geometric relation distribution for place recognition

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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/LRA.2019.2891432, IEEE Roboticsand Automation Letters

IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED DECEMBER 2018 1

Geometric Relation Distribution for PlaceRecognition

Dario Lodi Rizzini1,2, Francesco Galasso1 and Stefano Caselli1,2

Abstract—In this paper, we illustrate Geometric Relation Dis-tribution (GRD), a novel signature for place recognition and loopclosure with landmark maps. GRD encodes geometric pairwiserelations between landmark points into a continuous probabilitydensity function. The pairwise angles are represented by vonMises distribution whereas two alternative distributions, Erlangor biased Rayleigh, are proposed for distances. The GRD functionis represented through its expansion into Fourier series andLaguerre polynomial basis. Such orthogonal basis representationenables efficient computation of the translation and rotationinvariant metric used to compare signatures and find potentialloop closure candidates. The effectiveness of the proposed methodis assessed through experiments with standard datasets.

Index Terms—Mapping; Localization; Range Sensing

I. INTRODUCTION

AN important task in robot localization and mapping is therecognition of already visited regions during navigation.

When the recognition takes place after the robot has travelledlong paths, this operation is called loop closure. Loop closureenables to recover localization errors and map inconsisten-cies due to cumulative odometry error by estimating therelative transformation between the current robot frame andthe matching one. While this problem has been extensivelyinvestigated for several sensor and map models like occupancygrid maps from range finders or computer vision keypointfeatures, robust loop closure techniques for landmark mapshave been developed quite recently [1]–[5].

The keypoint features [3], [6] detected in range finder scansnaturally constitute local maps of landmarks. The landmarkcoordinates of each local map are referred to the correspond-ing local keyframe. Given this formulation, loop closure isachieved by associating a query landmark set and all the storedlandmark sets in the global map. Direct landmark-to-landmarkassociation is neither computationally efficient nor reliableover large collections of points. Thus, most loop closuremethods perform point-to-point matching, after selection ofcandidates by comparing signatures. Several signatures [1],

Manuscript received: September 10 2018; Revised: December 4 2018;Accepted December 24 2018.

This paper was recommended for publication by Editor Cyrill Stachnissupon evaluation of the Associate Editor and Reviewers’ comments. This workis supported by 3DPAL project, a spin-off of the SIMON project sponsoredby Regione Emilia-Romagna, CUP E18I17000110009.

1Authors are with RIMLab - Robotics and Intelligent MachinesLaboratory, Dipartimento di Ingegneria e Architettura, University ofParma, Italy, {dario.lodirizzini, francesco.galasso,stefano.caselli}@unipr.it.

2First and Third Authors are also with Centro Interdipartimentale perl’Energia e l’Ambiente (CIDEA), University of Parma, 43124, Italy.

Digital Object Identifier (DOI): see top of this page.

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Fig. 1: Examples of GRD signature. Left: the scans 368 and376 of dataset intel-lab (grey) and the SKIP keypoints (red)detected in the respective scan. Right: the corresponding GRDsignatures based on biased Rayleigh distribution obtained fromeach of the two point sets.

[5] are designed on bag-of-word (BoW) and explicitly re-quire features with descriptors. BoW approaches rely ondescriptors and their performance is affected when descriptorspoorly characterize landmark neighborhoods. Signatures likeGLARE [2] and GLAROT [4] encode the geometric relationsbetween the features like pairwise distances and relative an-gles into histograms. In particular, GLAROT is invariant totranslation and rotation in order to detect visited submapsregardless of viewpoint. These signatures capture distinctivedistribution of feature points and do not require neither specificdescriptors nor preliminary dictionary as for BoW approaches.However, their histogram-based representation is sensitive tothe resolution of the discretized geometric parameters.

In this paper, we propose the novel signature Geometric Re-lation Distribution (GRD) for place recognition and loop clo-sure detection in landmark maps. Like GLARE and GLAROT,GRD encodes geometric relations between landmarks, but itsrepresentation corresponds to the probability density function(PDF) of pairwise angles and distances between features. ThePDF is multi-modal with a mode for each pair of features. ThePDF mode is the product of two marginal distributions, respec-tively von Mises for angles and Erlang or biased Rayleighfor distances. Moreover, the GRD distribution is expandedinto a series of orthogonal functions based on Fourier series



2 IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED DECEMBER 2018

and Laguerre polynomials. The truncated series can effectivelyincorporate arbitrary number of modes using the same basisin order to scale with arbitrary number of features. The seriesis used to efficiently compute a rotation-invariant correlationmetric. The performance of GRD has been assessed throughexperiments with standard benchmark datasets and comparedwith the state-of-the-art signatures and loop closure detectionalgorithms.

The paper is organized as follows. Section II reviews thestate of the art of loop closure algorithms for landmark maps.Section III illustrates GRD and its efficient representation withorthogonal function basis. Section IV presents a transformationinvariant metric. Section V reports the experimental results.Section VI concludes the work.

II. RELATED WORKS

The landmark map model is based on the extraction ofpoint features from sensor measurements. Paradoxically, thedevelopment of robust keypoint features for planar rangefinders is quite recent acquisition. Tipaldi et al. [6] proposedFLIRT, the first keypoint feature using multiscale approachand descriptors inspired by computer vision features. FALKOfeatures [3] enable detection of more stable keypoints based oncareful neighborhood selection and cornerness score. FALKOkeypoints also support descriptors for encoding neighbor pointdistribution in order to improve data association and, possibly,loop closure.

During mapping, one of the problems arising is place-recognition used for loop closure action, in order to identifyalready visited poses. Loop candidates must be selected usinga robust signature to avoid recognition error. Gutmann etal. [7] proposed Local Registration and Global Correlation(LRGC), an incremental method for reconstruction of globalmaps from range data. Geometrical FLIRT phrases (GFPs) [1]were proposed as an efficient and reliable place recognitionexploiting FLIRT descriptors and the bag-of-words (BoW)approach. Deray et al. [5] improved BoW association usinga Viterbi algorithm and weak geometric check achievingbetter loop closure performance. Geometric LAndmark RE-lations (GLARE) [2], [8] have been proposed to transform2D laser scans into pose invariant histogram representationsevaluating invariant pairwise distances and angles betweenscan points. Nevertheless, this approach is not orientationinvariant. GLAROT [4] was introduced as an improvementover GLARE and designed to be rotation invariant. Point-to-point data association algorithms like maximum cliquemethods [9], [10] or joint compatibility test [11] are required tocheck the selected candidates and to estimate the transforma-tion. A different approach includes the loop closure methodscomparing complete scans or local maps with registrationalgorithms. Lu and Milios [12] proposed the first registration-based loop closure algorithm based on a variant of iterativeclosest point (ICP). More recent registration methods for loopclosure include ICP in pose graphs [13] and efficient alignmentof occupancy grid maps [14].

Similar problems are faced when building a map with 3Dpoint clouds. The work in [15] extends GLAROT to space

domain through an approximately regular binning of sphere.Tazaki et al. [16] proposed a loop detection method as apair of partial time series showing high similarity score. Kimet al. [17] introduced a new descriptor for 3D laser scansembedded with an efficient search algorithm based on KDtree online with Ring Key for fast search. Multiple pointcloud registration [18] can improve robustness of point-to-point association.

III. PAIRWISE POINT-SIGNATURE DISTRIBUTION

Given a set of 2D points P , the pairwise angles anddistances of two different points pi,pj ∈ P (assume thatpi,y > pj,y) are defined respectively as

θij = atan2(pi,y − pj,y, pi,x − pj,x) rij = ‖pi − pj‖ (1)

When the input points are acquired through sensors, their coor-dinates are noisy and uncertain. The point set P may representkeypoint features extracted from a given robot location. Whiletheir Cartesian coordinates are dependent on the viewpoint,their pairwise distances and angles are respectively invariantor invariant up to a rotational offset. Our goal is to definea signature using the distribution of pairwise angle θ anddistance r on P disregarding the exact pairs. Each pair (i, j)contributes to each mode Pij of the global distribution of θ andr. The probability density function (PDF) of [θ, r] associatedto P is

pθr(θ, r) =

np∑i=1

np∑j=i+1

pθr(θ, r|Pij) p(Pij) (2)

Hence after, we assume that all the modes are equally likelyand p(Pij) = 1/N , where N = np(np − 1)/2. Moreover, weassume that pθr(θ, r|Pij) can be factorized into the productof the independent PDFs of θ and r as

pθr(θ, r|Pij) = pθij (θ) prij (r) (3)

The two marginal distributions pθij (θ) and prij (r) must bechosen consistently with the random variable domain. Thevon Mises distribution is often used for angular statisticswith central mode in preference to the more mathematicallycumbersome wrapped normal distribution. Thus, it is used asmodel for θ with mean values θij . The probability densityfunction prij (r) must represent a random variable on interval[0,+∞) and single mode in rij . There are several models forthis purpose and in this work we discuss Erlang and biasedRayleigh distributions. The PDF pθr(θ, r) in eq. (3) has theform of weighted sum of uni-modal kernels, which is notconvenient for efficient algebraic manipulation. Moreover, ametric invariant to rigid transformation is required to comparetwo different signatures. The rest of this section is devoted toaddress these problems.

A. Pairwise Angle Statistics with Von Mises distribution

The probability density function of the von Mises distribu-tion is

pθij (θ) =1

2πI0 (κ)exp

(κ cos

(θ − θij

))(4)



LODI RIZZINI et al.: GEOMETRIC RELATION DISTRIBUTION FOR PLACE RECOGNITION 3

where I0 (κ) is the modified Bessel function of order 0,θij is the mean value and also the maximum of density ofprobability, and κ is the so called concentration parameter,a reciprocal measure of dispersion. We assume that the con-centration parameter κ is the same for all the angles. Thedistribution can be expanded into Fourier series [19, (9.6.34)]as

pθij (θ) =1

2π+

+∞∑kθ=1

(aijkθ cos (kθθ) + bijkθ sin (kθθ)

)(5)

aijkθ =Ikθ (κ) cos

(kθ θij

)πI0 (κ)

bijkθ =Ikθ (κ) sin

(kθ θij

)πI0 (κ)

(6)

The infinite Fourier series is usually truncated to term nθ assnθ (θ). A bound on the approximation error has been derivedin [20] as ∣∣pθij (θ)− snθ (θ)∣∣ 6 Inθ (κ)

πI0 (κ)

κ

nθ(7)

Such bound largely overestimates the error, but can be usedto compute the number of terms nθ needed to guarantee therequired maximum error.

B. Erlang distribution for Pairwise Distance

The random variable r representing the pairwise distancebetween two points is defined on domain [0,+∞) and hassingle mode corresponding to the measured one. The Erlangdistribution is among the probability distributions which sat-isfy such conditions. Given the scale parameter ηij > 0 andshape parameter dij ∈ N, the PDF of an Erlang distributionis

prij (r) =rdij−1 e

− rηij

ηdijij (dij − 1)!

(8)

The Gamma distribution is a generalization of Erlang whereparameter dij can be any positive real number. The mean valueηijdij and variance ηijd2ij of Erlang distribution depend on itstwo parameters. Since the shape parameter dij is integer not allthe values of mean and variance are simultaneously feasible.

Like in the case of von Mises, Erlang distribution can berepresented by a linear combination of orthogonal functions.Since the domain is the positive real line r > 0, the naturalbasis is given by Laguerre polynomials [21]. The Erlang canbe expanded in Laguerre polynomial series with coefficients{ckr} as

prij (r) =+∞∑kr=0

cijkr Lkr (r) (9)

where Lkr (r) is the Laguerre polynomial of order kr. The con-vergence of Laguerre series to a function is guaranteed, whenthe function satisfies proper condition [21, sec. 4.23]. Thiscondition is satisfied by all the probability density functions.Such polynomials are orthogonal w.r.t. integral inner productwith weight function e−r, i.e.

〈Li, Lj〉e−r =

∫ +∞

0

e−r Li(r) Lj(r) dr = δij (10)

where δij is the Kroenecker delta. Hence, the coefficient cijkris given as

cijkr = 〈prij , Lkr 〉e−r (11)

=

kr∑m=0

(krm

)(−1)m

ηdijij (dij − 1)!m!

∫ +∞

0

e−(1+ 1

ηij

)rri+k−1dr

=

kr∑m=0

(−1)m(krm

)(m+ dij − 1

m

)ηmij

(1 + ηij)m+dij(12)

The Erlang distribution suffers from some drawbacks. Theparameter dij is equal to the ratio between the square meanvalue and the variance of the distribution. Hence, when mod-elling a distribution with constant variance and arbitrary largemean value, the value of dij may significantly increase withpotential numeric inaccuracy.

C. Biased Rayleigh distribution for Pairwise Distance

The biased Rayleigh distribution is an alternative model weadopt to model the distribution of pairwise distances. The PDFof biased Rayleigh for r > 0 is

prij (r) =1

Kbrij

r exp

(− (r − µij)2

2σ2ij

)(13)

where µij > 0 is the mode and σij > 0 is the width. Themode and the width are parameters setting the shape of thedistributions and must not be confused with the mean valueand the standard deviation. If µij = 0, then eq. (13) becomesthe PDF of standard Rayleigh distribution. The unnormalizedmoment of order m is defined as

Mm(µ, σ) =

∫ +∞

0

tm+1 e− (t−µ)2

2σ2 dt

=

σ√

π2

(1 + erf

(µ

σ√2

))if m = −1

σµ√

π2

(1 + erf

(µ

σ√2

))+ σ2e

− µ2

2σ2 if m = 0

µMm−1(µ, σ) +m σ2 Mm−2(µ, σ) otherwise(14)

The value of unnormalized moment is given above in theform of recurrent definition. The moments can be used to findthe normalization constant Kbrij , the mean value µbrij andthe variance σ2

brijas

Kbrij =M0(µij , σij) µbrij =M1(µij , σij)

M0(µij , σij)(15)

σ2brij =

M2(µij , σij)

M0(µij , σij)−(M1(µij , σij)

M0(µij , σij)

)2

(16)

The biased Rayleigh PDF can be expanded into a series ofLaguerre polynomials as shown in eq. (9). The coefficientscijkr for this expansion are




cijkr =

∫ +∞

0

e−r1

Kbrijr exp

(− (r − µij)2

2σ2ij

)Lj(kr)dr

=

kr∑m=0

(n

m

)(−1)kreµij−

σ2ij2

Kbrij m!

∫ +∞

0

e−

(r−(µij−σ2ij))

2

2σ2ij rm+1dr

=

kr∑m=0

(n

m

)(−1)kreµij−

σ2ij2

Kbrij m!Mm(µij − σ2

ij , σij) (17)

D. Signature Distribution of Point Sets

There are all the elements to derive an efficient approxima-tion of the signature distribution in eq. (2). The series approx-imation is based on the approximation of marginals of eq. (5)and (9). The two series are arrested respectively to order nθfor random variable θ and nr for random variable r. Thus, thesignature distribution is approximated by pθr(θ, r) ' S(θ, r)defined as

S(θ, r) = 1

N

np∑i=1

np∑j=i+1

(nr∑kr=0

cijkr Lkr (r)

)

·

nθ∑kθ=0

(aijkθ cos (kθθ) + bijkθ sin (kθθ)

)=

nθ∑kθ=0

nr∑kr=0

ωkθkr (θ, r) (18)

where the terms ωkθkr (·) and their parameters are equal to

ωkθkr (θ, r) = AkθkrLkr (r) cos (kθθ) +BkθkrLkr (r) sin (kθθ)

Akθkr =

np∑i=1

np∑j=i+1

aijkθcijkr

NBkθkr =

np∑i=1

np∑j=i+1

bijkθcijkr

N(19)

The coefficients cijkr in the above equations refer to the Erlangor biased Rayleigh depending on the chosen model. Thenovel form of eq. (18) has the advantage of having a fixedform independently from the number of point pairs N . Thecomplexity of signature computation lies in the coefficientsof the above series. The basis functions Lkr (r) cos (kθθ) andLkr (r) sin (kθθ) are orthogonal. This makes the signature suit-able for analytical manipulation as illustrated in the followingsection. Figure 2 represents the two GRD signatures obtainedusing respectively biased Rayleigh and Erlang distributions torepresent range uncertainty. The signature obtained by Erlangis smoother and less detailed than the one by biased Rayleigh.In general, it is more difficult to shape Erlang kernel PDFaccording to the desired variance due to discretization andnumerical issues of integer exponent dij in eq. (13).

IV. A METRIC FOR SIGNATURE DISTRIBUTIONS

Signature distribution represents the geometric relations of alocal map of points. The main goal of signatures is detectionof already visited places or, at least, of potential candidatematches. A rotation-invariant metric should be used to com-pare a given target distribution signature St with a potential

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phi [deg]

GRD correlation between scans 368 and 376

correlcorrel max

Fig. 3: Example of GRD signature correlation between thescans 368 and 376 of dataset intel. Maximum correlation isachieved with angular shift ϕ = 345.8 deg (equivalent to−14.2 deg) consistent with point rotation (see Figure 1).

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scan 368 aligned

Fig. 4: Example of point-to-point association using correspon-dence graph (CG): the keypoint features of scans 368 (red) and376 (green) in Figure 1 with their pairwise relation graphs, andthe features 368 aligned to features 376 (blue).

matching one Ss. While the pairwise distances are invariantto translation and rotation, the pairwise angles depend onthe viewpoint. To achieve invariance, we follow the criterionadopted for GLAROT [4]. The similarity between the signaturedistributions St and Ss (apices or pedices s and t are usedhereafter) is defined as

d(Ss,St) = maxϕ

dϕ(Ss,St)√d0(Ss,Ss) d0(St,St)

(20)

where

dϕ(Ss,St) =

∫ 2π

0

dθ

∫ ∞0

dr e−r Ss(θ + ϕ, r) St(θ, r)

(21)

Instead of being a distance, the quantity d(·) is maximum whenthe two distributions represent the geometric relations of the




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theta [deg]

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theta 45 [deg]

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(a) (b) (c)

Fig. 2: Example of GRD signatures computed on a mit-csail dataset: (a) the input scan (grey) with the extracted FLIRT features (red); (b) theGRD-BR signature obtained using biased Rayleigh distribution for ranges (with parameters (κ, σij) = (1.5, 0.2)); (c) the GRD-Er signatureobtained using Erlang distribution for ranges (with parameters (κ, σij) = (1.5, 0.2)). The GRD-Er signature is smoother than GRD-BR oneand provides less details.

same point set. The integral in eq. (20), which consists of aproduct of two truncated series, could be difficult to solve. Letωskθkr (·) and ωtkθkr (·) be the terms of series Ss and St accord-ing to eq. (18), with coefficients respectively {Askθkr , B

skθkr}

and {Atkθkr , Btkθkr}. The integral of ωskθkr (·)ω

tk′θk′r(·) is zero

if kθ 6= k′θ or kr 6= k′r due to the orthogonality of the basisfunction. Thus, the only remaining terms are ωskθkr (·)ω

tkθkr

(·).In particular, the integral is solved as

ζkθkr (ϕ) =

∫ 2π

0

∫ ∞0

e−rωskθkr (θ + ϕ, r)ωtkθkr (θ, r)dθdr

=AskθkrA

tkθkr

+BskθkrBtkθkr

2cos (kθϕ)

+AskθkrB

tkθkr

−BskθkrAtkθkr

2sin (kθϕ) (22)

After the substitution of eq. (22) in the metric expression, weobtain

dϕ(Ss,St) =

nθ∑kθ=0

nr∑kr=0

ζkθkr (ϕ) (23)

=

nθ∑kθ=0

Aϕkθkr cos (kθϕ) +Bϕkθkr sin (kθϕ) (24)

where the coefficients {Aϕkθkr , Bϕkθkr} are obtained as

Aϕkθkr =

nr∑kr=0

AskθkrAtkθkr

+BskθkrBtkθkr

2(25)

Bϕkθkr =

nr∑kr=0

AskθkrBtkθkr−BskθkrA

tkθkr

2(26)

Figure 3 shows the correlation function dϕ(Ss,St) for thekeypoint features presented in Figure 1. It can be observedthat the harmonic component kθ = 1 dominates over higherfrequency components. The estimated angular shift ϕ, where

dϕ is maximum, provides a rather accurate estimation ofrotation angle between the two feature maps.

Signatures are used for retrieval of candidates matchinglocal maps. Next the candidates are checked by perform-ing point-to-point association. The Correspondence Graph(CG) [4] searches potential correspondences based on the samepairwise geometric relations used by GRD signature. Since CGexploits the consistency of matching points, it does not rely onthe estimation of local map reference frames w.r.t. the globalreference frame. Figure 4 shows the CG association for thepreviously discussed example.

V. EXPERIMENTS

In this section we present our experimental setup and resultsbased on loop closure. The experiments are assessed in fourpublic datasets: fr079, mit-csail, intel-lab and fr-clinic. Thedatasets provided by [6] contain both original scans andcorresponding corrected ground truth. Results are presentedwith precision-recall curves for loop closure tests.

The loop closure experiments are evaluated comparingeach scan Si with the other scans of the dataset Sj withj 6= i. The proposed loop closure algorithm GRD1, bothwith Erlang (GRD-Er) and biased Rayleigh (GRD-BR), hasbeen tested in each dataset and compared with two state-of-the-art algorithms, GFP and GLAROT. We evaluate theoriginal GFP algorithm using FLIRT keypoints and descrip-tors, since signature GFP was originally coupled with thespecific keypoint feature. The keypoint features SKIP [22]are used to build the landmark maps encoded by GLAROTand GRD. Signature evaluation requires that loop closurecandidates are tested with point-to-point association. GFPuses RANSAC algorithm as point-to-point association methodaccording to its original formulation and implementation. Theother signatures are evaluated with point-to-point matchingmethod Correspondence Graph (CG) [4].

1https://github.com/dlr1516/grd.




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Fig. 5: Precision-recall curves of loop closure based on GRD-BR, GRD-Er, GLAROT and GFP in datasets fr079.

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Fig. 6: Precision-recall curves of loop closure based on GRD-BR, GRD-Er, GLAROT and GFP in datasets mit-csail.

The loop closure evaluation has been performed as follows.For each scan S of a dataset the scan signatures have beencomputed and stored. Then, let SR be the reference scan, aset CS of 20 candidate scans (50 for GFP), whose signaturesare closest to the signatures of the reference scan SR accordingto the respective loop closure method, are extracted. The setof closest signatures CS is evaluated according to the corre-sponding distance function and thresholds of each loop closuremethod. The keypoints of each scan in CS are associated withSR using the selected point-to-point association algorithm.Then, the robot pose is estimated by solving the orthogonalProcrustes problem on associated point pairs. Finally, the scanselected for loop closure Si is the scan that, after alignment,has the greatest number of keypoints with a neighbor pointin SR within the 0.10 m range. The place recognition per-formance is measured by precision-recall curves. The robot isconsidered localized if the associated points are at least Nmin.The precision-recall curves are computed w.r.t. the thresholdNmin. A localization is considered correct when the positionerror of the aligned scan is less than 0.50 m and the angularerror less than 10◦.

Figures 5-8 depict the precision-recall curves obtained in

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GLAROTGFP

Fig. 7: Precision-recall curves of loop closure based on GRD-BR, GRD-Er, GLAROT and GFP in datasets intel-lab.

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pre

cisi

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recall

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GRD-BRGRD-Er

GLAROTGFP

Fig. 8: Precision-recall curves of loop closure based on GRD-BR, GRD-Er, GLAROT and GFP in datasets fr-clinic.

datasets fr079, intel, mit-csail and fr-clinic with the placerecognition algorithms previously discussed. GFP achievesresults comparable to the other features in dataset fr079, butin the other cases it is dominated by the other approaches. Itsperformance may be affected by the use of FLIRT features anddescriptors used in the local map. Previous works [3], [4] havealready observed the potential weakness of local descriptorsfor LIDAR measurements and the arising limitation of theresulting BoW detection. The curves of GLAROT, GRD-BRand GRD-Er tend to overlap in the case intel, but GRD-BRdominates GLAROT all the other approaches. While all suchsignatures are based on geometric relations, GRD-BR seems toprovide a more accurate representation of cumulated pairwisedistances and angles.

Indeed, GLAROT discretizes these parameters into his-togram bins whereas GRD-Er exploits Erlang distribution,which achieves smoother and less detailed description thanbiased Rayleigh, as shown in Figure 2. Thus, the proposedGRD signature is able to select potential loop closure candi-dates with similar or better performance compared to otherstate-of-the-art methods like GLAROT.

The efficiency of the signatures has been empirically as-sessed by signature matching with datasets fr079, intel, mit-




operation avg time [µs]

GFP construction 2722.0pairwise distance 3.7

GLAROT construction 138.7pairwise distance 4.2

GRD-Er construction 2052.4pairwise distance 275.2

GRD-BR construction 2445.0pairwise distance 274.6

CG association 538.6SKIP detection 97.5

TABLE I: Average times per scan required for creation and pairwisedistance of signatures (GFP, GLAROT, GRD-Er and GRD-BR) andfor CG point-to-point association. The times have been computed ondatasets fr079, intel, mit-csail and fr-clinic with an average numberof 13.47 detected features per scan.

csail and fr-clinic. The execution times presented in Table Iare obtained on an Intel i7-3630QM CPU @ 2.40GHz, 8 GBRAM processor. In particular, these tests measure the averageexecution time for the construction of signatures and for thecomputation of single pairwise distance between two signa-tures. A single loop closure query requires the computationof the pairwise distances between the query signature and allthe other signatures of the map. The construction time forGRD signatures is about one order of magnitude greater thanGLAROT, but comparable with GFP whose complexity is dueto the initial offline computation with BoW dictionary. Thecomplexity of GRD construction is quadratic with the numberof point features whereas the pairwise distance computationis computed using guaranteed branch-and-bound optimizationwith 1 deg tolerance on correlation (compared with 22.5 degbin size of GLAROT). Moreover, the computation time couldbe significantly improved by an optimized implementation andbetter parameter tuning.

VI. CONCLUSION

In this paper, we have presented the novel signature GRDfor landmark map location in order to recognize visited placesfor localization and loop closure. The signature correspondsto the probability density function of the pairwise angles anddistances among the landmarks of a local map. The distributionmodels for the two polar coordinates are respectively vonMises for angles and Erlang or biased Rayleigh for distances.The signature GRD has been expanded into series of or-thogonal function basis, namely Fourier series and Laguerrepolynomials. The coefficients of the series have been derivedand used to represent the signature. This representation enablesefficient analytic operations with multimodal distribution and,in particular, allows the definition of rotation and translationinvariant metric to compare the distribution. The GRD capabil-ity to detect visited places in the map has been compared withother state-of-the-art signatures in loop closure experiments.

In future works, we expect to apply GRD in the loopclosure component of a localization and mapping system.Moreover, we will investigate the extension of GRD signatureto 3D landmark maps and the design of scale-invariant metricssuitable for computer vision applications.

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