research article a monte-carlo algorithm for 3d fibre...

Research ArticleA Monte-Carlo Algorithm for 3D Fibre Detection fromMicrocomputer Tomography

Robert Gloeckner1 Stefan Kolling2 and Christian Heiliger1

1 Justus Liebig University Institute for Theoretical Physics Heinrich-Buff-Ring 16 35392 Giessen Germany2Technische Hochschule Mittelhessen Institute of Mechanics and Materials Wiesenstr 14 35390 Giessen Germany

Correspondence should be addressed to Stefan Kolling stefankollingmethmde

Received 14 March 2016 Revised 15 July 2016 Accepted 11 August 2016

Academic Editor Amine Ammar

Copyright copy 2016 Robert Gloeckner et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

A model-based approach to analyze fibre distributions in polymer composites applicable for high fibre content is suggested Thealgorithm is a four-step iterativemethod usingMonte-Carlo techniques in order to increase speed and robustness for fibre detectionSamples with up to 20 volume fraction of glass fibres and different matrix polymers (PP PBT) have been analyzed regardingdistributions of orientation and length and thickness of the fibres

1 Introduction

Short fibre reinforced polymer composites are appealing tomid-sized industry as their use promises low cost continuousproduction of parts with enhanced properties Local fibrelength distribution (FLD) and fibre orientation distribution(FOD) [1] determine mechanical properties [2ndash9] but arestrongly affected by production processes [10]Themost com-monly used manufacturing process for components made ofshort fibre reinforced plastics is injection moulding Herebymolten plastic material and glass fibres are filled into amould cavity under high pressure Particularly in thin walledstructural parts the fibres are oriented in tree layers twoboundary layers and a core layer whose thickness dependson the viscosity of the polymer matrix The fibres in theboundary layers are mostly oriented in filling direction whilethe fibres in the core layer are oriented perpendicular to thefilling direction This influences strongly the anisotropy ofthe composite Within the development process of mouldedplastic parts flow simulations are performed for examplein order to avoid weld lines in highly stressed areas byoptimization of the positions of the gates Moreover thelocal FOD can also be approximated within such flowsimulations This local FOD information can then be usedin anisotropic finite element analysis for the structural part

under mechanical loading However flow simulations arebased on idealized fibremodels and lead to uncertainty aboutFLD as well as FOD [11ndash13] A comparison of the FODdata computed by flow simulation with real FOD test datais indispensable Once the test data is available the materialparameters for example the fibre interaction coefficient in[12] can be calibrated and flow simulation becomes muchmore predictive Thus the usage of short fibre reinforcedpolymers in safety related components demands effective andaccurate quality testing

While other methods either lack a complete characteriza-tion [14 15] or require too much preparation and labour time[16ndash18] microcomputer tomography as a nondestructivetesting method is a suitable alternative for those needs[19ndash22] Common fibre contents of commercially availablecomposites vary between 10 and 70 weight Hence due toclusters and accumulations reliable methods will be neededthat are able to differentiate and locate single fibres therein

The system used in this work is a X-ray desktopmicrocomputer tomography (120583CT)Measurements were per-formed using a SkyScan 1172 Itsmicrofocus sealedX-ray tubeoperated at 20ndash100 kV0ndash250120583A below amaximumpower of10W Typical volume picture dimensions used in this workare 1024 times 1024 times 900 cubic voxels of 17 120583m length This120583CT together with the algorithm presented in this paper has

Hindawi Publishing CorporationJournal of Computational EngineeringVolume 2016 Article ID 2753187 9 pageshttpdxdoiorg10115520162753187

2 Journal of Computational Engineering

recently been used in [23] to compute the elastic propertiesof a short-fibre plastic based on realmeasuredmicrostructuredata by homogenization In the present work we set our focuson the numerical methodology of fibre detection and discussthe Monte-Carlo algorithm in detail

2 Model-Based Algorithm

A common approach of model-based image analysis is toextract image primitives and to establish a correspondencebetween these and the model primitives Probabilistic objectpattern matching speeds up the scanning process providesgood scaling capabilities if done in parallel and increasestolerance against noise and distortion For an overview ofimage analysis of materials structures the reader is referredto [24] and the references therein

The fibre model consists of a chord of cylinders ofconstant radii and various lengths The radius of scannablefibres is limited by the minimum cylinder length

All Monte-Carlo pattern-matching processes run inde-pendent of each other Their results are merged in a control-process which detects doublets and optimizes the vector dataSubsequently this process modifies the voxel data as wellas the parameters of the pattern-matching processes andinitiates their restart

21 Pattern Matching Core algorithm steps are Monte-Carlo scanning for separate fibre parts and cylinder integralextrusion for fibre length detection Scanning for separatefibre parts (index 119894) is done by calculating radial densityfunctions 120588lowast(119903

119894) equiv 120588lowast

119894in randomly chosen spherical volumes

(centres 119894)

120588lowast

119894=

| 119903|lt119903119894

sum

| 119903|ge119903119894minus1

119892lowast

( 119903) (1)

where

119892lowast

( 119903) fl

119892 ( 119903) if 119892 ( 119903) minus 119892 (minus 119903) le 120575

0 else(2)

and 120575 is a chosen toleranceTo approximate radial density the spherical volume is

divided into spherical shell lists of random point-symmetriccoordinate pairs within nonequidistant radius-intervals

0 = 1199030lt 1199031lt sdot sdot sdot lt 119877 lt sdot sdot sdot lt 119903

119878 (3)

where 119877 denotes the fibre radius and 119903119878is the pattern

radius of the final sphere Using correlating point-symmetriccoordinates according to (2) removes voxels from neighbourfibres as well as noise Figure 1 shows the radial density 120588

lowast

119894as

a function of the integral sphere radius 119903119894and the cylinder

radius 119877 which represents the fibre The density that is theprobability that a random point within the sphere is alsolocated within the fibre is 1 for 119903

119894lt 119877 For an increasing

sphere radius 119903119894gt 119877 we have a decreasing probability and

thus a decreasing density distribution which finally tends tozero

0 5 10 15 20 25 30 45

6 78

910

1112

Integral-sphere radius Cylinder-radius

0010203040506070809

1

Den

sity

Figure 1 Geometry detection of fibres using radial density distri-bution for different cylinder radii 119877 Note the strongly decreasinghyperbolic function for 119903

119894gt 119877

The volume119881119904of the spherical shell defined by radii 119903

119894and

119903119894minus1

is given by119881119904= (43)120587(119903

3

119894minus 1199033

119894minus1) And the corresponding

volume of the spherical cylinder can be approximated by119881119888=

1205871198772

(119903119894minus119903119894minus1

) for 119903119894gt 119877Thus the probability that a randomly

chosen point can be found within the cylinder shell is 120588lowast

119894=

119881119888119881119904and thus

4

3(1199033

119894minus 1199033

119894minus1) 120588lowast

119894997888rarr 119877

2

(119903119894minus 119903119894minus1

) for 119903119894gt 119877 (4)

which is used as a convergence criterion Only if this condi-tion is satisfied a separate fibre has been found and the centre119894of the scanned sphere volume is used for further fibre

detection Otherwise a spherical volume around anotherrandomly chosen within the sphere is checked

After locating separate fibre parts at 119894 the fibre orienta-

tion vector 119894is calculated either via diagonalizing the matrix

of second-order moments or using Jacobian matrices Thefibre length Δ119911fibre is determined by approximating cylinderintegrals using circular coordinate pairs as defined in (6)and using ordinary parameters as angle 120572 and radial (119903) andtransverse (119911) distances from the origin (

119894)

119892cyl (119903 119911 120572)

= 119892cyl [119894 + 119911119894+ 119903 (sin (120572) 119905

1+ cos (120572) 119905

2)]

(5)

where the normalized vectors 119894 1199051 and 119905

2are perpendicular

to each other and form a right hand system that is (119894times 1199051) sdot

1199052= 1

119892lowast

cyl (119903 119911 120572)

=

119892 (119903 119911 120572) if 119892 (119903 119911 120572) minus 119892 (minus119903 119911 120572) le 120575

0 else

(6)

For a given starting point and direction the fibre is madeup of small cylinders 119911

119894= 119911119894minus1

+Δ119911119894 until the last partial sum

Journal of Computational Engineering 3

falls below a certain fraction of the first partial sum 119878Δ119894

lt 119878Δ0

sdot

120576Δ119911 which determines the length of the fibre Δ119911fibre = sumΔ119911

119894

The partial sum is given by

119878119894fl 119878Δ119894

(119894 119911119894 Δ119911) fl

119911=119911119894+Δ119911

sum

119911=119911119894

119903=119877

sum

119903=0

120572=2120587

sum

120572=0

119892lowast

cyl (119903 119911 120572) (7)

After solving the cylinder integrals all necessary values(position length and orientation) for further fibre analysisare available At this point it is instructive to conclude thecomplete algorithm for the fibre identification

(1) 119896 = 1 chose 119896randomly within the CT-image

(2) Generate concentric shells with radii 119903119894at the centre

119896

(3) Generate point-symmetric random points within theshells

(4) Compute mass 119898119896+1

and centre of gravity 119896+1

of allpoints within the fibre

(5) Use 119896+1

as a new starting point for generatingspherical shells

(6) 119896 = 119896 + 1 go to (3) until119898119896+1

minus 119898119896lt 120575

(7) Compute the radial density distribution of the ran-dom points

(8) If Equation (4) is fulfilled a separate fibre is detectedElse 119896 = 1 and chose a new

119896randomly within the

shells and go to (2)(9) Compute fibre length and orientation by cylinder

extrusion

22 Duplicate Detection and Match Optimization Parallelindependent running pattern-matching processes intention-ally scan each fibre multiple times starting at randompositions In compounds with high filler fraction matchingquality strongly depends on the starting point of the scanThecontrol-process has to detect multiple scans of each fibre asduplicates and to select the best match of them

The pattern-matching processes deliver lists of matchinginformation bottom coordinates (

119894) directions (

119894) length

(Δ119911119894) radius (119903

119894) and included coloured voxels aliasmatching

hits 119878119894(1198940 ) of (7)

The control-process uses the geometric matching infor-mation (

119894 119894 119903119894 Δ119911119894) to generate vector representations

Each fibre-match is represented as a pair of nested prismsThe outer prism dimensions are equal to those of the fibre(119894 119894 119903119894 Δ119911119894) Scaling down radius (index harr) and length

(index ) of the inner prism is done independently keepinginner and outer prism-centre upon each other

119903119894119868fl 119904harr

sdot 119903119894

Δ119911119894119868fl 119904sdot Δ119911119894

119904harr 119904lt 1

119894+

Δ119911119894

2sdot 119894= 119894119868

+Δ119911119894119868

2sdot 119894

(8)

Y

X

50000

40000

30000

20000

8000

0

700

00

600

00

500

00

400

00

300

00

2000

0

1000

0 0

20000

30000

40000

50000

Z60000

70000

80000

90000

100000

110000

40

3000

0

Figure 2 Pairs of outer and inner prisms representing fibresembedded into a binning system

This prism-pairs are embedded into a 3D-space To detectduplicate scan-results of a fibre inner prisms are tested foroverlapping In order to prevent O(119873

2

) overlapping tests allfibres are managed by a binning system as seen in Figure 2The binning system consists of blocks subdividing the testvolume Each binning-cell contains information about allobjects touching it This reduces the number of overlappingtests to the objects touching the same subvolumes

In case of duplicates 119878119894(119894 119894) is used as heuristic to sort

out the best match

23 Iterative Strategy The goal of iterative scanning of thevolume is to remove well-found fibres in order to getresidual fibres more separated and therefore easier to detectin subsequent scans speeding up the first step of the Monte-Carlo scanning for separate fibresThe procedure is somehowsimilar to separating bundles in the game of mikado

As mentioned earlier the model-based algorithm iteratesthrough a loop of parallel distributed pattern-matching sub-processes controlled by a centralized main process mergingthe match-data and modifying the volume image

The central process also decides which fibres to deletefrom the volume image and how to adapt the parameters ofthe distributed pattern-matching before passing them to thesubprocesses via network

In the first iterations a descending fraction

120585119894=

119878119894( 119903 Δ119911)

120587 sdot 1198772 sdot Δ119911 (9)

of possible voxels of a matched fibre is usedIn the case |120585

119894minus 120585119894minus1

| lt Δ120585 radius boundaries and lengthare considered as well

Depending on how many fibres do contribute to therecent deletion-criteria parameters for pattern-matchingsubprocesses are adapted decrease the pattern radius 119903

119878in

order to find separate fibre parts accumulations increase 120575 in(2) in order to tolerate bigger colour-differences and increase120575 in (6) to enhance tolerance against blurring or cloudy fibres

The number of iterations needed to detect a specificfraction of fibres obviously depends on the image-quality thatis the contrast between matrix and filler as well on relation


065

07

075

08

085

09

095

1

105

11

115

0 1 2 3 4 5 6 7 8Iteration

pp-r-gf 80 (v) = 200 (w)pp-r-gf 125 (v)

pp-r-gf 20 (v)

pp-r-gf 40 (v)pp-r-gf 10 (v)

pbt-r-gf 100 (v)

Frac

tion

of g

lass

(vox

els)

Figure 3 Iterative detection fraction ofmatched voxels for differentmatrix-materials and different filler content

500

450

400

350

300

250

200

150

100

50

01 2 3 4 5

Iteration

Timing

Cyl-extrusionDir-analysisFind-match

Tim

e (se

c)

Figure 4 Iterative detection timing for different matching subpro-cesses high numerical effort for solving cylinder integrals

resolution and fibre radius Figure 3 shows for a voxel size of17 120583 and a fibre radius of 6 120583 that in case of poly-propylene(PP) containing up to 8-volume glass fibres 4 iterationsare needed to detect 90 of the fibre volume whereas poly-butylene terephthalate (PBT) compounds with 10-volumeof glass fibres need 7 iterations in order to detect at least 90of the fibres-volume

Figure 4 shows for PP with 4-volume glass fibres thetime fractions for each step of the pattern-matching of allsubprocesses Calculating the cylinder extrusion integrals

x y

z

Figure 5 Separation of virtually overlapping fibres Step 1 scanningfor fibres (spheres) and direction analysis (blue arrows)

x y

z

Figure 6 Separation of virtually overlapping fibres Step 2 calculat-ing cylinder integrals and refining orientation

is the most time consuming step The time consumed isproportional to the amount of undetected fibres The timeneeded for searching the volume image to find separate fibresis about constant in each iteration

24 Algorithm Robustness Reconstructed image volumequality depends on density contrast of filler to matrix fillercontent and image resolution Low density contrast resultsin blurred object boundaries High filler content yields toseemingly connected objects Low image resolution leads toextra or missing voxels on object boundaries

In order to separate seemingly connected objects eachobject is scannedmultiple times (see Figure 5)This automat-ically happens due to the fact that all pattern-matching pro-cesses are independent and randomly choose their startingpoints

Accuracy of orientation- and length-detection may varyfor each starting point (Figure 6) and so the controlling pro-cess decides upon heuristics regarding which combination of0 Δ119911 to keep and which one to drop (Figure 7)In order to reduce the effect of blurred object boundaries

cylinder integral directions are varied and the maximumof 119878(

0 ) is searched Using point-symmetric coordinates

allows us to separate fibres very efficientlyIn case of low image contrast between filler and matrix

120576Δ119911

can be reduced to tolerate missing or extra voxels withinthe cylinder-volume by increasing the acceptance of partialintegral sum 119878

Δ119894


z

x y

Figure 7 Separation of virtually overlapping fibres Step 3 optimiz-ing results by removing duplicate detection

z

x

yyy

Figure 8 Qualitative testing using virtual fibres plotted into avolume image Detected fibres are displayed as prisms surroundingtheir associated voxels

3 Results

First the algorithm was tested using computed volumeimages in order to estimate error magnitudes and effective-ness under various idealized circumstances After that shortglass fibre reinforced PP composites of 05 vol 10 vol20 vol 40 vol and 80 vol to 14 vol Finally shortglass fibre reinforced PBT composites including 20 vol wereanalyzed

31 Fibre Detection Quality According to the procedureproposed in [25] the algorithm was tested in several stepsIt was applied to virtual generated fibre volume images andlow filler content compounds and its results were comparedto those of manual fibre tracking using slices of tomographicvolume images as well as microscopic pictures of residualfibres after removing the matrix material by combustion

Plotting virtual fibres into a volume (see Figure 8) wasdone to produce a perfectly known system Additionally thisallows us to analyze the effects of different picture defects likeblurring and noise against detection efficiency separately

Compounds with reduced filler content (Figure 9) wereproduced to monitor each step of the detection algorithmapplied to real fibres Due to the low filler content the fibrescan be separated by the human eye and this allows manualcomparison of detection data to real fibre voxels In contrast

z

xy

Figure 9 Qualitatively testing using low fibre content compounds(poly-propylenewith 05 (vol) glass fibres) scanning for fibres andorientation analysis

z

xy

Figure 10 Qualitatively testing using low fibre content compounds(poly-propylene with 05 (vol) glass fibres) cylinder integralcalculation

to virtual fibres real fibre radiusmaynot be constant and theircurvatures may not vanish

As shown in Figure 10 aside from pieces of broken fibresmost fibres are detected correctly

In the next step 2D-slices (119909-119910 layers) of tomographicimages of commercial available fibre reinforced compositeswere used to manually track the fibres in order to comparerelative magnitudes of the main components 119886

11and 119886

22of

the orientation tensor in (11) between the tracking data andthe algorithm results

In Figure 11 results for 11988611

are shown for several slicesalong the 119911-axis In order to check whether fibre length maybe an issue for human tracking the tracked fibres were sortedby length and groups of upper 10 20 30 and 50and allof them were taken into account to calculate the orientationtensor Within Figure 11 different sizes of the marks ldquotimesrdquohave been used to illustrate the different fibre lengths Ascan be seen there is no significant influence on the fibreorientation In order to assign automatically detected fibredata well defined to 119911-slices the centre point of each fibre wastested to lie within To examine scattering of the orientationdata the thickness of the 119911-slices was refined to be 100 3320 and 11 of the gap between the tomographic 119909-119910 layerpictures


1

0

08

01

06

02

04

03

02

04

0

1

08

06

04

02

0

1

08

06

04

02

005 06 07 08 09 1

0 01 02 03 04 05 06 07 08 09 1

0 01 02 03 04 05 06 07 08 09 1Relative thickness zzmax

Relative thickness zzmax


efine 1aw refine 1efine 3efine 5efine 9

Manual u-frac 099

Manual u-frac 05

Manual u-frac 03

Manual u-frac 02

Manual u-frac 01

Orie

ntat

ion-

tens

or-e

lem

ent

Orie

ntat

ion-

tens

or-e

lem

ent

Orie

ntat

ion-

tens

or-e

lem

ent

0 01 02 03 04 05 06 07 08 09 1



120583CT r120583CT r120583CT r120583CT r120583CT r

Figure 11 Comparison of the main orientation tensor elementbetweenmanual tracking (tomographic slices) and automatic detec-tion of glass fibres (8-vol) in a poly-amide compound

Except some slices of the PBTdata (Figure 11) orientationinformation of tracked fibres and automatically detectedfibres are in good agreement

Manually tracking fibres usingmicroscope images of fibrecontent after probe combustion provides fibre length statisticsto match

32 Spatial Averaging The fibre vector representationswhich are generated using geometric matching information(119894 119894 119903119894 Δ119911119894) can be analyzed to get spatial distribution

of fibre information 119886119894119895(119909 119910 119911) One way to achieve this

is to divide the test volume into regions A fibre is takeninto account for analysis if its centre lies within a specificregion The region can be any combination of polyhedronandor spheres In most cases the volume is divided into

1000

0

800

50

600

100

400

150

200

2000 8004000

250 300

z

y

804

Figure 12 Equidistant subdividing a volume into cuboids in orderto spatialize fibre representations

equidistant boxes (119909-slices see Figure 12) in order to examineintermediate layer orientation of fibres within a part

33 Orientation Analysis In order to describe the fibreorientation state a second-order tensor

119860 = (

11988611

11988612

11988613

11988621

11988622

11988623

11988631

11988632

11988633

) (10)

is used [26]Therefore fibres are characterized as cylinders ofequal and constant radius119877 equal length 119897 starting points

119894

and directions 119894to determine the components of119860 in (10) as

119886119894119895=

1

119873

119899

sum

119896=1

(119886119896)119894119895=

1

119873

119899

sum

119896=1

(119899119896)119894sdot (119899119896)119895 (11)

Taking into account variations in radius 119903119894and length 119897

119894 this

yields to a volume weighted form

119886119894119895=

1

sum119873

119896=1Δ119911119896sdot 1199032119896

sdot

119873

sum

119896=1

Δ119911119896sdot 1199032

119896sdot (119899119896)119894sdot (119899119896)119895 (12)

Figure 13 shows an example of the main orientationtensor components which can be used directly in this formto validate numerical results from flow simulation

34 Length Analysis For basic validation of length analysisresults several mixtures of virtual fibres with fixed lengthsand correlated random orientation were plotted into a 3D-volume of voxels Figure 14 shows an example detection offibre-lengths with 200 300 and 400 voxels In general thepeak of the detected lengths lies within an error of 10depending on 120576

Δ119911 Overcompensated lowering of 120576

Δ119911due to

bad picture quality might yield to erroneous fibre elongationIn order to validate the resultant lengths on real world

probes obtained by the algorithm we used a crucible toremove matrix material by combustion and viewed theresidual fibres under an optical microscope (Figure 15) wherethe fibres were tracked manually Thereby fibres touching the


a11a22a33

0

02

04

06

08

1

Tens

or co

mpo

nent

05 1 15 2 250z-thickness (mm)

Figure 13 Main orientation tensor components

0

1000

2000

3000

4000

5000

100 150 200 250 300 350 400 450 500Length (voxels)

215

310

402

Occ

urre

ncetimes

leng

th

Figure 14 Example virtual fibre length analysis of a mixture of fibres with original lengths of 200 300 and 400 voxels

Figure 15 Stitched microscope pictures of residual fibres after combustion of the polymer material


0

10000

20000

30000

40000

50000

60000

70000

80000

0 250 500 750 1000 1250 15000

5000

10000

15000

20000

25000

30000

35000

Microscope

Length (120583m)

Occ

urre

ncetimes

leng

th (m

icro

scop

e)

Occ

urre

ncetimes

leng

th (120583

CT)

120583CT

Figure 16 Comparison of length analysis between manual fibretracking of microscope pictures and automatic 120583CT analysis

0

5

10

15

20

25

6 8 10 12 14 16 18

Occ

urre

nce

88

128

168

Diameter (voxels)

Figure 17 Virtual fibre diameter analysis distribution of a mixtureof virtual fibre configurations with original diameters of 80 120and 160 voxels Detected fibre diameters are within an error of onevoxel size

borders were not taken into account Figure 16 shows thecomparison of length analysis between manual fibre trackingand the computation As can be seen the length distributionobtained by the proposed algorithm is in a good agreementto the experimental results

35 Radius Analysis Validating radius results was analogousto the procedurewhichwas used to validate length detectionsFirst virtual fibres were analyzed Randomly chosen fibresfrom a small set of configurations (direction length andradius) was plotted at different positions inside a voxel-volume Using the geometric representatives (nested prisms)it is possible to avoid crossings between the fibres Invertednested prisms (soft shell inside hard shell) are used in orderto adjust minimal radial and tangential distances Figure 17shows detected diameter distributions for virtual fibres of 812 and 16 voxels In general diameters of perfectly plotted

0

500

1000

1500

2000

2500

12 14 16 18 20

Occ

urre

nce

= 168120583m= 133 120583m

Original = 140 120583m

Mean-stddevMean-mean

Diameter (1E minus 6m)

Figure 18 Several diameter analyses of fibre reinforced compoundusing the same glass fibre material Original value of the fibrematerial is marked with a dashed line

virtual fibres are overestimated by an error of about one voxelsize whereas real fibre diameters are slightly underestimatedabout one voxel size too

Subsequently different compounds which were rein-forced with the same glass fibre material (original diameter14 120583) were analyzed Figure 18 shows that the mean diametergenerally underestimates the original diameter by half ofa voxel size (09 120583) The mean relative error of the radiusdetection of fibres approximately corresponds to the pictureresolution For a standard resolution of 18 120583 and commercialavailable standard compounds (diameter 14120583) this results ina relative error of approximate lt7

4 Reproducibility

To test the reproducibility of the results for a given tomo-graphic volume image a poly-amide compound with 8volume fraction of glass fibres was analyzed 32 times and theprobe volume was divided into 15 slices For each of the 15slices all elements of the orientation tensor were calculatedand an average deviation of all elements for all slices of lt0015was achieved The deviation of all deviations was lt00033

Analyzing different tomographic volume images of thesame specimen again using 15 slices lead to an averagestandard deviation for all orientation tensor elements of lessthan 0025 with a standard deviation of all deviations of lessthan 0011

5 Conclusion and Outlook

Iterative model-based algorithms to analyze short fibre glassreinforced polymers yield a reproducible and robust methodto characterize fibre morphology The basic algorithm can beadapted to bad picture quality and high filler content so thatpartially crossed fibres can be separated directly

The model-based approach was successfully applied oncomputed volume graphics as well as on short glass fibrereinforced composites of poly-propylene poly-amide andpoly-butyl-terephthalate up to commercial filler contents of30 weight


In most cases 3 or 4 iterations are sufficient to detect atleast 90 of the total fibre volume

The results of the algorithm were validated by comparingthem to manual tracked fibres within tomographic slicesand microscopic images Reproducibility was verified forrepetitive analysis of tomographic volume images as wellas for different tomographic volume images of the samespecimen

Accuracy and speed of the analysis depend on the aspectratio of the fibres and on image contrast Because its statisticalbasis the algorithm is highly scalable Due to its file basedprotocol it can be used in heterogeneous clusters already

In future the geometric analysis will be extended to sup-port semiautomatic damage examination Therefore spatialinhomogeneities of the fibre distribution as well as additionalinformation about transversal and longitudinal distances ofthe fibres in combination with their length distribution willbe used to mark regions of possible fibre fraction Theseimplementations will additionally be ported to GPUs usingCUDA-techniques

Competing Interests

The authors declare that they have no competing interests

References

[1] G Fischer ldquoQuantitative Ermittlung der Orientierung vonKurzglasfasern mit der Bildanalyserdquo Kunststoffe vol 77 no 5pp 509ndash512 1987

[2] H L Cox ldquoThe elasticity and strength of paper and otherfibrous materialsrdquo British Journal of Applied Physics vol 3 no3 pp 72ndash79 1952

[3] J C Halpin and N J Pagano ldquoThe laminate approximation forrandomly oriented fibrous compositesrdquo Journal of CompositeMaterials vol 3 no 4 p 720 1969

[4] H Fukuda and T-W Chou ldquoA probabilistic theory of thestrength of short-fibre composites with variable fibre length andorientationrdquo Journal ofMaterials Science vol 17 no 4 pp 1003ndash1011 1982

[5] I M Robinson and J M Robinson ldquoThe influence of fibreaspect ratio on the deformation of discontinuous fibre-rein-forced compositesrdquo Journal of Materials Science vol 29 no 18pp 4663ndash4677 1994

[6] S-Y Fu and B Lauke ldquoThe elasticmodulus ofmisaligned short-fiber-reinforced polymersrdquo Composites Science and Technologyvol 58 no 3-4 pp 389ndash400 1998

[7] B Mlekusch ldquoThermoelastic properties of short-fibre-reinforced thermoplasticsrdquo Composites Science and Technologyvol 59 no 6 pp 911ndash923 1999

[8] SW Jung S Y KimHWNam andK S Han ldquoMeasurementsof fiber orientation and elastic-modulus analysis in short-fiber-reinforced compositesrdquoComposites Science and Technology vol61 no 1 pp 107ndash116 2001

[9] H R Lusti P J Hine and A A Gusev ldquoDirect numericalpredictions for the elastic and thermoelastic properties of shortfibre compositesrdquo Composites Science and Technology vol 62no 15 pp 1927ndash1934 2002

[10] R P Hegler G Mennig and C Schmauch ldquoPhase separa-tion effects in processing of glass-bead-and glass-fiber-filled

thermo-plastics by injection moldingrdquo Advances in PolymerTechnology vol 7 no 1 pp 3ndash20 1987

[11] G B Jeffery ldquoThe motion of ellipsoidal particles immersed in aviscous fluidrdquo Proceedings of the Royal Society of London SeriesA vol 102 no 715 pp 161ndash179 1922

[12] F Folgar and C L Tucker III ldquoOrientation behavior of fibersin concentrated suspensionsrdquo Journal of Reinforced Plastics andComposites vol 3 no 2 pp 98ndash119 1984

[13] U Mohr-Matuschek Auslegung von Kunststoff- und Elastomer-formteilen mittels Finite-Elemente-Simulationen [PhD thesis]Technische Hochschule Aachen Aachen Germany 1992

[14] R S Bay and C L Tucker ldquoFiber orientation in simple injectionmoldings Part II experimental resultsrdquo Polymer Compositesvol 13 no 4 pp 332ndash341 1992

[15] K K Kratmann M P F Sutcliffe L T Lilleheden R Pyrzand O TThomsen ldquoA novel image analysis procedure for mea-suring fibre misalignment in unidirectional fibre compositesrdquoComposites Science and Technology vol 69 no 2 pp 228ndash2382009

[16] L Zedler ldquoBildanalytische Rekonstruktion dreidmensionalerFullstoffstrukturenrdquo AIF 10833BVI Report German Institutefor Polymers Darmstadt Germany 1998

[17] C Eberhardt and A Clarke ldquoFibre-orientation measure-ments in short-glass-fibre composites Part I automated high-angular-resolution measurement by confocal microscopyrdquoComposites Science and Technology vol 61 no 10 pp 1389ndash1400 2001

[18] G Zak M Haberer C B Park and B Benhabib ldquoEstimation ofaverage fibre length in short-fibre composites by a two-sectionmethodrdquo Composites Science and Technology vol 60 no 9 pp1763ndash1772 2000

[19] J Goebbels HHeidt A Kettschau and P Reimers ldquoComputer-ized tomography of glass-fiber reinforced plastic componentsrdquoin Non-Destructive TestingmdashProceedings of the 4th EuropeanConference vol 3 pp 2111ndash2113 NDT International 1987

[20] C N Eberhardt and A R Clarke ldquoAutomated reconstructionof curvilinear fibres from 3D datasets acquired by X-raymicrotomographyrdquo Journal ofMicroscopy vol 206 no 1 pp 41ndash53 2002

[21] P J Schilling B P R Karedla A K Tatiparthi M A Vergesand P D Herrington ldquoX-ray computed microtomography ofinternal damage in fiber reinforced polymer matrix compos-itesrdquo Composites Science and Technology vol 65 no 14 pp2071ndash2078 2005

[22] J S U Schell M Renggli G H van Lenthe R Mullerand P Ermanni ldquoMicro-computed tomography determinationof glass fibre reinforced polymer meso-structurerdquo CompositesScience and Technology vol 66 no 13 pp 2016ndash2022 2006

[23] VMuller B Brylka F Dillenberger R Glockner S Kolling andT Bohlke ldquoHomogenization of elastic properties of short-fiberreinforced composites based onmeasuredmicrostructure datardquoJournal of Composite Materials vol 50 no 3 pp 297ndash312 2016

[24] J Ohser and K Schladitz 3D Images of Materials StructuresProcessing and Analysis Wiley-VCH Weinheim Germany2009

[25] H Shen S Nutt and D Hull ldquoDirect observation and mea-surement of fiber architecture in short fiber-polymer compositefoam through micro-CT imagingrdquo Composites Science andTechnology vol 64 no 13-14 pp 2113ndash2120 2004

[26] S G Advani andC L Tucker III ldquoThe use of tensors to describeand predict fiber orientation in short fiber compositesrdquo Journalof Rheology vol 31 article 751 1987

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



recently been used in [23] to compute the elastic propertiesof a short-fibre plastic based on realmeasuredmicrostructuredata by homogenization In the present work we set our focuson the numerical methodology of fibre detection and discussthe Monte-Carlo algorithm in detail

2 Model-Based Algorithm

A common approach of model-based image analysis is toextract image primitives and to establish a correspondencebetween these and the model primitives Probabilistic objectpattern matching speeds up the scanning process providesgood scaling capabilities if done in parallel and increasestolerance against noise and distortion For an overview ofimage analysis of materials structures the reader is referredto [24] and the references therein

The fibre model consists of a chord of cylinders ofconstant radii and various lengths The radius of scannablefibres is limited by the minimum cylinder length

All Monte-Carlo pattern-matching processes run inde-pendent of each other Their results are merged in a control-process which detects doublets and optimizes the vector dataSubsequently this process modifies the voxel data as wellas the parameters of the pattern-matching processes andinitiates their restart

21 Pattern Matching Core algorithm steps are Monte-Carlo scanning for separate fibre parts and cylinder integralextrusion for fibre length detection Scanning for separatefibre parts (index 119894) is done by calculating radial densityfunctions 120588lowast(119903

119894) equiv 120588lowast

119894in randomly chosen spherical volumes

(centres 119894)

120588lowast

119894=

| 119903|lt119903119894

sum

| 119903|ge119903119894minus1

119892lowast

( 119903) (1)

where

119892lowast

( 119903) fl

119892 ( 119903) if 119892 ( 119903) minus 119892 (minus 119903) le 120575

0 else(2)

and 120575 is a chosen toleranceTo approximate radial density the spherical volume is

divided into spherical shell lists of random point-symmetriccoordinate pairs within nonequidistant radius-intervals

0 = 1199030lt 1199031lt sdot sdot sdot lt 119877 lt sdot sdot sdot lt 119903

119878 (3)

where 119877 denotes the fibre radius and 119903119878is the pattern

radius of the final sphere Using correlating point-symmetriccoordinates according to (2) removes voxels from neighbourfibres as well as noise Figure 1 shows the radial density 120588

lowast

119894as

a function of the integral sphere radius 119903119894and the cylinder

radius 119877 which represents the fibre The density that is theprobability that a random point within the sphere is alsolocated within the fibre is 1 for 119903

119894lt 119877 For an increasing

sphere radius 119903119894gt 119877 we have a decreasing probability and

thus a decreasing density distribution which finally tends tozero

0 5 10 15 20 25 30 45

6 78

910

1112

Integral-sphere radius Cylinder-radius

0010203040506070809

1

Den

sity

Figure 1 Geometry detection of fibres using radial density distri-bution for different cylinder radii 119877 Note the strongly decreasinghyperbolic function for 119903

119894gt 119877

The volume119881119904of the spherical shell defined by radii 119903

119894and

119903119894minus1

is given by119881119904= (43)120587(119903

3

119894minus 1199033

119894minus1) And the corresponding

volume of the spherical cylinder can be approximated by119881119888=

1205871198772

(119903119894minus119903119894minus1

) for 119903119894gt 119877Thus the probability that a randomly

chosen point can be found within the cylinder shell is 120588lowast

119894=

119881119888119881119904and thus

4

3(1199033

119894minus 1199033

119894minus1) 120588lowast

119894997888rarr 119877

2

(119903119894minus 119903119894minus1

) for 119903119894gt 119877 (4)

which is used as a convergence criterion Only if this condi-tion is satisfied a separate fibre has been found and the centre119894of the scanned sphere volume is used for further fibre

detection Otherwise a spherical volume around anotherrandomly chosen within the sphere is checked

After locating separate fibre parts at 119894 the fibre orienta-

tion vector 119894is calculated either via diagonalizing the matrix

of second-order moments or using Jacobian matrices Thefibre length Δ119911fibre is determined by approximating cylinderintegrals using circular coordinate pairs as defined in (6)and using ordinary parameters as angle 120572 and radial (119903) andtransverse (119911) distances from the origin (

119894)

119892cyl (119903 119911 120572)

= 119892cyl [119894 + 119911119894+ 119903 (sin (120572) 119905

1+ cos (120572) 119905

2)]

(5)

where the normalized vectors 119894 1199051 and 119905

2are perpendicular

to each other and form a right hand system that is (119894times 1199051) sdot

1199052= 1

119892lowast

cyl (119903 119911 120572)

=

119892 (119903 119911 120572) if 119892 (119903 119911 120572) minus 119892 (minus119903 119911 120572) le 120575

0 else

(6)

For a given starting point and direction the fibre is madeup of small cylinders 119911

119894= 119911119894minus1

+Δ119911119894 until the last partial sum



lt 119878Δ0

sdot


119894


119878119894fl 119878Δ119894

(119894 119911119894 Δ119911) fl

119911=119911119894+Δ119911

sum

119911=119911119894

119903=119877

sum

119903=0

120572=2120587

sum

120572=0

119892lowast

cyl (119903 119911 120572) (7)




119896


(4) Compute mass 119898119896+1



(5) Use 119896+1


(6) 119896 = 119896 + 1 go to (3) until119898119896+1

minus 119898119896lt 120575





extrusion




119894) length

(Δ119911119894) radius (119903


hits 119878119894(1198940 ) of (7)





119903119894119868fl 119904harr

sdot 119903119894

Δ119911119894119868fl 119904sdot Δ119911119894

119904harr 119904lt 1

119894+

Δ119911119894

2sdot 119894= 119894119868

+Δ119911119894119868

2sdot 119894

(8)

Y

X

50000

40000

30000

20000

8000

0

700

00

600

00

500

00

400

00

300

00

2000

0

1000

0 0

20000

30000

40000

50000

Z60000

70000

80000

90000

100000

110000

40

3000

0



2



out the best match





120585119894=

119878119894( 119903 Δ119911)

120587 sdot 1198772 sdot Δ119911 (9)


119894minus 120585119894minus1



119878in




065

07

075

08

085

09

095

1

105

11

115

0 1 2 3 4 5 6 7 8Iteration

pp-r-gf 80 (v) = 200 (w)pp-r-gf 125 (v)

pp-r-gf 20 (v)


pbt-r-gf 100 (v)

Frac

tion

of g

lass

(vox

els)


500

450

400

350

300

250

200

150

100

50

01 2 3 4 5

Iteration

Timing


Tim

e (se

c)




x y

z


x y

z









120576Δ119911


Δ119894


z

x y


z

x

yyy


3 Results





z

xy


z

xy





11and 119886

22of





1

0

08

01

06

02

04

03

02

04

0

1

08

06

04

02

0

1

08

06

04

02

005 06 07 08 09 1

0 01 02 03 04 05 06 07 08 09 1





Manual u-frac 099

Manual u-frac 05

Manual u-frac 03

Manual u-frac 02

Manual u-frac 01

Orie

ntat

ion-

tens

or-e

lem

ent

Orie

ntat

ion-

tens

or-e

lem

ent

Orie

ntat

ion-

tens

or-e

lem

ent

0 01 02 03 04 05 06 07 08 09 1










1000

0

800

50

600

100

400

150

200

2000 8004000

250 300

z

y

804




119860 = (

11988611

11988612

11988613

11988621

11988622

11988623

11988631

11988632

11988633

) (10)


119894


119886119894119895=

1

119873

119899

sum

119896=1

(119886119896)119894119895=

1

119873

119899

sum

119896=1

(119899119896)119894sdot (119899119896)119895 (11)


119894 this


119886119894119895=

1

sum119873

119896=1Δ119911119896sdot 1199032119896

sdot

119873

sum

119896=1

Δ119911119896sdot 1199032

119896sdot (119899119896)119894sdot (119899119896)119895 (12)




Δ119911due to




a11a22a33

0

02

04

06

08

1

Tens

or co

mpo

nent



0

1000

2000

3000

4000

5000

100 150 200 250 300 350 400 450 500Length (voxels)

215

310

402

Occ

urre

ncetimes

leng

th




0

10000

20000

30000

40000

50000

60000

70000

80000

0 250 500 750 1000 1250 15000

5000

10000

15000

20000

25000

30000

35000

Microscope

Length (120583m)

Occ

urre

ncetimes

leng

th (m

icro

scop

e)

Occ

urre

ncetimes

leng

th (120583

CT)

120583CT


0

5

10

15

20

25

6 8 10 12 14 16 18

Occ

urre

nce

88

128

168

Diameter (voxels)




0

500

1000

1500

2000

2500

12 14 16 18 20

Occ

urre

nce

= 168120583m= 133 120583m







4 Reproducibility











Competing Interests


References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











lt 119878Δ0

sdot


119894


119878119894fl 119878Δ119894

(119894 119911119894 Δ119911) fl

119911=119911119894+Δ119911

sum

119911=119911119894

119903=119877

sum

119903=0

120572=2120587

sum

120572=0

119892lowast

cyl (119903 119911 120572) (7)




119896


(4) Compute mass 119898119896+1



(5) Use 119896+1


(6) 119896 = 119896 + 1 go to (3) until119898119896+1

minus 119898119896lt 120575





extrusion




119894) length

(Δ119911119894) radius (119903


hits 119878119894(1198940 ) of (7)





119903119894119868fl 119904harr

sdot 119903119894

Δ119911119894119868fl 119904sdot Δ119911119894

119904harr 119904lt 1

119894+

Δ119911119894

2sdot 119894= 119894119868

+Δ119911119894119868

2sdot 119894

(8)

Y

X

50000

40000

30000

20000

8000

0

700

00

600

00

500

00

400

00

300

00

2000

0

1000

0 0

20000

30000

40000

50000

Z60000

70000

80000

90000

100000

110000

40

3000

0



2



out the best match





120585119894=

119878119894( 119903 Δ119911)

120587 sdot 1198772 sdot Δ119911 (9)


119894minus 120585119894minus1



119878in




065

07

075

08

085

09

095

1

105

11

115

0 1 2 3 4 5 6 7 8Iteration

pp-r-gf 80 (v) = 200 (w)pp-r-gf 125 (v)

pp-r-gf 20 (v)


pbt-r-gf 100 (v)

Frac

tion

of g

lass

(vox

els)


500

450

400

350

300

250

200

150

100

50

01 2 3 4 5

Iteration

Timing


Tim

e (se

c)




x y

z


x y

z









120576Δ119911


Δ119894


z

x y


z

x

yyy


3 Results





z

xy


z

xy





11and 119886

22of





1

0

08

01

06

02

04

03

02

04

0

1

08

06

04

02

0

1

08

06

04

02

005 06 07 08 09 1

0 01 02 03 04 05 06 07 08 09 1





Manual u-frac 099

Manual u-frac 05

Manual u-frac 03

Manual u-frac 02

Manual u-frac 01

Orie

ntat

ion-

tens

or-e

lem

ent

Orie

ntat

ion-

tens

or-e

lem

ent

Orie

ntat

ion-

tens

or-e

lem

ent

0 01 02 03 04 05 06 07 08 09 1










1000

0

800

50

600

100

400

150

200

2000 8004000

250 300

z

y

804




119860 = (

11988611

11988612

11988613

11988621

11988622

11988623

11988631

11988632

11988633

) (10)


119894


119886119894119895=

1

119873

119899

sum

119896=1

(119886119896)119894119895=

1

119873

119899

sum

119896=1

(119899119896)119894sdot (119899119896)119895 (11)


119894 this


119886119894119895=

1

sum119873

119896=1Δ119911119896sdot 1199032119896

sdot

119873

sum

119896=1

Δ119911119896sdot 1199032

119896sdot (119899119896)119894sdot (119899119896)119895 (12)




Δ119911due to




a11a22a33

0

02

04

06

08

1

Tens

or co

mpo

nent



0

1000

2000

3000

4000

5000

100 150 200 250 300 350 400 450 500Length (voxels)

215

310

402

Occ

urre

ncetimes

leng

th




0

10000

20000

30000

40000

50000

60000

70000

80000

0 250 500 750 1000 1250 15000

5000

10000

15000

20000

25000

30000

35000

Microscope

Length (120583m)

Occ

urre

ncetimes

leng

th (m

icro

scop

e)

Occ

urre

ncetimes

leng

th (120583

CT)

120583CT


0

5

10

15

20

25

6 8 10 12 14 16 18

Occ

urre

nce

88

128

168

Diameter (voxels)




0

500

1000

1500

2000

2500

12 14 16 18 20

Occ

urre

nce

= 168120583m= 133 120583m







4 Reproducibility











Competing Interests


References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










065

07

075

08

085

09

095

1

105

11

115

0 1 2 3 4 5 6 7 8Iteration

pp-r-gf 80 (v) = 200 (w)pp-r-gf 125 (v)

pp-r-gf 20 (v)


pbt-r-gf 100 (v)

Frac

tion

of g

lass

(vox

els)


500

450

400

350

300

250

200

150

100

50

01 2 3 4 5

Iteration

Timing


Tim

e (se

c)




x y

z


x y

z









120576Δ119911


Δ119894


z

x y


z

x

yyy


3 Results





z

xy


z

xy





11and 119886

22of





1

0

08

01

06

02

04

03

02

04

0

1

08

06

04

02

0

1

08

06

04

02

005 06 07 08 09 1

0 01 02 03 04 05 06 07 08 09 1





Manual u-frac 099

Manual u-frac 05

Manual u-frac 03

Manual u-frac 02

Manual u-frac 01

Orie

ntat

ion-

tens

or-e

lem

ent

Orie

ntat

ion-

tens

or-e

lem

ent

Orie

ntat

ion-

tens

or-e

lem

ent

0 01 02 03 04 05 06 07 08 09 1










1000

0

800

50

600

100

400

150

200

2000 8004000

250 300

z

y

804




119860 = (

11988611

11988612

11988613

11988621

11988622

11988623

11988631

11988632

11988633

) (10)


119894


119886119894119895=

1

119873

119899

sum

119896=1

(119886119896)119894119895=

1

119873

119899

sum

119896=1

(119899119896)119894sdot (119899119896)119895 (11)


119894 this


119886119894119895=

1

sum119873

119896=1Δ119911119896sdot 1199032119896

sdot

119873

sum

119896=1

Δ119911119896sdot 1199032

119896sdot (119899119896)119894sdot (119899119896)119895 (12)




Δ119911due to




a11a22a33

0

02

04

06

08

1

Tens

or co

mpo

nent



0

1000

2000

3000

4000

5000

100 150 200 250 300 350 400 450 500Length (voxels)

215

310

402

Occ

urre

ncetimes

leng

th




0

10000

20000

30000

40000

50000

60000

70000

80000

0 250 500 750 1000 1250 15000

5000

10000

15000

20000

25000

30000

35000

Microscope

Length (120583m)

Occ

urre

ncetimes

leng

th (m

icro

scop

e)

Occ

urre

ncetimes

leng

th (120583

CT)

120583CT


0

5

10

15

20

25

6 8 10 12 14 16 18

Occ

urre

nce

88

128

168

Diameter (voxels)




0

500

1000

1500

2000

2500

12 14 16 18 20

Occ

urre

nce

= 168120583m= 133 120583m







4 Reproducibility











Competing Interests


References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










z

x y


z

x

yyy


3 Results





z

xy


z

xy





11and 119886

22of





1

0

08

01

06

02

04

03

02

04

0

1

08

06

04

02

0

1

08

06

04

02

005 06 07 08 09 1

0 01 02 03 04 05 06 07 08 09 1





Manual u-frac 099

Manual u-frac 05

Manual u-frac 03

Manual u-frac 02

Manual u-frac 01

Orie

ntat

ion-

tens

or-e

lem

ent

Orie

ntat

ion-

tens

or-e

lem

ent

Orie

ntat

ion-

tens

or-e

lem

ent

0 01 02 03 04 05 06 07 08 09 1










1000

0

800

50

600

100

400

150

200

2000 8004000

250 300

z

y

804




119860 = (

11988611

11988612

11988613

11988621

11988622

11988623

11988631

11988632

11988633

) (10)


119894


119886119894119895=

1

119873

119899

sum

119896=1

(119886119896)119894119895=

1

119873

119899

sum

119896=1

(119899119896)119894sdot (119899119896)119895 (11)


119894 this


119886119894119895=

1

sum119873

119896=1Δ119911119896sdot 1199032119896

sdot

119873

sum

119896=1

Δ119911119896sdot 1199032

119896sdot (119899119896)119894sdot (119899119896)119895 (12)




Δ119911due to




a11a22a33

0

02

04

06

08

1

Tens

or co

mpo

nent



0

1000

2000

3000

4000

5000

100 150 200 250 300 350 400 450 500Length (voxels)

215

310

402

Occ

urre

ncetimes

leng

th




0

10000

20000

30000

40000

50000

60000

70000

80000

0 250 500 750 1000 1250 15000

5000

10000

15000

20000

25000

30000

35000

Microscope

Length (120583m)

Occ

urre

ncetimes

leng

th (m

icro

scop

e)

Occ

urre

ncetimes

leng

th (120583

CT)

120583CT


0

5

10

15

20

25

6 8 10 12 14 16 18

Occ

urre

nce

88

128

168

Diameter (voxels)




0

500

1000

1500

2000

2500

12 14 16 18 20

Occ

urre

nce

= 168120583m= 133 120583m







4 Reproducibility











Competing Interests


References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










1

0

08

01

06

02

04

03

02

04

0

1

08

06

04

02

0

1

08

06

04

02

005 06 07 08 09 1

0 01 02 03 04 05 06 07 08 09 1





Manual u-frac 099

Manual u-frac 05

Manual u-frac 03

Manual u-frac 02

Manual u-frac 01

Orie

ntat

ion-

tens

or-e

lem

ent

Orie

ntat

ion-

tens

or-e

lem

ent

Orie

ntat

ion-

tens

or-e

lem

ent

0 01 02 03 04 05 06 07 08 09 1










1000

0

800

50

600

100

400

150

200

2000 8004000

250 300

z

y

804




119860 = (

11988611

11988612

11988613

11988621

11988622

11988623

11988631

11988632

11988633

) (10)


119894


119886119894119895=

1

119873

119899

sum

119896=1

(119886119896)119894119895=

1

119873

119899

sum

119896=1

(119899119896)119894sdot (119899119896)119895 (11)


119894 this


119886119894119895=

1

sum119873

119896=1Δ119911119896sdot 1199032119896

sdot

119873

sum

119896=1

Δ119911119896sdot 1199032

119896sdot (119899119896)119894sdot (119899119896)119895 (12)




Δ119911due to




a11a22a33

0

02

04

06

08

1

Tens

or co

mpo

nent



0

1000

2000

3000

4000

5000

100 150 200 250 300 350 400 450 500Length (voxels)

215

310

402

Occ

urre

ncetimes

leng

th




0

10000

20000

30000

40000

50000

60000

70000

80000

0 250 500 750 1000 1250 15000

5000

10000

15000

20000

25000

30000

35000

Microscope

Length (120583m)

Occ

urre

ncetimes

leng

th (m

icro

scop

e)

Occ

urre

ncetimes

leng

th (120583

CT)

120583CT


0

5

10

15

20

25

6 8 10 12 14 16 18

Occ

urre

nce

88

128

168

Diameter (voxels)




0

500

1000

1500

2000

2500

12 14 16 18 20

Occ

urre

nce

= 168120583m= 133 120583m







4 Reproducibility











Competing Interests


References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










a11a22a33

0

02

04

06

08

1

Tens

or co

mpo

nent



0

1000

2000

3000

4000

5000

100 150 200 250 300 350 400 450 500Length (voxels)

215

310

402

Occ

urre

ncetimes

leng

th




0

10000

20000

30000

40000

50000

60000

70000

80000

0 250 500 750 1000 1250 15000

5000

10000

15000

20000

25000

30000

35000

Microscope

Length (120583m)

Occ

urre

ncetimes

leng

th (m

icro

scop

e)

Occ

urre

ncetimes

leng

th (120583

CT)

120583CT


0

5

10

15

20

25

6 8 10 12 14 16 18

Occ

urre

nce

88

128

168

Diameter (voxels)




0

500

1000

1500

2000

2500

12 14 16 18 20

Occ

urre

nce

= 168120583m= 133 120583m







4 Reproducibility











Competing Interests


References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

10000

20000

30000

40000

50000

60000

70000

80000

0 250 500 750 1000 1250 15000

5000

10000

15000

20000

25000

30000

35000

Microscope

Length (120583m)

Occ

urre

ncetimes

leng

th (m

icro

scop

e)

Occ

urre

ncetimes

leng

th (120583

CT)

120583CT


0

5

10

15

20

25

6 8 10 12 14 16 18

Occ

urre

nce

88

128

168

Diameter (voxels)




0

500

1000

1500

2000

2500

12 14 16 18 20

Occ

urre

nce

= 168120583m= 133 120583m







4 Reproducibility











Competing Interests


References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














Competing Interests


References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article a monte-carlo algorithm for 3d fibre...

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