Tensor Valuations and Their Applications in Stochastic Geometry and Imaging PDF
Preview Tensor Valuations and Their Applications in Stochastic Geometry and Imaging
Lecture 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 LibraryofCongressControlNumber:2017942766 ©SpringerInternationalPublishingAG2017 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland 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
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