Sparse Optimization for Robust and Efficient Loop Closing

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by  Yasir Latif

Sparse Optimization for Robust and Efficient Loop Closing
YasirLatifa,GuoquanHuangb,JohnLeonardc,Jose´ Neirad
aARCCenterforRoboticVision,UniversityofAdelaide,AdelaideSA5005,Australia1
bDepartmentofMechanicalEngineering,UniversityofDelaware,Newark,DE19716,USA
cComputerScienceandArtificialIntelligenceLaboratoryMassachusettsInstituteofTechnology,Cambridge,MA02139,USA
dInstitutodeInvestigacio´nenIngenier´ıadeArago´n(I3A)UniversidaddeZaragoza,Zaragoza,Spain
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0 Abstract
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It is essential for a robot to be able to detect revisits or loop closures for long-term visual navigation.
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Akeyinsightexploredinthisworkisthattheloop-closingeventinherentlyoccurssparsely,i.e.,theimage
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1 currentlybeingtakenmatcheswithonlyasmallsubset(ifany)ofpreviousimages. Basedonthisobserva-
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tion,weformulatetheproblemofloop-closuredetectionasasparse,convex(cid:96)1-minimizationproblem. By

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O leveragingfastconvexoptimizationtechniques, weareabletoefficientlyfindloopclosures, thusenabling
R real-time robot navigation. This novel formulation requires no offline dictionary learning, as required by
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s most existing approaches, and thus allows online incremental operation. Our approach ensures a unique
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hypothesisbychoosingonlyasinglegloballyoptimalmatchwhenmakingaloop-closuredecision. Further-

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more,theproposedformulationenjoysaflexiblerepresentationwithnorestrictionimposedonhowimages
v
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shouldberepresented,whilerequiringonlythattherepresentationsare“close”toeachotherwhenthecor-
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9 responding images are visually similar. The proposed algorithm is validated extensively using real-world
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0 datasets.
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7 1. Introduction
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v Withagrowingdemandforautonomousrobotsinarangeofapplications,suchassearchandrescue[42,
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X 8],andspaceandunderwaterexploration[7],itisessentialfortherobotstobeabletonavigateaccuratelyfor
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a anextendedperiodoftimeinordertoaccomplishtheassignedtasks.Tothisend,theabilitytodetectrevisits
(i.e.,loopclosureorplacerecognition)becomesnecessary,sinceitallowstherobotstoboundtheerrorsand
uncertaintyintheestimatesoftheirpositionsandorientations(poses).Inthiswork,weparticularlyfocuson
1Correspondingauthor,email:{[email protected]}
PreprintsubmittedtoElsevier February1,2017
loopclosureduringvisual navigation, i.e., given acamerastreamweaimtoefficiently determinewhether
therobothaspreviouslyseenthecurrentplaceornot.
Eventhoughtheproblemofloopclosurehasbeenextensivelystudiedinthevisual-SLAMliterature(e.g.,
see[30,13,21]),avastmajorityofexistingalgorithmstypicallyrequiretheofflinetrainingofvisualwords
(dictionary)fromaprioriimagesthatareacquiredpreviouslyinvisuallysimilarenvironments. Clearly,this
isnotalwaysthecasewhenarobotoperatesinanunknown,drasticallydifferentenvironment. Ingeneral,
itisdifficulttoreliablyfindloopsin(visual)appearancespace. Oneparticularchallengeistheperceptual
aliasing–thatis,whileimagesmaybesimilarinappearance,theymightbecomingfromdifferentplaces.
Tomitigatethisissue,bothtemporal(i.e.,loopswillonlybeconsideredclosedifthereareotherloopsclosed
nearby) and geometric constraints (i.e., if a loop has to be considered closed, a valid transformation must
exist between the matched images) can be employed [21]. It is important to point out that the approach
of [21] decides on the quality of a match locally – If the match with the highest score (in some distance
measure)isawayfromthesecondhighest,itisconsideredavalidcandidate. However,thelocalinformation
mayleadtoincorrectloop-closuredecisionsbecausebothtemporalandgeometricconditionscaneasilyfail
inhighlyself-similarenvironmentssuchascorridorsinahotel.
Toaddresstheaforementionedissues,inthispaperweintroduceageneral,onlineloop-closureapproach
for vision-based robot navigation. In particular, by realizing that loops typically occur intermittently in a
navigationscenario,we,forthefirsttimeever,formulateloop-closuredetectionasasparse(cid:96) -minimization
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problemthatisconvex. Thisisopposedtothecurrentmethodsthatcastloopclosuredetectionasanimage
retrievalproblem[21,13].Byleveragingthefastconvexoptimizationtechniques,wesubsequentlysolvethe
problem efficiently and achieve real-time frame-rate generation of loop-closure hypotheses. Furthermore,
the proposed formulation enjoys flexible representations and can produce loop-closure hypotheses regard-
less of what the extracted features represent – that is, any discriminative information, such as descriptors,
Bag of Words (BoW), or even whole images, can be used for detecting loops. Lastly, we shall stress that
our proposed approach declares a loop that is valid only when it is globally unique, which ensures that if
perceptualaliasingisbeingcausedbymorethanonepreviousimage,noloopclosurewillbedeclared. Al-
thoughthisisconservativeinsomecases,sinceafalseloopclosingcanbecatastrophicwhilemissingaloop
closuregenerallyisnot,ensuringsuchglobaluniquenessisnecessaryandimportant,inparticular,inhighly
self-similarenvironments.
Therestofthepaperisorganizedasfollows:Afterreviewingtherelatedwork,weformulateloop-closure
detectionasasparse(cid:96) -minimizationprobleminSection3. InSection4wepresentindetailtheapplication
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of this formulation to visual navigation, which is validated via the real-world experiments in Section 5.
Finally,Section6concludesthisworkaswellasoutlinesthepossibledirectionsforfutureresearch.
2. RelatedWork
The problem of loop-closure detection has been extensively studied in the SLAM literature and many
different solutions have been proposed over the years (e.g., see [24, 30] and references therein). In what
follows,webrieflyoverviewtheworkthatcloselyrelatestotheproposedapproach.
In particular, the FAB-MAP [13] is a probabilistic appearance-based approach using visual BoW for
placerecognition,andwasshowntoworkrobustlyovertrajectoriesupto1000km. Similarly,theBinary-
BoW (BBoW)-based method [21] detects the FAST keypoints [38] and employs a variation of the BRIEF
descriptors [6] to construct the BoW. A verification step is further enforced to geometrically check the
featuresextractedfromthematchedimages. Itshouldbepointedoutthatbothmethods[13,21]arebased
onthesimilarideasoftext-retrieval[34,40]: ThesemethodslearntheBoWdictionariesbeforehand,which
are used later for detecting loop closures when the robots actually operates in the field. This restricts the
expressive power of the dictionary in cases where it has to operate in environments drastically different
fromwherethedictionarywasconstructed. Incontrast,theproposedapproachbuildsthedictionaryonline
astherobotexploresanunknownenvironment, whileatthesametimeefficientlydetectingloops(ifany).
Moreover,ratherthansolelyrelyingonthedescriptors-basedBoW,ourmethodisflexibleandcanutilizeall
pixelinformationtodiscriminateplaceseveninpresenceofdynamicobjects(encodedassparseerrors),any
descriptorthatcanrepresentsimilarplaces,oranycombinationofsuchdescriptors.
Somerecentworkhasfocusedonloopclosureunderextremechangesintheenvironmentsuchasdif-
ferent weather and/or lighting conditions at different times of the day. For example, Milford and Wyeth
[33]proposedtheSeqSLAMthatisabletolocalizewithdrasticlightingandweatherchangesbymatching
sequencesofimageswitheachotherasopposedtosingleimages. ChurchillandNewman[12]introduced
theexperience-basedmapsthatlearnthedifferentappearancesofthesameplaceasitgraduallychangesin
order to perform long-term localization. Building upon [12], Paul and Newman [36] also discovered new
imagestoattainbetterlocalization. Inaddition,Leeetal.[28,29]haveexploredgeometricfeaturessuchas
linesforthetaskofloopclosuredetectioninbothindoorandoutdoorscenarios. Notethatiftheinformation
invarianttosuchchangescanbeextractedasin[13,21,33,12,36,28,29], theproposedformulationcan
alsobeusedtoobtainloop-closurehypotheses. Essentially,inthisworkwefocusonfindingloopclosures
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givensomediscriminativedescriptionssuchasdescriptorsandwholeimages,assumingnospecifictypeof
imagerepresentations.
Morerecently,withtherediscoveryofefficientmachinelearningtechniques,ConvolutionalNeuralNet-
works (CNNs)[4, 27] have been exploited to address loop closure detection [44, 45]. These networks are
multi-layeredarchitecturesthataretypicallytrainedonmillionsofimagesfortaskssuchasobjectdetection
andsceneclassification. Theinternalrepresentationsateachlayerarelearnedfromthedataitselfandthere-
forecanbeusedasfeaturestoreplacehand-craftedfeatures. Basedonthisapproach,Su¨nderhaufetal.[44]
extract features from different layers in the network and identify the layers that are useful for view-point
andilluminationinvariantplacerecognition. Moreover,in[45]landmarksaretreatedasobjectsbyfinding
objectproposalsintheimagesandfeaturesareextractedforthemusingdeepnetworks. Thesefeaturesthen
allowforview-pointinvariantplacecategorizationbymatchingdifferentobjectsfromvariedviewpoints. In
these CNN-based place categorization techniques, the networks are used as feature extractors followed by
someformofmatching. Inthispaper,weshowthatthesedeepfeaturescanalsobeutilizedintheproposed
frameworkofloop-closuredetection.
Itshouldbenotedthatinourpreviousconferencepublication[26],wehavepreliminarilyshownthatthe
proposedloop-closingframeworkisgeneralandcanemploymosthand-craftedfeatures. Recently,Shakeri
andZhang[39]extendedthissparse-optimizationbasedframeworktoanincrementalformulationallowing
fortheuseoftheprevioussolutionofthesparseoptimizationtojumpstartthenextone,whileZhangetal.
[46]furtherextendedittoamulti-stepdelayeddetectionofloops(insteadofsingle-stepdetectionasinour
prior work [26]) in order to exploit the structured sparsity of the problem. In this paper, we present more
detailedanalysisandthoroughperformanceevaluations,includingnewexperimentsusingdeepfeaturesand
validations in challenging multiple-revisit scenarios, as well as new comparisons against the well-known
nearestneighbour(NN)search.
3. SparseOptimizationforLoopClosure
Inthissection,weformulateloop-closuredetectionasasparseoptimizationproblembasedonasparse
andredundantrepresentation. Suchrepresentationshavebeenwidelyusedincomputervisionforproblems
such as denoising [19], deblurring [18], and face recognition [11]. Similarly, Casafranca et al. [9, 10]
formulated the back-end of graph SLAM as an (cid:96) -minimization problem. However, no prior work has
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yet investigated this powerful technique for loop closure detection in robot navigation. The key idea of
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this approach is to represent the problem redundantly, from which a sparse solution is sought for a given
observation.
Suppose that we have the current image represented by a vector b ∈ Rn, which can be either the
vectorizedfullrawimageordescriptorsextractedfromtheimage. Assumethatwealsohaveadictionary
denoted by B = [b ··· b ] ∈ Rn×m, which consists of m basis vectors of the same type as b. Thus,
1 m
solving the linear system Bx = b yields the representation of b in the base B in the form of the vector
x ∈ Rm. Elementsofxindicatewhichbasisvectors,b ,bestexplainbandhowmuchthecorresponding
i
contributions are. A zero contribution (x = 0) simply implies that the corresponding basis vector b is
i i
irrelevanttob. Onetrivialexampleofadictionaryisthen×nidentitymatrix,B=I ,whichgivesusthe
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samerepresentation,i.e.,I x=b⇒x=b. Itisimportanttonotethatwehavemadenoassumptionabout
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whatthedictionarycontains,andingeneral,anyarbitrarybasesincludingrandommatricesorwaveletscan
beusedasadictionary.
We know that a general vector b can be represented by a basis matrix B if and only if it belongs to
the range space of B, i.e., ∃x (cid:54)= 0, s.t. Bx = b. Carefully choosing these bases may result in a sparse
representation,i.e.,xissparse. Veryoftenthisisthecaseinpractice,becausethesignalisnaturallysparse
when represented in a certain, often overcomplete, basis [19]. For instance, when representing an image
usingthewaveletbasis,thereareonlyafewcoefficientsthatarenonzero. Notealsothatavectormaynot
berepresentablebythebases,forexample,ifthebasismatrix(dictionary)isrank-deficientandthevectoris
intherightnullspaceofthematrix. Toexcludesuchsingularcases,inthiswork,weassumethattheimage
vectorbisalwaysrepresentablebythebasismatrixwhichisoffullrowrank(i.e.,rank(B)=n).Moreover,
we allow the bases to be redundant, that is, we may have more (not necessarily orthogonal) basis vectors
thanthedimensionoftheimagevector(i.e.,m>n). Notethatthisassumptioncanbesatisfiedbycarefully
designingthedictionary(seeSection4).Ingeneral,aredundantdictionaryleadstothesparserepresentation
ofthecurrentimage,whichiswhatweseekanddescribesthesparsenatureoftheloop-closureevents(i.e.,
occurringsparsely).
Considerthatwehavem > nbasisvectors,thenBx = bbecomesanunder-determinedlinearsystem
andhasinfinitelymanysolutions. Therefore,wehavetoregularizeitinordertoattainauniquesolutionby
specifying a desired criterion that the solution should satisfy. Typically, this regularization takes the form
of looking for a solution, x(cid:63), that leads to the minimum reconstruction error in the (cid:96) -norm sense, which
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correspondstotheleast-squaresformulation:
min (cid:107)Bx−b(cid:107)2 ⇒ x(cid:63) =BT(BBT)−1b (1)
x 2
Notethat(cid:96) -normiswidelyusedinpracticeinpartbecauseoftheclosed-formuniquesolution,whileleading
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toadenserepresentation,i.e.,almostalloftheelementsofx(cid:63)arenon-zeroandthusallthebasisvectorsare
involved in representing the current image vector. This goes against our prior knowledge of loop closures
occurringsparsely. Theoretically,theleast-squaressolutioncanlieveryfarfromthesparsesolutionforan
underdeterminedsystem[16],whichmotivatestheensuingsparseformulation.
Duetothefactthatloop-closureeventsoftenoccursparsely,weinsteadaimtofindthesparsestpossible
solutionundertheconditionthatitbestexplainsthecurrentimage. Intuitively, byassumingthatthebasis
dictionaryconsistsofallthepreviousobservedimagesandthusisredundant,wearelookingforthesmallest
possiblesubsetofpreviousimagesthatcanbestexplainthecurrentimage. Thesmallestsuchsubsetwould
containjustasingleimagewhichis“closest”tothecurrentimage(inappearanceordescriptorspace)under
theassumptionthatthereexistsauniqueimagefromthepastwhichmatchesthecurrentimage. Tothatend,
we employ the (cid:96) -norm to quantify the sparsity of a vector, i.e., (cid:107)x(cid:107) = Card(x : ∀i,x (cid:54)= 0), the total
0 0 i
numberofnon-zeroelementsinx. Notethatavectorwithdnon-zeroelementsiscalledd-sparse. Thus,the
problemofloopclosurecanbeformulatedasfollows:
min (cid:107)x(cid:107) subjectto Bx=b (2)
x 0
The above problem is a combinatorial optimization problem which in general is NP-hard [1], because all
of the possible d-sparse vectors have to be enumerated to check whether they fulfill the constraint. To
addressthiscomputational-intractabilityissue,werelaxthe(cid:96) -normin(2)to(cid:96) -norm,whichisdefinedas
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thesummationofabsolutevaluesoftheelementsofx,(cid:107)x(cid:107) =(cid:80)n |x |,sinceitisknownthat(cid:96) -normis
1 i=1 i 1
theclosestconvexapproximationto(cid:96) -normandalsoresultsinasparsesolution[15],i.e.,
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min (cid:107)x(cid:107) subjectto Bx=b (3)
x 1
Theproblem(3)assumestheperfectreconstructionwithoutnoise,whichclearlyisnotthecaseinprac-
tice. Hence, we introduce a sparse noise (error) term along with the basis vectors to explain the current
imageandreformulate(3)asfollows:
min (cid:107)x(cid:107) +(cid:107)e(cid:107) subjectto Bx+e=b (4)
x,e 1 1
⇒ min (cid:107)α(cid:107) subjectto Dα=b (5)
α 1
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 
(cid:104) (cid:105) e
whereD := I B andα :=  . Notethatwehavenormalizedthebasisvectorswhenbuilding
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x
the dictionary D, and thus I can be considered as the noise bases along each of the n directions of the
n
dataspace,andebecomesanindicationvariableforwhichnoisecomponentsdominatethereconstruction
error. Thisallowsustonormalizexandetogetherwhencomputingcontributionsofdataandnoisebases
(see Section 4.1). We stress that this new formulation of loop closure (5) takes advantage of the fact that
(cid:96) -norm automatically promotes sparsity, as opposed to the more commonly used (cid:96) -based least-squares
1 2
formulation[3]. Insolving(5)foraminimum(cid:96) -normsolution,weareineffectseekinganexplanationof
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thecurrentimagewiththefewestbasisvectorsfromtheredundantdictionary. Thisproblemisalsoknown
asatomicdecomposition[19],sincebisdecomposedintoitsconstituentatomsinthedictionary.
4. ClosingLoopsvia(cid:96) -Minimization
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Wehaveformulatedloopclosureasasparseconvex(cid:96) -minimizationproblem(5)intheprecedingsec-
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tion. Wenowpresentindetailhowthisnovelformulationcanbeutilizedinvision-basedrobotnavigation.
In what follows, we first explain how a dictionary can be constructed incrementally online, and then how
suchadictionarycanbeusedateachtimesteptogenerateloop-closurehypotheses.
4.1. BuildingDictionary
The first step in the proposed process of detecting loops is to build a set of basis vectors that make up
thedictionary.Unlikethestate-of-the-artloop-closuredetectionmethods(e.g.,[21]),theproposedapproach
doesnotlearnthedictionaryofflinebeforethestartoftheexperiment. Instead,thedictionaryisbuiltexclu-
sivelyforthecurrentexperiment,astherobotmovesandcollectsimages–thatis,incrementallyonlineas
imagesarrive.
As a new image becomes available, a mapping function, f : R(r,c) → Rn, transforms the image of
resolution r ×c to a unit vector of dimension n. Due to the flexibility of representation enjoyed by our
proposed approach, this function is general and can be either a whole image or descriptors such as HOG
[14]andGIST[35]computedovertheimage. Thatis,thebasisvectorscanrepresentanyinformationthat
helps distinguish between two images. The proposed method can be considered as data association in a
high-dimensionalspacecarriedoutby(approximately)reconstructingagivenvectorfromasubsetofunit
vectors in that space. As such, this approach is agnostic to what these vectors physically represent. For
thisreason,versatilityofrepresentationisinherenttoouralgorithm,allowingrepresentationsrangingfrom
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whole images to descriptors, BoW, or even mean and variance normalized images over time for matching
sequencesacrossdifferenttimesofdayorchangingweatherconditions. Also,sinceweusethe(cid:96) -normof
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theerror(i.e.,Euclideandistance)tomeasurehowgoodthereconstructionis,anydescriptorwhosedistance
betweentwovectorsismeasuredintermof(cid:96) -norm,canbenaturallyincorporatedintheproposedapproach.
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InordertoensurefullrowrankofthedictionarymatrixD,weinitializethedictionarywithanidentity
matrix I , which also accounts for the bases of the noise e [see (5)]. When the first image denoted by i
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arrives, b = f(i ) is added to the dictionary. In general, updating the dictionary at the i-th time step
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is simply appending b = f(i ) to the end of the current dictionary2. However, before augmenting the
i i
dictionary,weneedtodeterminewhetherornottherearesomepreviousimagesthatexplainthecurrentone,
i.e.,weneedtodetectanyloopthatcanbeclosedbasedonthecurrentimage.
4.2. Solving(cid:96) -Minimization
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OncethedictionaryDisavailableandwhenanimagearrivesateverytimestep, wearenowreadyto
solvetheconvex,sparse(cid:96) -minimizationproblem(5)inordertofind(ifany)loopclosures. Whilevarious
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approachessuchastheprimal-dualmethodareavailableforsolvingaconvexoptimizationproblem[5],we
here leverage the homotopy approach because of its efficiency [31, 17], which is specifically designed to
takeadvantageofthepropertiesof(cid:96) -minimization.
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Inparticular,relaxingtheequalityconstraintin(5)yieldsthefollowingconstrainedminimization:
min (cid:107)α(cid:107) subjectto (cid:107)Dα−b(cid:107) ≤(cid:15) (6)
α 1 2
where(cid:15)>0isapre-determinednoiselevel. Thisistermedthebasispursuitdenoisingproblemincompres-
sivesensing[15]. Anequivalentvariantof(6)isthefollowingunconstrainedminimizationproblemthatis
actuallysolvedbythehomotopyapproach:
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min λ(cid:107)α(cid:107) + (cid:107)Dα−b(cid:107) (7)
α 1 2 2
whereλisascalarweightingparameter. Tosolve(7),thehomotopymethodusesthefactthattheobjective
functionundergoesahomotopycontinuationfromthe(cid:96) cost(thesecondterm)tothe(cid:96) cost(thefirstterm)
2 1
asλincreasesfromzerotoone[17]. ThecomputationalcomplexityofthisapproachisO(dn2+dnm)for
2Althoughthissimpleaugmentationwouldmakethedictionarygrowunbounded,moresophisticatedupdatepolicies(e.g.,replacing
ormergingbasisvectorsforthesamelocations)canbedesignedinordertocontrolthesizeofthedictionarywhenarobotrepeatedly
operatesinthesameenvironment.
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Algorithm1ClosingLoopsvia(cid:96) -Minimization
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Input: DictionaryD ,Currentimagei ,Thresholdτ,Weightλ,Ignoring-timewindowt
i−1 i g
Output: Loop-closurehypothesesH,UpdateddictionaryD
i
1: bi :=f(ii)
2: Hypothesisgeneration:
3: Solvemαiin λ(cid:107)αi(cid:107)1+ 12(cid:107)Di−1αi−bi(cid:107)2usingthehomotopyapproach(seeSection4.2)
4: Normalizeαˆi := (cid:107)ααii(cid:107)2
5: FindhypothesesH:={j|αˆi,j >τ, (cid:107)i−j(cid:107)1 >tg}
6: Dictionaryupdate:
(cid:104) (cid:105)
7: Di := Di−1 bi
recoveringad-sparsesignalindstepsusingann×mdictionary,althoughintheworstcasewhenrecovering
a non-sparse solution in a high-dimensional observational space and large number of basis vectors, it can
performasworseasO(m3),whichfortunatelyisnotthecaseinthiswork.
The above homotopy solver is employed to determine loop closure for the i-th image represented by
(cid:104) (cid:105)
ii, by solving (7) for f(ii) using the up-to-date dictionary Di−1 = In f(i1) ... f(ii−1) .3
(cid:104) (cid:105)T
The solution αi = αi,1 ··· αi,n+i−1 at the i-th time step contains the contribution of all previous
bases in constructing the current image. To find a unique image to close a loop with, we are interested
in which basis vector has the greatest relative contribution, which can be found by calculating the unit
vector αˆ = α /(cid:107)α (cid:107) . Any entry greater than a predefined threshold, τ, is considered a loop-closure
i i i 2
candidate. In addition, due to the fact that in a visual navigation scenario, the neighbouring images are
typically overlapped with the current image and thus have great “spurious” contributions, we explicitly
ignoreatimewindow,t ,aroundthecurrentimage,duringwhichloop-closuredecisionsarenottaken. This
g
isadesignparameterandcanbechosenbasedonthecamerafrequency(fps)androbotmotion. Oncethe
(cid:104) (cid:105)
decision is made, the dictionary is updated by appending f(ii) to it, i.e., Di = Di−1 f(ii) . The
mainstepsofthisprocessaresummarizedinAlgorithm1.
3Thesubscripti−1ishereafterusedtodenotethetimeindex, thusrevealingtheonlineincrementalprocessoftheproposed
approach.
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Figure1: SampleimagesfromtheNewCollegedataset: Queryimages(top)andthecorrespondingmatch
images(bottom). Theimagesaredown-sampledto8×6pixels. Notethatinspiteofdynamicchangesand
motionblursoccurringintheseimageswhichdeterioratetheloop-closureproblem,theproposedapproach
stillprovidesreliableresults.
4.3. Remarks
4.3.1. Globaluniqueness
Itisimportanttonotethatthesolutionto(7),byconstruction,isguaranteedtobesparse. Intheideal
caseofnoperceptualaliasing,thesolutionisexpectedtobe1-sparse,becauseideallythereexistsonlyone
image in the dictionary that matches the current image when a revisit occurs. In the case of exploration
where there is no actual loop-closure match in the dictionary, the current image is best explained by the
lastobservedandthesolutionhenceisstill1-sparse,whichhoweverwillnotgenerateavalidloop-closure
detection because of the temporal constraint t enforced in our implementation. Note that if there are
g
significantdifferencessuchasilluminationordynamicobjects,thesolutionmaynolongerbe1-sparse(see
Section5.7).
Inageneralcasewherek > 1imagesthathavebeenpreviouslyobservedandthatarevisuallysimilar
tothecurrentimage,anaivethresholdingbasedmethod–whichsimplycomparesthecurrentimagetoeach
of the previous ones based on some similarity measure – would likely produce k loop-closure hypotheses
correspondingtothekimagesinthedictionary. Itisveryimportanttonotethatsuchanapproachindepen-
dently calculates the contribution of each previous image, without taking into account the effects of other
images or data noise, despite the fact that due to noise they may be correlated and thus is suboptimal. In
contrast, the proposed (cid:96) -minimization-based approach simultaneously computes the optimal contribution
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ofallthepreviousimagesandnoisebyfindingtheglobaloptimalsolutionαˆ oftheconvexproblem(7),and
i
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