Table Of ContentLecture Notes in Mathematics 2177
Eva B. Vedel Jensen
Markus Kiderlen Editors
Tensor Valuations and
Their Applications in
Stochastic Geometry
and Imaging
Lecture Notes in Mathematics 2177
Editors-in-Chief:
Jean-MichelMorel,Cachan
BernardTeissier,Paris
AdvisoryBoard:
MichelBrion,Grenoble
CamilloDeLellis,Zurich
MarioDiBernardo,Bristol
AlessioFigalli,Zurich
DavarKhoshnevisan,SaltLakeCity
IoannisKontoyiannis,Athens
GáborLugosi,Barcelona
MarkPodolskij,Aarhus
SylviaSerfaty,NewYork
AnnaWienhard,Heidelberg
Moreinformationaboutthisseriesathttp://www.springer.com/series/304
Eva B. Vedel Jensen (cid:129) Markus Kiderlen
Editors
Tensor Valuations and Their
Applications in Stochastic
Geometry and Imaging
123
Editors
EvaB.VedelJensen MarkusKiderlen
DepartmentofMathematics DepartmentofMathematics
AarhusUniversity AarhusUniversity
AarhusC,Denmark AarhusC,Denmark
ISSN0075-8434 ISSN1617-9692 (electronic)
LectureNotesinMathematics
ISBN978-3-319-51950-0 ISBN978-3-319-51951-7 (eBook)
DOI10.1007/978-3-319-51951-7
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Preface
Thepurposeofthisvolumeistogiveanup-to-dateintroductiontotensorvaluations
andtheirapplicationsonagraduatelevel.Avaluationisafinitelyadditivemapping.
Since Dehn’s use of real-valued valuations to give a negative answer to Hilbert’s
thirdproblemin1900,thetheoryofvaluationshasbeenextendedconsiderablyand
foundwidespreaduseinappliedscienceslikebiology,materialssciences,medicine
andphysics.Attheendofthetwentiethcentury,McMullenandseveralotherauthors
startedtoconsidertensor-valuedvaluations,buttheresearchonlygainedmomentum
when Alesker could show a characterization theorem for tensor valuations which
satisfy certain natural geometric and topological requirements, thus mirroring the
corresponding result for real-valued valuations by Hadwiger from the 1950s. In
the last decades, the intensive use of algebraic methodsand representationtheory
yielded a wealth of new results, including important integral geometric formulae
for tensor valuations. At the same time, the application of tensor valuations was
starting in stochastic geometry and a number of applied research areas, primarily
withthepurposeofquantifyingthemorphologyandanisotropyofcomplexspatial
structures.
In2011a veryclosecollaborationbetweentheCentre forStochasticGeometry
andAdvancedBioimaging(CSGB)andtheDFGfundedresearchgroupGeometry
and Physics of Spatial Random Systems (GPSRS) started, where one of the main
goalswastoadvancetheoreticalresearchandapplicationsoftensorvaluations.The
GPSRS,withresearchgroupsfromKarlsruheInstituteofTechnologyandtheUni-
versity of Erlangen-Nürnberg,combined fundamentalmathematical research with
studies in the physical sciences, enabling fruitful interdisciplinary collaborations.
The CSGB groupfromthe Universityof Aarhus,supportedbythe DanishVillum
Foundation,madeanumberofcontributionstothebasictheoryoftensorvaluations
and developed applications in biology and imaging. In view of the many new
developments,thesetworesearchgroupsjointlyorganizedtheWorkshoponTensor
ValuationsinStochasticGeometryandImagingduringSeptember21–26,2014,at
theSandbjergEstateinSouthernDenmark.Mostoftheeightinvitedspeakersofthis
workshop gave lectures of twice 45 min introducingtheir field of expertise on an
accessiblegraduatelevel.Extendedtranscriptsoftheselecturesformthebackbone
v
vi Preface
of thisbookand havebeen complementedbyinvitedcontributionsto broadenthe
scope.
The book develops around the central notion of Minkowski tensors, which
are tensor-valued valuations, typically defined on the family of convex bodies in
n-dimensionalEuclidean space. They are isometry covariant and continuouswith
respect to the Hausdorff metric. The most important special cases—also those
that have attracted most attention in the past—are the Minkowski tensors of rank
zero, the intrinsic volumes. Many of the results presented in this volume were
historically formulated for these scalar-valued valuations and later extended to
generalMinkowskitensors.
Since Blaschke introduced integral geometry as a subject of its own, integral
geometricformulaehavebeenplayingaprominentroleinthetheoryofvaluations.
Federer even stated that for a theory of curvature measures ‘to be worthwhile’, it
must contain versions of the principal kinematic formula (and the Gauss-Bonnet
theorem). Integral geometric formulae are also crucial for applications because
they can be used as tools in image reconstruction and stereology, as exemplified
inChap.14.Integralgeometricrelationsplayalsoaprominentroleinthisvolume.
InChaps.4and5,versionsoftheclassicalCroftonformulaarestatedforMinkowski
tensors. They are kinematic in nature, as they involve the intersection of a
convexbodywithaninvariantlytranslatedandrotatedflat.Importantstereological
applications in confocal microscopy are often based on flat sections through a
referencepointandthusrequirerotationalCroftonformulae,wheretheintegration
overtranslationsisomitted.RotationalCroftonformulaearepresentedinChap.7.
Hadwiger’sgeneralintegralgeometrictheoremallowstoderiveprincipalkinematic
formulae from (kinematic) Crofton formulae, and this is outlined in Chap.4. To
treatnon-isotropicBooleanmodels,atranslativeversionoftheprincipalkinematic
formulaisneededandthusprovidedinChap.11,eveninaniteratedform.Finally,
rotationsumformulaeforcertaintensorvaluationsaregiveninChap.4.
The book is organized as follows. The first two chapters lay the foundations by
introducing valuations and giving characterization theorems. Chapter 1 gives an
overviewofthestatusofthefieldpriortotherecentadvancesfirstbyAleskerand
later by others who exploited algebraic methods in integral geometry. It gives a
smooth introduction into the classical facts. It also introduces support, curvature
andareameasuresasimportantexamplesofmeasure-valuedvaluations.InChap.2,
Minkowskitensorsareintroducedandtheirpropertiesareexplained.Inparticular,
Alesker’s characterization theorem is given, stating that the vector space of all
continuous isometry covariant tensor valuations on convex bodies is spanned by
combinationsofMinkowskitensorsandthemetrictensor.Asimilarcharacterization
theorem is then established for local versions of the Minkowski tensors, where it
surprisinglyturnsoutthatthelatterclassisricherthanintheglobalcase.
The next four chapters give an introduction to aspects of ‘algebraic integral
geometry’. Chapters 3 and 4 introduce algebraic structures on certain subspaces
of tensor valuations.Significant in their own right, these conceptsare particularly
Preface vii
valuabletoderiveintegralgeometricresults.Chapter3introducesproduct,convo-
lution,theAlesker-Fouriertransformandpull-backandpush-forwardofvaluations
under linear maps. This chapter only treats the case of scalar-valued valuations.
Tensor-valuedversionsare given in Chap.4 and their close connectionto integral
geometric relations is revealed. This is then exploited to obtain rotation sum
formulae and a considerably simplified Crofton formula for translation invariant
tensor valuations.More generalCroftonformulae,also for surface area measures,
are outlined as well. Chapter 5 varies and extends the Crofton formulae of the
foregoingchapterin severalrespects. Firstly,itstates intrinsic versions,wherethe
intersection of the convex body with the integration plane E is considered as a
subsetofE.Secondly,itshowsthatCroftonformulaecanalsobestatedfortensorial
curvaturemeasures,which are versionsofthe Minkowskitensorslocalizedin Rn.
ThealgebraicmethodsoutlinedinChaps.3and4arealsoprincipaltoolsofChap.6.
Adecompositionofthespaceofcontinuousandtranslationinvariantvaluationsinto
asumofSO.n/-irreduciblesubspacesisdiscussed.ThisresultleadstoaHadwiger-
typetheoremfortranslationinvariantandSO.n/-equivariantvaluationswithvalues
in an arbitrary finite dimensional SO.n/-module. The class of these valuations
includes those with values in general tensor spaces. In Chap.7 rotational Crofton
formulaeandversionsofaprincipalrotationalformulaarepresented.Thischapter
also describes a Hadwiger-type characterization theorem for continuous, rotation
invariantpolynomialvaluations.
Chapters 8–10are devotedto valuationson domainsother than convexbodies,
although valuations on convex polytopes already played a role in the first two
chapters of the volume. In Chap.8, a theory of valuations on lattice polytopes is
outlined,includingaHadwiger-typecharacterization,theBetke-Knesertheoremfor
certainreal-valuedvaluationsandmorerecentresultsonvaluationswithvaluesin
the families of tensorsor convexbodies. In Chap.9, instead of treatingvaluations
on convex bodies of the Euclidean space Rn, (smooth) valuations on subsets of
n-dimensionalmanifoldsareconsidered.The roleof convexbodiesis nowplayed
by simple differential polyhedra. A theory, including local and global kinematic
formulae on space forms and a transfer principle, is explained with particular
emphasisontheHermitiancase.Chapter10generalizesthedomainsofvaluationsin
adifferentdirection:whilestillworkingwiththeflatcaseoftheEuclideann-space,
itinvestigateswhatregularityofitssubsetsactuallyisrequiredinordertodevelopa
theoryofvaluationsthatallowsforintegralgeometrickinematicformulae.Thevery
generalclass of WDC sets is introducedand the constructionof the normalcycle
forthesesetsisdiscussed.
Thelastfivechaptersaredevotedtoapplicationsoftensorvaluationsinstochastic
geometry, biology and imaging. Chapters 11 and 12 describe properties of tensor
valuations of Boolean models with convex or polyconvex grains. While isotropy
(together with stationarity) is often a standard assumption in the literature, none
of these chapters requires isotropy. In Chap.11 mean value formulae for scalar-
and tensor-valued valuations applied to Boolean models are given and explained
in the stationary case with an outlook to the newer developments and the non-
stationary case. Second-order formulae for valuations of the Boolean model in
viii Preface
an observation window, that is, covariances, can be derived in the asymptotic
regimewhenthewindowisexpanding.Thisisdonein Chap.12togetherwiththe
formulationofcentrallimittheorems.Chapter13describestheanalysisofrandom
tessellations with the help of tensor valuations applied to individual cells. The
goalis to assess propertiesof the underlyingstochastic processthat generatedthe
tessellation.Thechapterisatheory-basedsimulationstudyandcomparesVoronoi
tessellationsofstandardpointprocessmodels,STIT-andhyperplanemosaics,and
tessellations derived from hard sphere and hard ellipsoid models from particle
physics.Inappliedimageanalysisofstructuredsyntheticandbiologicalmaterials,
the described methods have been used to infer information about the formation
processfromspatialmeasurementsofanobservedrandomstructure.Chapter14is
motivatedbythe appliedproblemof estimatingvolumetensorsfromobservations
in planar sections in conventional microscopy using local stereological methods.
It presents a new estimator of mean particle volume tensors in three-dimensional
space from vertical sections. Also the last chapter is devoted to the determination
oftensorvaluationsin practicalapplications.Itgivesanoverviewoveralgorithms
thatapproximatetensorvaluationsofanobjectfrombinaryandgrey-valuedimages
anddiscussesin particulartheasymptoticpropertiesof these algorithmswhenthe
resolutionoftheimagestendstoinfinity.
Theeditorshaveaimedforacollectionofself-containedcontributionsallowing
thereadertoselectchapterswithoutthenecessityofreadingthewholevolumefrom
itsbeginning.Therefore,chaptersoftenstartwithashortsummaryofconceptsand
notionspossiblyalreadyintroducedearlierinthebook.This,andthecomprehensive
cross-referencingto other chapters, may give the reader with a sound background
adeeperunderstandingofaspecific subarea.Atthesametime,thenewcomercan
read these lecture notes as a comprehensive introduction to tensor valuations and
importantapplications.
AarhusC,Denmark EvaB.VedelJensen
September2016 MarkusKiderlen
Acknowledgements
Firstly, we would like to express our sincere gratitude to all contributors to this
book. The list of authors include a large number of the very top researchers in
valuationtheory,integralgeometryandstochasticgeometry.TheGPSRSgrouphas
playedaveryimportantrolefortheconceptandcontentofthisbook.Severalofthe
chaptersin this book are based on research performedwithin this research group.
ThefinancialsupportbytheVillumFoundationandtheGermanScienceFoundation
(DFG)isgratefullyacknowledged.ItenabledtheorganizationoftheWorkshopon
Tensor Valuations in Stochastic Geometry and Imaging in September 2014. The
invited overview lectures at this workshop were the basis for this book. We also
wanttothankLarsMadsenwhopatientlyunifiedthelayout,styleanduseofLaTex
inallthecontributions.Finally,weareindebtedtoUteMcCroryfromSpringer,who
counselledusinpracticalmattersfromtheverybeginningofthisbookproject.
ix
