GEOMETRIC METHODS FOR IMAGE REGISTRATION AND ANALYSIS by Anand Arvind Joshi A ...

120 Pages·2008·5.66 MB·English

GEOMETRICMETHODSFOR IMAGEREGISTRATION ANDANALYSIS
by
AnandArvindJoshi
A DissertationPresented tothe
FACULTYOF THEGRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
InPartial Fulfillmentofthe
RequirementsfortheDegree
DOCTOROF PHILOSOPHY
(ELECTRICAL ENGINEERING)
August2008
Copyright 2008 AnandArvindJoshi
Dedication
Idedicatethisthesistomyparents Arvindand SuvarnaJoshi.
ii
Acknowledgments
I would like to express my deepest respect and gratitude to my advisor, Prof. Richard
Leahy. His contribution has been invaluable; he supported me financially throughout
my graduate studies in USC, guided me in selecting and solving research problems,
introduced me to distinguished researchers and scientists, provided me a friendly and
welcome environment, gave me freedom to select my own research directions and let
me enjoy long holidays. He is a great mentor and a source of inspiration for me, and
it has really been an honor working with him. I also wish to thank the members of my
guidance committee: Dr. Krishna Nayak and Dr. Francis Bonahon for their sugges-
tions and valuable feedback. My appreciations go to Dr. David Shattuck and Dr. Paul
Thompsonat theUniversityofCalifornia,Los Angelesformanyfruitful collaborations
anddiscussions. IcouldnothaveselectedbettercolleaguesthanDr. AbhijitChaudhari,
Sangeetha Somayajula, Sanghee Cho, Joyita Dutta, Dr. Sangtae Ahn, Dr. Quanzheng
LiandDr. DimitriosPantazis. Wehavespendqualitytimetogether,discussingresearch
problems as well as life experiences. I must also mention that Abhijit, Quanzheng and
Dimitrios motivated me from time to time to look for research problems. I also would
liketo expressmygratitudeto Dr. IlyaEcksteinforfruitfulcollaborations.
In theUniversityofSouthern CaliforniaIhaveenjoyedthecozyand warmenviron-
ment of an excellent research lab. I want to thank its members: Sangtae Ahn, Abhijit
Chaudhari, Sanghee Cho, Belma Dogdas, Hua Hui, Zheng Li, Sangeetha Somayajula,
iii
Juan Luis Poletti Soto, Evren Asma, YuTeng Chang, David Wheland and Syed Ashra-
fullaforaddingpositivelytomyresearch and academicexperience.
Most importantly, I want to thank my family for providing unlimited support and
helpingmerealize mydreams.
iv
Table of Contents
Dedication ii
Acknowledgments iii
Abstract xii
Chapter 1: Introduction 1
Chapter 2: CorticalSurface Parameterization 5
2.1 Parameterizationand theCoordinateSystem . . . . . . . . . . . . . . . 6
2.1.1 Validationofp-harmonicmappings . . . . . . . . . . . . . . . 9
Chapter 3: CorticalSurface Registration 13
3.1 ThinPlateSplinesRegistrationintheIntrinsicGeometryoftheCortical
Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.1.1 MathematicalFormulation . . . . . . . . . . . . . . . . . . . . 16
3.1.2 DiscretizationAlgorithm . . . . . . . . . . . . . . . . . . . . . 20
3.1.3 BendingEnergy Minimization . . . . . . . . . . . . . . . . . . 21
3.1.4 ValidationTPS surfaceregistration . . . . . . . . . . . . . . . . 24
3.2 AFiniteElementMethodforSimultaneousRegistrationandParameter-
ization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.2.1 Surface Registration . . . . . . . . . . . . . . . . . . . . . . . 27
3.2.2 MathematicalFormulation . . . . . . . . . . . . . . . . . . . . 28
3.2.3 FiniteElementFormulation . . . . . . . . . . . . . . . . . . . . 29
3.2.4 Resultsand Validation . . . . . . . . . . . . . . . . . . . . . . 33
3.3 OptimumChoiceofSulcal SubsetforRegistration . . . . . . . . . . . . 35
3.3.1 RegistrationError . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.3.2 ProbabilisticModeloftheSulcal Errors . . . . . . . . . . . . . 39
3.3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.3.4 Discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
Chapter 4: Processing ofDatainthe Surface Geometry 49
4.1 ImageFilteringonSurfaces . . . . . . . . . . . . . . . . . . . . . . . . 50
v
4.2 MathematicalFormulation . . . . . . . . . . . . . . . . . . . . . . . . 51
4.2.1 Isotropicfiltering . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.2.2 Anisotropicfiltering . . . . . . . . . . . . . . . . . . . . . . . . 53
4.3 Discretizationand Numerical Method . . . . . . . . . . . . . . . . . . 55
4.3.1 DiscretizationAlgorithm . . . . . . . . . . . . . . . . . . . . . 55
4.3.2 TheHeat Equationin theIntrinsicGeometry . . . . . . . . . . . 61
4.4 TheHeat Kernelas aPDF . . . . . . . . . . . . . . . . . . . . . . . . . 62
Chapter 5: VolumetricRegistrationusing HarmonicMaps 66
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.2 ProblemStatement andFormulation . . . . . . . . . . . . . . . . . . . 68
5.3 IndirectMappingApproach . . . . . . . . . . . . . . . . . . . . . . . . 70
5.3.1 MathematicalFormulation . . . . . . . . . . . . . . . . . . . . 73
5.3.2 InitializationProcedure . . . . . . . . . . . . . . . . . . . . . . 75
5.3.3 MappingtotheUnitBall B(0,1) . . . . . . . . . . . . . . . . . 75
5.3.4 HarmonicMappingBetween theTwo Brains . . . . . . . . . . 77
5.3.5 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.4 Direct MappingApproach . . . . . . . . . . . . . . . . . . . . . . . . 79
5.4.1 MathematicalFormulation . . . . . . . . . . . . . . . . . . . . 80
5.4.2 HarmonicMapping . . . . . . . . . . . . . . . . . . . . . . . . 80
5.5 VolumetricIntensityRegistration . . . . . . . . . . . . . . . . . . . . . 81
5.5.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.5.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.6 Resultsand Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
Chapter 6: Conclusions andFuture Work 94
6.1 GeometricFeatures andManual LandmarksbasedSurface Registration 96
6.2 RegistrationofDTIimages . . . . . . . . . . . . . . . . . . . . . . . . 97
Bibliography 98
vi
List of Figures
1.1 ThecorticalsurfaceofthehumanbraindepictedonaMRdata(toprow)
andrendered as asurface(bottomrow). . . . . . . . . . . . . . . . . . 2
2.1 Sulcal TracingTools. . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Thefigureshowsthecorticalsurfaceanditsmaptoasquare. Thecorpus
callosumis constrainedtolieontheboundaryofthesquare. . . . . . . 9
2.3 Thep-harmonicmapsofthelefthemisphereofan individualcortex. . . 10
2.4 The figure shows smoothed histograms for angle distortion and area
distortion respectively. In the angle distortion plot, angle distortion
increases with the value of p. In the area distortion plot, the distortions
for p=4,6,8 are less than that for p=2 and most of the points have small
angledistortiononly. Howeverthereisnoobservabletrendforthevalue
ofpin eithercase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.1 (a) A cortical surface with hand labeled sulci; (b) A flat map of the
two cortical surface. The arrows show connectivity at points along the
boundary of the square. Due to the spherical topology of the cortical
surface,wecanassigntoitacoordinatesystemthatallowsustocompute
partial derivatives across the interhemispheric fissure. (c) Chessboard
texturemappedto thesurface usingthesquaremaps. . . . . . . . . . . 15
3.2 (upper) The figure shows the warping field computed on the surface.
Thedeformationfieldissmoothlyvarying. Thisisachievedbecausethe
bending energy regularization was performed in the intrinsic geometry
of the surface. The color indicates the magnitude of the deformation.
(lower) The thin-plate spline deformation field applied to a regular grid
representingleft and righthemispheres. . . . . . . . . . . . . . . . . . 22
vii
3.3 Alignment of the sulcal landmarks: 6 brains are registered to a com-
mon cortical surface using their p-harmonic maps in the plane. They
are approximately aligned by the p-harmonic maps justifying our small
deformation linear model (thin plate bending energy model) which is
used for landmark alignment. After applying the covariant TPS defor-
mation field to the surface parameterization, we can see that the sulci
showbetteralignment. . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.4 (a),(b)Thetwocorticalsurfaceswithhandlabeledsulciascoloredcurves;
(c),(d) flat maps of a single hemisphere for the two brains without the
sulcalalignmentconstraint;(e)overlayofsulcalcurvesontheflat maps
without alignment; (f),(g) flat maps with sulcal alignment; (h) overlay
ofsulcalcurveson theflat mapswithalignment. . . . . . . . . . . . . . 30
3.5 RMSerrorand percentage overlapin theflattened map as afunctionofσ. 33
3.6 Mapping of sulcal landmarks from 5 subjects to the atlas brain (left)
withoutand (right)withthesulcalalignmentconstraint. . . . . . . . . . 34
3.7 The complete set of candidate sulcal curves from which we select an
optimalsubsetforconstrainedcortical registration . . . . . . . . . . . . 37
3.8 (a)Registrationoftwocorticalsurfacesbasedontheflatmappingmethod;
(b) Parcellation of the cortex into regions surrounding the traced sulci;
(c) Registration error for two corresponding sulci where e (s) are sam-
n
plesoftheregistrationerror. . . . . . . . . . . . . . . . . . . . . . . . 38
3.9 Samplecovariance matrices for the x, y, and z componentsof the regis-
trationerror, represented as colorcoded images. . . . . . . . . . . . . . 42
3.10 Optimal subsets of sulci for cortical registration. Each row gives the
indicesoftheoptimalsubsetofsulcithatminimizetheregistrationerror
againstall other combinationswith equal numberof constrained curves
(also see Fig. 3.7). The three right columns show that the estimated
(est.) error is close to the calculated actual (act.) error when actual reg-
istrations with the same constrained curves are performed. Our method
predicts the registration error both for the training (trn) and the testing
(tst)set ofbrains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.11 Optimalsulcal setsfor5, 10,and 15curves. . . . . . . . . . . . . . . . 45
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3.12 Top row: subjectiveselection of 6 curves, with preference on long sulci
distant from each other that are expected to minimize cortical registra-
tion error; bottom row: optimal sulcal set with the 6 curves selected by
ourmethod. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.1 Theimpulseresponseofthe isotropicsmoothingfilters are displayedin
the parameter space and on the surface [JSTL05]. It can be seen that
when the surface metric is used to compute the Laplace-Beltrami, the
impulseresponsekernelisnotisotropicintheparameterspace,however
itisisotropiconthesurface. . . . . . . . . . . . . . . . . . . . . . . . 58
4.2 left: The mean curvature of the cortical surface plotted on a smoothed
representation (for improved visualization of curvature in sulcal folds;
right: The mean curvature plotted in 2D parameter space for a single
cortical hemisphere. Isotropic diffusion blurs the regions as well as the
edges separating them while while anisotropic diffusion reduces noise
whilepreservingedges. . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.3 The figures shows the heat kernels estimated to fit the two datasets for
MEG somatosensory data. For each of the datasets the estimated pdf is
displayedin theparameterspaceand onthecortical surface. . . . . . . 64
4.4 Theclassifier: Red andBlue regionsshowsthetwo decisionregions . . 64
5.1 Cortical surface alignment after using AIR software for intensity based
volumetricalignmentusinga168parameter5th orderpolynomial. Note
that although the overall morphology is similar between the brains, the
twocortical surfaces do notalignwell. . . . . . . . . . . . . . . . . . . 70
5.2 Illustration of our general framework for surface-constrained volume
registration. We first compute the map v from brain manifold (N,I)
totheunitballtoformmanifold(N,h). Wethencomputeamapu˜from
brain (M,I) to (N,h). The final harmonic map from (M,I) to (N,I)
isthen givenbyu = v−1 u˜. . . . . . . . . . . . . . . . . . . . . . . . 74
◦
ix
5.3 Initialization for harmonic mapping from M to N. First we gener-
ate flat square maps of the two brains, one for each hemisphere, with
pre-aligned sulci. The squares corresponding to each hemispheres are
mapped to a disk and the disks are projected onto the unit sphere. We
thengenerateavolumetricmapsfromeachofthebrainstotheunitball.
Sinceallthesemapsarebijective,theresultingmapresultsinabijective
pointcorrespondencebetween thetwobrains. However,thiscorrespon-
dence is not optimal with respect to the harmonic energy of maps from
thefirstbraintothesecond,butisusedasaninitializationforminimiza-
tionof(5.6). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.4 Illustration of the deformation induced with respect to the Euclidean
coordinates by mapping to the unit ball. Shown are iso-surfaces corre-
spondingtotheEuclideancoordinatesfordifferentradiiintheunitball.
Distortions become increasingly pronounced towards the outer edge of
thespherewhere theentireconvolutedcortical surface is mappedto the
surfaceoftheball. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.5 Schematic of the intensity alignment procedure. Once harmonic maps
uM and uN are computed, we refine these with intensity driven warps
wM and wN while imposing constraints so that the final deformations
areinverseconsistent. . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.6 Illustration of the effects of the two stages of volumetric matching is
shownby applyingthe deformationstoa regularmeshrepresenting one
slice. Since the deformation is in 3D, the third in-paper value is repre-
sented by color. (a) Regular mesh representing one slice in the subject;
(b) the regular mesh warped by the harmonic mapping which matches
the subject cortical surface to the template cortical surface. Note that
deformation is largest near the surface since the harmonic map is con-
strained only by thecortical surface; (c) Regular mesh representing one
slice in the harmonically warped subject; (d) the intensity-based refine-
ment now refines the deformation of the templateto improvethe match
between subcortical structures. In this case the deformation is con-
strained to zero at the boundary and are confined to the interior of the
volume. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.7 Examples of direct mapping approach. (a) Original subject volume; (b)
original template; (c) registration of subject to template using surface
constrainedharmonicmapping,notethatthesurfacematchesthatofthe
template;(d)intensity-basedrefinementoftheharmonicmapofsubject
totemplateto completeregistrationprocedure . . . . . . . . . . . . . . 88
x