length are used but just limited numbers of throws are done. Other possibility

is to evaluate the function just for a few specific vectors, but all data stored in the bitmap are

used for evaluation. In other words to throw the needles of just few chosen lengths but the

needles are thrown into every point of bitmap.

In the present work the second possibility was employed. The reason was that this

approach allows comparison with results obtained by Fast Fourier transform method, which

evaluates the function with use of all data stored in the image. This property is important as

the goal of this research is to compare different boundary conditions.

The step used through the bitmap was based upon the average size of one fiber. The

average diameter of fiber was c. 70 pixels and one quarter of the diameter is approximately 17

pixels. The used step was 15 pixels. The quarter of diameter of fiber is value determined from

results presented in [53]. For obtaining the step in horizontal direction the same relation was

used and since the fibers are circles, both steps were the same.

Another difference compare to the Fast-Fourier transform method is that the step

reducing the bitmaps sizes was bigger. Recall that in Fast Fourier transformation method the

step 70 pixels was used which in the Monte Carlo method was set to 140 pixels and just one

half of bitmap size reductions was done (Compare with Table 4.1). The resolutions and

number of reductions are summarized in Table 4.3.

Height of bitmap Width of bitmap Size of bitmap Number of bitmaps

1 148 1 148 1 317 904 1

1 008 1 008 1 016 064 10

868 868 753 424 20

728 728 529 984 30

588 588 345 744 40

448 448 200 704 50

308 308 94 864 60

168 168 28 224 70

Table 4.3: Sizes of binary images used in MCM. Units in table are pixels.

42

4.3.2. Periodic boundary conditions

1008 ×××× 1008pixels 868 ×××× 868pixels

1,12E-04

1,14E-04

1,16E-04

1,18E-04

1,20E-04

1,22E-04

1,24E-04

1,26E-04

1,28E-04

1,30E-04

1 2 3 4 5 6 7 8 9 10

Number of repeats

Mea

sure

0,00E+00

5,00E-05

1,00E-04

1,50E-04

2,00E-04

2,50E-04

1 3 5 7 9 11 13 15 17 19

Number of repeatsM

easu

re

728 ×××× 728pixels 588 ×××× 588pixels

0,00E+00

5,00E-05

1,00E-04

1,50E-04

2,00E-04

2,50E-04

3,00E-04

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

Number of repeats

Mea

sure

0,00E+00

1,00E-04

2,00E-04

3,00E-04

4,00E-04

5,00E-04

6,00E-04

1 4 7 10 13 16 19 22 25 28 31 34 37 40

Number of repeats

Mea

sure

43

448 ×××× 448pixels 308 ×××× 308pixels

0,00E+00

2,00E-04

4,00E-04

6,00E-04

8,00E-04

1,00E-03

1,20E-03

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49

Number of repeats

Mea

sure

0,00E+00

5,00E-04

1,00E-03

1,50E-03

2,00E-03

2,50E-03

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57

Number of repeats

Mea

sure

168 ×××× 168pixels

0,00E+00

1,00E-03

2,00E-03

3,00E-03

4,00E-03

5,00E-03

6,00E-03

7,00E-03

8,00E-03

9,00E-03

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69

Number of repeats

Mea

sure

44

4.3.3. Plain boundary conditions

1008 ×××× 1008pixels 868 ×××× 868pixels

1,08E-04

1,09E-04

1,10E-04

1,11E-04

1,12E-04

1,13E-04

1,14E-04

1,15E-04

1,16E-04

1,17E-04

1,18E-04

1,19E-04

1 2 3 4 5 6 7 8 9 10

Number of repeats

Mea

sure

0,00E+00

2,00E-05

4,00E-05

6,00E-05

8,00E-05

1,00E-04

1,20E-04

1,40E-04

1,60E-04

1 3 5 7 9 11 13 15 17 19

Number of repeatsM

easu

re

728 ×××× 728pixels 588 ×××× 588pixels

0,00E+00

5,00E-05

1,00E-04

1,50E-04

2,00E-04

2,50E-04

3,00E-04

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

Number of repeats

Mea

sure

0,00E+00

5,00E-05

1,00E-04

1,50E-04

2,00E-04

2,50E-04

3,00E-04

3,50E-04

4,00E-04

1 4 7 10 13 16 19 22 25 28 31 34 37 40

Number of repeats

Mea

sure

45

448 ×××× 448pixels 308 ×××× 308pixels

0,00E+00

1,00E-04

2,00E-04

3,00E-04

4,00E-04

5,00E-04

6,00E-04

7,00E-04

8,00E-04

9,00E-04

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49

Number of repeats

Mea

sure

0,00E+00

5,00E-04

1,00E-03

1,50E-03

2,00E-03

2,50E-03

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57

Number of repeats

Mea

sure

168 ×××× 168pixels

0,00E+00

2,00E-03

4,00E-03

6,00E-03

8,00E-03

1,00E-02

1,20E-02

1,40E-02

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69

Number of repeats

Mea

sure

46

4.3.4. Mirror boundary conditions

1008 ×××× 1008pixels 868 ×××× 868pixels

1,34E-04

1,35E-04

1,36E-04

1,37E-04

1,38E-04

1,39E-04

1,40E-04

1,41E-04

1,42E-04

1,43E-04

1 2 3 4 5 6 7 8 9 10

Number of repeats

Mea

sure

0,00E+00

5,00E-05

1,00E-04

1,50E-04

2,00E-04

2,50E-04

1 3 5 7 9 11 13 15 17 19

Number of repeatsM

easu

re

728 ×××× 728pixels 588 ×××× 588pixels

0,00E+00

5,00E-05

1,00E-04

1,50E-04

2,00E-04

2,50E-04

3,00E-04

3,50E-04

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

Number of repeats

Mea

sure

0,00E+00

1,00E-04

2,00E-04

3,00E-04

4,00E-04

5,00E-04

6,00E-04

1 4 7 10 13 16 19 22 25 28 31 34 37 40

Number of repeats

Mea

sure

47

448 ×××× 448pixels 308 ×××× 308pixels

0,00E+00

2,00E-04

4,00E-04

6,00E-04

8,00E-04

1,00E-03

1,20E-03

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49

Number of repeats

Mea

sure

0,00E+00

5,00E-04

1,00E-03

1,50E-03

2,00E-03

2,50E-03

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57

Number of repeats

Mea

sure

168 ×××× 168pixels

0,00E+00

1,00E-03

2,00E-03

3,00E-03

4,00E-03

5,00E-03

6,00E-03

7,00E-03

8,00E-03

9,00E-03

1,00E-02

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69

Number of repeats

Mea

sure

4.4. Comparison of periodic boundary conditions with others

In previous section statistical characteristics for different boundary conditions were

compared. In this section the two-point probability functions with periodic, mirror and plain

boundary conditions derived from one bitmap are shown. The difference between different

boundary conditions presented, too. The difference is provided in a absolute value, since it is

confusing to show negative and positive values with just one color. The two point probability

48

functions are shown in first quadrant. The enormous computational demand associated with

an exact evaluation of the function in every point is the main reason for this simplification.

Figure 4.4: Two point probability function with periodic boundary conditions.

Figure 4.5: Two point probability function with plain boundary conditions.

49

Figure 4.6: Two point probability function with mirror boundary conditions

Figure 4.7: Comparison of periodic and mirror boundary conditions.

The difference is plotted in an absolute value.

50

Figure 4.8: Comparison of periodic and plain boundary conditions.

The difference is plotted in an absolute value.

4.5. Conclusions

It is obvious from graphs in section 4.3 that all boundary conditions can be considered

as equal. The measure used for comparison is almost the same and behaves in a similar

manner. The suggested measure cannot fully contain the difference between functions set at

different resolutions of bitmaps. The reason is explained in the next paragraph at parable.

The two point probability function behaves little bit like surface of water in pond after

being hit by stone in the middle of the pond. If stones of different shapes, but quite similar

size are thrown in pond, the waves close to place of hit are quite different but in a farther

distance from the place of hits the waves are small. If the wave is small, the difference is

small too. If the observer is monitoring just small area of the surface close to the place of hit,

he will see that the differences between shapes of waves made by stones of different shapes

are bigger than if he monitors larger area. The results from this research are same. The

average values of measure are always smaller if determined from functions computed for

large bitmaps.

In section 4.4 it is shown that the difference between periodic and plain or mirror

boundary conditions are random and do not follow any regular pattern. Recall that the more

important values of the two-point probability functions are close to the origin. The origin of

51

graphs displayed in this work is in the left upper corner. All graphs look to be very similar in

this area and even the differences are close to zero (are bright) in this part.

Providing no boundary condition is superior to the other the only objective then

remains the speed of evaluation. The periodic boundary conditions can be determined with

the use of Fast Fourier transform, which is significantly faster than the classic Monte-Carlo

method. The largest bitmaps in this work have resolution about 1 000x1 000pixels. FFTM is

approximately 125000 times faster than MCM for these dimensions. In particular, when

using FFTM the evaluation lasted two minutes, while with MCM it would have taken almost

half a year.

The speed of evaluation with use of Monte-Carlo method is heavily influenced by the

dimensions of input bitmap. Out of the measures M it follows that it is faster to use a set of

smaller sections of the largest bitmap. The results of one big bitmap and set of sections are

comparable but the result for one big bitmap is nevertheless more accurate. Due to the Fast

Fourier method small time consumption I generally recommend to use this method on the

largest bitmap, because it is still very simple.

52

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56

5. Appendix: Design of object oriented code serving for

statistical research of microstructure

5.1. Introduction

The scientific community concerned with the research of microstructure needs to

explore microstructure statistically. This statistical research is predetermined by using

computer programs. There are many statistical functions we need to measure. Some of them

are simple while other need a lot of computer time. One of the basic approaches is to make an

image of some microscopic material and after some adjustment it is possible to use routines

for statistical evaluation of a sample.

Because the original few steps are generally the same, it is not necessary for each

scientific team dealing with statistical evaluation of samples to create their own code. It was

impossible to find any application, which could be extended just by adding new statistical

functions. It takes a lot of time to write service programs, which load a work with bitmap

instead of writing just the new part. Up to that it is hard to simulate the same conditions for

repeating the measurement in other scientific team.

Due to these facts the necessity of a suitable bitmap processing computational

environment is evident. Such computational environment must be able to reflect the

continuing development in terms of its wide and unrestricted extensibility, teamwork support

and portability.

The main goals can be summarized as follows:

Ability to work with various types of microstructures. (not just a two-phase material

system). That is why we need to be able to work with color bitmaps. The program should be

able to read and create generally used and multiplatform supported bitmaps.

Easy readability and extensibility of new functions without influencing the existing

code. The best way is if every researcher creates his own code file and does not have to

interfere with the rest of the code.

The code must provide output, which can be used both for the next research and for

the presentation of results.

57

Creating idealized models of a microstructure. Some functions have known graph for

specified model. This provides determination whether the used function is correct or not.

Even if the best organization of microstructure is searched the efficient modeling tool should

be prepared for use.

Computational performance. Bitmaps of microstructure samples often take many

MegaBytes and the run of program should be repeated many times.

The created application fulfils all mentioned requirements. It is written in C++ with

the use of object oriented features. This programming language was used, because it can be

used in various OS types and it is quite simple and powerful. Object organization of the code

provides easy addition of new functions.

The input and output graphical interface is standard windows bitmap (BMP). The 1-bit

(2 colors), 4-bit (16 colors) and 16-bit (256 colors) bitmap can be loaded and output is just in

the 16-bit mode. The function of program is tested in Windows XP and Linux. The code is

open and anybody can change any part of it.

5.2. General structure

The code is separated into independent files. Each file contains the part of code, which

performs specified tasks. The main part contains general functions. One part is aimed for the

bitmap analysis. Loading, saving and basic operation with bitmaps as is loading or storing

selected pixel, invert colors of the whole bitmap, add or substract two bitmaps etc. The bitmap

is loaded into an object bitmap. Since object bitmap stores also information about color table,

number of colors etc., it is better to work with a matrix. The matrix is a discrete description of

microstructure, where the value at any coordinates is the number of phases present at these

coordinates. The possibilities of work with matrix or with bitmap are the same. There are

some more operations due to mathematical essence of matrix. The last part contains the user

defined functions. So far, volume fraction function and the two-point probability function are

implemented.

58

5.3. The two-point probability function

One of the basic statistical functions describing microstructure is the two point

probability function. It describes many aspects of a microstructure. It is necessary to find the

influence of boundary conditions on the shape of the two-point probability function. There

exist two computational models for this function. The simple direct Monte Carlo method or

Fast Fourier transform of matrix is available. The direct way is quite slow for big parts of

bitmap, but it provides many possibilities for changing behavior of bitmap boundary

conditions. The Fast Fourier transform is faster, but builds up on the bitmap periodicity.

5.4. Output possibilities

The output is available in the form of text file with function values stored in the matrix

or in the graphical format. Some sample outputs are displayed in the following part. It is

possible to change properties of the output depending on the shape of graph.

Figure 5.1: Plan view of the graph. The darker places have the higher value than the lighter

colors representing the smaller values. The tag on the right gives information about the values

in the scheme.

59

Figure 5.2: Isometric view of the graph. The darker places have the higher value than the

lighter colors representing the smaller values. The tag on the right gives information about the

values in the scheme.

Figure 5.3: Isometric view of the wire frame graph. In some cases this image can supply the

better evidence value than the previous two images.