Poisson models for extended target and group tracking

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1 Poisson models for extended target and group tracking Kevin Gilholm a, Simon Godsill b, Simon Maskell c and David Salmond a a QinetiQ, Cody Technology Park, Farnborough, Hants GU1 0LX, UK; b Engineering Department, University of Cambridge, Trumpington Street, Cambridge, UK; c QinetiQ, St Andrews Road, Malvern, Worcs, WR14 3PS, UK ABSTRACT It is common practice to represent a target group (or an extended target) as set of point sources and attempt to formulate a tracking filter by constructing possible assignments between measurements and the sources. We suggest an alternative approach that produces a measurement model (likelihood) in terms of the spatial density of measurements over the sensor observation region. In particular, the measurements are modelled as a Poisson process with a spatially dependent rate parameter. This representation allows us to model extended targets as an intensity distribution rather than a set of points and, for a target formation, it gives the option of modelling part of the group as a spatial distribution of target density. Furthermore, as a direct consequence of the Poisson model, the measurement likelihood may be evaluated without constructing explicit association hypotheses. This considerably simplifies the filter and gives a substantial computational saving in a particle filter implementation. The Poisson target-measurement model will be described and its relationship to other filters will be discussed. Illustrative simulation examples will be presented. Keywords: Target tracking, Extended targets, Poisson models, Particle filters 1. INTRODUCTION In the classical target tracking problem, it is assumed that at most a single measurement is received from a point target at each time step. However, in many cases, high resolution sensors may be able to resolve individual features or measurement sources on an extended object. A natural way to approach this problem is to model the target as a rigid (or semi-rigid) set of point sources, each of which may be the origin of a sensor measurement. The tracking problem is then to infer the structure of the target (i.e. the geometry of the measurement sources) given a sequence of measurement frames and prior information on structure and dynamics. This is the type of approach adopted by Brodia 1, Huang 2 and Dezert. 3 A similar model may be employed for a group of point targets. 4 If the problem is tackled from a Bayesian standpoint, a key aspect of the solution is the construction of feasible measurement-source assignment hypotheses and the evaluation of the probabilities of these hypotheses 4,5. Clearly such an explicit approach is most challenging if the number of sources and their disposition is highly uncertain. In this paper we propose a different approach that leads to a representation of the measurements over the sensor observation region as a spatial point process. The scenario is modelled as a known number of of measurement generating entities. Each of these entities might be a target (point or extended), a cluster of objects or a clutter process. Each entity is characterized by a state vector which determines the measurement generation process for that entity in terms of number of measurements issuing from that entity and their spatial distribution. In particular, it is convenient to model this as a non-homogeneous Poisson point process. 6 (An obvious consequence of this model is that multiple measurements may originate from an entity.) It will be shown that this leads to an exact expression for the overall measurement likelihood (for the full scenario) that does not involve explicit assignments between the measurements and the entities. This is especially relevant for particle Send correspondence to djsalmond@qinetiq.com SPIE CONFERENCE 5913: SIGNAL AND DATA PROCESSING OF SMALL TARGETS 2005 (August), Edited by O E Drummond

2 filter implementations as the most expensive computational element of the filter is almost always the likelihood evaluation: the possibility of avoiding explicit summation over assignment hypotheses is most attractive. Conversely, the proposed model can only be exploited using a particle filter (or possibly a numerical grid based scheme). Furthermore, it is argued that in many cases the spatial Poisson model is a good representation of real target-sensor processes Relationship to other filters A direct consequence of the Poisson assumption is that the multiple measurements may originate from the same target. This is an underlying assumption of the Probabilistic Multiple Hypothesis Tracker (PMHT) of Streit and Luginbuhl, 7, 8 where the measurement-target association variables are assumed to be statistically independent. The PMHT is usually implemented as a batch filter, but Hue, Le Cadre and Perez 9 have used the PMHT assumption to develop a particle filter for multiple target tracking. The likelihood model they employ is essentially the same as the likelihood derived in this paper from the spatial Poisson assumption. This is discussed further in Section 2.5. Point process models are also employed in the Finite Set Statistics (FISST) approach developed by Mahler. 10 In FISST, point process ideas are used to describe the general multiple target transition and observation densities. The Probability Hypothesis Density (PHD) filter is an implementation of FISST where the multitarget filtering distribution is approximated as a Poisson point process. For a fixed discrete state space, the PHD filter is an exact multi-target algorithm, however, it is usually applied in continuous spaces where an adaptive discretisation is performed via a particle filter. 11 The sufficient statistic that is passed from one iteration to the next is the product of the Poisson rate for a given discrete point and the probability of this point. Therefore, the PHD filter is unable to distinguish between different pairs of Poisson rate and probability that have the same product; this ambiguity regarding target number and distribution of each target is exactly what the PHD filter exploits (and stems from the fact that the sum of two Poisson distributed variables is itself Poisson distributed). The PHD does not propagate the target labelling Structure of the paper In the following section we present a statement of the problem and derive the joint likelihood for the number and values of measurements in the observation region under the spatial Poisson model. An extended target likelihood model is also described and particle filter implementation is discussed. Simulation examples illustrating the operation of the scheme are presented in Section Problem statement 2. THEORETICAL DEVELOPMENT Scenario model The parameters to be estimated are contained in the state vector X T k = (x T 0 k,x T 1 k,...,x T N T k,b T k ), where k denotes time step and component vector x i k corresponds to entity i. It is assumed that the number of entities N T + 1 is known, and we reserve i = 0 for the background clutter process. As discussed above, the entities could be point targets, extended targets, diffuse clusters of objects or concentrations of clutter. In each case, the component state vector will usually include the standard descriptors of centroid position and velocity, with the possible addition of parameters that specify the spatial extent or other characteristics. The component B k is a bulk term which may be common to some or all entities. For example, it might described the common 4, 12 motion of a group of objects to facilitate tracking of target formations. Dynamics The state vectors of the entities are assumed to evolve according to known Markov models: p(x i k x i k 1 ) for i = 0,1,...,N T and p(b k B k 1 ).

3 Measurements At each time step k a set or frame of sensor measurements Z k = {z 1,...,z n } k becomes available. Each of these n measurements originates from one of the N T + 1 entities, but the assignment is unknown - i.e. the measurements are not labelled. More specifically, we assume that the received measurements over the observation volume result from the (conditionally independent) superposition of measurements from the N T + 1 entities (so no measurement can originate from more than one entity). For each entity i, we model the production of measurements over the observation space at a particular time step k as an inhomogeneous (or nonhomogeneous) Poisson point process. (A spatial point process is a stochastic model describing the location of events in some subset of R d.) So, the probability of n measurements from entity i falling in a region A of the observation space is given by 6 Pr{N(A) = n} = exp( µ i(a))(µ i (A)) n (1) n! where µ i (A) = λ i (z x i,b)dz A is the expected number of measurements falling in A and λ i (z x i,b) is the spatially dependent rate parameter or intensity of the process (where the time subscript k has been omitted). So each measurement z from entity i produced in region A has a pdf proportional to the intensity parameter λ i (z x i,b): p(z x i,b) = λ i(z x i,b) µ i (A), for z A, (2) i.e. given x i and B, any measurement z originating from entity i is an independent random draw from this pdf - we identify A with the observation region (sensor field-of-view). This concept will be clarified in Section 2.4 and in the examples of Section 3. In this paper we assume that µ i (A) is known, but if this is not so, the state vector x i can be augmented to include it. (Note that to some extent, this Poisson mean has a similar role to detection probability in a more standard approach.) Initial conditions To complete the problem description, we assume that the prior distribution p(x 1 ) of the state vector is available at time step k = 1. Requirement Given a sequence of measurements Z k = {Z 1,...,Z k } and prior distribution p(x 1 ), the problem is to determine the posterior distribution p(x k Z k ) Formal Bayesian filter On the basis of the above model, a formal Bayesian recursive filter may be derived (see for example 12 ). Suppose that the posterior pdf p(x k 1 Z k 1 ) at time step k 1 is available. It is required to construct the posterior pdf p(x k Z k ) at the following time step k. In the usual fashion, the prior at time k may be obtained from the dynamics model via the Chapman-Kolomogorov equation: p(x k Z k 1) = p(x k X k 1 )p(x k 1 Z k 1) dx k 1. (3) This prior pdf may be updated with information from the latest set of measurements Z k via Bayes rule: p(x k Z k) p(z k X k )p(x k Z k 1). (4) These two relations (3) and (4) are the general form of the formal Bayesian recursive filter. To complete the definition, the likelihood p(z k X k ) must be specified (the dynamics p(x k X k 1 ) is assumed to be available from the problem statement).

4 2.3. Measurement likelihood for a nonhomogeneous Poisson process As already noted, in this development, we assume that the measurements received over the observation volume result from the superposition of measurements from N T + 1 conditionally independent inhomogeneous Poisson point processes. Since the superposition of independent Poisson processes with parameters λ 0,λ 1,...λ NT is a Poisson process with parameter N T i=0 λ i, the totality of measurements received over the observation region is also an inhomogeneous Poisson process where the spatially dependent rate parameter or intensity of the process given by N T λ(z X) = λ i (z x i,b). i=0 Each term in this summation is the intensity parameter corresponding to entity i and which depends on its state vector x i and the bulk term B. In this case, the joint likelihood of the number n of measurements falling in the observation volume A and their values (z 1,z 2,...,z n ) is given by 6 p(z X) = p((z 1,z 2,...,z n ),n X) = e µ(a) n! n j=1 λ(z j X) = e µ(a) n! n N T λ i (z j x i,b) (5) j=1 i=0 where the product should be interpreted as unity if n = 0, and µ(a) = N T i=0 µ i(a) is the expected total number of measurements received in this frame. Using relation (2), the likelihood (5) may be written p(z X) = e µ(a) n! n N T µ i (A)p(z j x i,b). (6) j=1 i=0 Equation (6) is the measurement likelihood for the spatial Poisson model. It should be emphasized that this is an exact result - it is a direct consequence of the spatial Poisson model assumptions. Also note that it is identical to the result given in 13 (following equation (37)) that was obtained by directly evaluating the sum over all possible assignments. If entity i = 0 is identified with clutter uniformly distributed over the sensor s field-of-view, then p(z x 0 ) = 1/ A dz (i.e. a homogeneous Poisson process) and µ 0(A) = ρ dz, so (6) becomes A { } n p(z X) = e µ(a) N T ρ + µ i (A)p(z j x i,b). (7) n! j=1 This is proportional to the likelihood given in equation (13) of 14 for the special case of N T = 1 - i.e. single (possibly extended) target in uniform clutter. i= Extended target likelihood model In this section we consider the measurement model for a target that may have spatial extent. The pdf p(z x i,b) of measurements originating from a target (say, entity i) depends on the target model (parameterized by (x i,b)) and the measurement error process. As suggested in 14 (from which this section is summarized), it may be convenient to represent these as two separate processes as a route to the required likelihood. The target model describes how measurement sources are distributed over the target: so the pdf of a source y given target parameters (x i,b) may be written p(y x i,b). This may be viewed as a model of the spatial extent of the target. The measurement z arising from the source y is described by the sensor error model p(z y) - i.e. the measurement values depend only on y. p(z y) is typically a Gaussian perturbation about a function of y. Therefore, the pdf of a measurement z originating from the target is given by the convolution p(z x i,b) = p(z y)p(y x i,b)dy. (8)

5 Note that for a classical point target, where the origin of measurements is a deterministic function of the state vector, p(y x i,b) reduces to a delta functional on the relevant components of the state vector. There are many possible extended target models, ranging from the classical point target to complex representations of particular target structures. The range includes models corresponding to sets of point sources (as studied in 1 to 5 ) and more diffuse representations. 14 This spatial model could be a bounded distribution such as a uniform pdf or an unbounded distribution such as a Gaussian. In many cases, the principal axis of the target extent will be (approximately) aligned with the target s velocity vector. For example, a 1-D uniform stick model may be an adequate representation for certain targets (see Section 3). It is also possible to model source concentrations via mixture models of the form: N C p(y x i,b) = w j (x i,b)p j (y x i,b), (9) j=1 where the w j (x i,b) are weights that sum to unity, and the component pdfs and the weights may depend on the target state vector (x i,b). For this mixture model, the probability of a target source originating from component j is w j (x i,b), and since each measurement is independent, multiple sources may arise from particular components Implementation of the filter Particle filtering 12,15 is a means of directly mechanising the formal Bayesian recursive filter (equations (3) and (4)). Essentially, the required posterior is represented and propagated as a (typically large) set of random samples or particles. This approach does not rely on the linear-gaussian assumptions that underpin Kalman filtering schemes: it may be applied directly given an expression for the measurement likelihood. In particular, a particle filter can exploit the likelihood model (6). As already noted, the likelihood (6) is especially convenient for a multiple object particle filter as it avoids the computational effort of explicitly enumerating assignment hypotheses. This fact has been previously exploited by Hue, Le Cadre and Perez 9 who used effectively the same likelihood, but started from the PMHT assumption rather than invoking a Poisson model. In their approach, the mixture coefficients in the likelihood are interpreted as association probabilities - which are proportional to the Poisson means µ i in our formulation. In the spirit of PMHT, Hue et al treat these association probabilities as unknown stochastic coefficients to be estimated (they employ a Gibbs sampling scheme). In our approach the µ i are either assumed to be known or are added to the state vector and included in the dynamics model. In all but the most trivial cases, it is not possible, or is at best very inconvenient, to derive a Kalman filter implementation from the likelihood (6). Even when such a filter can be derived it requires explicit construction of assignment hypotheses (see 14 ), so loosing one of the benefits of the approach. 3. ILLUSTRATIVE SIMULATION EXAMPLES The following two simulation examples are presented to illustrate the operation of the filter and to show how it can be implemented. The results are not intended to provide a performance assessment, although they are typical of a wide range of randomly generated data sets that have been studied Example 1: passing extended targets This first example is an extension of a single target case presented in 14 to two targets, moving in a common plane, which pass close to one another.

6 Measurement likelihood Each target is a stick or rod of known length: target 1 is l = 2 units long and target 2 is l = 1 unit long. Each stick is aligned with its velocity vector and measurement sources are randomly and uniform distributed along the lengths (frame-to-frame independent). A source produces a Cartesian measurement z = (z x,z y ) which is a zero-mean Gaussian perturbation about the source with a variance of σ 2 = 0.01 in the x and y directions (independent). The resulting pdf for a target-related measurement is given by (8), the convolution of the 1-D uniform pdf with the Gaussian error. If the centre of a target is located at the origin and aligned with the x-axis (i.e. x = y = ẏ = 0 and ẋ > 0), the pdf of a target-related measurement z = (z x,z y ) is given by 14 : p T (z x,l) = p(z y)p(y x, l)dy = [ 1 2l 2πσ exp 1 2 ( zy ) ] { ( 2 zx erf l ) ( zx 1 2 erf l )} σ 2σ 2σ The required general form for an inclined target away from the origin can be obtained by rotation and translation of (10). The numbers of measurements originating from the targets are Poisson distributed with known means: µ 1 = 3 for target 1 and µ 2 = 1.5 for target 2. Uniform random clutter measurements of density ρ = 0.2 per unit area are superposed on the target-related measurements - as will be seen this is a substantial clutter density. Thus, { the measurement likelihood for this problem is given from (7) by: p(z X) n j=1 i=1 (µ i/ρ)p T (z j x i,l i ) where x 1 and x 2 correspond to the two targets - there is no bulk term B as they are assumed to be independent. (10) } Dynamics model For each target, the motion of its centre is represented by a discrete second order (constant velocity) model with unit time interval between measurement frames: x( k+1 = Φx ) k + Γw k where ( x ) = 1 1 1/2 (x,ẋ,y,ẏ) T, w = (w X,w Y ) T, Φ = diag{φ 1,Φ 1 }, Γ = diag{γ 1,Γ 1 }, Φ 1 = and Γ =. 1 The acceleration process w k is modelled as zero mean, white Gaussian noise with covariance E[w k wk T] = qi 2 where I 2 is the 2 2 identity matrix. For this simulation example, the variance of the acceleration noise is q = 0.01, the same for each target. (So the targets cannot be distinguished via prior information on dynamics.) Results Target 1 follows a constant speed (about 0.42 units per time step) path, initially moving in a straight line and then turning at a constant rate of about 8.5 degrees per time step. The second target moves in a straight line with the same constant speed and passes target 1 coincidentally and tangentially during the turn - thus there is substantial scope for track switching. The actual target paths and the filter estimates of the target centroid (mean of the particle set) are shown in fig 2. Note that the filter is able to exploit the differences between measurement characteristics of the targets (i.e. the differences between l 1 and l 2 and between µ 1 and µ 2 ) to maintain the correct tracks after the targets diverge. These results were obtained from a filter using 2000 particles. Fig 1 shows the number of measurements being processed by the filter after gating (see 14 ): on average the filter processes about 11.5 measurements per time step, of which on average 7 are clutter - so, at least subjectively, this is a stressing problem. Fig 3 shows clusters of stick positions for the two targets corresponding to 30 particles chosen randomly from the full set at every fifth time step. It can be seen that, especially for target 2, there is at times a substantial uncertainty in target location. This is also illustrated by fig 4 which shows the posterior pdfs of the target positions at time steps 18 and 20 (just after the targets begin to diverge). These posterior pdfs are obtained from the full set of posterior particles via a (Gaussian based) kernel density estimate and are shown as contour plots with a logarithmic contour spacing. Local target and clutter measurements are also plotted. Note that the posterior pdfs may be significantly multi-modal, although in all cases the actual target reference points are within the plotted distributions. The average processing time for one iteration of the filter with 2000 particles was just over a second using a 3 GHz Pentium 4 processor. However, this is for a non-optimized Matlab implementation and includes various ancillary monitoring and analysis operations Example 2: tracking a manoevring target using a variable rate particle filter In this second example we consider a single spatially distributed target manoeuvring in heavy clutter. Here we envisage a single group of spatially-distributed objects that try to evade detection by a random combination of sharp manoeuvres and periods of little activity. The dynamical models adopted for the manoeuvring target fit

7 20 Number of measurements processed by filter Total in gates From target 1 From target 2 From clutter 14 Number of measurements Figure 1. Numbers of measurements accepted by filter Target paths and estimates Target path and samples of paticles Y 4 2 Y Actual path of tgt 1 Estimate for tgt 1 Actual path of tgt 2 Estimate for tgt X Actual path for target 1 Particle filter estimate (mean) for target 1 Sample stick positions for target X Figure 2. Paths and track estimates for the two targets Figure 3. Samples of particles for the two targets - target 2 dashed (shown at every fifth time step)

8 Posterior pdf at time step 18 (logarithmic contour spacing) Posterior pdf at time step 20 (logarithmic contour spacing) Posterior pdf of target 1 Posterior pdf of target 2 Measurements from target 1 Measurements from target 2 Clutter measurements Centre of target 1 Centre of target Posterior pdf of target 1 Posterior pdf of target 2 Measurements from target 1 Measurements from target 2 Clutter measurements Centre of target 1 Centre of target Y Y X X Figure 4. Contours of posterior pdfs of target positions at time steps 18 and 20: logarithmic contour spacing Figure 5. Intrinsic coordinate system into the general variable rate particle filtering framework introduced in 16 and further developed with specific dynamical models in 17. These models introduce an additional time variable into the state-space, which models the random times at which a new manouevre commences. The variable rate state is defined as as x k = (θ k,τ k ), where k N is the discrete state index, τ k R + denotes the state arrival time, and θ k denotes the vector of state variables (essentially position and velocity in this case). We assume that the variable rate state sequence follows a Markovian process such that successive states are independently drawn as follows x k p(x k x k 1 ) = p θ (θ k θ k 1,τ k,τ k 1 )p τ (τ k τ k 1 ). (11) Notice that the times τ k need not be synchronised with the observation times, and indeed this is the essence of the variable rate model. In between state arrival times τ k, the target state is deterministically obtained from a continuous time differential equation model based on an intrinsic coordinate system, as in 17, from which we summarise the model here. In an intrinsic coordinate system applied forces can be represented relative to the heading of the object, rather than relative to the more standard Cartesian or polar fixed coordinate frame. This we postulate is a

9 more realistic representation of the thrusts applied when turning a vehicle. Distance travelled along the path of motion is denoted s, while angle of the path relative to east in the horizontal plane is denoted ψ. Accelerations tangential to and perpendicular to the motion are then given in fig 5. In the variable rate model we now assume that a piecewise constant thrust, relative to the direction of heading, is applied between any two times τ k and τ k+1, with tangential component T T and perpendicular component T P. A resistance term λ ds dt, assumed to apply in the opposite direction to the heading adds some damping to the system. Resolving forces tangentially and perpendicularly: T T = λ ds dt + s md2 dt 2 T P = m ds dt These equations are readily integrated to give equations for s(t) and ψ(t) during any time period with fixed thrusts T T and T P, from which the cartesian position x(t) can be obtained by numerical integration for any time t between τ k and τ k+1. We thus have the variable rate state variables as follows: dψ dt θ k = [T T,k, T P,k, v(τ k ), ψ(τ k ), x(τ k )] where v() is ds dt, from which the position, speed and angle at any time between τ k and τ k+1 may be obtained deterministically. To complete the dynamical model we specify the distribution of thrusts in the time interval τ k to τ k+1 : and the distribution of time points: T T,k N(µ T,σ 2 T), T P,k N(0,σ 2 P) τ k+1 τ k G(α τ,β τ ) where N is the normal and G is the gamma distribution. The model is initialised with random time points and thrusts drawn from the prior, position x(0) drawn from a circularly symmetric Gaussian with standard deviation 60 and mean [0,0], heading uniformly distributed in [0,2π], and speed uniformly distributed in [0,28]. The observation model is a Poisson model as detailed earlier. Specifically, at each observation time, the number of target-originating measurements is a Poisson random variable having mean µ 1 (A), and each having spatial distribution given by a normal distribution in each Cartesian co-ordinate: p(z x) = N(z x,σ 2 z) The number of clutter observations is also a Poisson random variable with mean µ 0 (A). The clutter may itself be a nonhomogenous Poisson process, and to give a specific case, we consider a Gaussian mixture clutter distribution having 4 equally weighted components, spatially located at [ ], [ ], [ ], [ ], plus a uniform background component over A, having weight 0.6. These represent clumps of persistent clutter that may correspond to certain fixed features in the land/seascape, which targets may be expected to traverse at various times during each trajectory. Here we have a surveillance region A of size , clutter mean µ 0 (A) = 80 and target mean µ 1 (A) = 0.5 (i.e. the target is very often unobserved), and with observation standard deviation σ z = this figure is considered to include both the distributed nature of the target and sensor noise, hence for this example we do not need a bulk term B. A manoeuvring target is generated using the intrinsic dynamic model with m = 500, λ = 0.3, µ T = 50, σ T = 50 and σ P = 1000, initialised with a random velocity and position [00]. The data are plotted in figure 6. Note that the target is entirely buried in clutter to the human eye. Note also the clutter hotspots where tracking is expected to be particularly difficult. The target passes through one of these regions while carrying out its looping trajectory. Despite the evident difficulty of the scenario, the variable rate filter is able to track its manoeuvres with good success, see figure 7. In this figure the output of the filter after 200 time steps is displayed. 400 particles were used, and we display the final smoothed trajectories at the end of the filtering run.

10 Figure 6. Data set for variable rate filtering example. Top left: raw data, 200 time scans; top right: true manoeuvring trajectory; bottom left: raw data, x coordinate as a function of time; bottom right: raw data, y coordinate as a function of time

11 Figure 7. Particle filter: smoothed trajectories Particles were randomly initialised from the initial prior distribution and the (variable rate) dynamical model was adopted as importance function. An important issue for future consideration in this type of application is how one might display these results in a user-friendly way. Note that with a distributed target under these detection and clutter conditions we cannot expect a very accurate tracking of the precise position - rather it is remarkable that the filters are sensitive enough to make any useful tracking effort and not to lose track with this type of data. We have found this performance to be consistent, and to outperform similar dynamical models expressed in a standard (fixed rate) setting. As would be expected, performance gets significantly better in less clutter and with higher target mean values, while we have obtained meaningful results with target means as low as and clutter means of CONCLUSIONS The proposed Poisson spatial model is an efficient representation for handling measurement association uncertainty with a particle filter. In particular, the model leads to an (exact) measurement likelihood which avoids the need to construct explicit measurement-target assignment hypotheses. Promising simulation results have been obtained and we believe that there is considerable scope for exploitation in other applications. The approach is well matched to the tracking of extended targets, closely spaced groups with objects passing in and out of resolution cells and modelling of lumpy clutter. ACKNOWLEDGMENTS This work was carried out as part of the Weapons & Platform Effectors Domain and the DIF-DTC of the UK MOD Research Programme. REFERENCES 1. T. Broida and R. Chellappa, Estimating the kinematics and structure of a rigid object from a sequence of monocular images, IEEE Transactions on Pattern Analysis and Machine Intelligence 13(6), pp , 1991.

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