Originally published in: Research Collection Permanent Link: …7774/eth... · cess, an unsupervised learning algorithm is required. Sophis-ticated clustering algorithms usually depend

Page 1

Research Collection

Conference Paper

DP-FusionA generic framework for online multi sensor recognition

Author(s): Liu, Ming; Wang, Lujia; Siegwart, Roland

Publication Date: 2012

Permanent Link: https://doi.org/10.3929/ethz-a-010034783

Originally published in: http://doi.org/10.1109/MFI.2012.6343031

Rights / License: In Copyright - Non-Commercial Use Permitted

This page was generated automatically upon download from the ETH Zurich Research Collection. For moreinformation please consult the Terms of use.

ETH Library

Page 2

DP-Fusion: A generic framework of online multi sensor recognition

Ming Liu∗, Lujia Wang†, Roland Siegwart∗∗ Autonomous Systems Lab, ETH Zurich, Switzerland

† Department of Electronic Engineering, The Chinese University of Hong Kong[ming.liu,lujia.wang]@mavt.ethz.ch, [email protected]

Abstract—Multi sensor fusion has been widely used in recog-nition problems. Most existing work highly depend on thecalibration between different sensor information, but less onmodeling and reasoning of co-incidence of multiple hints. Inthis paper, we propose a generic framework for recognition andclustering problem using a non-parametric Dirichlet hierarchicalmodel. It enables online labeling, clustering and recognition ofsequential data simultaneously, while taking into account multipletypes of sensor readings. The algorithm is data-driven, which doesnot depend on prior-knowledge of the data structure. The resultshows the feasibility and its reliability against noise data.

I. INTRODUCTION

Perception is the process that converts raw sensor readingsto expedient information. As we know, human are good atperception. One important reason is that we use multiplesensors, such as eyes, nose and ears together, trying to gatherinformation from different perspectives. Luo et al in [1]provided an interesting biological explanation of multi sensorintegration for animals. Inspired by this fact, in past decades,multi sensor fusion [2] has shown its importance impact indifferent engineering fields, such as monitoring of complexstructure, fault diagnosis and especially robotics. Most recentwork treated multi sensor fusion problem in a decentralizedfashion [3]. In brief words, they first considered multi sensorreadings separately, reason/infer them, then fuse the conclu-sion of each sensor in the end. This pattern is potentially robustto failure of any one of sensors. Nevertheless, we observe thefollowing drawbacks. First, it highly depends on the calibrationprecision between sensors; second, high believes on certainsensors may cause false positives in final results as well; lastbut not the least, coincidence of multiple information hints issimply ignored. However, it is the primary pattern how humanrecognize the world, i.e. through appearance, smell, audition,tactility etc. simultaneously.

In order to elevate this, we introduce a generic frameworkwhich allows recognition tasks to take multiple sensor read-ings simultaneously. It is proved to be low in computation.Meanwhile, the sensor measures are coherently linked togethervia clustering. As a primary feature, the proposed algorithmuses non-parametric statistics to discover the inner relationsamong data from different subjects. It starts from zero prior-knowledge and takes sequence of concurrent data from differ-ent sensors as input. No specific training is required during

This work was supported by the EU FP7 project NIFTi (contract # 247870)and EU project Robots@home (IST-6-045350).

the process. It enables the fused data to autonomously buildnew clusters and recognize existing cluster in real-time.

The proposed model is stimulated by Dirichlet ProcessMixture Model (DPMM) [4], which is nowadays widely usedin texts classification and segmentation. The original algorithmtakes only one type of input, such as words or letters. More-over, the inference of a DPMM is computationally expensive,because sampling algorithms are usually required [5] from thelarge test set. We extend the model to multiple observationfrom different sensors and develop an online approximationalgorithm which enables fast inference in real-time.

A. Pattern of Multi sensor Fusion

The taxonomy of data fusion algorithms varies. We only listseveral related elements that are generally used in surveys.

1) Decision Fusion: Decision making is the most criticalproblem for intelligent systems. It is a general concept andis usually embedded into specific paradigms, such as failuredetection, object recognition, pedestrian detection etc. Severalwork regarding decision fusion have been proposed in thescope of decentralized multi sensor state representation. In[6], the authors introduced a decision fusion framework to fusemulti sensors by using confidence regions of the sensor model.Fauvel et al [7] uses fuzzy set theory to fuse the decisionfrom multiple classifiers. [8] introduced a force aggregationand classification model by fusing information from sensorswith different resolutions.

2) Sensory State: The purpose of multi sensor fusion is toobtain information more robustly than single sensor. In mostcases, the target information can be considered as goal states.At early stage, Extended Kalmann Filter (EKF) was widelyused, where the perception outcomes from multi sensors aretaken as a unified state. This model is usually named ascentralized state estimation. Several robotic applications areproposed, such as [9] fuses vision and haptic sensor for objectrecognition; [10] used neural networks to fuse the sensorinformation of intelligent vehicles etc. However, these earlywork do not treat the multi sensor fusion mathematicallyefficiently. Moreover, the robustness to sensor failure is abig problem. Durrant-Whyte et al proposed a decentralizedarchitecture named Decentralized Kalmann Filter (DKF) in[3], which handles multi sensor data separately then fusethe conclusions from each filter. The obvious advantage ofDFK lies in its robustness to single sensor failure. Somerecent researches still follow the same concept, such as objectrecognition by [11], segmentation problem by [12], pose

Page 3

estimation problem by [13], [14]. The proposed algorithm doesnot show explicitly decentralized characteristics. However, thejoint probability given in section II depicts the independenceof all sensor readings. It indicates that the confidence of eachsensor is propagated to the posterior directly, which meanssensor readings are not centralized as a single system state.

B. ClusteringIn order to automate the classification and recognition pro-

cess, an unsupervised learning algorithm is required. Sophis-ticated clustering algorithms usually depend on iterative cal-culation such as K-means, spectral clustering [15] or affinity-propagation [16]. A representative of online reasoning is chow-liu tree based segmentation [17] for static data and changepoint detection [18], [19] for sequential data. For extremecases, the synchronization of multi sensor data need to betaken care of [20] or spatial and temporal hints must be jointlyconsidered [21]. In this paper, an online naive change pointdetection algorithm is implemented, which is validated throughsimulation in section IV.

C. Recognition and inferenceRecognition is the core of most robotic applications. For

example, robot topological mapping requires detection andrecognition of loop-closure; semantic mapping usually re-quires recognition of objects; human-machine interfaces re-quire recognition of human behaviors etc. Researches targetingat these core problems attempt to seek the best algorithmsto build efficient models which can represent this perceptionprocess efficiently.

Regarding inference approaches, hierarchical probabilisticmethods based on statistical techniques won a great success intext mining and biological information processing [22], [23].In this work, we alternate the classical mixture model to fitthem with multiple types of observations. At the same time, weallow infinite increment of the number of labels. Furthermore,the model is to be learned, updated, inferred in real-time on-line.

In most of the related works, change-point detection [24],[19], [25] is the basis to segment a data sequence. In thiswork, as we are targeting at a lightweight method, the change-point detection is not feasible when using multiple hypothesismethods, such as particle filtering [19]. Instead, we use non-parametric statistic test to evaluate the labeling for each frameseparately. This may cause instability in the output label.However, it relief the requirement of saving all the previousdata of the sequence.

The theoretical advances in hierarchical probability frame-works, such as LDA [23] and HDP [22], provide a goodsupport for our algorithm. The Dirichlet Process MixtureModel (DPMM) enables countable infinite clusters for themeasures, which can be used to represent the process of staterecognition.

D. Assumptions and ContributionsNot withdraw the generality, the proposed algorithm deals

with data with the following assumptions.

• The multi sensor readings are synchronized, or they canbe treated as a complete observation unit when they havedifferent sampling rate;

• Features of the sensor readings are observable and com-putational feasible in near real-time;

• As an assumption of DPMM, multi sensor readings in thedata set must be exchangeable, which indicates that thelabeling of a reading does not depend on whether suchreading appears earlier or later.

The objectives that we want to achieve in this paper aredouble folded.• Modeling multi sensor recognition process using hierar-

chical probability model. The model of the recognitionprocess depends on parameter set with small cardinality.

• A concise approach for on-line inference of the proposedDirichlet Process Mixture Model;

The remainder of this paper is organized as follows. Wewill start with proposing the hierarchical model for onlinerecognition using multi sensor data. The full inference ofthe model will also be introduced. Then we introduce anapproximate method for fast inference of the model. Weexplain the evaluation of the model using simulation in sectionIV. The conclusion and future steps of this work are given inthe end.

II. MODEL FORMULATION

We propose a DPMM model as shown in figure 1, wherethe parameters are depicted in rectangles, and random vari-ables are in circles. Especially, the following components aredesigned in the proposed model.

φ t N

xtp

Gα

H

θ kp

βp

K

P

t ~ 1:N Datak ~1:K Clustersp ~1:P Sensors

Fig. 1. Directed Acyclic Graph (DAG) of the proposed model for recognitionby multi sensor fusion

A. Chinese Restaurant Process (CRP)

G is a Dirichlet process distributed with base distributionH and concentration parameter α. The base distribution is the

Page 4

mean of the DP and the concentration parameter α is as aninverse variance. The distribution G itself has point masses,and the draw from G will be repeated by sequential drawsconsidering the case of an infinite sequence. Additionally, φtis an indicator of the cluster identity which the current dataset at time t belongs to. We could see that φt is the targetvariable of inference. If the process is considered as a partitionproblem, a CRP model is usually used. It uses a prior obtainedfrom a stick-breaking process [26]. By integrating over G, thedrawing of φt ’s can be depicted as:

φt | φ1:t−1 ∼∑t−1n=1 δφn

+ αH

t− 1 + α

where δφnis an indicator of a certain frame n is labeled as

φn, i.e. a mass point function locates at φn. We must noticethat this assumption implies that the more we see a certaincluster of data, the high a prior that data from such clustermay be observed again. The target problem is then convertedto an estimation of

P (φt | φ \t, G,x,θ;α, β)

where φ \t is the full set of indicators excluding the currentone, namely the history labels. Sets of random variables andsets parameters are shown in bold.

B. Multi Sensor Data Perception

The multi sensor data of P different types of readingsare modeled as the orange plate (encircled by dashed line)shown in figure 1. For all N readings in the sequence, xptrepresents the perceived information acquired at time-stampt from sensor p. Taking discretized readings as an example,perceived information from raw sensor data can be representedas histograms [27]. Assuming there are K different clusters,θk is then a matrix of K × Zp, where Zp is the number ofpossible histograms for sensor p. xt’s of dimension Zp aredrawn from θk. In general cases, Zp represent the number ofpossible readings from sensor p.

On one hand, xpt is inherently determined by its label φt,as defined in section II-A; on the other hand, we can alsoconsider the sensor reading as a draw (sample) from a sensormodel θpk for cluster k, with a sensor model prior βk. So far,we build a model of two sub-processes, namely the sensoringprocess and perception process, which serves as a basis tobuild data-driven inference model of the recognition problem.

C. Model Inference

As a summary of the proposed model,

G ∼ Dir(αH)

φt | G ∼ Gxpt ∼ F (φt, θ

pφt

)

F represent the generation function of the measurements fromthe base models, regarding label φt. The joint probability can

be written directly as,

p(φ G θ x; β) =

P∏p=1

K∏k=1

p(θpk ; βp)

N∏t=1

p(G ; H,α) p(φt | G)

P∏p=1

p(xpt | θpφt

)

In order to factorize it to independent components, weintegrate the joint probability over θ1, θ2 . . . θP and G,

p(φ x ; β) =

∫θ1. . .

∫θP

∫G

p(φ G θ x ; β) dG dθ1 . . . dθP

=

∫θ1

K∏r=1

p(θ1k ; β1)

N∏t=1

p(x1t | θ1φt) dθ1

. . .∫θP

K∏r=1

p(θPk ; βP )

N∏t=1

p(xPt | θPφt) dθP

∫G

∫H

N∏t=1

p(φt | G)p(G ; Hα) dH dG

(1)The last component is an exception of G, i.e.

EG [p(φ1 φ2 φ3 φ4 · · · φN | G)]. According to thefeatures of the Dirichlet process, it is proportional tothe product

∏Nt=1 p(φt | φ \t) ∝ p(φN | φ \N ). Therefore,∫

G

∫H

N∏t=1

p(φt | G)p(G ; Hα) dHdG ∝∑N−1t=1 δφt + αδφk

N − 1 + α(2)

where δφnis a mass point function located at φn. k is the

indicator for a new cluster.The first parts can be treated in a similar manner. Take

the integral of θp for an instance, using nkv representing thenumber of measures who is the v-th element in θp withincluster k. ∫

θp

K∏k=1

p(θpk ; λ)

N∏t=1

p(wt | θφt) dθp

=

K∏k=1

∫θpk

Γ(∑Zp

v=1 βpv)∏Zp

v=1 Γ(βpv)

Zp∏v=1

θβpv−1

k,v

Zp∏v=1

θnkvk,v dθ

pk

=

K∏k=1

∫θpk

Γ(∑Zp

v=1 βpv)∏Zp

v=1 Γ(βpv)

Zp∏v=1

θβpv+nk

v−1

k,v dθpk

(3)

since from the integral of Dirichlet distribution,∫θpk

Γ(∑Zp

v=1 βpv + nkv)∏Zp

v=1 Γ(βpv + nkv)

Zp∏v=1

θβpv+n

kv−1

k,v dθpk = 1 (4)

The joint probability is represented as follows.

p(φ x ; β)

∝P∏p=1

K∏k=1

Γ(∑Zp

v=1 βpv)∏Zp

v=1 Γ(βpv)

∏Zp

v=1 Γ(βpv + nkv)

Γ(∑Zp

v=1 βpv + nkv)(∑N−1

t=1 δφt+ αδφk

N − 1 + α

) (5)

Page 5

When we consider a collapsed Gibbs sampling process onthe cluster indicator φt at time t, we have

p(φt | φ\t x ; β) ∝ p(φt φ\t x ; β) (6)

However, the huge size of Zp makes the direct inference notpossible. Usually sampling methods [5] is used to estimate theposterior. Nevertheless, the sampling based algorithm usuallyis computational expensive as well. It is required to findan online approximation algorithm, in order to make thealgorithm work in real-time.

III. APPROXIMATION

In this section, we introduce the approximation algorithm toinfer the proposed DPMM. For the case where measurementφt = k, for simplicity, we rewrite equation 5 as follows.

p(φt = k | φ \t x)

∝P∏p=1

∏Zp

v=1 Γ(βpv + nkv)

Γ(∑Zp

v=1 βpv + npv)

(∑N−1t=1 δk + αδφk

N − 1 + α

)

=

P∏p=1

ξp(xpt | θpφt

)p(φt | φ \t)

(7)

We could see from equation 7 that the first P compo-nents ξp()s calculate the gamma function of the count ofa certain observation over all possibilities. In another word,they represent the probability of a certain measure showingup in a sequence of observations. Therefore, it can also beconsidered as a measure of the similarity of current obser-vation to all the predefined models. As a result, we don’tneed sampling methods to estimate this measure if we canapproximate the underlying similarity between current obser-vation and reference models. This conclusion leads to veryflexible means to recognition problems, since the similaritybetween observation and model can be obtained by variouscriteria, e.g. number of matched key-point features, result ofspectrum analysis, dot product of observation vectors etc. Inthe end, a scalar will be used to indicate this similarity. Theresulting scalar s can further represent the observation as asample from a distribution of exponential family, such as zero-mean Gaussian distributions [e.g. C ·e−s2 ] or Beta distribution[e.g. Be(1, S) where S > 1].

However, another factor much be considered. It is theweighting factor among all sensors. As for equation 5, thisfactor is modeled by prior β. Joining with the approximationby exponential family distribution,

ξp(xpt | θpφt

) ≡ e−(ωps2(xp

t ,θpk))

A set of weights for sensors can be used as follows.

p(φt = k | φ \t x)

∝

(∑N−1t=1 δk + αδφk

N − 1 + α

)e−(

∑p ωps

2(xpt ,θ

pk))

P∑p=1

ωp = 1.

(8)

where θpk is the incrementally estimated model, and s() depictsthe matching result between the current observation and themodel. One example of the incremental estimation of themodel is given in section V.B of [27].

IV. SIMULATION

The simulation with multi sensor inputs for online clusteringand recognition is introduced in this section. We simulatethree synchronized sensor readings, which are observed froma system with change states. The ground truth of the changingstate is shown in first block of figure 2. Subplot A,B and Cshow the readings from three different sensors. Please note thatsensor reading C provides only noisy signal, independentlyto the state change. We use the case C to simulate that lowinformation sensor readings could be successfully omitted bythe proposed decentralized framework. The sensor modelsare zero-biased Gaussian distributions. We first check the

0 100 200 300 400 500 6000

5

Sta

te

Ground Truth

0 100 200 300 400 500 6000

5

A

0 100 200 300 400 500 600

0

2

4

B

0 100 200 300 400 500 6000

1

2

C

Multi Sensor Data Sequence

Fig. 2. Simulated multi sensor readings

likelihoods of the joint probability regarding state 1,2 and3 without considering change-point detection, in order tovalidate the distinctness drawn from equation 7. The jointlikelihood for three sensors are shown in figure 3. It shows

0 100 200 300 400 500 6000

0.2

0.4

0.6

0.8

1

Multi Sensor Data Sequence

Lik

elih

ood

cluster1

cluster2

cluster3

Fig. 3. Likelihood of readings

that if the cluster of the data is given, the model couldvalidate sensor readings as samples from each cluster. Theremaining problem is that the clusters of date (respectingeach state of the system) need to be automatically detectedand incrementally generated. To this end, we use a naive

Page 6

change-point detection algorithm, since the noise level of thesimulated data is low. A time-stamp is considered as change-point when the posterior of the observations is lower than athreshold in conjugated 5 readings. For sophisticated change-point detection algorithm such as particle filter, please referto [25], [19], [28]. The result is shown in figure 4. The first

0 100 200 300 400 500 6000

0.5

1

Poste

rior

giv

en m

odel

0 100 200 300 400 500 6000

2

4

6

8

Changepoin

t D

ete

ction

0 100 200 300 400 500 6001

2

3

4

5

Ouput

Multi Sensor Data Sequence

Fig. 4. Simulation of change point detection and clustering result

subplot shows the posterior of MAP (Maximize-a-Posterior)result. The change-point detection is shown in the second part.In the end, we present the resulting labeling (in blue) againstthe ground truth (in red).

The results indicate that the proposed DPMM model isable to detect and register new clusters of data online, whileperforming recognition task simultaneously. Experiment re-sults on a real-time scene recognition problem, by fusing twodifferent types of readings, can be obtained from our previousreport [27].

V. DISCUSSION AND REASONING

A. Independence and decentralized state

We observe from equation 8 that the inference of the DPMMmodel falls back to a product of the likelihood of each sensorreading and a CRP process. It shows that given the observationxpt , all the sensor perception model are independent. This isconsistent with the original model design of figure 1. Thereforethe system state can be easily written as a decentralized way. Itmeans that a DFK filter is also applicable as post-processing.

B. Complexity

We instantiated a similar model as scene recognition prob-lem in [27] for dual-sensor perception. Based on further study,we draw the following properties of the model.

1) Cardinality: Recognition algorithm usually leads to abig set of parameters. The choice of parameters especiallythresholds will lead to dramatically change in the final result.Equation 8 shows that the proposed algorithm depends on theweighting factors for sensors and prior of the CRP process.

The influence of prior α for the CRP process can be ignoredwhen the number of measurements N goes large. The weight-ing factors can either be chosen empirically (e.g. vision usuallyplays a more important role in object recognition than laser)or enable them to be adaptively tuned by variance analysis(e.g. greater variance among different measures leads to higherweight) etc.

2) Computational complexity: As the number of mea-surement grows, new models are automatically detected andupdated. The complexity of the algorithm only rises linearlyalong with the number of models. Based on the dual-sensormodel in [27], we record the computation time over the test.The result is shown in figure 5. This result implies the potentialof the proposed method can be extended to large scale datasetwithout jeopardizing the real-time ability.

●

●

●

●

●

●

●●●●

●

●

●●

●

●

●●●

●

●

●

●

●

●●●●●

●

●

●●

●●●●●●●●●●●●●

●

●●●

●

●●● ●

●

●●●

●●● ●●●

●●

●

●

●

●

●●●●

●●●

●

●●●

●●●●●

●●●●

●

●

●

●

●

●●

●●

●

●

●

●

●●

●●

●●

●

●●

●

●●●●●●●●●●

●

●●

●●

●

●

●

●

●●●●

●●

●

●

●

●●●●●●●●●●●●●●

●

●

●●

●●●

●●●

●●

●●●●

●●●●●

●

●●●●●

●

●●

●

●

●●

●●●●

●●

●●●

●

●

●

●●

●●●●●●

●●●

●

●●

●

●

●●●●●●

●●●●

●

●●

●

●●

●

●●●●●●●

●

●

●

●

●●

●●●

●●●

●

●●●

●●●

●●

●●●

●

●●●●●●●●●

●●●

●

● ●

●

●●

●

●

●●●●●●●●●●●

●

●●●●●●●●

●

●●●●●●●●●

●

●●●●

●

●●●●

●●●

●

●

●

●

●

●

●

●

●

●

●

●●●●●

●●●●●

●

●

●●

●●

●

●●●

●

●●●●●●

●

●●

●

●●●

●

●

●● ●

●

●

●●

●

●

●

●

●

●●●

●

●●●

●

●●●●

●●

●

●

●●

●

●●●●●●●●●

●●●

●

●

●

●●

●

●

●●●●●●

●

●

●

●

●●

●●

●

●

●

●●●

●●

●●●

●

●●●

●●

●●

●●●●●●

●

●

●

●●

●

●

●

●●●

●

●●●●●●●

●●●●●

●

●

●

●●●

●

●●

●●●

●

●

●●

●

●

●

●

●

●●

●●●

●

●

●●●●●●

●

●

●

●

●●

●● ●

●

●●

●

●

●

●

●

●

●

●

●

●●

●●●●

●

●

●●●

●●

●

●

●

●

●●

●

●

●

●●

●

●

●●●

●

●

●

●

●

●

●

●

●●●

●●

●

●●

●

●

●

●●

●

●●●●●●●

●

●●●●●●

●

●

●●●

●

●●

●

●

●

●

●●

●

●●

●

●●●●

●

●

●

●

●●

●

●●

●●●●

●●●

●●

●

●

●

●

●●

●●

●●●

●●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●●

●

●

●

●

●

●●●●●

●

●

●

●

●

●

●

●●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●●

●

●●●●

●

●●

●

●

●

●

●●

●

●

●

●●

●

●●

●

●

●●●●●●●●●●

●●●

●●●●●●●

●

●●

●

●

●

●●

●

●●●

●

●

●

●

●●●

●

●

●●

●

●●●●

●

●●

●

●

●

●

●

●●●

●

●

●●●●

●

●

●●

●●

●●

●

●●

●

●

●

●●●

●

●●

●

●●●●●●●

●

●

●

●●

●

●

●

●

●

●●●

●

●●●

●

●

●●●●●●

●●

●

●

●●●●●●

●

●

●

●

●●

●●●

●●●

●

●●

●

●

●●

●

●

●

●

●

●

●●●

●

●

●

●

●●●●●

●

●●

0 5 10 15 20 25 30 35

2.0

2.5

3.0

3.5

4.0

4.5

Number of nodes in the map

Infe

renc

e tim

e in

ms

● raw measurefitted curve

● raw measurefitted curve

Fig. 5. Inference time vs number of nodes

VI. CONCLUSION AND FUTURE WORK

In this paper, we present DP-Fusion, an on-line informationfusion framework for multi sensor data based on Dirichlet Pro-cess Mixture Model. It combines synchronized sensor readingsto automatically cluster data into models, while recognizingdata from existing models simultaneously. Results show itsadvantage of on-line computing mode and low computationalcost. This study also implies that the inference of a DPMM canbe approximated by the product of the conditional probability.We envision that similar concept can be borrowed to solveother inference problem as well. For further study, we willexplore automatic learning of the parameter sets required bythe model using kernel density estimation. Extended experi-ments on large dataset will also be carried out.

REFERENCES

[1] R. Luo and M. Kay, “Multisensor integration and fusion in intelligentsystems,” Systems, Man and Cybernetics, IEEE Transactions on, vol. 19,no. 5, pp. 901–931, 1989.

[2] D. Hall and J. Llinas, “An introduction to multisensor data fusion,”Proceedings of the IEEE, vol. 85, no. 1, pp. 6–23, 1997.

Page 7

[3] H. Durrant-Whyte, B. Rao, and H. Hu, “Toward a fully decentralizedarchitecture for multi-sensor data fusion,” in Robotics and Automation,1990. Proceedings., 1990 IEEE International Conference on. IEEE,1990, pp. 1331–1336.

[4] S. MacEachern and P. Muller, “Estimating mixture of dirichlet processmodels,” Journal of Computational and Graphical Statistics, pp. 223–238, 1998.

[5] R. Neal, “Markov chain sampling methods for dirichlet process mixturemodels,” Journal of computational and graphical statistics, pp. 249–265,2000.

[6] S. Thomopoulos, R. Viswanathan, and D. Bougoulias, “Optimal decisionfusion in multiple sensor systems,” Aerospace and Electronic Systems,IEEE Transactions on, no. 5, pp. 644–653, 1987.

[7] M. Fauvel, J. Chanussot, and J. Benediktsson, “Decision fusion for theclassification of urban remote sensing images,” Geoscience and RemoteSensing, IEEE Transactions on, vol. 44, no. 10, pp. 2828–2838, 2006.

[8] B. Yu, J. Giampapa, S. Owens, and K. Sycara, “An evidential modelof multisensor decision fusion for force aggregation and classification,”in Information Fusion, 2005 8th International Conference on, vol. 2.IEEE, 2005, pp. 8–pp.

[9] P. Allen, “Integrating vision and touch for object recognition tasks,”Multisensor integration and fusion for intelligent machines and systems,pp. 407–440, 1995.

[10] S. Nagata, M. Sekiguchi, and K. Asakawa, “Mobile robot control bya structured hierarchical neural network,” Control Systems Magazine,IEEE, vol. 10, no. 3, pp. 69–76, 1990.

[11] D. Klimentjew, N. Hendrich, and J. Zhang, “Multi sensor fusion ofcamera and 3d laser range finder for object recognition,” in Multisen-sor Fusion and Integration for Intelligent Systems (MFI), 2010 IEEEConference on. IEEE, 2010, pp. 236–241.

[12] N. Hendrich, D. Klimentjew, and J. Zhang, “Multi-sensor based seg-mentation of human manipulation tasks,” in Multisensor Fusion andIntegration for Intelligent Systems (MFI), 2010 IEEE Conference on.IEEE, 2010, pp. 223–229.

[13] G. Nuetzi, S. Weiss, D. Scaramuzza, and R. Siegwart, “Fusion of imuand vision for absolute scale estimation in monocular slam,” Journal ofIntelligent and Robotic Systems, vol. 61, p. 287299, 2011.

[14] A. Martinelli, “Vision and imu data fusion: Closed-form solutions forattitude, speed, absolute scale, and bias determination,” Robotics, IEEETransactions on, vol. 28, no. 1, pp. 44 –60, feb. 2012.

[15] I. S. Dhillon, Y. Guan, and B. Kulis, “Kernel k-means , SpectralClustering and Normalized Cuts,” in Proceedings of the tenth ACMSIGKDD international conference on Knowledge discovery and datamining. ACM, 2004, pp. 551–556.

[16] B. Frey and D. Dueck, “Clustering by passing messages between datapoints,” science, vol. 315, no. 5814, p. 972, 2007.

[17] M. Liu, F. Colas, and R. Siegwart, “Regional topological segmentationbased on mutual information graphs,” in Proc. of the IEEE InternationalConference on Robotics and Automation (ICRA), 2011.

[18] J. Pillow, Y. Ahmadian, and L. Paninski, “Model-based decoding,information estimation, and change-point detection techniques for multi-neuron spike trains,” Neural Computation, vol. 23, no. 1, pp. 1–45, 2011.

[19] A. Ranganathan, “PLISS: labeling places using online changepointdetection,” Autonomous Robots, no. 32, pp. 351–368, 2012.

[20] N. Kaempchen and K. Dietmayer, “Data synchronization strategiesfor multi-sensor fusion,” in Proceedings of the IEEE Conference onIntelligent Transportation Systems. Citeseer, 2003.

[21] B. Douillard, D. Fox, and F. Ramos, “A spatio-temporal probabilisticmodel for multi-sensor multi-class object recognition,” Robotics Re-search, pp. 123–134, 2011.

[22] Y. Teh, M. Jordan, M. Beal, and D. Blei, “Hierarchical dirichletprocesses,” Journal of the American Statistical Association, vol. 101,no. 476, pp. 1566–1581, 2006.

[23] D. Blei, A. Ng, and M. Jordan, “Latent dirichlet allocation,” The Journalof Machine Learning Research, vol. 3, pp. 993–1022, 2003.

[24] L. Orvath and P. Kokoszka, “Change-point detection with non-parametric regression,” Statistics: A Journal of Theoretical and AppliedStatistics, vol. 36, no. 1, pp. 9–31, 2002.

[25] B. Ray and R. Tsay, “Bayesian methods for change-point detectionin long-range dependent processes,” Journal of Time Series Analysis,vol. 23, no. 6, pp. 687–705, 2002.

[26] J. Sethuraman, “A constructive definition of dirichlet priors,” StatisticaSinica, vol. 4, pp. 639–650, 1994.

[27] M. Liu and R. Siegwart, “DP-FACT: Towards topological mapping andscene recognition with color for omnidirectional camera,” in Proc. ofthe IEEE International Conference on Robotics and Automation (ICRA),2012.

[28] V. Moskvina and A. Zhigljavsky, “An algorithm based on singular spec-trum analysis for change-point detection,” Communications in Statistics-Simulation and Computation, vol. 32, no. 2, pp. 319–352, 2003.