On the Induction of Topological Mapsfrom Sequences of Colour Histograms

Felix Werner and Joaquin Sitte and Frederic MaireNICTA, Queensland Lab

andFaculty of Information Technology, Queensland University of Technology

Brisbane, QLD 4000, Australia{felix.werner, joaquin.sitte, frederic.maire}@nicta.com.au

Abstract

This paper presents an appearance–based methodto automatically determine places from vision datafor topological mapping. The approach exploits thecontinuity of the visual appearance of consecutive im-ages when a robot traverses the environment. Placesare determined by clustering colour histograms, anda probabilistic filtering strategy eliminates spuriousplaces with weak evidence.

Further, we discuss steps towards the induction ofthe topology of an environment from a sequence of vis-ited places. Particularly, our system faces the problemof physically different places which appear identical inperception space.

We present results from experiments on two datasets, one consist of panoramic images and another oneincludes images from a standard camera.

1 Introduction

Mapping an unknown environment is considered asone of the most difficult problems in robotics [20].Robotic mapping addresses the problem of mobilerobots autonomously acquiring spatial models ofphysical environments. For true autonomy and to con-duct higher level navigation tasks, autonomous mobilerobots must be capable of building an environmentmap from sensor perceptions. In robotics, two kindsof maps have emerged – metric and topological maps.Metric maps capture the fine–grained geometric struc-ture of the environment [16, 7, 21] and are suitable ifthe accurate metric position in terms of Cartesian co-ordinates is required. However, the correct metric po-

sition is not needed for many applications and tasks inmobile robotics and there are no examples in natureusing this strategy. In contrast, biological systemsnavigate in a partially reactive manner if they roughlyknow where they are and where to go [1]. Topologi-cal maps are inspired by this strategy and representthe environment in a more abstract way as a graph ofconnected vertices. The vertices refer to places andthe edges represent the connectivity of the environ-ment [22, 11]. By place we mean not a point like posi-tion but a set of physically neighbouring positions thatshare the same appearance. An appearance is a viewwhich can be represented through global image fea-tures such as colour histograms or Fourier coefficients.Topological maps exhibit attractive properties such asan efficient and a compact representation. Further-more, an other advantage of this representation is theability to apply graph algorithms to higher–level op-erations such as finding the shortest path between twovertices.

Places are described through a set of landmarkswhereby a landmark represents a prominent artificialor natural object in an environment. Artificial land-marks are easy to detect reliably but require modifi-cations to the environment. Natural landmarks usu-ally consist of predefined template models like corners,doors, junctions and floors [10, 5]. Systems which relyon predefined templates for landmark determinationare specific to particular surroundings and cannot beeasily utilised in different kinds of environments. Re-cently, image based local features [2, 13, 4, 15, 10]have been successfully applied to topological mappingand navigation tasks. A place is described througha set of local image features which can be interpretedas weak landmarks. Although these approaches are

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very reliable they are complex and computationallyexpensive and hence not applicable on systems withlow computational power.

Another image based approach is to use an omni–directional camera and extract low computation costrotation invariant features such as Fourier coefficientsor colour histograms as visual landmarks. Clearly,visual appearance is usually not independent of theheading of the robot and can only be achieved by us-ing a panoramic camera. Menegatti et al. suggest anappearance based method for topological mapping us-ing panoramic images [8]. The panoramic images areunwarped and the 15 lowest frequency Fourier coeffi-cients of each row of the image are stored as landmarkfor a place.

Ulrich and Nourbakhsh introduced a systemwhich uses R,G,B and H,S, V colour histograms ofpanoramic images as landmarks which are associatedwith places [23]. The actual places are determinedby hand. For localisation the histogram of thecurrent image is matched to the database. However,this system requires an external supervisor for thelabelling of the places.

In this paper we use an appearance–based methodfor automatically determining places [25] which ex-ploits the continuity of the visual appearance whena robot traverses the environment (Section 2). Wecompare the results using a data set which consists ofpanoramic images and another data set which consistsof images from a standard camera. Furthermore, weconsider the problem of inducing the topology fromsequences of visited places as they are perceived whena mobile robot traverses the environment and ap-ply a probabilistic filtering to cope with measurementnoise and spurious places (Section 3).

2 Appearance–based Place Determi-nation

This section describes the approach we use forappearance–based place determination from visiondata. The method is based on the assumption thatthe visual appearance is a continuous function of therobots position. This leads to the assumption thatpositions with similar appearance can be embracedto a place. A place is considered to be a region ofpositions that generate similar appearances. Abruptchanges in the visual appearance of consecutive po-sitions can indicate a passage between two adjacentplaces. Hence, it should be possible to realise the ab-straction process with a clustering method. The clus-

tering will associate positions with similar appearanceto particular places. In feature space the centroids ofthe clusters can be interpreted as automatically deter-mined landmarks. In the physical domain positionswith similar visual appearance should fall into thesame cluster and hence represent a physically cohe-sive place. In contrast to methods employing templateshapes [10, 5] this place determination strategy doesnot rely on any predefined model or previous knowl-edge about the environment, instead it relies only onthe visual appearance.

2.1 Place Description

Colour histograms are simple image features whichexhibit the property of varying smoothly as the fieldof vision sweeps through the scene. Histograms areeasy to calculate and can represent salient colour in-formation of images in a very compact manner. Inearlier work we introduced entropy–constrained colourhistograms as a clustering method in colour spacewhich adapts to the colour distribution of the envi-ronment [25]. This method allows histograms with alow number of bins (e.g. 24 or 32) without significantloss of information compared to standard histogramswith 256 bins for each R,G,B colour band [25] whichreduces the computation costs in the place determi-nation process.

2.2 Place Determination

We assume the robot has explored the environ-ment sufficiently and has stored the histograms im-ages along the travelled path. As described above, themethod we use for automatically determining placesis based on the continuity property of the visual ap-pearance when a robot traverses the environment. Ifthis assumption holds, it should be possible to applyclustering methods to find similar appearing positionsand cluster them to a place. One of the most popu-lar clustering methods is k–means [6] which aims tominimise the distortion error function

e =k∑

i=1

∑x∈Ci

||x− ci||2 (1)

where xi ∈ S ⊆ <n and ci are the centroids of k clus-ters Ci. The clustering associates similar input vec-tors to the same cluster whereby the centroids canbe interpreted as single landmarks describing places.The histograms which fall into a cluster were taken onpositions which appear similar to this landmark and

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Figure 1. left: Example panoramic image.right: Predefined mask to cut out the usedpixel ring (white).

due to the continuity property of the visual appear-ance these positions should be physically close.

Another clustering method is the self–organisingmaps (SOM), also known as the Kohonen map [9].SOMs are unsupervised single–layer winner–take–allneural networks that learn to categorise input pat-terns and to associate them to different output neu-rons. Its principle is to map the input space S ⊆<n onto a regular two–dimensional array of neurons,where each neuron is associated with a weight vectorwi(t) ∈ <n. When an input x ∈ <n is presented theclosest weight vector wi(t) is computed by

c = arg mini

(d∗(x,wi(t))) (2)

where d∗(., .) denotes the dissimilarity measurementrespective a particular metric. The weight vectors areadapted during the learning process by

wi(t + 1) = wi(t) + α(t)hci(t) [x− wi(t)] (3)

where α(t) is a decreasing learning rate and hci(t) theneighbourhood function. In our experiments we usea Gaussian neighbourhood function

hci(t) = e−d(i,c)2

β(t)2 (4)

where β(t) denotes a time dependent linear decreas-ing neighbourhood size. The term d(i, c) is the mea-sure on the grid from neuron i to the winning neu-ron c. After the clustering, the output space of theSOM reflects neighbourhoods of input similarity, i.e.similar input vectors are mapped to the same out-put neuron. Hence, in this case the neuron representsa landmark which refers to a place with similar in-put vectors (histograms from images). Furthermore,because of the topology conserving property, wherebydata which is close in input space is mapped to neigh-bouring neurons [9], there may also be a benefit ofbeing able to induce the topological map of the envi-ronment from the SOM.

Figure 2. Desk environment and the robot’strajectory (red).

2.3 Results from Experiments

We evaluate the method for appearance–basedplace determination using two different data sets. Thefirst data set consists of 602 panoramic images, odom-etry measurements and laser range scans [24]. It wasrecorded using a remote controlled mobile robot inan indoor environment. Our system uses only a pre-viously defined ring of pixels in the panoramic im-ages (see Figure 1). For evaluation purposes odome-try measurements and laser range scans are processedwith a SLAM algorithm to provide reliable referencelocations in the real world domain and a 2D environ-ment map (see Figure 3).

The second data set contains 548 images froma normal camera (non–panoramic) and odometrysamples. This data was recorded with a Kheperarobot which was driven by remote control through anartificial environment on a desk (see Figure 2). Again,only the image data is used for the place determina-tion process, however, the odometry samples are usedfor evaluation and visualisation purposes.

The images were processed with entropy–constrained colour histograms [25] for differentnumbers of clusters and in HS colour space. Thetransformation to the HS colour space may resultin a certain illumination invariance and reduces thedimensionality of the input data significantly. Anextensive analysis of the effect of different numbersof clusters and a comparison of applying differentcolour spaces for the place determination process canbe found in [25].

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Figure 3. Results of the place determination process for the dataset with images from thepanoramic camera (left) and the standard camera (right). Regions A and B are shown enlargedin Figure 5.

Because the number of clusters determines thenumber of possible places we analysed the two clus-tering methods with different numbers of clusters. Asexpected, we found that increasing the number of clus-ters results in more distinct places. However, if thereare too many clusters, it may happen that places con-sist of a few positions only which is contrary to theaim of abstracting many positions to a place.

Figure 3 illustrates the results of the place determi-nation process using k–means clustering to five clus-ters. Five clusters were chosen for the ease of visualis-ing the results, however, the figures demonstrate thegeneral results saliently. Positions whose histogramsfall into a particular cluster are marked as /, .,�, ◦, 2. It is apparent that the clustering is capableof finding homogenous places with similar appearancewhich justifies the assumption of continuity in visualappearance when a robot moves through the environ-ment. However, as Figure 3 shows it sometimes hap-pens that there are physically different places whichfall into the same cluster. Furthermore, it also hap-pens that within a place or between two places thereare some spurious places of only one or a few positionsas it is exemplarily shown in the zoomed frames Aand B in Figures 4 and 5.

It is useful to consider that using the data from astandard camera increases the difficulty for the placedetermination task as the robots orientation also af-fects the visual appearance (see Figure 3, right). Wefound, that in our experiments rotating a robot onspot usually breaks the continuity property of visualappearance as the rotation is too fast and hence theoverlap of two consecutive images is too small.

Evaluating the results of the clustering demon-strates that exploiting the continuity of the visual ap-pearance of a moving robot is successful in general andit is possible determine places based on the similarityof colour histograms. Figure 3 shows that there areregions in the environment which appear similar andhence represent places which can be described througha landmark. Clearly, increasing the number of clus-ters allows more specific places and hence decreasesthe size of the places. However, if there are too manypossible places used for clustering, the landmarks be-come very specific, which avoids exploiting the conti-nuity property of consecutive colour histograms andalso increases the occurrence of spurious places as dis-played in Figures 4 and 5.

Furthermore, it is plain to see that colour his-togram based landmarks are not unique descriptors

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for places. It may happen that physically differentregions are associated with the same landmark andhence the landmark is ambiguous (see Figure 3). Thisoccurrence does not depend on the number of possibleclusters. Clearly, increasing the feature complexity ofthe landmarks may help to find distinctive descrip-tions for each place but this comes with a significantincrease in computational cost and is not applicableon small robot platforms. Furthermore, this problemmay also originate in the structure of the environment.In a hotel for example, there might be several identicalrooms which can not be distinguished directly.

Comparing the performance of the SOM vs. k–means we have found no significant difference in placedetermination quality, however, training the self–organising map is much more time consuming. Also,the neurons of a self–organising map adapt to thedistribution of the input space which allocates morelandmarks in more extensively explored regions. It ispossible to reason that this expresses the increase ofknowledge for a more explored part of the environ-ment. However, as a side–effect of this it may happenthat the appearance of regions which have been ex-plored just once may not be represented at all in theSOM. Furthermore, the connectivity of the SOM doesnot necessarily reflect the connectivity of the places inthe environment, as it seems the topology of the his-tograms space is not related to the physical topologyof the environment. Also, of course, the occurrence ofambiguous landmarks disrupts the possible topologi-cal consistency of the histogram space and the physi-cal domain.

2.4 Discussion

In conclusion determining places in terms of ho-mogenous positions with similar visual appearanceachieves satisfying results. One reason for the con-tinuity of colour histograms is that the backgroundoften occupies a large area of the image and the back-ground shifts slowly as the robot moves. However, forthe induction of a topological map from these regionsof positions there are still several problems to solve.As the experiments show, it is possible to exploit thecontinuity property of the visual appearance of cohe-sive positions to find places in the environment. Un-fortunately the number of places is not determined bythe number of clusters and there are landmarks whichrepresent more than one place. Hence, a topologicalmap and the mapping method must be capable of han-dling two different places with identical appearancewhich means two vertices in the topological environ-

ment map are described with an identical attribute.Furthermore, spurious places of only a few posi-

tions unnecessarily increase the number of vertices fora topological map, as a mapping algorithm may con-sider these as independent places. Analysing their oc-currence in the indoor data set shows that this hap-pens mostly when the robot passes through a door-way. In fact, in this case tiny places are of minorimportance as they can be interpreted as some mea-surement noise between two adjacent places. An ap-proach to solve the latter problem is presented in thefollowing section.

3 Towards Inferring the Topology ofthe Environment

When a robot traverses the environment it per-ceives a sequence of visited places. The previous sec-tion has shown that in our system it is only possible todetermine regions with similar visual appearance but,there are different regions in the environment whichappear similar and hence are associated with the samelandmark. Thus, the robot’s perceptions are ratherappearances of places. This ambiguity and also spu-rious places cause trouble to a mapping process.

3.1 Filtering the Sequence of Appear-ances

In the border region of two clusters a small varia-tion in the histogram may cause a change in the as-signed cluster. A small histogram variation may becaused by measurement noise. We can apply a fil-ter method to the sequence of observations to avoidchanges of clusters from measurements with weak ev-idence.

In the following, ot denotes the observation and st

the filtered state at time t. The observation oi,t rep-resents the evidence of a histogram being associatedwith a particular centroid ci (landmark). We use aevidence measure which depends on the distance

di,t = ||ci − oi,t|| (5)

to the centroids ci and the probability distribution isgiven by

P (ot) ∝(

1d1,t

, · · · , 1dk,t

)T

. (6)

Note, we assume that the observation probability onlydepends on the distance of the measurement to the

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Figure 4. The effect of the filtering is demon-strated on the two enlarged regions (see Fig-ure 3, left) A (left) and B (right). The top rowshows the regions before the filtering andthe bottom row afterwards.

centroids of the clustering. So far, we do not knowthe topological map and hence the transitions betweenany two states have the same probability. However,due to the continuity of visual appearance it is morelikely that several consecutive states appear similarand hence it is more likely to stay in the same state.Thus, the transition probability between two consec-utive landmarks at time t and t+1 which fall into theclusters ci,t and cj,t+1 is given by

P (cj,t+1|ci,t) ∝{

α ifi = j1 otherwise

(7)

where α ≥ 1 denotes a regularisation parameter whichrepresents the influence of the continuity assumption.Given the observation probability distribution and thetransition probability distributions the state st givenall observations o1:t can be calculated [17] as

P (st|o1:t) ∝ P (ot|st, o1:t−1)P (st|o1:t−1) (8)

and simplified using Bayes’ rule and the Markov prop-erty [17] to

P (st|o1:t) ∝ P (ot|st)P (st|o1:t−1). (9)

Figures 4 and 5 show the results of the filteringwith α = 2 for the frames A and B in Figure 3.We can see that places with only a few positions arefiltered out. Though, the filtering causes a delaying

Figure 5. The effect of the filtering is demon-strated on the two enlarged regions (seeFigure 3, right) A (first and second row)and B (third and forth row). The first andthird row show the regions before the filter-ing and second and fourth rows afterwards.The black lines display the robot’s pathswhen it visited the region. Rows three andfour are subject to a 90° rotation for visuali-sation purposes.

low pass filter effect which depends from which di-rection the sequence of appearances is filtered. How-ever, the fact that after the filtering states differ fromthe corresponding observations indicates that the ob-servations were not attributed with strong evidenceand hence are probably situated between two adjacentplaces. Clearly, the continuity of the visual appear-ance usually implies a smooth change between twoplaces hence the evidence of measuring a particularlandmark is weaker. Thus the delay caused by the fil-tering is of minor importance as it happens usually inregions with minor importance in terms of topologicalplaces. Furthermore, for the non–panoramic imagesthe right part of Figure 3 clearly displays the effect oftraversing a region with two different orientations (in-dicated by 2 and .) of the robot. The orientationsappear different and hence are associated to differentplaces.

3.2 Induction of the Topology

The final task in the mapping process is the induc-tion the topology of the environment from a sequence

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of visited appearances. A major concept for buildingtopological maps autonomously from a sequence of de-scriptions of visited places is to connect vertices thatrefer to similar places [24, 12, 19, 18, 28, 3]. Anotherpopular method is to link vertices that are consecu-tive in the sequence [14, 5, 27, 26]. However, bothof these strategies rely on the assumption that thereare only distinctive places in the environment. If theenvironment includes physically different places withare associated with identical landmarks as it occurs inour system, these strategies would result in distortedmaps and hence fail.

We consider the sequence of appearances as theonly knowledge about the environment. As describedbefore, due to the ambiguous landmarks, the sequencecontains appearances which originate from physicallydifferent places. Hence, given this sequence, the taskof inducing the topology requires solving the corre-spondence problem in terms of which of these appear-ances are to fuse as they originate from the same place,and, which must be distinguished.

A strategy which may be appropriate is to con-sider the adjacencies of a particular appearance. Forexample, if an appearance B is located always betweenthe appearances A and C, we can conclude that at theplace is described by the appearance B must be ad-jacent to places with appearances A and C. However,there might be more than one region in the environ-ment which has this structure which requires the con-sideration of a bigger neighbourhood. Furthermore, ifthe sequence does not contain any transitions betweentwo appearances there must not be any two adjacentplaces described by these two appearances in the en-vironment map.

A valid map of the environment must be consistentwith the sequence of appearances. That is, it mustbe possible to derive all possible sub–sequences oflength n contained in the initial sequence by travers-ing the reconstructed environment graph. Further-more, it must not be possible to derive sub–sequenceswhich are not contained in the initial sequence.

An obvious and valid reconstruction is given by agraph which contains a vertex for each occurrence of aparticular appearance in the sequence of appearances.However, this graph would contain too many verticesand hence would not properly represent the environ-ment.

The induction of topological maps which rely onthese constraints and further reduce the number ofvertices will be investigated in future research.

4 Conclusion

In this paper we proposed an appearance–basedmethod for automatically determining places in an en-vironment. This method exploits the continuity of thevisual appearance when a robot moved through theenvironment. We use colour histograms as an imagefeature that is simple and fast to calculate and variessmoothly when the field of vision sweeps through thescene. For determining places we analysed two clus-tering methods, k–means and SOMs and no signifi-cant difference in the quality of the determination wasfound. However, self–organising maps were computa-tionally much slower than k–means.

The system was analysed using two data sets, oneincluding panoramic images and another including im-ages from a standard camera. With both data sets wecould justify the assumption of the continuity of thevisual appearance when a robot moves through an en-vironment.

Two problems emerged in the place determina-tion process which hinder the induction of topologicalmaps from sequences of visited places. Spurious placeswhich originate in measurements with weak evidencecan be counteracted by probabilistically filtering thesequences of appearances which are obtained by therobot.

Furthermore, we found that colour histograms arenot a unique description for places and hence there aredifferent places in the environment that appear similarso the robot can not distinguish at which of severalpossible places it is. We have discussed the constraintsa map induction process and the topological map itselfmust satisfy to be consistent to the knowledge givenfrom the sequence of appearances. Future work willbe concerned with the map induction process.

5 Acknowledgements

NICTA is funded by the Australian Government’sBacking Australia’s Ability initiative, in part throughthe Australian Research Council and the QueenslandState Government.

The authors would like to thank Peter Biber (Tue-bingen University) and Henrik Andreasson (Univer-sity of Orebro) for providing the dataset and alsoChristian Weiss (Tuebingen University) for the calcu-lation of the reference positions and the environmentmap.

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