Table Of ContentLecture Notes in Computer Science 3753
CommencedPublicationin1973
FoundingandFormerSeriesEditors:
GerhardGoos,JurisHartmanis,andJanvanLeeuwen
EditorialBoard
DavidHutchison
LancasterUniversity,UK
TakeoKanade
CarnegieMellonUniversity,Pittsburgh,PA,USA
JosefKittler
UniversityofSurrey,Guildford,UK
JonM.Kleinberg
CornellUniversity,Ithaca,NY,USA
FriedemannMattern
ETHZurich,Switzerland
JohnC.Mitchell
StanfordUniversity,CA,USA
MoniNaor
WeizmannInstituteofScience,Rehovot,Israel
OscarNierstrasz
UniversityofBern,Switzerland
C.PanduRangan
IndianInstituteofTechnology,Madras,India
BernhardSteffen
UniversityofDortmund,Germany
MadhuSudan
MassachusettsInstituteofTechnology,MA,USA
DemetriTerzopoulos
NewYorkUniversity,NY,USA
DougTygar
UniversityofCalifornia,Berkeley,CA,USA
MosheY.Vardi
RiceUniversity,Houston,TX,USA
GerhardWeikum
Max-PlanckInstituteofComputerScience,Saarbruecken,Germany
Ole Fogh Olsen Luc Florack
Arjan Kuijper (Eds.)
Deep Structure,
Singularities,
and Computer Vision
First International Workshop, DSSCV 2005
Maastricht, The Netherlands, June 9-10, 2005
Revised Selected Papers
1 3
VolumeEditors
OleFoghOlsen
ITUniversityofCopenhagen,DepartmentofInnovation
RuedLanggaardsVej7,2300CopenhagenS,Denmark
E-mail:[email protected]
LucFlorack
TechnicalUniversityofEinhoven
DepartmentofBiomedicalEngineering
DenDolech2,Postbus513,5600MBEindhoven,TheNetherlands
E-mail:[email protected]
ArjanKuijper
ITUniversityofCopenhagen,DepartmentofInnovation
RuedLanggaardsVej7,2300Copenhagen,Denmark
E-mail:[email protected]
LibraryofCongressControlNumber:2005935534
CRSubjectClassification(1998):I.4,I.5,I.3.5,I.2.10,I.2.6,F.2.2
ISSN 0302-9743
ISBN-10 3-540-29836-3SpringerBerlinHeidelbergNewYork
ISBN-13 978-3-540-29836-6SpringerBerlinHeidelbergNewYork
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Preface
Whatisactuallytheinformation directlyrepresentedinthescale-space? Istarted
to wonder about this shortly after Peter Johansen, 15 years ago, showed me
his intriguing paper on how uniquely to reconstruct a band-limited 1D signal
fromitsscale-spacetoppoints.Still,Ihavenotfullyunderstooditsimplications.
Merely recording where structure vanishes under blurring is sufficient to fully
reconstruct the details. Of course, technicalities exist, for example, you must
also know negative scale toppoints. Nevertheless, I find it surprising that we
may trade the metric properties of a signal with the positions of its inherent
structure.Theresulthasbeengeneralizedtoanalyticsignals,shownalsoforthe
zero crossings of the Laplacean, but has not yet been generalized to 2D. This
remains an open problem.
In 2003, Peter Giblin, LiverpoolUniversity, Luc Florack,Eindhoven Univer-
sity of Technology, Jon Sporring, University of Copenhagen, my colleague Ole
Fogh Olsen, and severalothers started the project collaborationDeep Structure
and Singularities in Computer Vision under the European Union, IST, Future
and Emerging Technologies program, trying to obtain further knowledge about
whatinformationisactuallycarriedbythesingularitiesofshapesandgray-scale
images. In this project, we probed from several directions the question of how
much of the metric information is actually encoded in the structure of shapes
and images. We, and many others, have given hints in this direction. We have
shownthat—to a verylarge degree—youmay reconstruct2D images fromtheir
toppoints, and—to a very large degree—you may identify images in a database
based solely on the toppoints. Likewise, we have shown that—to a very large
degree—youmayindex shapes basedontheir singularities,aswasshownearlier
byBenjaminKimiaandcolleagues.Hence,thestructuremaybeuseful.Butstill,
we do not really know its limitations.
This current volume of LNCS is the proceedings from the workshop held
in Maastricht, June 10–11, 2005. This workshop was based on invited speakers
and contributed papers subjected to peer review. From these, 22 papers were
selected for this volume.They representthe year 2005state of the artin under-
standingtherelationbetweenstructural,topologicalinformationrepresentedby
singularities and metric information of signals, shapes, images, and colors. The
concise results like “the toppoints encode all metric information” still remain,
butprogressandinsighthavebeengainedoverthelast15years.Inthis volume,
the reader will find papers by many of the people who have contributed to the
discussion in the past: Does structure matter?
Mads Nielsen
The IT University of Copenhagen
Organization
Organizing Committee
Luc Florack Ole Fogh Olsen
Mads Nielsen Camilla Jørgensen
Arjan Kuijper
Technical Committee
Bram Platel Mac Wendelboe
Frans Kanters
Program Chairs
Ole Fogh Olsen Luc Florack
Program Committee
Mads Nielsen Frans Kanters
Kim Steenstrup Pedersen Bram Platel
Martin Lillholm Peter Giblin
Arjan Kuijper Andre Diatta
Anna Pagh Ali Shokoufandeh
Philip Bille Benjamin B. Kimia
Jon Sporring James Damon
Kerawit Somchaipeng
Referees
Remco Duits Evguenia Balmachnova
Marco Loog
Invited Speakers
James Damon (University of North Carolina)
Ali Shokoufandeh (Drexel University)
Benjamin B. Kimia (Brown University)
VIII Organization
Sponsoring Organizations
NWO, the Netherlands Organisation for Scientific Research, and the IST Pro-
gramme of the European Union are gratefully acknowledged for financial sup-
port.
Table of Contents
Oral Presentations
Blurred CorrelationVersus Correlation Blur
Jan J. Koenderink, Andrea van Doorn............................ 1
A Scale Invariant Covariance Structure on Jet Space
Bo Markussen, Kim Steenstrup Pedersen, Marco Loog .............. 12
Essential Loops and Their Relevance for Skeletons and Symmetry Sets
Arjan Kuijper, Ole Fogh Olsen .................................. 24
Pre-symmetry Sets of 3D Shapes
Andr´e Diatta, Peter Giblin ...................................... 36
Deep Structure of Images in Populations Via Geometric Models in
Populations
Stephen M. Pizer, Ja-Yeon Jeong, Robert E. Broadhurst, Sean Ho,
Joshua Stough ................................................. 49
Estimating the Statistics of Multi-object Anatomic Geometry Using
Inter-object Relationships
Stephen M. Pizer, Ja-Yeon Jeong, Conglin Lu, Keith Muller,
Sarang Joshi .................................................. 60
Histogram Statistics of Local Model-Relative Image Regions
Robert E. Broadhurst, Joshua Stough, Stephen M. Pizer,
Edward L. Chaney ............................................. 72
The Bessel Scale-Space
Bernhard Burgeth, Stephan Didas, Joachim Weickert ............... 84
Linear Image Reconstruction from a Sparse Set of α-Scale Space
Features by Means of Inner Products of Sobolev Type
Remco Duits, Bart Janssen, Frans Kanters, Luc Florack ............ 96
A Riemannian Framework for the Processing of Tensor-Valued Images
Pierre Fillard, Vincent Arsigny, Nicholas Ayache, Xavier Pennec .... 112
From Stochastic Completion Fields to Tensor Voting
Markus van Almsick, Remco Duits, Erik Franken,
Bart ter Haar Romeny ......................................... 124
X Table of Contents
Deep Structure from a Geometric Point of View
Luc Florack ................................................... 135
Maximum Likely Scale Estimation
Marco Loog, Kim Steenstrup Pedersen, Bo Markussen .............. 146
Adaptive Trees and Pose Identification from External Contours of
Polyhedra
Yannick L. Kergosien .......................................... 157
Poster Presentations
Exploiting Deep Structure
Arjan Kuijper ................................................. 169
Scale-Space Hierarchy of Singularities
Tomoya Sakai, Atsushi Imiya ................................... 181
Computing 3D Symmetry Sets; A Case Study
Arjan Kuijper, Ole Fogh Olsen .................................. 193
Irradiation Orientation from Obliquely Viewed Texture
Sylvia C. Pont, Jan J. Koenderink ............................... 205
Using Top-Points as Interest Points for Image Matching
Bram Platel, Evguenia Balmachnova, Luc Florack,
Frans Kanters, Bart M. ter Haar Romeny ........................ 211
Transitions of Multi-scale Singularity Trees
Kerawit Somchaipeng, Jon Sporring, Sven Kreiborg,
Peter Johansen ................................................ 223
A Comparison of the Deep Structure of α-Scale Spaces
Remco Duits, Frans Kanters, Luc Florack, Bart ter Haar Romeny ... 234
A Note on Local Morse Theory in Scale Space and Gaussian
Deformations
Jan–Cees van der Meer ......................................... 249
Author Index................................................... 259
Blurred Correlation Versus Correlation Blur
Jan J. Koenderink and Andrea van Doorn
Universiteit Utrecht
[email protected]
Abstract. We discuss the topic of correlation in a scale space setting.
Correlation involves two distinct scales. The “outer scale” is the scale
of the region over which the correlation will be calculated. Classically
this is the whole space of interest, but in many cases one desires the
correlation over some region of interest. The “inner scale” is the scale
at which the signals to be correlated are represented. Classically this
meansinfiniteprecision.Forourpurposeswedefine“correlation” asthe
point–wise product of two signals, “blurred correlation” as the integra-
tionofthiscorrelation overtheregionofinterest,and“correlation blur”
as this point–wise correlation applied to the signals represented at the
inner scale. For generic purposes we are interested in “blurred correla-
tion blur”. We discuss a well known (and practically important) exam-
pleofblurredcorrelationforessentiallyzeroinnerscale.Suchasituation
leadstoapparentlyparadoxicalresults.Wethendiscusscorrelationblur,
which can beunderstood as a form of “regularized” correlation, leading
tointuitivelyacceptableresultsevenforthecaseofpointsets(e.g.,tem-
poraleventsorpointsetsinspace).Wedeveloptheformalstructureand
present a numberof examples.
1 Blurred Correlation
We will speak of a “correlation” r(s) of two signals f(r) and g(r) if r(s) is
a blurred version of the product of blurred versions of f(r) and g(r). This is
a slight generalization of the usual concept where the correlation is the fully
blurredproductofthe unblurredfunctions.Notice thatthere existtwoessential
scaleparametershere,namelythe“outerscale”,whichisthescaleoftheblurring
of the product, and the “inner scale”, which is the scale of the blurring of the
components.
Firstwe willconsiderthe caseofverysmallinner scale,i.e., the components
are fully resolved atthe point where they enter the multiplication process.Such
cases are comparatively rare.
A key instance of this “blurred correlation” occurs in human vision[10].
The causal chain in vision goes as follows: An illuminant causes an illumina-
tion spectrum f(λ) on a surface with spectral reflectance g(λ), causing a beam
to be scattered to the eye whose luminance—apart from an inessential con-
stant factor—has a spectrum that equals the product f(λ)g(λ). The product is
blurred in the retinal transduction process and gives rise to the blurred correla-
tionk(λ;σ)◦(f(λ)g(λ)) ofwhichthreepointsamplesenterthe optic nerve.The
O.F.Olsenetal.(Eds.):DSSCV2005,LNCS3753,pp.1–11,2005.
(cid:1)c Springer-VerlagBerlinHeidelberg2005
