On Hausdorf f Distance Measures - UMass Amherst

21 Pages·2009·0.87 MB·English

On Hausdorff Distance Measures
MichaelD.Shapiro MatthewB.Blaschko
PalaceofDreams ComputerVisionLaboratory
120GarfieldPl. UniversityofMassachusetts
Brooklyn,NY11215,USA Amherst,MA01003,USA
email: [email protected] email: [email protected]
UM-CS-2004-071
ABSTRACT
A number of Hausdorff-based algorithms have been proposed for finding objects in
images.WeevaluatedifferentmeasuresandarguethattheHausdorffAveragedistance
measure outperformsother variants for model detection. This method has improved
robustness properties with respect to noise. We discuss the algorithms with respect
to typical classes of noise, and we illustrate their relative performances through an
exampleedge-basedmatchingtask. Weshowthatthismethodproducesamaximuma
posterioriestimate. Furthermore,weargueforimprovedcomputationalefficiencyby
tree-likesubdivisionsofthemodelandtransformationspaces.
KEYWORDS
Hausdorff,ObjectRecognition,SpatiallyCoherentMatching
1 Introduction
Objectdetectionoftenreliesonmetricsthatdescribethedegreeofdifferencebetween
two shapes. The one-sided Hausdorff distance, h A B , in this context is a measure
betweenthesetoffeaturepointsdefiningamodel,A,andthesetofpointsdefininga
targetimage,B,where
h A B maxmin a b
a A b B
and is a normof thepoints ofA andB. ImagematchingusingHausdorff-based
distances has been applied to many domains including astronomy [10], face detec-
tion[5][13],andwordmatching[8].
Severalvariationsofthissetmetrichavebeenproposedasalternativestothemax
oftheminapproachintraditionalHausdorffmatchingthatarelesspronetooutliersin
thedata. TheseincludeHausdorfffraction,Hausdorffquantile[4],SpatiallyCoherent
Matching[1], and Hausdorffaverage[3]. We analyzethe formof these metrics and
argueabouttheirbehaviorunderseveralclasses ofnoise. Specifically, we determine
whether there are discontinuities of the measure with respect to minor perturbations
ofthepointsets. WeshowanalyticallyandexperimentallythatHausdorffaveragehas
superiorperformanceundertheseconditionsasin[12].
2 A General Schema
Our general schema is as follows. We have an image I and a model M. We wish
to discoverif the objectrepresentedbyM occursin I, andif so where. Our concept
of “where” is encapsulated in a set of transformations T which we think of
as positioninga copyT M of M in I. For eachsuch position, we computea score
s T M . Amongalltransformations,wepicktheone Tˆ withthe“best”score. Ifthis
scoreissufficiently“good”,weannouncethatM isatpositionT inI. Otherwise,we
announcethatMisnotinI.
3 Notation
Letusfixnotation.OurmodelMconsistsofavectoroffeatures,
M S S
1 NM
AfterapplicationofatransformationT eachfeatureS acquiresaposition
i
x y x y T S
i i T i
We will drop the T subscript when this is clear from context. Strictly speaking S
i
itselfdoesnothaveaposition.Itacquiresoneifweimagineanidentitytransformation
placingitatsomepredeterminedpositioninI.Ingeneral, neednotactontheimage
plane.Forexample,ifMisathree-dimensionalmodel, mayconsistofrigidmotions
followedbyprojection.Wewillbeinterestedinthecasewherethetransformationsare
translationsintheplane.Then
x y T S T x y S
i i
If welike, we mayassume thatI consists ofa similar vectorof features, butthis
is not necessary. We onlyneedto enquirewhethera featureS of M is matched at a
i
position x y inI. WhenI andM arebothsetsofedgepixels,everyfeatureofM is
matchedateveryfeatureofI.
ForeachfeatureS, the matchlocus, S is theset ofpositionsat whichS is
i I i i
matchedinI.
S x y S ismatchedat x y
I i i
We will drop the subscript I when there is no ambiguity and that neither I nor M is
empty.
WhenIandMarepointsets(forexample,pixelsfoundbyanedgedetector), S
i
consistsofallpointsofI. However,theremaybemorethanonetypeoffeature. For
example, one type may consist of edge pixels, whereas another type may consist of
pointswhere a filter exceedssome threshold. If the features S and S are bothedge
i j
2
pixels, then S S . Likewise if S and S are features for the same filter
i j p q
then S S . However,weshouldnotexpectthat S S . Thus
p q i p
wemaybuilddifferentaspectsoftheobjectintoourmodelandattempttomatchthem
intheimage.
Wechooseadistancefunctiond x y x y intheplane. Thenthematchset
1 1 2 2
inducesadistanceforeachtranslatedfeature,
d d T S min d x y x y x y S
Ti i i i T i
ForeachT thisgivesusavectorofdistances,
d d d d T S d T S
T T1 TNM 1 NM
Eachd tellsushowfarthefeatureS isfromitsnearestmatchwhenMisplaced
Ti i
atpositionT. Eachinstanceofourschemacanbedescribedintermsofthisvector. In
eachinstance,weassumeafixedmodelandimage.
4 Distance measures
4.1 One-sided Hausdorffdistance
For each T, the Hausdorff score, s T , is max d . We pick Tˆ to minimize this
H i Ti
score.Weacceptitifs Tˆ islessthansomethresholdvalue,s .
H 0
4.2 Hausdorfffraction
Wefixaminimumacceptabledistancek.ForeachT,Hausdorfffractionscore,s T
HF
iscardinality
# S d k
i Ti
s T
HF
N
M
WepickTˆ tomaximizes T .WeacceptTˆ ifthefractions Tˆ exceedsathreshold
HF HF
values .
0
4.3 Hausdorffquantile
Wefixaquantileq.ForeachT,theHausdorffquantilescore,s T istheqthquantile
HQ
valueofdistancesind . WechooseTˆ tominimizes T andacceptTˆ ifs Tˆ is
T HQ HQ
lessthansomefixedvalues .
0
4.4 SpatiallyCoherent Matching
TodefineSpatially CoherentMatching,we needtodefinea neighbor relationonthe
featuresofM. We takethefeaturesofM tobetheverticesofagraph. Twofeatures
areneighborsifthereisanedgebetweentheminthisgraph. Weareinterestedinthe
casewherethefeaturesareedgepixels.Wewillconsiderpixelstobeneighborsifthey
areadjacent,horizontallyorvertically. Wefixaminimumacceptabledistancek. For
3
eachT,theSpatiallyCoherentMatchingscore,s T isdefinedasfollows.Wetake
SCM
G T S d k . WetakeB T tobethesetofedgeswithonevertexinG T
i Ti
and one vertex not in G T . This counts the numberof edges on the “boundary”of
G T .Wetake
#G T #B T
s T
SCM
N
M
We chooseTˆ tomaximizes T . WeacceptTˆ ifs Tˆ exceedssomefixed
SCM SCM
values .
0
4.5 HausdorffAverage
ForeachT,theHausdorffaveragescore,s T is
HA
s T !idTi
HA
N
M
WechooseTˆ tominimizes T . WeacceptT ifs Tˆ islessthansomefixedvalue
HA HA
s .
0
We will want to generalize this a little bit. Given a function f, we will want to
consideraHausdorfffunctionalaveragescore,s T ,
HA f
1
s T !f d
HA f Ti
N
M i
Note that Hausdorff average score is a special case of Hausdorff functional average
scorewith f d d .
Ti Ti
5 Maximum a Posteriori Estimate
TheHausdorfffunctionalaveragealgorithmhas asimplejustificationas amaximum
aposterioriestimate. Wefollowtheargumentof[1]inasimplifiedform. Wehavea
collectionofhypothesesandmustchoosetheonewhichmaximizestheprobability
p h p I h
p h I
p I
SinceIisfixed,weseek
h argmaxp h p I h
h
Wetakeasourhypothesisspace 0/ whereeachhypothesisT corresponds
totheassertionthattheobjectmodeledbyMoccursatpositionT inI,andthehypoth-
esis0/ istheassertionthattheobjectdoesnotoccurinI. Ourpriorknowledgeconsists
oftheassumptionthat p 0/ "andthatallpositionsT haveequalprobabilityC .We
1
assumethatgivenapositionT,thelikelihoodofIgivenT is
p I T C #e f dTi
2
i
4
Weassumethatwhentheobjectisnotthere,allimageshaveequalprobability,C .This
3
gives
p h p I h 1 " C1C2#ie f dTi ifh T
"C ifh 0/
3
InorderforthepresenceoftheobjectatpositionT tobemorelikelythantheabsence
oftheobject,weneed
1 " C C #e f dTi "C
1 2 3
i
Sincelogarithmsaremonotonic,theargmaxofthisfunctionistheargmaxofitsloga-
rithm.Takingnegativelogarithmsreversestheinequality,turnstheargmaxintoargmin,
andyieldstheinequality
ln 1 " lnC1 ln C2 !f dTi ln " ln C3
i
The best position Tˆ is the one minimizings T , and this represents the best hy-
HA f
pothesisifthatvalueislessthan
ln 1 " lnC ln C ln " ln C
1 2 3
s
0
N
M
6 Noise and stability:
Wewouldliketoinvestigatethebehaviorofthesemeasureswhenthereisnoise. We
willdealwithtwoclassesofnoise,distortionandstrayfeatures,typically,straypixels.
Bydistortionwemeandisturbanceofthecoordinatesofafeatureorthepointsof
itsmatchset. Thus,wemayhaveM withfeatures S S andM withfeatures
1 NM
S S . ThedistortioniscapturedinthefactthatforeachT,wehave x y
1 NM i i
x y T S and x y x y T S . WecapturedistortionbetweenimagesI and
i i i i
I byassumingthatthereisafunctiongsothatforeachi S g S .
I i I i
Straypixelsshowupasextrafeaturesinthemodelortheimage.Thus,wemayhave
M with features S S and M with features S S S S .
1 NM 1 NM NM 1 NM n
Weassumethatforeach j N thereisi N sothat S S .Thusweare
M M j i
notthrowinginanynewtypesoffeatures,andnoneofthesehaveemptymatchsets.
Straypixelsintheimageshowupasextrapointsinthematchset. Thus,wemay
haveanimageIwith S andanimageI withmatchset S S X.
I i I i I i i
Wewouldliketoknowhowrobustourscoresarewithrespecttonoise. Arobust
score should only change a small amount if we introduce a small amount of noise.
Whatwewantisforthescoretobeacontinuousfunctionofnoise. Sincepositionis
typicallydiscrete, we will take a less puristapproachto this. We will act as if noise
canbemadearbitrarilysmall,thoughthismaynotbethecase.Whatisasmallamount
ofnoise? Adistortionissmallwhenforeachpairofpoints x y and x y asabove,
thedistanced x y x y issmall. Onemightchooseastricterdefinitionhere.One
mightaskthatthesupportofthedistortionissmall,thatis,thatonlyasmallnumberof
pairsofpointshaved x y x y 0. Wewillarguethatthisistoorestrictive. We
willsaythatthesetofstraypixelsissmallwhentheyarefewinnumbercomparedto
5
thetotalnumberoffeaturesorpixelsinthematchset. Wewillrestricttostraypixels
lyingafixedboundeddistancefromtheremainingpixels.Thisisanotherwayofsaying
thatourimagehasafixedsizeandthestraypixelslieinthatimage.
These are realistic classes of noise. If the model is derived from measurements,
thesearesubjecttoinaccuracy.Ifthemodelcomesfromaphotograph,distortioncould
resultfrominaccuracyofthephotographicsetup,forexample,astoangleordistanceto
theobject.Thissortofdifferencecorrespondstotakingaphotograph,applyingarigid
motiontotheobjectandtakingasecondphotograph. Whenthatmotionissmall,the
resultingdifferencebetweenthetwophotographsisasmalldistortion. Thisdistortion
extendstotheentiremodel.Thisiswhywerejectrestrictingthesupportofadistortion
asarequirementforitbeingsmall.
Differingedgedetectorsarelikelytodisagreeaboutexactplacementofedge,pro-
ducinganothersourceofdistortion. Likewisea modelthat is the resultofextracting
edgesfromanimagecaneasilyacquirestraypixels.
Stability withrespectto distortioncarriesoverto stability withrespectto change
oftransformationT. IfT andT areclosethensoare x y T S and x y T S ,
i i
andthus,soared andd .
T T
7 Stability results
Letusfixnotation. WearegivenanimageI,andamodelM. SupposethatI andM
differfromthese by distortion, and that I and M differ fromthese by stray pixels.
ForeachfixedT have
d d I M d d
T T T1 TNM
d d I M d d
T T T1 TNM
d d I M d d d d
T T T1 TNM TNM 1 TNM n
d d I M d d
T T T1 TNM
Toinvestigatestability,wemustobservehowthesevectorsdifferandthediffering
resultsthatourscores(andhenceouralgorithms)produceasaresult.
Thereareseveralsourcesofinstabilitythatapplytoallofourmethodsabove.The
firstisthatevenasmallchangeinthescores Tˆ canpushitaboveorbelowthethresh-
oldvalues . Thesecondisthattheoperationsargmaxandargminarediscontinuous.
0
Thatis,ifs T ands T arenearoptimalandcloseinvalue,theneithercanbepro-
motedtoTˆ byasmallchangeins ,eventhoughT andT maybewidelyseparated.
Third,allthesescoresarehighlysensitivetostraypixelsintheimage. Thereason
hereisthattheadditionofasinglepointtoImaycaused todifferfromd byalarge
T T
amountinalargenumberofcoordinates.ThisoccurswhentherearemanyfeaturesS
i
sothat x y T S isclosertothenewpointthananyofthepointsofI. Thishappens
i
when T fails to match many features of I in M. Since every pixel which is not part
oftheobjectisinsomesenseastraypixel,wemusthopethatthisdoesnotoccurtoo
oftenforoptimalvaluesofT.
However,therequirementthatwehaveasmallamountofnoisedoesplacemean-
ingfulrestrictionsonthedifferencesbetweend ,d andd andwecantracetheseto
T T T
theirpotentialimpactonouralgorithms.Therestrictionthatwehaveasmallamountof
distortionensuresthatd andd differatmostbyasmallamountineachcoordinate.
T T
6
TherequirementthatwehaveasmallnumberofstraypixelsinM ensuresthat n is
NM
small.
7.1 One-sided Hausdorffdistance
Thescores T isstablewithrespecttodistortion. Thisisbecauseasmalldistortion
H
resultsinsmalldifferencesinthecoordinatesofd andd ,andthusasmalldifference
T T
betweenmax d andmax d .
i Ti i Ti
One-sidedHausdorffdistanceisunstablewithrespecttostraypixelsinthemodel.
ThisisbecauseasinglestraypixelS mayproduced max d .
NM 1 TNM 1 i Ti
7.2 Hausdorfffraction
Thescores isunstablewithrespecttodistortion.Thisisbecauseofthediscontinuity
HF
involvedinthresholding. Hereis asimpleexample. SupposethatM is theboundary
ofasquareofsidenandIistheboundaryofasquareofsiden 2k 2.Supposethat
T placesM squarelyinthecenterofI. ThenallofT MiswithinkofI. IfM isthe
boundaryofasquareofsiden 2,thennoneofT M iswithinkofI.Indeed,forthe
optimaltransformationTˆ onlyhalfofTˆ M iswithinkofI.
Hausdorfffractionisstablewithrespecttostraypixelsinthemodel. Assumewe
have matchedm pixels, so that s T m . After the addition of stray pixels, we
HF NM
havethescore m j with j n. Since n issmall,soisthedifferencebetweenthese
NM n NM
twoscores.
7.3 Hausdorffquantile
ThescoreS isstablewithrespecttodistortion. Thisfollowsfromthefactthatthe
HQ
coordinatesofd andd arecloseandthatthequantileoperationiscontinuouswhen
T T
restrictedtoaspaceoffixeddimension.
Hausdorff quantile is unstable with respect to stray pixels in the model. This is
because the quantile operatoris discontinuouswith respect to additionaldata points.
Toseethis,considerthesituationinwhichthereisalargegapinthevaluesofourdata.
Suppose also that the quantile we seek is representedby the datum at the bottom of
thisgap.Addinganadditionaldatapointabovethisgapcausesthequantilemeasureto
jumpuptothetopofthisgap.
7.4 SpatiallyCoherent Matching
Thescores T isunstablewithrespecttodistortionforthesamereasonsasHaus-
SCM
dorfffraction.
Spatiallycoherentmatchingisstablewithrespecttostraypixels.Toseethis,apply
theargumentforHausdorfffractionseparatelytoboththematchingsetG T andthe
boundarysetB T .
7
7.5 Hausdorffaverage
Thescores T isstablewithrespecttodistortion. Toseethisobservethatd and
HA T
d arecloseandtheoperationoftakingtheaverageiscontinuous.
T
Hausdorffaverageisstablewithrespecttostraypixelsinthemodel. Toseethis,
recall that we have bounded the distance of our stray pixels. This ensures that for
i N wehaved d forsomefixedboundd .Beforetheadditionofthestray
M Ti max max
pixels,ourscoreis
!NiM1dTi
N
M
Aftertheadditionofthestraypixels,thescoreisnow
!NiM1dTi !iNMNMk 1dTi
N n
M
Thelattersummandsarebounded,so when n is sufficientlysmall, thesescoresare
NM
ascloseaswelike.
Thesetheoreticalresultscanbeseenexperimentallyasthepresenceorabsenceof
noiseinthemeasure.Seefigures9-14andtheexplanationinsection9below.
8 Algorithmic advantages of stability
All of the previous methods search for an optimum value Tˆ . The most naive
implementation of this search would iterate through all values of T . For each
such T, it would iterate througheach featureS of M to compute x y T S . For
i i
each S it must then computethe distance from x y T S to S for which it
i i i
woulditeratethroughthepointsof S . Thus,ifM consistsofN edgepixels,
i M
consistofN transformationsandI consistsofN edgepixels,thisrequiresN N N
T I M T I
individualdistancecomputations.Whatisworse,theimagesaretwo-dimensionaland
is at least two-dimensions (and could easily be four or five dimensional). This
contemplatesa computationwhich is at least n6 . Fortunately,we neednot be so
naive!
8.1 Efficiencies inthe image
AsHuttenlocherpointsout[4],theiterationoverthepointsofI canbedoneonceand
recordedintheVoronoifunction.IfIrepresentsanimageofwidthw andheighth ,the
I I
Voronoifunctionisthemapfrom 0 w 1 0 h 1 whichgivesthedistancefrom
I I
eachpixelinthedomainofI tothenearest(actual)pixelofI. IfM has f different
M
typesoffeatures,thenthereare f differentmatchlociinI andweneed f different
M M
Voronoifunctions.
8.2 Efficiencies in
For s , Huttenlocher shows that one can improve the efficiency of the search for Tˆ
H
by subdividing into regions. Given a region R, one picks T R (perhaps at its
R
8
center)andcomputessH TR .Oneisalsoabletocomputeabound,$RsothatifT R,
sH T sH TR $R. Thus, if sH TR is sufficiently bad (say sH TR s0 $R),
onecanremovefromfurtherconsiderationnotonlyT ,butalltransformationsT R.
R
If we are seeking Tˆ and we have already computed s T , then we can reject all
H R
T RifsH T sH TR $R. Regionswhicharenoteliminatedfromconsideration
aresubdividedandtheprocessrepeats. Theprocessendsbyeitherdiscoveringthatall
transformationshavebeeneliminated(inwhichcasethereisnotransformationT with
s T s )orbyfindingTˆ withs Tˆ s .
H 0 H 0
Thisprocessdependsontheexistence$R.Theexistenceofthisboundisguaranteed
bystabilitywithrespecttodistortion. Toseethisingeneral,chosetheregionRtobe
compact. Foreachfeature,S , x y T S iscontinuouswithrespecttoT. Hence,the
i i
vectord iscontinuouswithrespecttoT. Finally,stabilitywithrespecttodistortion
T
ensuresthats T iscontinuouswithrespecttoT. ThisensuresthattransformationsT
whichareclosetoT havescoress T whichareclosetos T .Inparticularthesetof
R R
scoress T withT Risbounded.
Inpractice,ofcourse,weneednotonlytheexistenceof$R butaneffectivewayto
computeit.Thisisfeasibleformanyreasonablechoicesof andR.Inthecasethats
iss ,s ors , consistsoftranslations,Risarectangle,andR isitscenter,then
H HQ HA T
$R isonehalfthediagonalofR. Themostnaturalwaytosubdivide inthiscaseis
byusingaquadtree[11].
8.3 Efficiencies inthe model
Tofurtherspeedupthecomputation,HuttenlochersuggestsquickMonteCarloelim-
inationofregionsbycomputingthedistanceforrandomlychosenpointsofM. This
mayfailtoeliminatearegionwhichwouldhavebeeneliminatedifthedistancewere
computedforallpointsofM. However,allofitssub-regionswilleventuallybeelimi-
nated.
Thisworksonlyinthecaseofs . Itworksbecauses T max d . HenceT
H H Ti
canbeeliminatedonthebasisofasinglecoordinateofd .
T
We proposeanadditionalefficiencywhichappliestoanymeasurewhichhassta-
bility with respectto distortion. It applies most easily to s , s ands where the
H HQ HA
errorboundsareonceagainthelengthsofhalf-diagonals.
We require the additional assumption that each feature S of M has a position
i
x y S andthatforeachT, x y T S isacontinuousfunctionof x y S . (These
i i i
methodscanbeappliedtothreedimensionalmodelsundergoingout-of-planerotation,
butwewillexposittheminthecasewhereMistwodimensional.)
Supposenow,thatweimposeaquadtreeT onourmodelM.Ateachlevel ,Mis
M
composedinto4 regions,R R . Letc c betheircenters. Supposethat
1 4 1 4
theseregionscontainn n featuresofM.WemayreplaceourmodelMwithan
1 4
approximationM consistingofthepointsc c withmultiplicitiesn n .
1 4 1 4
Ofcoursewewillomitanypointsofmultiplicity0.
Now,foragiventransformationT,wemaycomputethevector
d d T c d T c d T c d T c
T 1 1 4 4
9
whereeachcoordinated T c isrepeatedn times.Noweachofthesecoordinateis
i i
within$ ofthecorrespondingcoordinateofdT.(Here$ isthehalf-diagonalthesere-
gions.)Noticealsothats ,s ands areinvariantunderpermutationofcoordinates.
H HQ HA
Itfollowsthatforeachofthese,
s T s d $
T
This observation combines with that of the previous subsection, so that if R is a
regionof the subdivisionof and is a level of the subdivisionof our model, and
T Rthen
s T s d TR $R $
Noticethatcomputings d requiresatmost4 distancelookupsintheVoronoi
T
diagramratherthanthefullN ofthem. Inthecasewherewearecomputings d
M H T
ors d weneednotevenwriteoutmultiplecopiesofthevalued T c since
HA T i
wearecomputingeitherthemaxoftheseoraweightedaverageofthem.
8.4 Implementation issues
Thetree-likenatureofthedecompositionof intoregionsraisesquestionsastowhat
isthebestwaytosearchthistree.Introducingthelevels raisesaquestionastowhich
levelisbestforqueryingagivenregionR. Weconjecturethattheoptimumstrategyis
tokeep$R and$ assimilarinmagnitudeaspossibleoncetheregionsRbecomesmall
enough.
9 Experimental results
Forourexperimentswesynthesizeddatausing11arbitrarybackgroundswithacropped
objectfromtheCOIL20dataset[9]superimposed. Wesynthesized100targetimages
intotalbyrandomlychoosingaforegroundobjectandrandomlyplacingitwithinthe
background. We thensampled10modelsforeachtargetincludingtheobjectknown
tobeinthescene. WeusedtheresultsofmatchingthesemodelstoestimateReceiver
OperatingCharacteristic(ROC)curvesforeachmeasureasin[1].
EachtargetimagewasprocessedusingtheCannyedgedetectorwithMatLabde-
faultparametersforthresholdandsigma[2]. TheCOIL imageswerealso processed
usingthesameedgedetectionalgorithm. Werefertotheresultsofedgedetectionon
theCOILimagesasthemodelset.
Detection consisted of selecting the maximum value and transform into a target
for a given model of the distance measure for the Hausdorff Fraction and Spatially
Coherent Matching algorithms, and of selecting the minimum value of the distance
measure for the One-sided Hausdorff, Hausdorff Quantile, Hausdorff Average, and
HausdorffFunctionalAveragedistancemeasures. Thetransformspacewaslimitedin
these experimentsto translationin x andy by onepixel increments. In the case that
thereweremultiplepointsinthetransformationspacethattiedforthebestmatch,we
detected8-connectedregionsandtookthecentroidofthe largestsuchregion. In the
casethatthereweremultipleregionswiththesamesize,wechoseoneatrandom.Thus
10