Table Of ContentIEEETRANSACTIONSONPATTERNANALYSISANDMACHINEINTELLIGENCE, VOL.X,NO.X,AUGUST2016 1
Accurate Motion Estimation through Random
Sample Aggregated Consensus
Martin Rais, Gabriele Facciolo, Enric Meinhardt-Llopis,
Jean-Michel Morel, Antoni Buades, and Bartomeu Coll
Abstract—Wereconsidertheclassicproblemofestimatingaccuratelya2Dtransformationfrompointmatchesbetweenimages
containingoutliers.RANSACdiscriminatesoutliersbyrandomlygeneratingminimalisticsampledhypothesesandverifyingtheir
consensusovertheinputdata.Itsresponseisbasedonthesinglehypothesisthatobtainedthelargestinliersupport.Inthisarticlewe
showthattheresultingaccuracycanbeimprovedbyaggregatingallgeneratedhypotheses.ThisyieldsRANSAAC,aframeworkthat
improvessystematicallyoverRANSACanditsstate-of-the-artvariantsbystatisticallyaggregatinghypotheses.Tothisend,we
7
introduceasimplestrategythatallowstorapidlyaverage2Dtransformations,leadingtoanalmostnegligibleextracomputationalcost.
1
Wegivepracticalapplicationsonprojectivetransformsandhomography+distortionmodelsanddemonstrateasignificantperformance
0
gaininbothcases.
2
n IndexTerms—RANSAC,correspondenceproblem,robustestimation,weightedaggregation.
a
(cid:70)
J
9
1 1 INTRODUCTION
] SEVERAL applications in computer vision such as im- theparametersoftheunderlyingtransformationbyregres-
V
age alignment, panoramic mosaics, 3D reconstruction, sion. RANSAC achieves good accuracy when observations
C motion tracking, object recognition, among others, rely on are noiseless, even with a significant number of outliers in
. feature-basedapproaches.Theseapproachescanbedivided theinputdata.
s
c inthreedistinctsteps.First,characteristicimagefeaturesare Instead of using every sample in the dataset to per-
[ detected on the input images. They can come either in the form the estimation as in traditional regression techniques,
1 form of special image points [1] or as distinguished image RANSAC tests in turn many random sets of sample pairs.
v regions [2], and should meet some repeatability criteria to Since picking an extra point decreases exponentially the
8 ensure they can be detected on images, regardless of their probabilityofselectinganoutlier-freesample[9],RANSAC
6 location, pose, illumination conditions, scale, etc. Second, takes the minimum sample size (MSS) to determine a
2
a unique description is given to each detected feature. unique candidate transform, thus incrementing its chances
5
In the case of feature points, this is done by describing of finding an all-inlier sample, i.e., a sample exclusively
0
its surroundings [1], [3], [4]. For regions, their silhouettes composed of inliers. This transform is assigned a score
.
1
and/orthecontainedtextureinformationareemployed[5], based on the cardinality of its consensus set. Finally the
0
[6]. Finally, the descriptors of both images are matched method returns the hypothesis that achieved the highest
7
together. Ideally, these matches correspond to samples of consensus.Torefinetheresultingtransformation,alast-step
1
: the underlying model to be estimated. Unluckily, the pres- minimizationisperformedusingonlyitsinliers.
v
ence of outliers (i.e. matches that do not correspond to RANSAC weaknesses.TheRANSACmethodisableto
i
X this model) complicates the estimation task. Moreover, the give accurate results even when there are many outliers
locations of matching points are frequently contaminated in the input data. However, it leaves some room for im-
r
a withnoisecommonlyreferredtoasmeasurementnoise[7]. provement. First, the probability of RANSAC obtaining a
Introduced by Fischler and Bolles [8], Random Sample reasonable result increases with the number of iterations,
Consensus (RANSAC) is an iterative method that deals however it may never reach the optimal solution. In ad-
with outliers, and is used to simultaneously solve the cor- dition, RANSAC results have a high degree of variabil-
respondence problem while estimating the implicit global ity for the same input data, and this variability increases
transformation.Byrandomlygeneratinghypothesesonthe with the amount of input points and their measurement
transform matching the points, it tries to achieve a maxi- noise. Second, although robust to outliers, RANSAC is not
mumconsensusintheinputdatasetinordertodeducethe particularly immune to measurement noise on the input
matches according to the transformation, known as inliers. data [7]. Furthermore, the final minimization step weighs
Oncetheinliersarediscriminated,theyareusedtoestimate uniformlytheassumedinliermatch,ignoringwhetherthey
are real inliers or how strongly they may be affected by
• M.Rais,A.BuadesandT.CollarewiththeDepartamentdeMatema`tiques noise. Third, the maximum tolerable distance parameter δd
i Informa`tica, Universitat de les Illes Balears, Balearic Islands, Spain. should be sufficiently tight to obtain a precise transforma-
E-mail:{martin.rais,toni.buades,tomeu.coll}@uib.es.
tion, however it must also be loose to find enough input
• G.Facciolo,E.Meinhardt-Llopis,andJ.-M.MorelarewiththeCMLA, samples [10]. Because of such considerations, setting this
EcoleNormaleSupe´rieureParis-Saclay,94230,France.E-mail:{facciolo, parameter is a difficult task, even under low measurement
enric.meinhardt,morel}@cmla.ens-cachan.fr.
noise.Fourth,eventhoughthetotalamountofiterationscan
IEEETRANSACTIONSONPATTERNANALYSISANDMACHINEINTELLIGENCE, VOL.X,NO.X,AUGUST2016 2
be set according to the theoretical bound proposed by the The parameter δd is a critical parameter of the algorithm
original authors [8], more iterations are typically required and will be discussed below. Finally, the optional last-step
[7].Finallyandmostimportantly,theaccuracyofRANSAC minimizationrefinesthepreviousresultbycomputing
is based on the single sample where the best model was
M
found. Although this may be accurate in some cases, it θˆ=argmin(cid:88)dist(cid:0)φθ(X˜i),Y˜i(cid:1), (3)
nevertheless ignores the other all-inlier models that may
θ i=1
have been generated throughout the iterations, and which
couldbeusedtoimprovetheresultingaccuracy. wheretheM inliermatches(X˜i,Y˜i)aredefinedby
RanCdoonmtrSibamutpiolenAs.gTgoremgaakteeduCpofnosretnhseusse(iRssAuNesS,AwAeCp),raopsiomse- {(X˜i,Y˜i)|X˜i ∈X,Y˜i ∈Y,ρ(dist(cid:0)φθˆ(X˜i),Y˜i(cid:1))=1}. (4)
pleyetpowerfulmethodthatcombinestherandomsample
2.1.1 RANSACiterationsandmodelerror
consensus scheme with a statistical approach. By aggregat-
ingtherandomhypothesesweightedbytheirconsensusset TheauthorsofRANSAC[8]showthatbyuniformlydraw-
cardinalities, the proposed approach improves systemati- ingksamples,with
callyonRANSAC.Wegivepracticalimplementationsofthis
log(1−η )
idea on 2D parametric transformation models, by propos- k ≥ 0 , (5)
log(1−(cid:15)m)
ing a simple strategy that allows to rapidly aggregate 2D
parametric transformations using negligible computational wheremistheminimumsamplesize(MSS),onecanensure
resources. with confidence η0 that at least one outlier-free sample
ItsmainbenefitsovertraditionalRANSACarefourfold. is obtained from a dataset having inlier rate (cid:15). Therefore,
First,resultsarebothmoreaccurateandwithlessvariability the total number of iterations could be set according to
(in terms of standard deviation of the error). The improve- Eq. (5). However, since the inlier rate (cid:15) is in general not
mentinaccuracyoverthetraditionalapproachisonaverage knownbeforehand,thisvalueisupdatedeverytimeabetter
by a factor of two to three, and is even more important hypothesis is found, by using Eq. (5) and taking (cid:15) as the
withhighernoise,moreinliersavailable,andhigheroutlier currentinlierratio.
ratios.Moreover,itimprovessystematicallyoverotherstate- However, as noted by Chum et al. [9], the number of
of-the-artRANSACextensions.Second,aswiththeoriginal iterations of Eq. (5) is overly optimistic, so that an outlier-
RANSAC method, the accuracy is dramatically improved free sample does not guarantee an accurate estimation ei-
by adding a local optimization step, which in fact seems ther.Thisisduenotonlytothenoiseonthepositionofthe
suited for our approach because it avoids discarding the inputpoints,butalsototheunderlyingnoiseonthemodel
potentially useful intermediate models generated during hypothesesthemselves[11],[12].Thisothernoiseiscaused
its execution. Third, by including this step, the theoretical byseveralfactorssuchaslimitednumericalprecision,poor
adaptativestoppingcriterionproposedin[8]becomesmore conditioning of the model estimation method, or by other
realistic and could be used to effectively stop the itera- sometimes non-avoidable reasons such as the rounding of
tions without affecting the final accuracy. Finally, using the pixel intensities or point coordinates. In practice, the num-
proposed 2D transformation aggregation method adds an berofsamplesrequiredinRANSACshouldbesettypically
almost negligible extra computational cost to the RANSAC totwoorthreetimesthetheoreticalnumber[7].
algorithm.
Therestofthisarticleisorganizedasfollows.Webegin 2.1.2 Distanceparameter
by giving a review on RANSAC and its variants in section
The parameter δd depends on the measurement noise of
2. In section 3 we detail the proposed algorithm. We thor-
the input samples. Assume that these are 2D points with
oughly evaluate it in section 4 to finally conclude and give
Gaussiannoiseofzeromeanandstandarddeviationσ.Then
someremarksregardingthefutureworkinsection5.
thedirecttransfererror,computedas
dist(φθ(Xi),Yi)=(cid:107)Yi−φθ(Xi)(cid:107)2, (6)
2 RANSAC AND ITS VARIANTS
i.e. the distance between the projected point on the first
image onto the second image and its match, is the sum of
2.1 TheRANSACmethod
squaredGaussianvariableswhichfollowsaχ2 distribution
m
Givenanunknowntransformationφwithparametersθand with m = 2 degrees of freedom. The probability that this
adatasetofpairsXi,Yi consistingofi=1,...,N samples, error is lower than a given value k can be computed from
RANSACcomputes the cumulative χ2m distribution Fm. Capturing a fraction
α of the inliers is ensured by setting δd = Fm−1(α)σ2,
N
θˆ=argmax(cid:88)ρ(cid:16)dist(cid:0)φθ(Xi),Yi(cid:1)(cid:17), (1) w1−hicαhpmerecaennstothfattheattirmuee.iTnhlieenriwftihllebeerrionrcoorfrtehcetlyesrtiemjeactteedd
θ i=1 transformation φ for a particular set of matches (X,Y) is
where dist is usually the squared L2 distance and the cost measured as the direct transfer error, this implies setting
functionρisdefinedas δd =5.99σ2 forα=0.95[13].However,iftheerrorisgiven
bythesymmetrictransfererror,namely
(cid:26)
ρ(e)= 01 oitfheer≤wδisde,. (2) dist(φθ(Xi),Yi)=(cid:107)Yi,φθ(Xi)(cid:107)2+(cid:13)(cid:13)Xi,φ−θ1(Yi)(cid:13)(cid:13)2, (7)
IEEETRANSACTIONSONPATTERNANALYSISANDMACHINEINTELLIGENCE, VOL.X,NO.X,AUGUST2016 3
then the χ2 distribution has four degrees of freedom (i.e. PROSAC algorithm [22] exploits the ordering structure of
m
m=4).Therefore,onemustsetδd =9.4877σ2 forα=0.95 thesetoftentativecorrespondencesandbasesitssampling
orδd =13.2767σ2 forα=0.99.However,sinceinpractice, onthesimilaritycomputedonlocaldescriptors.GroupSAC
the standard deviation of the noise is not available, this [23]combinesbothstrategiesbysegmentingtheinputdata
parameter is usually set experimentally. Throughout this basedonfeaturecharacteristics.Suterandhiscollaborators
article, this parameter was fixed to α = 0.99 and the error show that taking random non-minimal subsets of samples
wasassumedtobemeasuredasinEq.(7). considerablyreducesthebiascausedbytakingminimalsize
samples over noisy input data [24], [25]. Another approach
2.1.3 Finalrefinementstep [26]istoperformaninitial,notsoextenseroundofrandom
The original RANSAC method reduces the effect of mea- samplestogetherwiththeirhypothesestobuildsomeprior
surementnoisebyre-estimatingthetransformusingalast- knowledge, followed by guiding the subsequent sampling
stepminimizationontheresultinginliers.Dependingonthe using the residuals of these hypotheses, i.e. the residual
underlyingmodel,bothlinearandnon-linearminimization sortinginformation.
methodsexist[13],[14],withthelatteroneachievingbetter Anotherstrategyistoperformasimpletestbypartially
results with a higher computational cost. Although this evaluating the dataset to discard hypotheses before computing
final stage does in general improve RANSAC results, it the score, thus reducing the overall computational cost. The
givesequalimportancetoeachmatchfortheminimization. Td,d test [27] checks whether d randomly drawn points are
ThenifRANSACisnotabletocompletelydiscriminatethe inliers to the hypothesis, discarding it otherwise. In [28],
outliers within the input data, this introduces a bias in the theauthorsproposetouseasequentialprobabilityratiotest
computationoftenresultinginaperformancedrop. (SPRT)todiscard“bad”hypotheses,basedonWald’stheory
Throughout this article, we will refer to RANSAC as of sequential testing [29]. Finally, instead of evaluating one
the method described by Eq. (1) without performing the model at a time, Preemptive RANSAC [30] filters hypothe-
last refinement step, and RANSAC+M or more directly ses by scoring several of them in parallel over the same
RANSAC with last-step minimization to the full approach subsetofsamples,permittingreal-timeapplications.
followingEq.(3). To improve on accuracy, instead of computing the cardi-
nality of the set of sample points having smaller residuals
thanaspecifiedthreshold,severalmethodsmodifyhowsam-
2.2 RANSACAlternativesorImprovements
ples are scored by weighting inliers based on their residual
Oneoftheaimsofrobuststatisticsistoreducetheimpactof error. The MSAC method [31], a simple redescending M-
outliers.Incomputervision,arobustestimatormustbeable estimator [32], takes the maximum between the error and
tofindthecorrectmodeldespiteofthenoisymeasurements the threshold to score each datum to finally minimize over
andthehighpercentageofoutliers. their sum. Conversely, MLESAC [33] maximizes the log
In the context of regression analysis, the Least-Median- likelihood of the given data by assuming the inlier error
of-Squares (LMedS) and the Least Trimmed Squares (LTS) hasGaussiandistributionwhiletheoutliererrorfollowsan
are well-known classical methods and still widely used uniformlaw.
[15]. Both approaches resemble RANSAC by generating One strategy used to improve both accuracy and speed
hypothesesfromminimalisticsamples.Howeverinsteadof istoperformalocaloptimizationstepaftera“good”enough
discarding the outliers, both methods sort the input points modelhasbeenfound.ThelocallyoptimizedRANSAC(LO-
based on their residual error. The LMedS then returns the RANSAC) [9], as well as its variants [7], [34], performs a
hypothesiswiththelowestresidualofthemedianpoint.The fixed amount of iterations by taking non-minimal samples
LTSmethodreturnsinsteadthemodelforwhichthesumof of the inliers of each newly found best hypothesis, in a so-
a fixed percentage of best ranked input data is minimized. calledinner-RANSACprocedure.Foreachmodelgenerated
AlthoughtheLMedsmethoddoesnotrequiretospecifythe bytakinganon-minimalsample,alargerdistancethreshold
δdparameterofRANSAC,itimplicitlyassumesatleast50% is used to compute its inliers and an iterative procedure is
ofinliersinthedataset.Thismethodwassuccessfullyused performed to refine this model by progressively shrinking
toestimatetheepipolargeometrybetweentwoimages[16]. the threshold. The recently published Optimal RANSAC
Since its introduction and due to the massive prolifer- methodbyHastetal.[35]alsoexploitsthistechnique,how-
ation of panorama generation [17], 3D estimation [13] and ever they add small modifications such as dataset pruning
invariant feature points [1], [18], several modifications of to remove outliers among iterations along with adaptive
RANSAC were proposed [12], [19], [20], mainly address- terminationofthemethodwhenthesameinliersetisfound
ing three major aspects: computational cost, accuracy, and againaftertheinner-RANSACprocedure.
robustness against degenerate cases, i.e., cases where it is ToincreasetherobustnessofRANSACagainstdegenerate
notpossibletorecognizeanincorrecthypothesisbysimply data in an epipolar geometry estimation context, DEGEN-
scoringthedrawnminimalsample.Interestingly,thesetech- SAC [36] performs a test aimed at identifying hypotheses
niquescanbegroupedbasedonthestrategytheyadopt. wherefiveormorecorrespondencesintheminimalsample
One of such strategies is to modify the random sampling are related by a homography, to avoid bad solutions on
process of RANSAC to generate better hypotheses, improv- images containing a dominant plane in the scene. In cases
ing both speed and accuracy. In this group, NAPSAC [21] where most of the data samples are degenerate and do not
segments the sample data by assuming that inliers tend to provideauniquesolution,QDEGSAC[37]performsseveral
bespatiallyclosertooneanotherthanoutliers,thuspicking RANSAC executions, iteratively, by adding constraints to
pointslayingwithinahypersphereofaspecificradius.The theinliersofthepreviousresults.
IEEETRANSACTIONSONPATTERNANALYSISANDMACHINEINTELLIGENCE, VOL.X,NO.X,AUGUST2016 4
Based on these approaches, Raguram et al. [12] studied that two images are related by a homography H, each
several RANSAC extensions and proposed an algorithm with some detected feature points and a certain amount
that intelligently combined them into a single modular of matching pairs between both images. Let us consider a
method called Universal RANSAC (USAC). This method singlefeaturex0 fromthefirstimageanditscorresponding
obtains state-of-the-art results using few computational re- point y0 = H(x0) on the second image (the match (x0,y0)
sources. It does so by combining ideas from LO-RANSAC, does not necessarily need to be a detected match). At
DEGENSAC,theSPRTtestandPROSAC. iteration k RANSAC randomly picks matches, generates a
Anotherrecenttrendistoapproachtherobustestimation hypothesizedhomographyHkandcomputesitsscoreusing
problem in an inverse fashion [38], [39], [40], [41]. The idea thewholedataset.Thishypothesisinfactyieldsatentative
is to use the residuals for each data point obtained from matchyˆ0 =Hk(x0).Ifthematchesrandomlyselectedwere
several random models to generate the preference matrix, all inliers, then, with high probability, yˆ0 will be close to
which relates input points with the generated hypotheses. therealmatchingpointy0 onthesecondimage.Thus,after
Thismatrixisfinallyclusteredtodistinguishbetweeninliers severalluckyall-inlieriterations,manyestimatesclosetoy0
and outliers. The methods in the literature mainly differ areactuallyfound.
in how to sample, define, and cluster this matrix. While J- Allof theseestimatesare affectedby twonoisesources:
Linkage[38]usesabinarymethodtodefinetherelationship the underlying noise of the model hypothesis as explained
betweeneachdatapointandeachhypothesisinthematrix, before, and the measurement noise of the input points.
T-Linkage [39] uses a continuous thresholded value, con- Provided both of them have zero mean and a symmetric
ceptually similar to the difference between RANSAC and distribution function, aggregating these estimates should
MSACscores.Toperformtheclustering,bothmethodsrely yield a value closer to the true point y0. However, since
on an agglomerative clustering technique: each iteration of there may be outliers in the input samples, it is better to
the algorithm merges the two clusters with the smallest assignaweightforeachgeneratedhypothesisreflectingits
distance. J-Linkage uses the Jaccard distance metric while reliability. Luckily, RANSAC already computes this mea-
T-LinkagereliesontheTanimotodistance. sure, which is its so-called score. Therefore, by weighting
The approach of Tepper and Sapiro [41] performs bi- each of the resulting estimated points using the RANSAC
clustering on the preference matrix by using non-negative score and aggregating them by, for example, taking their
matrix factorization (NMF). A similar approach is used in weighted mean, the aggregated point will, with a high
[40],wheretheycombineRobustPCAwithSymmetricNMF probability,beclosertoitstruevaluey0.
toclusterthepreferencematrix.Itshouldbenotedthat,op- Therefore, in order to aggregate the transformations,
positetoRANSACwhereasinglemodelisestimated,these the proposed approach requires the possibility to perform
approaches target multimodel estimation and therefore are operations, such as averaging, directly on the models. Of-
notthefocusofthepresentwork. ten, the set of these models has typically the structure of
Finally, to address the selection of the δd parameter, a Riemannian manifold [43], [44]. So they could be di-
ORSA[42]proposesana-contrariocriteriontoavoidsetting rectlyaggregatedinsidethismanifold,byunderstandingits
hard thresholds for inlier/outlier discrimination. On the geodesics[45].However,forthecaseof2Dtransformations
other hand, Raguram [7] estimates the covariance of the mapping points in one image to points on another, instead
underlying transform and propagates this error due to the ofaggregatingonthismanifold,asimplestrategybasedon
noisebymodifyingtheregionsforwhicheachdatapointis pre-selectingpointscouldbeused,virtuallyaddingnocost
consideredaninlier. totheoverallcomputation.
Let {Xi,Yi}i=1,...N be a set of matching points where
Xi ∈ C1, Yi ∈ C2, both C1,C2 ⊂ R2 and {xj}j∈1,...,n ∈ C1
3 RANDOM SAMPLE AGGREGATED CONSENSUS withn<<N predefinedpointsonthefirstimage.Foreach
As explained above, the RANSAC algorithm as well as iterationk withhypothesisHk = φθk RANSAACcomputes
its extensions discard the current best hypotheses when a theprojectionofthepredefinedpoints
ntheewsehhyyppooththeessiseswciothulldarbgeerucsoefnuslentosuosbitsaifnouanmd.orHeopwreecviesre, yˆjk =φθk(xj) for1≤k ≤n (8)
estimation of the underlying model. In fact, if hypotheses andtheRANSACscore
generated by all-inlier noisy samples are similar enough, N
(cid:88) (cid:16) (cid:0) (cid:1)(cid:17)
i.e., their variance is bounded, then it makes sense to ag- wk = ρ dist φθk(Xi),Yi , (9)
gregate them to improve the accuracy of the method. The i
idea behind the proposed RANSAAC approach is to take wheredistisagainthesquaredL2distanceandρisthecost
advantageofeverygeneratedhypothesis,usingitsscoreas function. After K iterations, for each predefined point xj,
a confidence measure allowing not only to discard outliers RANSAACaggregatesthedifferentestimationsyˆk usingthe
j
but also to give more importance to better models. In this weightswk,resultingintrustworthymatches(xj,yˆj)
section we first discuss the aggregation strategy employed,
followedbypresentingthelocaloptimizationimprovement. yˆj =aggregate(yˆji,wi) (10)
i=1,...,K
This procedure, in fact, could be regarded as a way to
3.1 Aggregationof2Dparametrictransformations
generate denoised inlier matches, since xj could be any
In the context of RANSAC, we shall first illustrate this point on the first image, while yˆj will be placed, with a
idea through a simple but classic example. Assume again highprobability,closetothetruematchingpointyj.
IEEETRANSACTIONSONPATTERNANALYSISANDMACHINEINTELLIGENCE, VOL.X,NO.X,AUGUST2016 5
Fig. 2. Two images related by a homography. Each of the points on
the corner of the first image are evaluated with 5 hypotheses. These
destinationpointsareaggregatedtocomputethefinaltransform.
Fig.1.Suggestedpredefinedsourcepoints(inred).Top:forhomogra-
phyestimation.Bottom:forhomography+distortion.
It can be calculated using the Weiszfeld’s algorithm [48],
whichisaformofiterativelyre-weightedleastsquares[49]
3.1.1 PredefinedSourcePoints whichiterativelycomputes
TpitehyneodfsotohunirstchbeeapsgiolsoinsbhtasolutthrldaantbseaforaertmlaeabpseatirneagqmueeasttleitmrooathtfeetdhM.eTSmhSe.eWcthaerodsdeinldeaecl--t yt+1 =j(cid:88)|=C|1(cid:107)xwjjp−xjyt(cid:107)(cid:44)j(cid:88)|=C|1(cid:107)xjw−jpyt(cid:107), (13)
pointswhichcanuniquelydeterminethedesiredtransform
whereagainpisaparameterofthemethod.
and minimize the noise caused by model generation, as
depicted in Fig. 1. This strategy could be applied to any
model that can be correctly estimated from points sampled 3.2 LocalOptimization
on the image extents. For a homography, for example, the One of the major improvements of the original RANSAC
four points on the corners of the image should be selected. method, in terms of accuracy, is to include an intermediate
For a transformation involving a homography and two ra- step every time a new best hypothesis has been found.
dialdistortionparameterssuchastheonerecentlyproposed This step, known as the local optimization step, achieves
byKukelovaetal.[46],anadditionalpointonthecenterof improved results by using the fact that models with good
thetopand/ortheleftimagebordermustbeadded. enough inlier supports are probably not distant from the
true underlying model. Therefore, by taking non-minimal
3.1.2 AggregationStrategies samplesfromtheinliersofthisnewbestmodelfollowedby
For the case of 2D parametric transformations, instead of a greedy strategy to refine the results, the LO-RANSAC al-
simply returning the best hypothesis, RANSAAC gathers, gorithm[9]anditsvariants[12],[34]improvesystematically
for each point of the predefined basis, a set of estimates overRANSAC.
of its location on the second image. These points form a The local optimization step is shown in Algorithm 1.
weightedcluster,eachonebeingassociatedthescoreofthe The method begins by taking non-minimal samples of the
hypothesisfromwhichitwasderived.Alloftheseestimates inliersofthenewbesthypothesisfound,andcomputesthe
canthenbeaggregatedtocomputeamoreaccuratelocation model by performing a least squares fit (lines 10-11). Then,
for each point of the basis. This yields a minimum sample it computes the inliers of this model for the whole dataset
thatpermitstorecoverthetransform.Intheexampleshown using a larger distance threshold (line 12) and uses these
inFig.2,fivepossiblehypothesesareshown.Sincethefifth inliers to calculate a second model by least squares fitting
hypothesis H5 (green arrow) obtained a lower score, it has (line 15). This second model is later refined in an iterative
lessinfluenceontheresultingaggregation. fashion, by progressively shrinking the distance threshold
Two aggregation techniques were considered, namely until its original value is attained (lines 16-21). The refined
the weighted mean and the weighted geometric median modelwiththelargestamountofinliersisthenreturned.
[47]. To aggregate the points xi of the set C, with weights In practice, if an all-inlier sample is picked during a
wi,withi∈{1,...,|C|},theweightedmeancomputes RANSAC iteration, this method obtains excellent approxi-
mations to the optimal solution. However, since RANSAC
|C| (cid:30) |C|
wmean(C)=(cid:88)wipxi (cid:88)wip, (11) olansltymkaeyepnsotthenebceesstsmaroildyebledethfieneodptbiymiatlsminolideerls,uapspaorbte,tttheer
i=1 i=1
modelmayhaveappearedduringtheseinneriterationsand
wherepisaparameteroftheaggregationprocedure. was subsequently discarded. This suggests that RANSAAC
Theweightedgeometricmedianofasetisdefinedasthe may significantly benefit from this procedure, and it turns
pointy withtheminimumsumofdistancestotheothers out that it does. Therefore, saving and returning these
intermediatemodelsistheonlyextraworkneededtoadapt
|C|
wgmed(C)=aryg∈mR2in(cid:88)i=1wip(cid:107)xi−y(cid:107)2. (12) tbhaicskmgreotuhnodditnoARlAgoNrSitAhmAC1,.as seen by the lines with pink
IEEETRANSACTIONSONPATTERNANALYSISANDMACHINEINTELLIGENCE, VOL.X,NO.X,AUGUST2016 6
Algorithm1LocalOptimization RANSAAC procedure includes green and grey lines. As
Require: X1,X2 ∈ N2 × N: N matching points, seenwithinthelineswithpinkbackground,RANSAACfirst
Xi1,Xi2 ∈N2×K:K inliermatches,δd ∈R:RANSAC verifiesforeachiterationifthehypothesisisacceptable(i.e.
distance threshold, sis ∈ N: size of inner sample, thattheamountofinliersislargerthantheMSS),andthen
rmepδdet∈itioRn:st,hlrseIstheorlsd∈mNu:ltliepalsiters,qrueaprses∈iteNr:atinionnesr,sample awpepiglihetss.itThtoeLthOepsrooucercdeurpeo,icnotdsewdhinilegrseteonribnagckigtsrosucnodre,sjuasst
1: ifK <2sis then performsthelocaloptimizationstepwhenanewbestmodel
is found, and stores its results. Finally, after iterating, if no
2: return0
acceptable transform was found, it returns the traditional
3: endif
54:: m∆δadx←Sco(rmeδ←d ·δKd−δd)/lsIters RsiAdeNreSdAvCalrieds,utlht.eHcoomwpeuveter,dipfrsoejveecrteadl hpyopinottsheasreesagwgerreegacotend-
6: bestModel←[] usingtheirweightsandthealgorithmreturnsthemodelthat
7: models←[] bestfitstheseestimates.
8: weights←[]
Algorithm2Computinga2Dtransformationusingthe(LO)
9: forit=1torepsdo
RANSAACalgorithm.
10: Xss1,Xss2 ←GetSample(Xi1,Xi2,sis)
11: hnonMSS ←GetModelLS(Xss1,Xss2) Require: X1,X2 ∈ N2 × N: N matching points,
1132:: AXdiid1(,mXoidie2l←s,hEnovnaMlHSS(h)nonMS(cid:46)SS,aXve1,cXom2,pδudt·emdδmd)odel mcoimnpSuatmepthleestr∈ansNfo:rmm,insirmcPumts a∈mNo2un×tKof: vmeacttcohrewsittho
14: Add(weights,#Xii1) (cid:46)anditsweight K = minSamples input points, δd ∈ R: RANSAC
15: hRLS ←GetModelLS(Xii1,Xii2) distance threshold, iters ∈ N: amount of iterations,
16: forj =1tolsItersdo p∈R:aggregationparameter.
17: Xii1,Xii2←EvalH(hRLS,X1,X2,δd·mδd−j·∆δd) 1: maxScore←0
18: Add(models,hRLS) (cid:46)Savecomputedmodel 2: forit=1toitersdo
19: Add(weights,#Xii1) (cid:46)anditsweight 3: Xss1,Xss2 ←GetSample(X1,X2,minSamples)
20: hRLS ←GetModelLS(Xii1,Xii2) 4: h←GenerateHypothesis(Xss1,Xss2)
21: endfor 5: Xi1,Xi2 ←EvalH(h,X1,X2,δd)
22: Xii1,Xii2 ←EvalH(hRLS,X1,X2,δd) 6: if#Xi1 >maxScorethen
23: Add(models,hRLS) (cid:46)Savecomputedmodel 7: maxScore←#Xi1
24: Add(weights,#Xii1) (cid:46)anditsweight 8: ransacRes←h
25: if#Xii1 >maxScorethen 9: models,weights←LO(X1,X2,Xi1,Xi2,δd)
26: maxScore←#Xii1 10: Add(dstPtsLO,Project(srcPts,models))
27: bestModel←hRLS 11: Add(weightsLO,weights)
28: endif 12: endif
29: endfor 13: if#Xi1 >minSamplesthen
30: returnmaxScore,bestModel,models,weights 14: Add(dstPts,Project(srcPts,h))
15: Add(weights,#Xi1)
16: endif
17: endfor
As in its original formulation, the local optimization is 18: dstPts←dstPtsLO
executedeverytimeanewbesthypothesisisfoundduring 19: Ws←weightsLO
RANSAC iterations, working as an “algorithm within an 20: if#dstPts>0then
algorithm”. This means that the local optimization step 21: resPts←Aggregate(dstPts,Ws,p) (cid:46)See3.1.2
does not update RANSAC’s internal best-so-far variables 22: returnGenerateHypothesis(srcPts,resPts)
(maxScore and ransacRec), but has its own. In the lo-
23: endif
cally optimized version of RANSAAC (i.e. LO-RANSAAC), 24: returnransacRes
instead of aggregating all generated hypotheses, we only
aggregate hypotheses obtained from the local optimization
step,sincethesemodelsarelikelytobeclosertotheground
4 EXPERIMENTS
truth. We evaluated the inclusion of all other generated
modelsintheaggregationstepandobservedthattheresults We compared the proposed method and its variants (LO,
were slightly less accurate. Furthermore, excluding these wmean, wgmed) with several state-of-the-art approaches
hypothesesintheaggregationreducestheprocessingtime. both qualitative and quantitatively. Two transformation
ThefinalRANSAACalgorithmanditslocallyoptimized types were considered, namely the projective transform
versionaddafewlinestothetraditionalRANSACmethod estimatedusingtheDLTalgorithm[50],andahomographic
and an almost negligible computational cost. Algorithm 2 model that allows distortions on both images. The latter
shows RANSAC together with both RANSAAC and LO- was estimated using the H5 method of Kukelova et al.
RANSAAC approaches, coded in colors. Each variant is [46], which, given five matches, computes a homography
understood by its identifying color. The original RANSAC and one radial distortion coefficient per image (using the
is coded with white background. The RANSAAC method division model and assuming the distortion center is on
is obtained by including the pink and grey lines. The LO- the center of the image). Since homographies are, along
IEEETRANSACTIONSONPATTERNANALYSISANDMACHINEINTELLIGENCE, VOL.X,NO.X,AUGUST2016 7
form, we used the SIFT algorithm, followed by the ap-
plication of RANSAC with 100000 iterations between two
images. To generate the matches, we randomly selected
a fixed set of points from the first image as inliers and
we matched them to the points obtained by applying the
groundtruthtransformtothem.Finally,toincorporateout-
liers in the dataset, we randomly picked other points from
the first image and matched them with random positions
on the second image. Measurement noise up to σ = 5
was added afterwards, since such high noise levels may
appear in real-life applications. A possible reason for this
is noisy point matches coming from higher scales of the
Laplacian pyramid, which translate to large noisy values
ontheoriginalscale.Anotherreasonisduetoanimperfect
Fig.4.Resultingpointdistributionsfor1000iterations. model selection that, by setting large values for δd, accepts
matching points far away from the selected model. For ex-
withfundamentalmatrices,themostfrequentglobalmodel ample,whensearchingforaplanarhomographyonradially
estimated in computer vision, we focus our evaluation on distortedimagesfromthesameplanarscene.
it.Nevertheless,weshowresultsprovingthattheproposed On this dataset, we compared several state-of-the-art
approach also works on the other transform types as well, methods and all variants of RANSAAC. To measure the
and could be potentially generalized to any parametric error, we averaged the symmetric transfer error in pixels
model obtained from points provided the possibility of for every inlier. If φθ is the obtained model, Xi is an inlier
averagingthosemodelsisavailable. point in the first image, Yi its match and K the amount of
Matlab implementations of RANSAAC and all its vari- inliers,thenthemeanerrorE isdefinedas
antsareavailableathttp://dev.ipol.im/∼mrais/ransaac/.
E = 1 (cid:88)K (cid:107)φθ(Xi)−Yi(cid:107)2+(cid:13)(cid:13)φ−θ1(Yi)−Xi(cid:13)(cid:13)2, (14)
4.1 Qualitative Evaluation: Paintings Registration with K 2
ProjectiveTransformations i=1
We first evaluated both RANSAC and RANSAAC with whereXi andYi representtheoriginalinputpoints,before
addingthenoise.
weighted mean aggregation on pictures of paintings taken
By varying the number of inliers, the percentage of
inamuseumunderpoorlightingconditions,whichresulted
outliers, the input noise and the number of iterations per-
in considerable noise visible on the output images. The
formed,wecomputedforeachexperimenttheaverageerror
objectivewasthentoregistereachphotographofapainting
detail taken from a shorter range, with the whole painting perexperimentE¯ andthestandarddeviationsσE over100
evaluations.
[51].ByusingSIFTfeaturesandmatchingitsdescriptors,we
proceeded to register both images using both methods. An Note that for all presented results, the very same ran-
example can be seen in Fig. 3. The camera distortion was dom samples for each iteration were used for RANSAC,
corrected beforehand so the images are related by a pure LO-RANSAC and RANSAAC, thus ensuring equal chance
homography. After registering both images, the resulting for those methods. We used the implementation of USAC
difference is mainly noise, however RANSAAC performs availablefromtheauthor[12].
better, as evidenced in the flamingo on the bottom part of Weevaluatedtheperformanceoftheproposedapproach
thepainting(middlecolumnofFig.3).Whenmeasurement under distinct conditions. Since the noise level was known
noiseisaddedtotheinputpointsonbothimages(σ=5),the beforehand, the RANSAC δd parameter was set according
difference in the registration performance becomes much tosection2.1.2,andforRANSAACthisparameterwasfixed
more evident. Edges due to misalignments appear all over to 2 ∗ δd due to the higher tolerance of the method with
the difference image in RANSAC, but are dimmer on the respect to this parameter. Unless explicitly mentioned, the
RANSAACresults(rightrowofFig.3). parameterpwassetdependingontheamountofiterations
To get a better understanding of the behaviour of the performed and on the used aggregation scheme, indicated
method, figures 4 and its zoomed in version 5 shows the bytheresultsobtainedinthefollowingsection.
point distribution with their sizes according to their ob-
tained weights using p = 5. As can be seen, the RANSAC 4.2.1 AggregationWeights
algorithm always returns the model given by the points Because of outliers in the input data, the weighting of the
withthehighestweight,ignoringeveryotherestimation.By hypothesesontheaggregationismandatory.Increasingthe
incorporating other weighted estimations and aggregating value of the weight parameter p leads to eliminate the
them,acloserpointtothegroundtruthisobtainedforevery influence of poor hypotheses, but a too high value will
corner,thereforeyieldingamoreprecisehomography. implyusingtoofewtransformsforthefinalaggregation.
An experiment was performed to measure the impact
4.2 Quantitative Evaluation by a Simulated Ground of the parameter p for RANSAAC and its LO-RANSAAC
Truth.Application1:EstimatingProjectiveTransforms.
variant, shown in Fig. 6. For the case of RANSAAC, it
We built a valid ground truth, composed of point matches turns out that the value of p depends on both the outlier
and a transform. In order to obtain a valid real-life trans- percentage, but most importantly, on the number of drawn
IEEETRANSACTIONSONPATTERNANALYSISANDMACHINEINTELLIGENCE, VOL.X,NO.X,AUGUST2016 8
Fig.3.RANSAC (top) and RANSAAC (bottom) registration results. Dynamic ranges were stretched to highlight differences. Left:
input images. Center: registration difference using input images. Right: registration difference by adding measurement noise of
σ = 5tothe keypointpositionson bothimagesin pixels.Edges duetomisalignments canbeperceived inthe difference image,
particularlybyusingthetraditionalRANSACmethod.Painting:L’Air,ouL’Optique.JanIBRUEGHEL,1621.LouvreMuseum.
Fig.5.PointdistributionofeachcornerofFig.4resizedaccordingtotheirweightsusing1000iterations.Pointswithlowweightswereexcludedfor
visualizationpurposes.wmeanandwgmedaretheweightedmeanandtheweightedgeometricmedianaggregation.gmedand2dmedrepresent
aggregatedresultsusing(unweighted)geometricmedianand1Dmedianonbothdimensions.Thereaderisadvisedtozoomintoseebetterthe
positionsofthevariousestimates.
IEEETRANSACTIONSONPATTERNANALYSISANDMACHINEINTELLIGENCE, VOL.X,NO.X,AUGUST2016 9
8 Error by varying P parameter (wmean) (1000 inliers with 20% outliers) 10 Error evolution by iteration
6677....4826 89..8955 RRRAAANNNSSSAAACAACC ((wwmgmeaend))
6 7.5
5.6 7
)slexip( rorrE233445......4826482 )slexip( rorrE3456....4565555
2.4 3
2 2.5
0011....4826 111000000 00it 0.it .it. 01..1255
0 0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2
Parameter p Iteration #104
1.5 Error by varying P parameter (LO wgmed) (1000 inliers with 20% outliers) 100 Error evolution by iteration
11..124.1723.5552 88990505 RRRAAANNNSSSAAACAACC ((wwmgmeaend))
1.125 75
1.05 70
)slexip( rorrE0000....05689.0027277..5555569 )slexip( rorrE 34455665050505
0.45 30
0.375 25
00..002.0721.5553 111000000 00it 0.it .it. 1120505
0 0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0 25 50 75 100125150175200225250275300325350375400425450475500
Parameter p Iteration
Fig.6.Error(inpixels)andstd.dev.byvaryingthepexponent,using Fig.7.Comparisonoftheerrorbyiteration(inpixels)betweenRANSAC
100, 1000 and 10000 iterations averaged over 100 realizations. Dots andtheproposedalgorithmusingbothaggregationmethodsaveraged
indicate lowest errors. Top: RANSAAC with wmean. Bottom: LO- over1000realizations.Noise:σ =5.100inliersand100outliers.Top:
RANSAACwithwgmed.Inlier/Outlierratio:1000/1250.Noise:σ=5. Upto20000iterations.Bottom:First500iterations.
hypotheses. Indeed, as more hypothesis are drawn, they
This is expected, since RANSAAC gains from aggregating
should be better discriminated, suggesting higher values
all generated samples, thus only having a single “good”
of p. As soon as the value of p is high enough so that it
hypothesis and several “bad” ones will always benefit a
allowstocorrectlydiscriminatebetweeninliersandoutliers,
method such as RANSAC that just considers the single
then the resulting accuracy does not vary significantly. A
hypothesis with the highest obtained score. However, as
similar trend was observed using wgmed aggregation and
more all-inlier hypotheses are sampled, RANSAAC starts
thereforeitsresultswerenotincluded.However,fortheLO-
improving over traditional RANSAC (in the figure this
RANSAACvariantusingwgmedaggregation,varyingthep
happensat125iterations).
parameter does not affect the resulting accuracy. This can
beexplainedintwoways.First,themodelsreturnedbythe
4.2.3 RANSAACvariantsevaluation
LOstephaveusuallyfeweroutliers.Secondly,thegeometric
median is more robust to outliers than the mean, therefore WeevaluatedbothaggregationvariantsforRANSAACwith
not requiring to completely avoid them in the aggregation. and without local optimization, by varying the number
When using wmean aggregation, results were similar as in of inliers, the percentage of outliers, noise level, and the
thestandardRANSAACwithlowvaluesofp.However,the numberofiterationsperformed.Table1showsresultsunder
curvesremainedalmostconstantlater,aswiththewgmed. threedifferentnoiselevels(σ =0.5,2or5).Foreachmethod
andnoiselevel,both1000and10000iterationsweretested,
4.2.2 Accuracyevolutionalongiterations each row corresponding to situations presenting either 0%,
To give insight about the evolution of the error under 5%,20%or50%outliersrespectively.
RANSACandundertheproposedapproach,theiraccuracy Several conclusions could be drawn from these results.
was computed while iterating. To this end, we simulated As can be seen, by using the local optimization procedure,
input points from a predefined real homography (i.e. com- every aggregated method improved, and in every case,
puted from two images of a landscape), corrupted them the best performing aggregation scheme was the weighted
withwhiteGaussianNoiseofσ =5andadded50%outliers. geometricmedian.Interestingly,thisdidnotoccurwhenno
Using this input, we ran 20000 iterations of RANSAC and localoptimizationstepswereperformed.Thereasonbehind
ourmethodandcalculatedtheerrorforeveryiteration(for this is evident, as both the outlier ratio and the number of
RANSAAC, this implied performing aggregation). Finally, inliersincrease,theprobabilityofobtaininggoodmodelsby
we computed the average error per iteration from 1000 randomsamplingbecomeslower,thereforeaveragingthese
repetitions of the experiment. In Fig. 7 left, we observe not-so-close-to-the-optimal models will achieve better re-
howRANSACdoesnotonlyconvergeslowercomparingit sultsthantakingtheirmedian,whichwillbefarawayfrom
with both aggregation schemes of RANSAAC, but also that theoptimalsolution.However,whenbetter(i.e.closertothe
it never reaches their accuracy. However, when observing optimalsolution)hypothesesareavailable,thencomputing
the first 500 iterations, RANSAC usually achieves better their median is not only more accurate but also drastically
results at the beginning, while RANSAAC usually requires reducesthepossiblebiascausedbyincludingoutliersinthe
more all-inlier samples to produce more accurate results. estimation.Indeed,performingLOstepsprovidesexcellent
IEEETRANSACTIONSONPATTERNANALYSISANDMACHINEINTELLIGENCE, VOL.X,NO.X,AUGUST2016 10
TABLE1 TABLE2
AverageerrorsforthedifferentversionsofRANSAACusingboth ExecutiontimesforRANSAC,thelaststepminimization(LSM)
proposedaggregationschemes.Fornoisesσ=0.5,2and5,each usingDLT,RANSAAC andboththewmeanandwgmed
methodwasevaluatedwith100and1000inliersusingboth1000and aggregationproceduresusing100000iterations,1000/1500
10000iterations.Thefourerrors(rows)representfouroutlierratios: inlierratioandσ=2.Resultsareaveragesof100realizations.
0%,5%,20%and50%.Bolddenotesthebestperformers.
RANSAC LSM RANSAAC wmeanagg. wgmedagg.
σ wmean LO(wmean) wgmed LO(wgmed) 52.810s 0.007s 53.269s 0.007s 0.052s
1k 10k 1k 10k 1k 10k 1k 10k
100inliers
0.23 0.21 0.20 0.19 0.25 0.23 0.19 0.20 Average error with 50% outliers and 1000 inliers
0.24 0.21 0.20 0.20 0.27 0.23 0.20 0.20
0.5 0.26 0.21 0.21 0.20 0.29 0.23 0.21 0.20 L(wOm-ReAanN)SAAC WNoit hA dAadpatpivtiev eT eTremrminiantaiotinon
0.45 0.26 0.21 0.21 0.61 0.30 0.21 0.21
0.89 0.87 0.75 0.81 1.02 0.92 0.75 0.81 LO-RANSAAC
0.91 0.86 0.76 0.81 1.01 0.91 0.76 0.81 (wgmed)
2 1.03 0.89 0.83 0.83 1.21 0.97 0.83 0.82
1.76 1.03 0.84 0.82 2.39 1.23 0.82 0.81 USAC
2.20 2.07 1.90 1.90 2.42 2.29 1.88 1.91
2.39 2.22 2.02 2.07 2.72 2.40 2.01 2.06
5 2.58 2.25 2.05 2.01 2.95 2.47 2.03 2.03 RANSAC+M
4.60 2.47 2.01 1.94 6.00 2.98 1.98 1.97
1000inliers RANSAC
0.11 0.07 0.07 0.07 0.15 0.09 0.07 0.06
0.12 0.07 0.07 0.07 0.16 0.09 0.06 0.06 0 2.5 5 7.5 10 12.5 15 17.5 20 22.5 25
0.5 0.18 0.08 0.07 0.07 0.22 0.10 0.06 0.06 Average error (in pixels)
0.42 0.15 0.08 0.07 0.54 0.22 0.07 0.06
0.47 0.30 0.27 0.27 0.57 0.35 0.24 0.26 Fig.8.Comparisonbyusingadaptiveterminationtotheproposedap-
2 00..5627 00..3313 00..2288 00..2287 00..6942 00..3460 00..2265 00..2265 proach.
1.68 0.61 0.32 0.28 2.21 0.83 0.26 0.25
1.18 0.75 0.71 0.68 1.54 0.82 0.63 0.62
1.25 0.80 0.74 0.70 1.55 0.92 0.65 0.67 4.2.5 Resultsbyusingadaptivetermination
5 1.66 0.88 0.70 0.69 2.15 1.04 0.63 0.64
4.22 1.55 0.80 0.70 5.50 2.15 0.66 0.63 WecomparedtheLO-RANSAACalgorithmusingbothpro-
posedaggregationschemestogetherwithUSAC,RANSAC
and RANSAC+M. A maximum of 1000 iterations was al-
models, which justifies why wgmed aggregation improves lowedforallmethods,althoughthemethodswithadaptive
overwmean. termination varied this parameter dynamically during the
It is remarkable how the wgmed aggregation benefits execution, based on Eq. (5) using η0 =0.99. Results show
from local optimization, evidenced by the increase in ac- average errors for 100 executions, where the noise value
curacyuptoanorderofmagnitudefor1000iterations,1000 was fixed to σ = 5 and the distance values were set for
inliers, 50% outliers and σ = 5. Finally, performing more every method according to section 2.1.2. USAC results are
RANSAC iterations when using local optimization does duplicatedsinceadaptiveterminationwasalwaysactivated.
not improve the resulting accuracy. Indeed, results could Several conclusions are drawn from results shown in
becomeslightlyworsewhenperformingmorethanenough Fig. 8. First, while the weighted mean aggregation slightly
iterationssincetherearemorepossiblebadhypothesesthat suffersfromusingadaptivetermination,theweightedgeo-
have to be correctly discriminated. In conclusion, adding metric median aggregation in fact slightly benefits from it.
local optimization to RANSAAC does not only improves The reason for this is again the same as mentioned before,
results but also enables the method to perform fewer iter- and is related with the fact that making more iterations
ations,makingitlesscomputationallydemanding. impliesalsohavingtobetterdiscriminatebetweensamples.
To sum up our findings, LO-RANSAAC using wgmed is
4.2.6 Comparisonwithstate-of-the-artmethods.
our recommended algorithm whenever local optimization
isapplicable.Whenthisisnotpossible,thebestRANSAAC We compared the proposed LO-RANSAAC approach with
results are obtained by using the wmean aggregation. An RANSAC and several state-of-the-art variants, namely LO-
exampleofthisisshowninsection4.3. RANSAC, MSAC [33], ZHANGSAC [16] and the recent
USAC algorithm [12]. We included in the comparison two
minimization methods applied directly on the inliers using
4.2.4 Executiontimescomparison
an oracle: the linear least squares methods using the DLT
We compared the execution times for both RANSAC and algorithm and the non-linear method that minimizes the
RANSAAC together with their extra steps. By performing Sampson’s approximation to the geometric reprojection er-
100000iterationsonadatasetcomposedof1000inliersand ror [13]. For every method, we also evaluated adding the
500outlierswithmeasurementnoiseofσ=2,wecomputed finalminimizationstep(usingtheDLTalgorithm)andkept
the total running time of each approach. Execution times onlythebestbetweenboth.
are indicative and were measured on a 3.1GHz Intel Core With 1000 inlier matches available, the average errors
i7 5557U processor. As seen from Table 2, the standard togetherwiththeirstandarddeviationsareshowninFig.9
RANSAAC method took 1% more time than the original for 50% and 75% outlier ratios. All approaches not using
RANSAC. Moreover, if the LO-RANSAAC variant is used, local optimization steps have much higher errors proving
since only the hypotheses coming from the LO step are theimportanceofthismethod.TheLO-RANSACapproach
aggregated, the difference with respect to LO variant of was able to attain good results, although it never achieves
RANSACisalmostnegligible. theaccuracyoftheUSACmethod.EveryRANSAACvariant
