An Integrated Probabilistic Model for Scan-Matching, Moving Object ...

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The detection algorithms address the problem in terms ofseparating the data into static and dynamic clusters. Dynamicpoints are used for tracking of dynamic objects while thestatic points are used to obtain better motion estimates forthe moving platform.Hähnel et al. [9] detect moving points in range datausing an Expectation-Maximisation (EM) based approach.The algorithm maximises the likelihood of the data using ahidden variable expressing the nature of the points (static ordynamic).In [26], two separate maps are maintained; one for thestatic part of the environment and one for the dynamic. Themaps employ a modified occupancy grid framework whichalso infers if points are static or dynamic.Rodriguez-Losada and Minguez [20] improve data associationfor the ICP algorithm. They introduce a new metricwhich models dynamic objects to better reflect the realmotion of the robot. Their approach however, does notdistinguish moving objects from outliers.The second category of algorithms focuses on objectsegmentation and tracking. Anguelov et al. [1], [4] detectmoving points using simple differencing and then apply amodified EM algorithm for clustering the different objects.In [25] an integrated solution to the mapping and trackingproblem is defined: static points are used for mappingand dynamic points for tracking. Data differencing andconsistency-based motion detection [24] is used for thedetection and segmentation of dynamic points. Points areclassified as static or dynamic and clustered into segments;when a segment contains enough dynamic points is considereddynamic. Montesano et al. [16] subsequently improvethe classification procedure of [25] by jointly solving it in asequential manner.Schulz et al. [22] use a feature based approach to detectmoving objects; the features used are the local minima ofthe laser data. The objects are then tracked using a jointprobabilistic data association filter (JPDAF).The focus for most of these approaches is on trackingdifferent objects under different hypothesis. The detectionpart, however, is mainly based on heuristics such as scan differencing.The detection routines observe the actual positionof the object and the velocities are computed by the trackingalgorithm. CRF-Clustering [23] on the other hand, appliesa feature based graphical model to the data. This allows itto reason about the underlying motion of points; it detectsand tracks the objects in a single framework, iterativelyimproving the estimates.The work presented in [10] allows different, state-ofthe-artalgorithms, to be combined in a cascaded/sequentialframework. The approach treats each algorithm as a ”blackbox”though it allows limited unidirectional interactionthrough feature vectors. The model proposed in this paperon the other hand, integrates and extends the CRF-Clusteringand CRF-Matching approaches into a single framework.Motion estimates are computed in-line with motion detectionand data association, allowing for joint reasoning enhancedestimates.III. PRELIMINARIESA. Conditional Random FieldsConditional Random Fields [12] (CRFs) are undirectedgraphical models that represent probability distributions; thevertices of the graph index random variables while the edgesof the graph capture relationships between variables. In thecase of CRFs the distribution is conditional (a probabilisticdiscriminative model); the hidden variables x = 〈x 1 ,x 2 ,...,x n 〉are globally conditioned on the observations z; p(x | z).Let C be the set of all cliques of the graph. The distributionmust then factorise as a product of clique potentialsφ c (x c ,z), where c ∈ C and x c are the hidden variables ofthe clique. Clique potentials are commonly expressed bya log-linear combination of feature functions, φ c (x c ,z) =exp ( w T c · f c (x c ,z) ) , resulting in the definition of a CRF as:(p(x | z) = 1Z(z) exp ∑ w T c · f c (x c ,z)c∈C). (1)Here Z(z) = ∑ x exp(∑ c∈C w T c · f c (x c ,z)) is the partitionfunction; it normalises the exponential ensuring Equation 1is a proper distribution. C is again the set of all cliquesin the graph. w c are parameters (or weights). The featurefunctions f c extract feature vectors given the value of theclique variables x c and observations z.B. Parameter LearningThe weights w c express the relative importance of eachfeature function. As such they play an important role indetermining the shape of the distribution. The weights arelearnt by maximising the conditional likelihood (Equation1) given labelled training data. In our case, this is computationallyintractable as the partition function Z(z) sums overthe entire state space of all hidden variables. A typical graphin the experiments contains on average 600 hidden variableswith an approximate total number of states of 100000.We therefore use maximum pseudo-likelihood learning[3]. Maximum pseudo-likelihood learning approximates thejoint by considering, for each hidden variable, only itsneighbours in the graph: the Markov blanket. As a result,the partition function is approximated by summing over thestate space of individual hidden variables.C. InferenceThe model defined in the next section is a joint distributionover data association, motion detection and motion clusteringhidden variables. The desired outcome of inference is aconfiguration of the hidden variables for which the jointdistribution achieves its maximum; a maximum a posteriori(MAP) inference problem. The proposed algorithm is avariation on Max-Product Loopy Belief Propagation.Belief Propagation (BP) [18] is a class of inferencealgorithms in which each node sends messages to each ofits neighbours in the graph. The messages convey what anode believes its neighbours’ state should be given its ownstate. The received messages together with a node’s ownbelief are then used to compute the MAP configuration.
Construction of the messages is performed using the Max-Product algorithm [18] as follows:)m i j (x i ) = max(φ(x j ,z)φ(x i ,x j ,z)x j∏ m k j (x j ) , (2)k∈N ( j)\iwhere m i j (x i ) is the message node j sends to node i.φ(x j ,z) is the local clique potential for node j. Local cliquepotentials represent what a node believes its state is, giventhe observations. Computation of the local potential can bevisualised as passing a message from the observation to thenode, see m Z in the left hand side of Figure 2. φ(x i ,x j ,z) isthe pairwise clique potential. Pairwise potentials relate nodesi and j on either end of an edge. They transform the belief ofnode j into a form suitable for node i and vice versa. Lastlya product of all incoming messages is computed, except fromthe node to which the message is being sent. In the left handside of Figure 2, the incoming messages are represented bym a/c and m RT , while the outgoing messages m i j is visualisedas m out .BP generates exact solution for graphs such as treesor polytrees. The proposed model is neither a tree nor apolytree. We therefore use Loopy Belief Propagation (LBP)[17], an approximation to BP. In LBP, message propagationcontinuous until the difference in messages falls below somethreshold, or until a maximum number of iterations has beenreached. Once message propagation is finished, the maximalconfiguration can be found by applying Equation 3 to eachnode in turn,)xi ∗ ∈ argmax(φ(x i ,z)x i∏ m ji (x i ) . (3)j∈N (i)Here x i indicates the node while xi∗ is its maximal configuration.As can be seen, maximisation is performed ona node’s own belief (local potential) combined with allincoming messages. The maximal configuration for the nodeis then simply a state for which the combined belief ismaximal.IV. JOINT MODEL FOR SCAN MATCHING ANDCLUSTERINGA. Model DefinitionAs discussed in Section II, solutions for motion detection/estimationand scan matching are generally defined independently.In reality though, both scan matching and movingobject detection are strongly interdependent; laser pointsbelonging to the same object will have similar associationsand vice versa.Following [23], motion detection is formulated as a clusteringproblem. The environment consists of one or more dynamicobjects (one cluster for each dynamic object) togetherwith a cluster representing the background. Laser points areclustered based on the object they belong to (i.e. those withsimilar motion patterns). Additionally, here we also includean outlier cluster for those points which can not reasonablybe considered part of an object. Given the clusters for staticx c 1x c 2x c Nx RT x a Nx a 2zx a 1Fig. 1. Graphical model for combined scan matching, motion clusteringand motion estimation.and dynamic objects, motion estimation then determines therotation and translation of each object.A model (Equation 4), is proposed to solve moving objectdetection, motion estimation and data association simultaneously.p(x | z) = p(x a ,x c ,x RT | z). (4)Equation 4 expresses a conditional distribution of threesets of hidden variables (x a , x c , x RT ) given an observation z.The observation consists of two laser scans for which associationand motion detection/estimation is to be determined.The variables x a consists of N association nodes, where Nis the number of points in the first scan. Each of these nodesis discrete with M+1 states; M being the number of points inthe second scan. The states of x a at node i have the followinginterpretation: The first state indicates the likelihood thatpoint i in the first scan associates to point 1 in the secondscan. The second state is the likelihood of association to thesecond point in the second scan, etc. Finally, the M+1 staterepresents the likelihood that point i is an outlier.The variables x c consists of N motion detection nodes,with D discrete states each. Here D represents the numberof clusters a point can belong to. The interpretation isanalogous to that of association. The first state represents thelikelihood that point i belongs to the first object (cluster),etc. By convention, the last state D represents the clusterfor outliers. The choice on number of clusters D may beguided using a priori knowledge. If no such knowledge isavailable then D can be set to M+1, i.e. start with a clusterfor each point. Once the inference algorithm has clusteredthe points, any vacuous clusters can safely be discarded. Theinference algorithm will assign probability mass to each nodeaccording to the likelihood of belonging to a cluster. Clustersthat have no significant probability mass for all nodes aretherefore not likely, and can be removed. For computationalreasons it is advisable to reduce the number of clusters.Lastly, the variable x RT represents the rotational R andtranslational T parameters of each object in the systemexcept the outlier cluster. As such this is a multi-dimensionalcontinuous variable. Note that the robot motion can beobtained from the rotation and translation parameters of thebackground object.
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Page 5 and 6: C. Feature DescriptionCRFs owe much
Page 7 and 8: KMeans (ICP) ResultsKMeans (CRF−M

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