Table Of ContentBarbara Kaltenbacher
Thomas Schuster
Anne Wald Editors
Time-dependent
Problems in
Imaging and
Parameter
Identification
Time-dependent Problems in Imaging
and Parameter Identification
Barbara Kaltenbacher • Thomas Schuster
Anne Wald
Editors
Time-dependent Problems
in Imaging and Parameter
Identification
Editors
BarbaraKaltenbacher ThomasSchuster
DepartmentofMathematics DepartmentofMathematics
Alpen-Adria-Universita¨tKlagenfurt SaarlandUniversity
Klagenfurt,Austria Saarbru¨cken,Saarland,Germany
AnneWald
DepartmentofMathematics
SaarlandUniversity
Saarbru¨cken,Saarland,Germany
ISBN978-3-030-57783-4 ISBN978-3-030-57784-1 (eBook)
https://doi.org/10.1007/978-3-030-57784-1
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Preface
Inverseproblemsdealwiththereconstructionofunknownquantitiesfromindirect
observations and have a large number of applications ranging from medical
imaging via nondestructive testing to geophysical prospection, to name just a
few exemplary areas. These problems are by their nature ill-posed in the sense
that small perturbations in the given data, e.g., due to measurement noise, can
lead to large deviations in the reconstructions, unless appropriate mathematical
tools—regularization methods—are developed and used. In addition, the question
of uniqueness, i.e., whether the given observations unambiguously determine
the searched-for quantity, is essential and often challenging to answer. Another
important aspect in inverse problems is mathematical modeling, i.e., the correct
description of the underlying relations and processes governing the connection
betweentheparameterstobereconstructedontheonehandandtheobserveddata
ontheotherhand.
Whileinvestigationsonthesequestionshavesofarmainlybeenfocusedonstatic
problems,thedependenceofparametersand/orstatesnotonlyonspacebutalsoon
time plays an increasingly important role in inverse problems applications: Time-
resolved observations on the one hand allow to image evolutionary phenomena
such as blood flow or a motion of the object, for example, a beating heart. On
the other hand, they enable super-resolution in microscopy. Correspondingly, the
underlying physical mechanisms are instationary and thus require modeling with
time-dependentpartialdifferentialequations(PDEs)—typicallywave,diffusion,or
transportequations.Theaimofthisbookistocollectsomenovelcontributionson
time-dependentparameteridentificationandimagingproblems,therewithproviding
anoverviewonrecentdevelopmentsaswellasastimulusonfurtherresearchinthis
area.
A key step for working with novel time-resolved imaging techniques is their
proper modeling based on the underlying physics. Consequently, a considerable
emphasisofthecontributionsinchapters“JointPhaseReconstructionandMagni-
tudeSegmentationfromVelocity-EncodedMRIData”,“QuantitativeOCTRecon-
structionsforDispersiveMedia”,“InverseProblems ofSingleMolecule Localiza-
tion Microscopy”, and “Parameter Identification for the Landau–Lifshitz–Gilbert
vii
viii Preface
Equation in Magnetic Particle Imaging” is on describing or actually even deriving
suchmodels.
Chapter“JointPhaseReconstructionandMagnitudeSegmentationfromVeloc-
ity-Encoded MRI Data” provides a complete description of the inverse problem
of velocity-encoded MRI from the acquisition process to the spin proton density
estimation and devises a joint variational model to simultaneously estimate phase
and magnitude reconstruction and its segmentation. The resulting non-convex and
nonlinearoptimizationproblemisthensolvedbyaBregmaniteration.
In chapter “Quantitative OCT Reconstructions for Dispersive Media”, optical
coherencetomography(OCT)forlayeredmediaisdescribedasaninverseproblem
for a one-dimensional wave equation, which can be explicitly solved in the
nondispersivecase,thusleadingtoalayerstrippingtypereconstructionalgorithm.
Thisapproachisthenextendedtothepracticallyrelevantdispersivesettingandthe
caseofanabsorbingmediumisaddressedaswell.
Also, chapter “Parameter Identification for the Landau–Lifshitz–Gilbert Equa-
tion in Magnetic Particle Imaging” contains a modeling section, describing the
physics of magnetic particle imaging (MPI) and yielding the Landau–Lifshitz–
Gilbert (LLG) equation as a mathematical model. The task of calibration thus
becomesaparameteridentificationproblemfortheLLGequationandisconsidered
from two different inverse problem perspectives: A classical reduced approach
relyingonaparameter-to-statemapandanall-at-onceapproachconsideringmodel
andobservationsasasimultaneoussystemofoperatorequationsforparameterand
state.
Finally,chapter“InverseProblemsofSingleMoleculeLocalizationMicroscopy”
is entirely devoted to the derivation of a model for a novel super-resolution
microscopy technique, namely single-molecule localization microscopy (SMLM).
Here the resolution limit determined by Abbe’s limit of diffraction is overcome
by replacing the simultaneous acquisition of a microscopy image with a sequence
of frames recording single blinking events. The chapter provides a complete
mathematical description of the measurement setup by Maxwell’s equations and
adaptations thereof, thus finally establishing SMLM as an inverse problem for a
coupledsystemofPDEs.
One of the key issues in the context of time-dependent imaging is motion of
the target. This may be of periodic nature or irregular, and the actual motion
may be the object of interest or an unwanted side effect—think of tracking flow
in a pipe on the one hand and the movement of a patient during a recording
on the other hand. Thus, detection and estimation of motion, as well as its
compensationforproperimagingarekeytopicsinchapters“MotionCompensation
Strategies in Tomography”, “Microlocal Properties of Dynamic Fourier Integral
Operators”, “Joint Motion Estimation and Source Identification Using Convective
Regularisation with an Application to the Analysis of Laser Nanoablations”,
and“TomographicReconstructionforSingleConjugateAdaptiveOptics”.
Chapter “Joint Motion Estimation and Source Identification Using Convective
Regularisation with an Application to the Analysis of Laser Nanoablations” con-
siders spatially one-dimensional imaging of a certain morphogenesis process and
Preface ix
proposesavariationalmethodforjointmotionestimationandsourceidentification
in the underlying one-dimensional mechanical model of tissue formation. Here
the data misfit term in the variational model is defined as the L2 residual of the
nonhomogeneouscontinuityequationandregularizationisbasedontheconvective
derivativeofthesource.
Chapter “Tomographic Reconstruction for Single Conjugate Adaptive Optics”
deals with the reconstruction of atmospheric turbulence profiles from wavefront
measurements. Here the wind speeds, which cause the motion of the different
turbulent layers, play a crucial role for uniqueness: It is proven that mutually
different velocities lead to unique identifiability of the turbulence profiles. This
resultisextendedtothepracticallyrelevantsettingoffinitetelescopeaperturebased
onFourierexpansions.
Chapter “Motion Compensation Strategies in Tomography” provides a frame-
work for compensating known motion in tomographic imaging in order to avoid
artifacts in the reconstruction. The motion is described as a sequence of diffeo-
morphisms, corresponding to deformation vector fields. Moderate deformations,
where the dynamic forward operator results as an appropriate concatenation of
the static forward operator with a (multiplicative and superposition type) function
of a diffeomorphism, allow for an explicit reversal of motion which, combined
with a static reconstruction, yields a correct image from dynamic data. In the
complementarycaseofstrongdeformations,themethodoftheapproximateinverse
isshowntoprovideefficientmotioncompensation.
In chapter “Microlocal Properties of Dynamic Fourier Integral Operators”,
methodsfrommicrolocalanalysisareusedtoanswerthequestionwhichfeatures(in
thesenseofsingularities)oftheimagedobjectcanberecoveredfromthegivendata
andinwhichlocationsmotionartifactscanarise.Theanalysisisbasedonthetheory
of Fourier integral operators, which covers many relevant dynamic tomographic
methods based on integral transforms, such as the classical Radon transform in
X-ray CT or the circular Radon transform in a particular setting of photoacoustic
tomographybutisprobablyalsoapplicabletoPDE-basedmodels.
Reconstruction schemes are often built on existing paradigms known from
staticinverseandimagingproblemsandneedtobeadaptedtothetime-dependent
setting in order to work efficiently. In view of practical applications, the solution
oftime-dependentproblemsofteninvolveselaboratecomputations,whichdemands
fast reconstruction schemes. In the case of ill-posed inverse problems, they also
need to incorporate regularization. Iterative reconstruction schemes, as consid-
ered, e.g., in chapters “Joint Phase Reconstruction and Magnitude Segmentation
from Velocity-Encoded MRI Data”, “Dynamic Inverse Problems for the Acoustic
WaveEquation”,“SequentialSubspaceOptimizationforRecoveringStoredEnergy
Functions in Hyperelastic Materials from Time-Dependent Data”, and “Parameter
IdentificationfortheLandau–Lifshitz–GilbertEquationinMagneticParticleImag-
ing”, generate successive approximations of the searched-for quantity and achieve
regularization by a well-chosen stopping criterion. To guarantee convergence of
these methods, conditions on the forward problem need to be verified, such as
x Preface
differentiability of the forward operator and certain restrictions on its nonlinearity
suchastheso-calledtangentialconecondition.
The latter is investigated in chapter “The Tangential Cone Condition for Some
Coefficient Identification Model Problems in Parabolic PDEs”, first of all in an
abstract setting of evolution models and then for several concrete inverse source
or coefficient problems for linear and nonlinear parabolic PDEs, contrasting the
reducedandtheall-at-onceapproachofformulatingtheinverseproblem.
For the identification of space- and time-dependent coefficients in an acoustic
wave equation, chapter “Dynamic Inverse Problems for the Acoustic Wave Equa-
tion” provides results on the derivative of the forward operator in appropriate
function spaces, as well as its adjoint, as required for iterative regularization.
Likewise, this is done in chapter “Parameter Identification for the Landau–Lif-
shitz–GilbertEquationinMagneticParticleImaging”formagneticparticleimaging
and in chapter “Sequential Subspace Optimization for Recovering Stored Energy
Functions in Hyperelastic Materials from Time-Dependent Data” for structural
healthmonitoringwithhyperelasticwaves.
Thelatter,i.e.,chapter“SequentialSubspaceOptimizationforRecoveringStored
Energy Functions in Hyperelastic Materials from Time-Dependent Data”, also
devises an accelerated iteration scheme, namely sequential subspace optimization,
which uses, instead of only one gradient step, a linear combination of directions
with optimized step sizes. This method is applied to a dictionary discretization of
thestoredenergyfunctioncharacterizingthehyperelasticmaterialandthus,viaits
spatialvariability,allowingtodetectdamageintheinspectedmedium.
Whenever a quantity is to be determined from indirect measurements, the
questionariseswhetherthesedatauniquelydeterminethehiddenobject.Themore
complex the underlying model that connects the searched-for parameters with the
givenobservations,themorechallengingistheanswertothisquestion.Uniqueness
proves are provided in chapters “Holmgren-John Unique Continuation Theorem
for Viscoelastic Systems” and “An Inverse Source Problem Related to Acoustic
NonlinearityParameterImaging”.
Chapter “Holmgren-John Unique Continuation Theorem for Viscoelastic Sys-
tems” establishes the unique continuation property for a viscoelastic system, i.e.,
a hyperbolic PDE with a memory term, based on the Holmgren-John concept of
proving uniqueness for problems with analytic coefficients and noncharacteristic
boundaries. This is essential for solving inverse problems such as the detection of
obstaclesorinclusions.
For a nonlinear acoustic wave equation (the Moore–Gibson–Thompson equa-
tion), the uniqueness of an inverse source problem as well as its conditional
stabilityisestablishedinchapter“AnInverseSourceProblemRelatedtoAcoustic
Nonlinearity Parameter Imaging”, as a first step into the mathematical analysis of
ultrasonicnonlinearityimaging.
Segmentation and Registration, i.e., the decomposition of image regions
accordingtotheirdifferentintensities,aswellasthealignmentofdifferentframes
ormodalitiestoacommoncoordinatesystem,areimagingtasksthatplayarolein
manyapplicationsandrequirespecialadaptationinthetime-dependentcontext.
Preface xi
Here similarity measures are crucial in order to identify objects of the same
type on the one hand and to distinguish between different ones on the other hand.
Chapter“ReviewofImageSimilarityMeasuresforJointImageReconstructionfrom
MultipleMeasurements”providesabroadoverviewonsimilaritymeasureswitha
particularviewonjointmulti-modalityimaging,asoftenalsoencounteredintime-
dependentsettings.
Inthecontextofimagingflowmotionbyvelocity-encodedMRI,segmentationis
implementedinchapter“JointPhaseReconstructionandMagnitudeSegmentation
fromVelocity-EncodedMRIData”bymeansofaspatiallyresolvedpenaltytermin
avariationalmodel.
Future research on time-dependent parameter identification and imaging prob-
lemswillontheonehandbecertainlydirectedintocomplementationandunification
ofthetheory,asithastoacertainextentalreadybeenaccomplishedforstaticinverse
problems. On the other hand, many new tailored methods for specific dynamic
inverse problems as well as innovative general solution paradigms to tackle time-
dependencearejustabouttoemerge.Inparticular,questionsconcerningtheoptimal
sampling of data or the promotion of special features of solutions such as spatial
and/or temporal sparsity will be further investigated. Last but not least, in the
explodingfieldofmachinelearning,evolutionarymodelsturnouttoplayacrucial
role.
1. Joint Phase Reconstruction and Magnitude Segmentation from Velocity-
Encoded MRI Data (Veronica Corona, Martin Benning, Lynn F. Gladden,
AndiReci,AndrewJ.Sederman,andCarola-BibianeSchönlieb)
2. DynamicInverseProblemsfortheAcousticWaveEquation(ThiesGerken)
3. MotionCompensationStrategiesinTomography(BernadetteN.Hahn)
4. Microlocal Properties of Dynamic Fourier Integral Operators (Bernadette N.
Hahn,Melina-L.KienleGarrido,andEricToddQuinto)
5. The Tangential Cone Condition for Some Coefficient Identification Model
Problems in Parabolic PDEs (Barbara Kaltenbacher, Tram Thi Ngoc Nguyen,
andOtmarScherzer)
6. SequentialSubspaceOptimizationforRecoveringStoredEnergyFunctionsin
Hyperelastic Materials from Time-Dependent Data (Rebecca Klein, Thomas
Schuster,andAnneWald)
7. JointMotionEstimationandSource Identification UsingConvective Regular-
isation with an Application to the Analysis of Laser Nanoablations (Lukas
F. Lang, Nilankur Dutta, Elena Scarpa, Bénédicte Sanson, Carola-Bibiane
Schönlieb,andJocelynÉtienne)
8. QuantitativeOCTReconstructionsforDispersiveMedia(PeterElbau,Leonidas
Mindrinos,andLeopoldVeselka)
9. ReviewonImageSimilarityMeasuresforJointMulti-modalityImageRecon-
struction(MingJiang)
10. Holmgren-John Unique Continuation Theorem for Viscoelastic Systems
(MaartenV.deHoop,Ching-LungLin,andGenNakamura)
