From Global to Local Statistical Shape Priors: Novel Methods to Obtain Accurate Reconstruction Results with a Limited Amount of Training Shapes

266 Pages·2017·11.35 MB·English

by  Carsten Last

Carsten Last
From Global to Local
Statistical Shape Priors
Novel Methods to Obtain Accurate
Reconstruction Results with a Limited
Amount of Training Shapes
123
CarstenLast
Institut für Robotik undProzessinformatik
Technische UniversitätBraunschweig
Braunschweig
Germany
ISSN 2198-4182 ISSN 2198-4190 (electronic)
Studies in Systems,DecisionandControl
ISBN978-3-319-53507-4 ISBN978-3-319-53508-1 (eBook)
DOI 10.1007/978-3-319-53508-1
LibraryofCongressControlNumber:2017930419
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Preface
ThepresentthesisoriginatesfrommyscientificworkasemployeeattheInstitutfür
Robotik und Prozessinformatik, Technische Universität Braunschweig, Germany.
However, the completion of this thesis would not have been possible without the
help of others.
Atfirst,IwouldliketothankProf.Dr.-Ing.FriedrichM.Wahl fordirectingme
towards the interesting topic of medical image segmentation and giving me the
freedomtodevelopmyownideas withinthisfield.He providedmewith excellent
technical resources and a great working environment, and he gave me the oppor-
tunitytopresentmyideasatvariousnationalandinternationalconferences.Iwould
also like to thank Prof. Dr. Thomas Vetter for his effort to review this thesis as a
secondrefereeandforprovidingthemorphablefacemodelaswellastheadditional
face scans that are used for evaluation.
My thanks go especially also to Dr.-Ing. Simon Winkelbach for his continuous
support in developing and discussing new ideas, but as well for proofreading this
thesis.Hisinputhadagreatinfluenceonthefinaloutcome.Ialsohavetothankthe
whole staff and all students of the Institut für Robotik und Prozessinformatik for
creating such a nice and friendly working atmosphere.
Additionally,IwouldliketothanktheGermanResearchFoundation(DFG)for
funding parts of my research within the project Roboterunterstützte,
erwartungskonformeEndoskopführunginderendosalenChirurgie.Inthisregard,I
would like to thank Prof. Dr. med. Dr. h.c. Friedrich Bootz from the Klinik und
Poliklinik für Hals-Nasen-Ohrenheilkunde/Chirurgie at the Universitätsklinikum
Bonnforprovidingmewiththeparanasalsinusesdatabasethatisbeingusedwithin
this thesis. Also, I would like to thank my former colleagues Dr.-Ing. Ralf
Westphal, Markus Rilk, and Dr. med. Klaus Eichhorn for the fruitful and friendly
cooperation within this project.
vii
viii Preface
Specialthanksgotomyfamilyfortheircontinuousmoralandfinancialsupport
throughoutallmyyearsofstudyandtomywifeCarolinforthecountlesshoursthat
she spent discussing and proofreading this thesis.
Braunschweig, Germany Carsten Last
December 2016
Contents
1 Basics. .... .... .... .... .... ..... .... .... .... .... .... ..... 1
1.1 Naming Conventions . .... ..... .... .... .... .... .... ..... 1
1.2 Vector Calculus . .... .... ..... .... .... .... .... .... ..... 2
1.2.1 Pixels and Voxels.. ..... .... .... .... .... .... ..... 2
1.2.2 The Gradient and Necessary Condition for a Minimum ... 3
1.2.3 Directional Derivative.... .... .... .... .... .... ..... 3
1.2.4 Descent Direction.. ..... .... .... .... .... .... ..... 3
1.2.5 Descent Methods .. ..... .... .... .... .... .... ..... 4
1.3 Calculus of Variations .... ..... .... .... .... .... .... ..... 4
1.3.1 The Functional Derivative. .... .... .... .... .... ..... 6
1.3.2 Differentiation Rules..... .... .... .... .... .... ..... 7
1.3.3 Functionals of Functions with Multiple Arguments . ..... 7
1.3.4 Functionals of Multiple Functions .. .... .... .... ..... 8
1.3.5 The Functional Differential.... .... .... .... .... ..... 9
1.4 Level Set Methods... .... ..... .... .... .... .... .... ..... 9
1.4.1 Level Set Function. ..... .... .... .... .... .... ..... 9
1.4.2 Level Set Evolution ..... .... .... .... .... .... ..... 11
1.4.3 Level Set Methods for Image Segmentation Problems .... 13
1.5 Variational Image Segmentation.. .... .... .... .... .... ..... 16
1.5.1 Geodesic Active Contours .... .... .... .... .... ..... 16
1.5.2 Active Contours Without Edges.... .... .... .... ..... 18
2 Statistical Shape Models (SSMs) .... .... .... .... .... .... ..... 21
2.1 Definition of Shape .. .... ..... .... .... .... .... .... ..... 22
2.2 An Explicit Linear Parametric Statistical Shape Model .... ..... 24
2.3 An Implicit Linear Parametric Statistical Shape Model .... ..... 31
2.4 Rigid Shape Alignment ... ..... .... .... .... .... .... ..... 39
2.4.1 Similarity Transformation. .... .... .... .... .... ..... 41
2.4.2 Rigid Alignment of Two Shape Configurations .... ..... 41
2.4.3 Rigid Alignment of Multiple Shape Configurations . ..... 43
ix
x Contents
2.5 Problem: Limited Training Data.. .... .... .... .... .... ..... 44
2.5.1 Artificial Enlargement of Shape Variations.... .... ..... 46
2.5.2 Relaxation of Model-Constraints.... .... .... .... ..... 48
2.5.3 Model Partitioning . ..... .... .... .... .... .... ..... 49
3 A Locally Deformable Statistical Shape Model (LDSSM) .... ..... 53
3.1 Motivation . .... .... .... ..... .... .... .... .... .... ..... 53
3.2 Mathematical Formulation of Our LDSSM . .... .... .... ..... 55
3.3 An Iterative Framework to Determine the Weight Fields... ..... 58
3.3.1 General Idea: Demons-Based Approach .. .... .... ..... 58
3.3.2 A Side Note to Optimal Weights for the SSM. .... ..... 60
3.3.3 Determination of the Weights for a Known
Target Shape.. .... ..... .... .... .... .... .... ..... 62
3.3.4 Determination of the Weights for Segmentation
Problems. .... .... ..... .... .... .... .... .... ..... 65
4 Evaluation of the Locally Deformable Statistical Shape Model..... 75
4.1 Segmenting the Nasal Cavity and the Paranasal Sinuses ... ..... 75
4.1.1 Description of the Paranasal Sinuses Database. .... ..... 78
4.1.2 Segmentation with Global Model Information . .... ..... 81
4.1.3 Experimental Setup and Results .... .... .... .... ..... 88
4.2 Segmenting the Bones in the Human Knee . .... .... .... ..... 100
4.2.1 Motivation ... .... ..... .... .... .... .... .... ..... 100
4.2.2 Experimental Setup. ..... .... .... .... .... .... ..... 100
4.2.3 Results .. .... .... ..... .... .... .... .... .... ..... 104
4.3 Fitting the LDSSM to Range Scans of Faces.... .... .... ..... 106
4.3.1 The Basel Face Model: A Meta-Database for 3D Faces... 107
4.3.2 Evaluating the Shape Approximation for Known
Target Shapes. .... ..... .... .... .... .... .... ..... 109
4.3.3 Application: Reconstructing Incomplete Face Scans. ..... 116
5 Global-To-Local Shape Priors for Variational Level
Set Methods.... .... .... .... ..... .... .... .... .... .... ..... 125
5.1 Problems of the Iterative Segmentation Framework... .... ..... 126
5.2 Existing Variational Image Segmentation Approaches
with Shape Priors.... .... ..... .... .... .... .... .... ..... 127
5.2.1 Linear Subspace-Constrained Approaches. .... .... ..... 127
5.2.2 Approaches with Additional Flexibility... .... .... ..... 132
5.2.3 Drawbacks of Existing Approaches . .... .... .... ..... 137
5.3 Variational Formulation for a Global Shape Prior .... .... ..... 138
5.3.1 Problem Formulation .... .... .... .... .... .... ..... 139
5.3.2 An Exemplary Segmentation Problem ... .... .... ..... 141
Contents xi
5.3.3 General Solution for Arbitrary Segmentation Problems ... 143
5.3.4 Extension of the Approach by the Trained Weight
Distribution... .... ..... .... .... .... .... .... ..... 144
5.3.5 Consideration of Rigid Transformations.. .... .... ..... 146
5.4 Variational Formulation for a Local Shape Prior . .... .... ..... 147
5.4.1 Motivation ... .... ..... .... .... .... .... .... ..... 147
5.4.2 Problem Formulation .... .... .... .... .... .... ..... 148
5.4.3 Extension of Our Approach by the Trained Weight
Distribution... .... ..... .... .... .... .... .... ..... 150
5.4.4 Finding a Solution to the Problem by Functional
Gradient Descent .. ..... .... .... .... .... .... ..... 152
5.4.5 Obtaining the Functional Gradient .. .... .... .... ..... 154
5.5 Global-to-Local Variational Formulation ... .... .... .... ..... 162
5.5.1 Problems of the Variational Gradient Descent
Approach .... .... ..... .... .... .... .... .... ..... 163
5.5.2 Connection Between the Global Approach
and Our Local Approach . .... .... .... .... .... ..... 164
5.5.3 Combining the Global Approach and Our Local
Approach .... .... ..... .... .... .... .... .... ..... 168
6 Evaluation of the Global-To-Local Variational Formulation.. ..... 173
6.1 Extracting the Nasal Cavity and the Paranasal Sinuses
from Two-Dimensional CT Slices .... .... .... .... .... ..... 173
6.1.1 New Global-To-Local Approach Versus Global
Approach by Tsai et al. .. .... .... .... .... .... ..... 174
6.1.2 New Global-To-Local Approach Versus Former
Segmentation Framework. .... .... .... .... .... ..... 185
6.2 Extracting the Nasal Cavity and the Paranasal Sinuses
from Three-Dimensional CT Data .... .... .... .... .... ..... 191
6.2.1 New Global-To-Local Approach Versus Global
Approach by Tsai et al. .. .... .... .... .... .... ..... 192
6.2.2 New Global-To-Local Approach Versus Former
Segmentation Framework. .... .... .... .... .... ..... 199
7 Conclusion and Outlook.. .... ..... .... .... .... .... .... ..... 207
7.1 Open Topics and Future Work... .... .... .... .... .... ..... 209
Appendix A: Results from Sect. 4.1.3 ... .... .... .... .... ..... .... 211
Appendix B: Results from Sect. 6.1.1 ... .... .... .... .... ..... .... 221
Appendix C: Results from Sect. 6.1.2 ... .... .... .... .... ..... .... 231
Appendix D: The Sample Covariance Matrix. .... .... .... ..... .... 241
xii Contents
Appendix E: More Examples for the Global Variational
Formulation.... ..... .... .... .... .... .... ..... .... 245
Appendix F: Rigid Shape Alignment.... .... .... .... .... ..... .... 249
Own Publications and References .. .... .... .... .... .... ..... .... 253
List of Figures
Figure 1.1 Depiction of the brachistochrone curve on which the blue
bodyreachesthepointP inshortesttimewhenstartingwith
2
zero velocity in point P .... .... .... .... .... .... ..... 5
1
Figure 1.2 Exemplary level set representations of different curves. ..... 10
Figure 1.3 Zero level set and normalized gradient vectors of the level
set function U. .. .... ..... .... .... .... .... .... ..... 11
Figure 1.4 Example of an edge-indicator function g ... .... .... ..... 14
Figure 1.5 Curve moving under the Euclidean curve shortening flow ... 15
Figure 1.6 Zero level curve C and corresponding narrow
band NBðCÞ.... .... ..... .... .... .... .... .... ..... 16
Figure 1.7 Exemplary level set segmentation of an implanted knee
prosthesis .. .... .... ..... .... .... .... .... .... ..... 18
Figure 2.1 Exemplary CT cross-section and range scan
of a human head. .... ..... .... .... .... .... .... ..... 22
Figure 2.2 Exemplary explicit shape representations of a hand and a
dolphin..... .... .... ..... .... .... .... .... .... ..... 23
Figure 2.3 Mean shape overlaid with the trainings shapes before and
after they have been rigidly aligned with regard to rotation,
translation, and scale. . ..... .... .... .... .... .... ..... 24
Figure 2.4 Distribution of the training shapes, projected onto the two
main modes of variation. ... .... .... .... .... .... ..... 27
Figure 2.5 Cumulative variance for a growing number of eigenvectors. . 28
Figure 2.6 Screegraph,showingtheeigenvaluesr2indescendingorder
i
of importance ... .... ..... .... .... .... .... .... ..... 29
Figure 2.7 Variation of the mean shape by (cid:2)3 standard deviations
along the five most important eigenvectors. . .... .... ..... 30
Figure 2.8 Exemplaryschematicoverviewofthetrainingprocessofthe
implicit linear parametric statistical shape model from
Eq. (2.34).. .... .... ..... .... .... .... .... .... ..... 35
xiii
xiv ListofFigures
Figure 2.9 First four main modes of variation of the implicit training
shapes. .... .... .... ..... .... .... .... .... .... ..... 36
Figure 2.10 Screegraph,showingtheeigenvaluesr2indescendingorder
i
of importance ... .... ..... .... .... .... .... .... ..... 37
Figure 2.11 Cumulative variance for a growing number of eigenvectors. . 37
Figure 2.12 Variation of the mean shape by (cid:2)3 standard deviations
along the five main modes of variation. .... .... .... ..... 38
Figure 2.13 Distribution of the implicit training shapes, projected onto
the first three main modes of variation.. .... .... .... ..... 39
Figure 2.14 Two-dimensional projections of the plot shown
in Fig. 2.13a. ... .... ..... .... .... .... .... .... ..... 40
Figure 3.1 Benefits of the local adaptation of global shape parameters... 54
Figure 3.2 Schematic visualizations of the global SSM
and the LDSSM.. .... ..... .... .... .... .... .... ..... 57
Figure 3.3 Exemplary registration of two hands with a demons-based
approach. .. .... .... ..... .... .... .... .... .... ..... 59
Figure 3.4 General idea of the demons-based approach to the
determination of a smooth field of weight vectors for our
LDSSM.... .... .... ..... .... .... .... .... .... ..... 60
Figure 3.5 Original shape and best possible approximation that can be
obtained with the SSM from Eq. (2.34) .... .... .... ..... 62
Figure 3.6 Demons-based approach to the determination of a smooth
field of weight vectors for our LDSSM that is constraint by
the trained shape distribution .... .... .... .... .... ..... 64
Figure 3.7 Original image I, edge image I , and unsigned distance
bin
image Idist . .... .... ..... .... .... .... .... .... ..... 66
Figure 3.8 Weight field update loop for an unknown target shape. ..... 67
Figure 3.9 Weight update target estimation for an image edge outside
the modeled shape ... ..... .... .... .... .... .... ..... 68
Figure 3.10 Weightupdatetargetestimationforanimageedgeinsidethe
modeled shape .. .... ..... .... .... .... .... .... ..... 69
Figure 3.11 Weight update target estimation based on the smoothed
gradient magnitude... ..... .... .... .... .... .... ..... 71
Figure 4.1 Exemplary slices in frontal view from two different CT
datasets. ... .... .... ..... .... .... .... .... .... ..... 77
Figure 4.2 Datasets of the paranasal sinuses database .. .... .... ..... 79
Figure 4.3 Eight iterations of the Nelder–Mead simplex method,
searching for a minimum of Himmelblau's two-dimensional
test function..... .... ..... .... .... .... .... .... ..... 84
Figure 4.4 Gradient magnitude image jrIj, convolved with Gaussian
smoothing kernels Kgauss that differ in the standard
deviation rgauss. .... ..... .... .... .... .... .... ..... 86
Figure 4.5 Intermediate steps of our processing chain for segmenting
the paranasal sinuses, applied to the CT data in Fig. 4.1a.... 87