Fast and Accurate 3D X ray Image Reconstruction for Non Destructive Test Industrial Applications

186 Pages·2017·5.04 MB·English

by  Li Wang

Fast and Accurate 3D X ray Image Reconstruction for
Non Destructive Test Industrial Applications
Li Wang
To cite this version:
Li Wang. Fast and Accurate 3D X ray Image Reconstruction for Non Destructive Test Industrial Ap-
plications. ImageProcessing[eess.IV].UniversitéParis-Saclay,2017. English. ￿NNT:2017SACLS362￿.
￿tel-01695872￿
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Fast and Accurate 3D X-ray Image
NNT : 2017SACLE362
Reconstruction for Non Destructive
Testing Industrial Applications
Thèse de doctorat de l'Université Paris-Saclay
préparée au Laboratoire des Signaux et Systèmes (L2S)
École doctorale No. 580 : Sciences et technologie de l’information et
de la communication
Spécialité de doctorat : traitement du signal et des images
Thèse présentée et soutenue à Gif-sur-Yvette, le 01/12/2017, par
Li Wang
Composition du Jury :
Bruno Sixou, MCF (HDR), INSA Lyon
Laboratoire CREATIS Rapporteur
Jan Sijbers, Professeur, Université Antwerp
Department Physique Rapporteur
Udo von Toussaint, DR
Institut de Max-Planck Examinateur
Thomas Rodet, Professeur, ENS Cachan
Laboratoire SATIE Président
Ali Mohammad-Djafari, DR, CNRS
Laboratoire des Signaux et Systèmes (L2S) Directeur de thèse
Nicolas Gac, MCF, Université Paris-Sud
Laboratoire des Signaux et Systèmes (L2S) Co-Encadrant de thèse
Université Paris-Saclay
Espace Technologique / Immeuble Discovery
Route de l’Orme aux Merisiers RD 128 / 91190 Saint-Aubin, France
Acknowledgement
How time flies! It has been more than six years since I came to France! After one year of
college, two years of master and three years of Ph.D., the little ”Orsay Ville” has become my
secondhometown.
The three years of Ph.D. finished in the blink of an eye. Still, I’ve learned a lot from my
supervisorsandmycolleaguesandhavemetalotoffriendsinthewarmfamilyof”Laboratoire
des Signaux et Syste`me”. I would like to express my deepest gratitude and appreciation to all
themembersofthebigL2Sfamilyandallmyfriends.
First and foremost I would like to express my deep gratitude to my advisors Prof. Ali
MOHAMMAD-DJAFARI and Dr. Nicolas GAC. It has been an honor to be one of their Ph.D.
students.IwanttofirstlythankProf.DJAFARIforhiscontinuoussupportandpatientguidance,
for leading me to the charm and profound Bayesian methods. I want to gratefully thank Dr.
GAC,forhiskindnesstoalwaystryhisbesttohelpmesolvetheimplementationproblemsand
opensthedoortothe”GPUworld”forme.
I would also like to thank the members of the Groupe Proble`me Inverse (GPIers). I am gra-
teful to Dr. Aurelia Fraysse and Prof. Charles Soussen for their suggestions and help for my
presentation of my Ph.D. defense. I would like to thank all the GPIers : Dr. Franc¸ois Orieux,
Dr. Mathieu Kowalski, Dr. Mircea Dumitru, Camille Chapdelaine, Amine Hadjyoucef, Alina
Meresescu, Dr. Fangchen Feng, Guillaume Revillon, Thi Thanh Nguyen, Maxime Martelli and
theancientGPIerswhohavealreadyleftbutwillbemembersoftheGPIfamilyforever:Yous-
souf Sameut, Dr. Thomas Boulay, Dr. Long Chen, Dr. Yuling Zheng and Dr. Leila Gharshalli,
Dr. Caifang Cai, for their friendship and help, and a lot of interesting and helpful discussions
duringtheseyears.
I would like to express my deep gratitude for the reviewers of my thesis, Prof. Jan Sijbers
and Dr. Bruno Sixou and the examiners of my Ph.D. defense, Prof. Thomas Todet and Prof.
UdovonToussaint.ItwasagreathonortohavethemasjurymembersandIamdeeplygrateful
fortheirsuggestionsformyworkandthetimetheyinvestedforreadingmymanuscriptandfor
comingtomydefense.
MysincerethanksalsogoestoallmyfriendsandcolleaguesfromCentraleSupe´lecwhoare
verykindwithmeandtakecaresofmeallthetime:MaryvonneGiron,Fre´de´ricDesprez,Ce´line
Labrude, Laurence Antunes, Delphine Maucherat, Dr. Patrice Brault, Dr. Giuseppe Valenzise,
Prof. Dominique Lesselier, Prof. Gilles Duc, Prof. Michel Kieffer, Dr. Zicheng Liu, Yanqiao
Hou, Chen Kang, Dr. Wenjie Li, Shanshan Wang, Jian Song, Xiaojun Xi, Dr. Chi Jin, Weichao
Liang,ZhenyuLiao,PeipeiRan,PauloPrezotti,SaraBerri,Dr.ZhengChen,Dr.ChaoHe...
I also deeply thank all my dearest Chinese friends, who have been together with me for
already ten years : Hui Hua, Dr. Jie Wei, Mengsu Guo, Yao Liu, Kai Wan, Dr. Yunsong Wang,
I
SiqiWang,Dr.XinWang,NanGuan,Dr.XiangLiu.NomatterwhereweareintheworldIwill
alwaysrememberthemforeverandalways.
IwanttothankallthoseIknowandallthosewhoknowme...
But the most important, I devote this thesis to my parents, Feng Wang and Xiuying Li, for
their endless support and love! At last, this thesis should be devoted to my husband, Dr. Chao
ZhangforalwaysbeingbymysidewheneverI’mhappyorsad.
Great thanks to the China Scholarship Council (CSC), this work would not have been pos-
siblewithouttheirsupport.
Table des matie`res
Tabledesfigures VII
Listedestableaux 1
1 Re´sume´ enfranc¸ais 3
1.1 Motivationetcontexte . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Organisationdethe`se . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Lame´thodeHHBM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.5 Lame´thodROCC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.6 Conclusionsetpersipectives . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 Introduction 17
2.1 MotivationandContext . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3 Organizationofthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3 X-rayCTandthestate-of-the-artmethods 23
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2 TheX-rayComputedTomography . . . . . . . . . . . . . . . . . . . . . . . . 24
3.3 TheRadonTransformandAnalyticalreconstructiontechniques . . . . . . . . 24
3.3.1 Thecentralslicetheorem . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3.2 TheBack-projectionfiltering(BPF)algorithm . . . . . . . . . . . . . . 27
3.3.3 TheFilteredBack-projection . . . . . . . . . . . . . . . . . . . . . . . 28
3.4 Thealgebraicmethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.5 TheRegularizationmethods . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.5.1 Differentregularizationcriterion . . . . . . . . . . . . . . . . . . . . . 33
III
3.5.2 Optimizationalgorithms . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.5.3 Theregularizationparameter . . . . . . . . . . . . . . . . . . . . . . . 35
3.6 Thestatisticalmethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.7 TheBayesianinference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.7.1 Bayesianpointestimators . . . . . . . . . . . . . . . . . . . . . . . . 39
3.7.2 LinkbetweenBayesianandRegularizationmethods . . . . . . . . . . 41
3.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4 Bayesianmethodandsparseenforcingpriordistributions 43
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.2 GradientandMarkovianmodels . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.2.1 HomogeneousMarkovianmodel . . . . . . . . . . . . . . . . . . . . . 45
4.2.2 InhomogeneousMarkovianmodels . . . . . . . . . . . . . . . . . . . 46
4.3 Sparsetransformationbasecoefficients . . . . . . . . . . . . . . . . . . . . . . 47
4.4 Sparsityenforcingpriordistributions . . . . . . . . . . . . . . . . . . . . . . . 49
4.4.1 Distributionsforsparsevariables . . . . . . . . . . . . . . . . . . . . . 49
4.4.2 GeneralizedStudent-tdistribution . . . . . . . . . . . . . . . . . . . . 50
4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5 SparsityEnforcingUnsupervisedHierarchicalModel 55
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.2 Thechoiceofthesparsetransformation . . . . . . . . . . . . . . . . . . . . . 55
5.3 AsynthesishierarchicalBayesianmodel . . . . . . . . . . . . . . . . . . . . . 59
5.3.1 TheHierarchicalModel. . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.3.2 Bayesianpointestimation . . . . . . . . . . . . . . . . . . . . . . . . 63
5.3.3 TheJointMAPestimation. . . . . . . . . . . . . . . . . . . . . . . . . 63
5.3.4 Algorithmandimplementation . . . . . . . . . . . . . . . . . . . . . . 67
5.3.5 Simulationresults . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.4 HHBMmethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.4.1 JointMaximumAPosteriorestimation . . . . . . . . . . . . . . . . . 72
5.4.2 Analysisofhyper-parametersandinitialization . . . . . . . . . . . . . 75
5.4.3 Simulationresultswithlimitednumberofprojections . . . . . . . . . . 80
5.4.4 Simulationresultswithlimitedangleofprojections . . . . . . . . . . . 81
5.5 Dual-tree complex wavelet transform (DT-CWT) Based Hierarchical Bayesian
Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.5.1 ComparisonwithHaartransform . . . . . . . . . . . . . . . . . . . . . 89
5.5.2 Truncatingpartsofcoefficientsz . . . . . . . . . . . . . . . . . . . . 89
5.5.3 Addingnoisetocoefficientsz . . . . . . . . . . . . . . . . . . . . . . 91
5.6 Testondifferentimages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.6.1 Test1:Truncationinthetransformcoefficientsz . . . . . . . . . . . . 92
5.6.2 Test2:Addingnoisetothetransformcoefficientsz . . . . . . . . . . 93
5.7 TheuseofDT-CWTinBayesianmethod . . . . . . . . . . . . . . . . . . . . 93
5.7.1 Simulationresults . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.7.2 Conclusionsandperspectives . . . . . . . . . . . . . . . . . . . . . . 94
5.8 VariableSplittingAlgorithmforHHBM . . . . . . . . . . . . . . . . . . . . . 97
5.8.1 VariableSplitting(VS) . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.8.2 VSinBayesianinference . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.8.3 Gaussian-St priormodelforthevariablesplittingmethod(HHBM-GS)100
g
5.8.4 St -St priormodelforthevariablesplittingmethod(HHBM-SS) . . 103
g g
5.8.5 InitializationofhyperparametersforHHBM-GSandHHBM-SSmethods106
5.8.6 Simulationresultsofthenoisesplittingmethod . . . . . . . . . . . . . 107
5.9 Conclusionsandperspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6 SimultaneousObject-ContourReconstructionModel 111
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.2 Forwardmodel:RadonTransformandnotations . . . . . . . . . . . . . . . . 112
6.3 BasicpropertiesofRadontransformation . . . . . . . . . . . . . . . . . . . . 112
6.4 Proposed Method of Reconstruction of Object while Considering Contours
(ROCC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.4.1 Noisemodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
¨
6.4.2 SparsityenforcingStudent-tpriormodelforf . . . . . . . . . . . . . 116
6.4.3 Thepriormodelforobject . . . . . . . . . . . . . . . . . . . . . . . . 117
6.5 Semi-supervisedROCCmethod . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.5.1 Thesystemmodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.5.2 JMAPEstimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.6 Experimentresults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.6.1 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
6.6.2 Simulationresultsofreconstructing3DphantomwithROCCmethod . 124
6.6.3 Initializations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
6.6.4 Simulationresults . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6.7 Comparisonofcomputationtime . . . . . . . . . . . . . . . . . . . . . . . . . 132
6.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
7 ConclusionsandPerspectives 139
7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
7.2 Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
A Posterior Mean estimation via Variational Bayesian Approach (VBA) algorithm
forHHBMmethod 143
A.1 q (f ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
1j j
A.1.1 Computationof(cid:10)(cid:107)g −Hf(cid:107)2(cid:11) . . . . . . . . . . . . . . . . . 145
2 q1,m (fm)
m(cid:54)=j
A.1.2 Computationof(cid:10)(cid:107)f −Dz(cid:107)2(cid:11) . . . . . . . . . . . . . . . 146
2 q1,m (fm)q2(z)
m(cid:54)=j
A.1.3 Computationofq (f ) . . . . . . . . . . . . . . . . . . . . . . . . . . 146
1j j
A.2 Computationofq (z ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
2j j
(cid:0) (cid:1)
A.3 Computationofq v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
3j zj
A.4 Computationofq (v ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
4 (cid:15)
A.4.1 Computationof(cid:10)(cid:107)g −Hf(cid:107)2(cid:11) . . . . . . . . . . . . . . . . . . . 149
2 q1(f)
A.5 Computationofq (v ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
5 ξ
A.6 TheiterativeoptimizationbyusingtheVBAalgorithm . . . . . . . . . . . . . 150
B Mycontributions 153
C Author’spublicationsduring3yearsofPhD 157
Bibliographie 159
Table des figures
2.1 TopviewoftheX-rayCTprojectionssystem. . . . . . . . . . . . . . . . . . . 18
3.1 The industrial components tested by X-ray CT reconstruction. Top : the
outward appearance of the components, bottom : the corresponding recons-
tructed objects. (Image on the left : http ://www.materialstoday.com/hardmetals-and-
ceramics/features/2c-ceramics-moves-into-the-industrial-reality-zone/; image on the right :
http://www.npl.co.uk/science-technology/dimensional/x-ray-computed-tomography.) . . . . 24
3.2 The medical images detected by X-ray CT reconstruction. Brain, bone, teeth,
etc. (Image on the top : http ://www.southlakeregional.org/Default.aspx?cid=784; image
on the bottom left : http ://emedicine.medscape.com/article/355892-overview; image on
thebottomright:http://texas-dental-implants.com/wp-content/uploads/2015/02/PAN-X-rays-
Small.jpeg.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3 TheRadontransformofcontinuousfunction. . . . . . . . . . . . . . . . . . . 26
3.4 TheRadontransformandBackprojectionoperator. . . . . . . . . . . . . . . . 26
3.5 Thecentralslicetheorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.6 TheBack-projectionFilteringreconstructionalgorithm. . . . . . . . . . . . . . 28
3.7 TheFilteredBack-projectionreconstructionalgorithm. . . . . . . . . . . . . . 28
3.8 The Original Shepp Logan figure, the projection sinogram without noise by
using64projectionsandthereconstructionresultsbyusingtheBack-projection
andFilteredBack-projectionmethods. . . . . . . . . . . . . . . . . . . . . . . 29
3.9 Thereconstructedfigurebyusing64projectionsdistributedin[0,π],32projec-
tionsdistributedin[0,π]and64projectionsdistributedin[0, π]. . . . . . . . . 30
2
3.10 TheRadontransformofdiscretesystem. . . . . . . . . . . . . . . . . . . . . . 30
3.11 Reconstruction of 2D Shepp Logan phantom of size 2562 by using the SIRT
method,withrespectively180projections(left)and60projections(right). . . . 32
3.12 The influence of the regularization parameter on the relative mean square error
ofthereconstructionresultsintheQuadraticRegularizationmethodwith128
projections(top)or64projections(bottom)withanoiseofSNR=40dB(left)or
SNR=20dB(right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
VII