global optimization and approximation algorithms in computer vision

144 Pages·2007·3.79 MB·English

GLOBAL OPTIMIZATION AND APPROXIMATION
ALGORITHMS IN COMPUTER VISION
CARL OLSSON
FacultyofEngineering
CentreforMathematicalSciences
Mathematics
Mathematics
CentreforMathematicalSciences
LundUniversity
Box118
SE-22100Lund
Sweden
http://www.maths.lth.se/
LicentiateThesesinMathematicalSciences2007:1
ISSN1404-028X
ISBN978-91-628-7268-7
LUTFMA-2026-2007
(cid:13)CarlOlsson,2007

PrintedinSwedenbyKFS,Lund2007
Organization Document name
Centre for Mathematical Sciences LICENTIATE THESES IN MATHEMATICAL
Lund Institute of Technology SCIENCES
Mathematics Date of issue
Box 118 September 2007
SE-221 00 LUND Document Number
LUTFMA-2026-2007
Author(s) Supervisor
Carl Olsson Fredrik Kahl, Kalle Åström
Sponsering organisation
Title and subtitle
Global Optimization and Approximation Algorithms in Computer Vision
Abstract
Computer Vision is today a wide research area including topics like robot vision, image analysis, pattern
recognition, medical imaging and geometric reconstruction problems. Over the past decades there has been a
rapid development in understanding and modeling different computer vision applications. Even though much
work has been made on modeling different problems, less work has been spent on deriving algorithms that
solve these problems optimally. Generally one is referred to local search methods such as bundle adjustment.
In this thesis we are interested in developing methods that are guaranteed to find globally optimal solutions.
Typically the considered optimization problems are non-convex and may have many local optima.
The thesis consists of an introductory chapter followed by five papers. The introduction gives a motivation for
the thesis, and a brief introduction to some concepts from optimization that are used throughout the thesis.
Furthermore a summary of the included papers is given.
In the first paper we study different kinds of pose and registration problems involving Euclidean
transformations. We develop an efficient branch and bound algorithm, that is guaranteed to find the global
optimum. In the second paper the theory of L -optimization is applied to 1D-vision problems.
∞
We show that the problems considered can be simplified considerably when using the L -norm instead of the
∞
standard L -norm. In the third paper necessary and sufficient conditions for a global optimum for a class of
2
L -norm problems is derived. Based on these conditions a more efficient algorithm, compared to the usual
∞
bisection method, is presented. The fourth paper deals with large scale binary quadratic optimization. Two
alternatives to semidefinite programming, based on spectral relaxations, are given.
In the final paper we present a reformulation of the classical normalized cut method for image segmentation.
We show that using this formulation it is possible to incorporate contextual information.
Key words
Keywords: quasiconvex optimization, multiple view geometry, registration, camera pose, image segmentation,
normalized cuts, branch and bound, spectral relaxations, computer vision, binary quadratic optimization.
Classification system and/or index terms (if any)
Supplementary bibliography information
ISSN and key title ISBN
1404-028X 978-91-628-7268-7
Language Number of pages Recipient’s notes
English 142
Security classification
The thesis may be ordered from the Department of Mathematics, under the adress above.

Preface
Thislicentiate’sthesisconsidersoptimizationmethodsusedincomputervision. Numer-
ousproblemsinthisfieldaswellasinimageanalysisandotherbranchesofengineering
can be formulated as optimization problems. Often these problems are solved using
greedyalgorithms that find locallyoptimal solutions. In this thesis we areinterested in
developingmethodsforfindingsolutionsthataregloballyoptimal. Forcertain(NP-hard)
problemsitcanbeshownthatitisnotpossibletofindtheglobaloptimuminreasonable
time. Inthiscaseonewishestofindapproximationalgorithmsthatyield,notjustlocally
optimal, butsolutions that are close to the global optimum. The thesis consists of five
papersandanintroductorychapter. Intheintroductionsomebackgroundmaterialand
anoverviewofthepapersisgiven.
Thethesisconsistofthefollowingfivepapers:
• C.Olsson,F.Kahl,M.Oskarsson,BranchandBoundMethodsforEuclideanReg-
istrationProblems,submittedtoIEEETransactionsonPatternAnalysisandMachine
Intelligence,2007.
• K. Åströ m, O. Enqvist, C. Olsson, F. Kahl, R. Hartley, An L Approach to
∞
Structure and Motion Problems in 1D-Vision, Proc. International Conference on
ComputerVision(ICCV),RiodeJaneiro,Brazil,2007.
• C.Olsson,A.P.Eriksson,F.Kahl,EfficientOptimizationforL -problemsusing
∞
Pseudoconvexity, Proc. InternationalConference on Computer Vision (ICCV), Rio
deJaneiro,Brazil,2007.
• C.Olsson,A.P.Eriksson,F.Kahl,ImprovedSpectralRelaxationMethodsforBi-
naryQuadraticOptimization Problems,submittedtoComputerVisionandImage
Understanding,2007
• A.P.Eriksson,C.Olsson,F.Kahl,NormalizedCutsRevisited:AReformulationfor
SegmentationwithLinearGroupingConstraints,Proc. InternationalConferenceon
ComputerVision(ICCV),RiodeJaneiro,Brazil,2007.
The first three papers concernproblems whereit is possible to find the globalopti-
mum,whilethelasttwopapersdealwithapproximationtechniquesforknownNP-hard
problems.
Mostofthematerialiscoveredbythesepapersordescribedintheintroduction. The
thesisisalsobasedonthefollowingpapers:
• C. Olsson, F. Kahl, M. Oskarsson, Optimal Estimation of Perspective Camera
Pose, Proc. International Conference on Pattern Recognition (ICPR), Hong Kong,
China,2006.
3
• C.Olsson,F.Kahl,M.Oskarsson,TheRegistrationProblemRevisited: Optimal
SolutionsFromPoints,LinesandPlanes,Proc. ComputerVisionandPatternRecog-
nition(CVPR),NewYorkCity,USA,2006.
• A.P.Eriksson,C.Olsson,F.Kahl,ImageSegmentationwithContext,Proc. Scan-
dinavianConferenceonImageAnalysis(SCIA),Åhlborg,Denmark,2007.
• C. Olsson, A. P. Eriksson, F. Kahl, Solving Large Scale Binary Quadratic Prob-
lems: SpectralMethodsvs.SemidefiniteProgramming,Proc. ComputerVisionand
PatternRecognition(CVPR),Minneapolis,USA,2007.
• A.P.Eriksson,C.Olsson,F.Kahl,EfficientSolutionstotheFractionalTrustRegion
Problem,Proc. AsianConferenceonComputerVision(ACCV),Tokyo,Japan,2007.
• I.Dressler,C.Olsson,K.Åström,A.Robertsson,R.Johansson,AutomaticKine-
maticCalibrationofaRobotUsingVision, tobesubmitted.
Acknowledgements
First of all, I would like to thank my supervisors Fredrik Kahl and Kalle Åström for
givingmeguidance,supportandpatience. Furthermore,theircarefulexaminationofthis
manuscriptaswellasothermanuscriptshasimprovedthequalityconsiderably. Iwould
alsoliketothankFredrikforintroducingmetooptimizationincomputervisionandfor
many discussions on the subject. The time he has devoted to me has been crucial, his
helpandguidancehasleadtoanumberofacceptedpapers;thankyouFredrik.
Ihavehadthe privilege ofworkingwithin the MathematicalImagingGroupatthe
Centre for Mathematical Sciences. I am indebted to the members of the groupas well
as other colleagues within the department for interesting discussions. Most notably to
Anders P. Eriksson for fruitful collaborations on different problems. Furthermore, I
would like to acknowledge Olof Barr for not so interesting, but necessary, discussions
on the spelling and grammatics of the English language, Gunnar Sparr for reading the
manuscriptandAnkiOttosson for helpingme with allkinds of administrative difficul-
ties.
Finally,Iwouldalsoliketothankmyfamilyforgeneralsupportinallaspectsofreal
life.
4
Introduction
1.1 Motivation
ComputerVisionhasevolvedintoawideresearchareaincludingtopicslikerobotvision,
image analysis, pattern recognition and multiple view geometry. Over the pastdecades
therehasbeenarapiddevelopmentin understandingandmodelingdifferentcomputer
vision applications. While much workhasbeenmade on modelingdifferent problems,
lessworkhasbeenspentonderivingalgorithmsthatsolvetheseproblemsoptimally. Typ-
icallyoneisreferredtolocalsearchmethods suchasgradientdescentsearchorNewton
basedmethods.
Consider for instance the two view structure and motion problem. The goal is to
compute the 3D scene geometry (structure) and the positions and relative orientations
of the cameras (motion) from corresponding image features. A common approach for
solvingthisproblemisbycalculatingthesocalledessentialmatrix. Thisisdonebyusing
analgebraicsolversuchasthesix-pointsolver(see[11]). Eventhoughthisworksfineif
the data is wellbehaved, it often fails in the presence of significantmeasurementnoise.
Thisisbecausetheabovemethodisonlyabletosolvethe problemforexactlysixpoint
correspondences,andhenceanyremainingdataisdisregarded.
Instead we would like to optimize a geometrically meaningful quantity such as the
reprojectionerrors. Itiseasytoformulateameaningfuloptimization criterionforgood
reconstructions, however, we are referred to local search methods for solving it. The
successofthese methods arehighlydependenton good initialization, which is typically
donewithalgebraicsolvers.
In contrast to the methods mentioned above, we would like to find a meaningful
optimizationcriterionthatallowsustodesignalgorithmsthatcanbeshowntoconverge
totheglobaloptimum. Thegoalofthisthesisistodevelopmethodsthatareguaranteed
to find globally optimal solutions if possible. For certain problems this is not possible
to doin reasonabletime. In these caseswe wantto find approximatesolutions thatare
notjustlocallyoptimal, butwith objective valueclose to the globaloptimum. Inmost
cases,goodformulationsoftheproblemsarereadilyavailable. Thederivationofmodels
falloutsidethescopeofthisthesis;wearemerelyinterestedinhowtosolvetheexisting
formulationsoptimally.
Our aim has been to improve the state-of-the-art for a large class of optimization
problems in computer vision. Applications include multiview geometry problems, reg-
istration problems, image segmentation, partitioning, binary restoration and subgraph
5
INTRODUCTION
matching. Ourapproachis basedon globaloptimization methods, suchas Branchand
Bound, convex andquasiconvexoptimization. Some ofthe problemscan besolvedex-
actlywithglobalmethodswhileothershavetobeapproximatedtoobtaingoodsolutions
inreasonabletime.
Theremainderofthischapterisorganizedasfollows: Insection1.2and1.3wegive
ashortintroductiontosomebasicconceptsformoptimizationthatareusedthroughout
thethesis. Insection1.4wegiveanoverviewoftheincludedpapersandsummarizethe
maincontributions.
1.2 Convex Optimization
In this section we will review some basic concepts used in optimization, that are used
throughout this thesis. For a more complete introduction of convex optimization see
[4,2].
1.2.1 Convex Sets
AsetS ∈RniscalledconvexifthelinesegmentjoininganytwopointsinSiscontained
inS. Thatis,ifx,y ∈S thenλx+(1−λ)y ∈S forallλwith0≤λ≤1.
Wecallapointxoftheform
n
x= λ x , (1.1)
i i
Xi=1
n
where λ = 1, 0 ≤ λ ≤ 1 and x ∈ S, a convex combination of the points
i=1 i i i
x1,...,Pxn.Aconvexsetalwayscontainseveryconvexcombinationofitspoints. Further-
more,itcanbeshownthatasetisconvexonlyifitcontainsallitsconvexcombinations.
TheconvexhullofasetS,denotedconvhull(S),isthesetofallconvexcombinations
ofpointsinS. Sincethissetcontainsallitsconvexcombinationsitisaconvexset. Itis
alsothesmallestconvexsetcontainingS. Figure1.1showssomesimpleexamplesofthe
notionsintroduced.
Nextwewillstatethreespecialcasesofconvexsetsthatareusedextensivelythrough-
outthethesis.
Thehalfspace. Ahalfspaceisasetoftheform
{x∈Rn;aTx≤b}, (1.2)
where a 6= 0, i.e., it is the solution set of a nontrivial affine inequality. The
boundary of the halfspace is the hyperplane {x ∈ Rn;aTx = b}. It is straight
forwardtoverifythatthesesetsareconvex.
6
1.2. CONVEXOPTIMIZATION
Figure1.1: Left: Aconvexset. Middle: Anon-convexset. Right: Theconvexhullofthe
middleset.
Thesecondordercone. Let||·||beanynormonRn. ThenormconeinRn+1associ-
atedwiththenorm||·||istheset
{(x,t)∈Rn+1;||x||≤t}. (1.3)
Fromthegeneralpropertiesofnormsitfollowsthatthenormconeisaconvexset
inRn+1. ThesecondorderconeisthenormconefortheEuclideannorm
{(x,t)∈Rn+1;||x||2 ≤t}. (1.4)
Thepositivesemidefinitecone. Let Sn be the set of symmetric matrices. The set Sn
canbeviewedasavectorspaceofdimensionn(n+1)/2.ByX (cid:23)0wemeanthat
the matrix X is positive semidefinite. The setof symmetric positive semidefinite
matrices
Sn ={X ∈Rn×n;X =XT,X (cid:23)0}, (1.5)
+
isaconvexsetinSn. ThiscanbeseenbynotingthatifyTAy ≥0,andyTBy ≥
0,then
λyTAy+(1−λ)yTBy ≥0. (1.6)
Iff :Rm 7→RnisanaffinemappingthenthesetS′ ={x;f(x)∈S}isconvexin
Rm ifS isconvexinRn. Thatis,convexityispreservedunderaffinemappings. When
appliedtothesecondorderconewegetthesetsoftype
{x;||Ax+b||2 ≤cTx+d}. (1.7)
WecandefineanaffinemappingA(x):Rn 7→Sm by
A(x)=x1A1+...+xnAn+B (1.8)
whereA1,...,An,B ∈ Sm. Whenappliedtothe positive semidefinite conewegetthe
linearmatrixinequalities
x1A1+...+xnAn+B (cid:23)0. (1.9)
7
INTRODUCTION
Convexityis alsopreservedunderintersection. ThusasetS thatisgivenbyseveral
ofthe constraintsabove(halfspaces,cone-constraintsandlinearmatrix inequalities)isa
convexset.
1.2.2 Convex functions
A function f : S 7→ R is calledconvex if S is a convex set, and for all x,y ∈ S and
0≤λ≤1,wehave
f(λx+(1−λ)y)≤λf(x)+(1−λ)f(y). (1.10)
Thegeometricinterpretationofthisdefinition,isthatthelinesegmentbetweenthepoints
(x,f(x)) and (y,f(y)) should lie above the graph of f. The function f is said to
be concave if −f is convex. For differentiable functions convexity can alternatively be
definedbytheinequality
f(x)≥f(y)+∇f(y)T(x−y). (1.11)
Iff is differentiablethenf is convexifandonlyif(1.11)holdsfor allx,y ∈ S. Geo-
metricallythis meansthatf is convexifandonlyiff liesaboveits tangentplaneatall
points. Figure 1.2 shows the geometrical interpretation of the definitions. It is easy to
x y y
Figure 1.2: Graph of a convex function. Left: The line segment joining the points
(x,f(x)) and (y,f(y)) lies above the graph. Right: The graph of the function lies
aboveitstangentplaneaty.
see whyconvexityis importantin optimization in view of (1.11). Theinequalitystates
that the tangent plane is a global underestimator of the function. That is, from local
information(thegradient)itispossibletoobtainglobalinformation(anunderestimator)
for the function. For example, if ∇f(y) = 0 then f(x) ≥ f(y) for all y ∈ S. That
is,anystationarypointisalsoaglobalminimum. Whendealingwithmaximizationthe
samepropertiesapplyifconvexityisreplacedwithconcavity.
8