The Delta-Generalized Labeled Multi-Bernoulli Tracking Filter ... - arXiv

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The Delta-Generalized Labeled Multi-Bernoulli Tracking Filter with Target Spawning

arXiv:1705.01614v1 [stat.CO] 3 May 2017

Daniel S. Bryant, Ba-Tuong Vo, Ba-Ngu Vo, and Brandon A. Jones

Abstract—In its initial development, the δ-Generalized Labeled Multi-Bernoulli (δ-GLMB) filter was formulated with a target motion model that accounts for survival and birth. The filter is capable of addressing a wide variety of multi-target tracking challenges with birth as the only means of new target track instantiation, still it is not readily equipped to address scenarios in which the appearance of new targets is conditional on the behavior of preexisting ones. In this paper, we extend development of the δ-GLMB filter to incorporate target spawning. Derivation yields a predicted multi-target density in labeled random finite set form, then exploiting the versatility of the family of GLMB distributions, the density is approximated as δ-GLMB and processed in a joint prediction and update, facilitating efficient implementation while preserving its cardinality and probability hypothesis density (PHD). A key development is the inclusion of a spawn track’s ancestry in its label, which is jointly estimated along with its state, an outcome with applications across multiple fields. Results are verified through simulation. Index Terms—Random finite sets, generalized labeled multiBernoulli filter, target spawning, multi-target tracking, Bayesian estimation.

I. I NTRODUCTION Multi-target tracking is concerned with estimating the number of targets and their states in a given surveillance region with the presence of clutter and uncertainty regarding detections, measurements, and data associations. The field covers a wide variety of applications which include aviation [1], astrodynamics, [2]–[4], defense [5], robotics [6], [7], and cell biology [8]. Three of the most prominent approaches to multitarget tracking are Multiple Hypotheses Tracking (MHT) [9]– [12], Joint Probabilistic Data Association (JPDA) [13], and Random Finite Set (RFS) [14], [15]. The random finite set (RFS) approach provides a principled method for multi-target filter formulation within a Bayesian framework. Accordingly, central to this approach is the multitarget Bayes recursion [14] which propagates multi-target densities forward in time. For practical application, the multitarget Bayes recursion is typically considered intractable due its numerical complexity, hence its various approximations including: the probability hypothesis density (PHD) [16], Cardinalized PHD (CPHD) [17], [18], and multi-Bernoulli D. S. Bryant is with the Department of Aerospace Engineering Sciences, University of Colorado Boulder, Boulder, CO 80309 USA (e-mail: [email protected]). B.-T. Vo and B.-N. Vo are with the Department of Electrical and Computer Engineering, Curtin University, Bentley, WA 6102, Australia (e-mail: [email protected]; [email protected]). B. A. Jones is with the Department of Aerospace Engineering and Engineering Mechanics, The University of Texas Austin, Austin, TX, 78712 USA (e-mail: [email protected]).

[14], [19], [20] filters. Note, however, a fundamental aspect of these filters is their inability to distinguish targets. An analytic solution to the multi-target Bayes recursion capable of distinguishing targets has recently been presented in [21], [22]. Known as the δ-Generalized Labeled MultiBernoulli (δ-GLMB) filter, it is capable of estimating target states and their identities through the introduction of labeled RFSs. Key to its tractability is the truncation of filter terms in the interest of mitigating their exponential growth. Consisting of a two-stage prediction and update operation, each filter iteration requires truncation to be performed twice, which is prone to inefficiencies. Hence, the development of an efficient δ-GLMB implementation that employs a joint prediction and update, thereby eliminating such inefficiencies [23]. Additionally, as it is relevant to this work, it has been shown that δ-GLMB densities can be used to approximate a general labeled RFS density while preserving its cardinality distribution and PHD, as well as maintaining efficacy in challenging tracking scenarios [24]. In its original presentation [21], [22], the δ-GLMB filter was not formulated with target spawning, thus its motion model accounts only for the survival of tracks from one time step to the next and the spontaneous appearance of new tracks via target birth. A δ-GLMB implementation formulated with a motion model that additionally includes target spawning could potentially address more complex multi-target tracking challenges where a track’s origin is conditional on the behavior of a distinct preexisting target. Examples of such tracking challenges can be found in research pertaining to: Space Sitational Awareness (SSA), where estimating the source of debris fragments generated by satellite collision or explosion is desired [25]–[27]; and (biological) cell tracking, where construction of a cell’s lineage, or ancestry, is desired [8], [28]–[30]. In this paper, we propose a new δ-GLMB based filter that formally incorporates spawning, in addition to birth, for track instantiation of newly appearing targets. Each filter iteration involves multi-target prediction and update densities. Using a principled formulation, we derive an arbitrary multitarget prediction density, then approximate it as a best fitting δ-GLMB while performing a joint prediction and update, in order to exploit the correlation between a new spawn and its parent. A key outcome of incorporating target spawning is the capability of track labels to capture a target’s ancestry. When a track is instantiated by a birth model, its label contains information pertaining to when a target is born and from which birth region [21]. Similarly, a track instantiated by a

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spawn model contains information pertaining to when and from which parent a new target is spawned. The remainder of this paper is arranged as follows: In Section II background is provided on the δ-GLMB filter, its use in approximation of general labeled RFS densities, and its efficient implementation. In Section III the derivation, approximation, and joint prediction and update of target spawning inclusive δ-GLMB densities is developed. Simulation results are presented in Section IV and in Section V concluding remarks are given. II. BACKGROUND This section provides background on δ-GLMB filter implementation and approximation pertinent to the formulation of our multi-target filtering problem.

This work is concerned with a multi-target filtering recursion that propagates a posterior distribution forward through time to form a prior distribution. Then, using Bayes’ rule, a new posterior distribution is formed from the normalized product of the prior distribution and a likelihood function derived from the current observation. Instead of a single state x ∈ X, we consider a more general case of a finite set of states X ⊂ X called the multi-target state, distributed according to the multi-target posterior π. A given multi-target state X evolves to the multi-target state X+ at the next time via the multi-target transition kernel f (·|·), which encapsulates all information pertaining to loss of objects via death, the movement of surviving objects via Markov transitions, and the appearance of new objects via superposition. Evolution of the multi-target state is formulated using the Chapman-Kolmogorov equation given by [14] Z π+ (X+ ) = f (X+ |X)π(X)δX, (1)

f (X)δX =

Z ∞ X 1 f ({x1 , . . . , xi })d(x1 , . . . , xi ) i! i=0

is the set integral of f : F(X) → R and F(X) denotes all of the finite subsets of X. Note that π is a multi-target probability density function (pdf). Detected multi-target state elements and false observations are represented by a finite set of points Z ⊂ Z, called the multi-target observation, which is modeled by the multi-target likelihood g(·|·). The multi-target likelihood encapsulates all information pertaining to missed detections, false observations or clutter, and the observation of objects via Markov shifts. Given the multi-target prior π of form (1), using Bayes’ rule the multi-target posterior becomes g(Z|X)π(X) . π(X|Z) = R g(Z|X)π(X)δX

An RFS is a finite-set-valued random variable [16]. Unlike a random vector consisting of a single point, an RFS consists of a random number of points; the points themselves are random and unordered. A labeled RFS is an RFS where each element is augmented with a unique label ` ∈ L = {αi : i ∈ N}, where N denotes the set of positive integers and the αi ’s are distinct. Consider the single-object state space X and let L : X × L → L be the projection L((x, `)) = `. The labels of realization X ⊂ X × L are then L(X) = {L(x) : x ∈ X}, where x = (x, `). With |X| denoting the cardinality of set X, the realization is said to have distinct labels if and only if |X| = |L(X)|, a concept that is compactly formulated by the distinct label indicator defined by [21], [22] ∆(X) = δ|X| (|L(X)|) .

A. Bayesian Multi-Object Filtering

where Z

B. Labeled Random Finite Sets

(2)

Implicitly, multi-target states and multi-target observations are modeled as RFSs which we characterize following the Finite Set Statistics (FISST) notion of integration and density [14], [15].

Throughout this paper we adhere to the convention that lower case letters represent single-object states, e.g., x, x, while upper case letters represent multi-target states, e.g., X, X. Bold symbols represent labeled states and their distributions/statistics, e.g., x, X, π, etc., to distinguish them from unlabeled ones. Blackboard letters represent spaces, e.g., X, Z, L, N. We denote a generalization of the Kroneker delta that takes arbitrary arguments such as integers, sets, vectors, etc., by ( 1, if X = Y, δY (X) , 0, otherwise, and the inclusion function, a generalization of the indicator function, by ( 1, if X ⊆ Y, 1Y (X) , 0, otherwise. We use the standard inner product notation hf, gi , R f (x)g(x)dx and denote the multi-target exponential as Q hX , h(x), where h is a real-valued function and x∈X ∅ h = 1 by convention. Additionally, where it is convenient we let the symbol + denote the time index at the next time and its absence denote the time index at the current time, e.g., the state xk at the current time and the state xk+1 at the next time can equivalently be denoted as x and x+ , respectively. C. The Delta-GLMB Filter A δ-GLMB RFS is a labeled RFS with state space X and (discrete) label space L distributed according to [21], [22] h iX X π(X) = ∆(X) w(I,ξ) δI (L(X)) p(ξ) (3) (I,ξ)∈F (L)×Ξ

where Ξ is a given discrete space of association histories, each Rp(ξ) (x, `) = p(I,ξ) (x, `) is a probability density on X (i.e., p(ξ) (x, `)dx = 1 with each x ∈ X denoting a singletarget state and each ` ∈ L denoting a distinct label), and each w(I,ξ) δI (L) = w(I,ξ) (L) is non-negative such that X X w(I,ξ) (L) = 1. (4) I∈F (L) ξ∈Ξ

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Each δ-GLMB density component (I, ξ) in (3) consists of a weight w(I,ξ) (L(X)) that depends solely on the labels of the multi-target state X and a product of single-object probability densities that depends on the labels and kinematics of X. Relevant to this work, a Labeled Multi-Bernoulli (LMB) RFS X defined on X × L is an RFS with parameter set {(r(ζ) , p(ζ) ) : ζ ∈ Ψ} distributed according to [21] Ψ

π(X) = ∆(X)1α(Ψ) (L(X)) [Φ(X, ·)]

(5)

where α : Ψ → L is a 1-to-1 mapping and X Φ(X, ζ) = δα(ζ) (`)r(ζ) p(ζ) (x) (x,`)∈X

+ 1 − 1L(X) (α(ζ))


 1 − r(ζ) .

(6)

Both δ-GLMB and LMB RFSs are members of the larger family of Generalized Labeled Multi-Bernoulli (GLMB) RFSs and are closed under the Chapman-Kolmogorov equation (1), whereas the δ-GLMB is closed under Bayes’ rule while the LMB is not [21], [22], [31]. 1) Prediction Model: Using the convention detailed in [21], a label ` = (k, i) is an ordered pair where the first term k denotes time of birth, the second term i ∈ N is a unique index distinguishing objects born at the same time, and ` belongs to the label space at the current time L. Birth labels at the next time belong to the space B+ , hence the label space at the next time is L+ = L ∪ B+ , where L ∩ B+ = ∅. Given a single-object state x ∈ X at the current time, an object either survives to the next time with probability pS (x, `) and moves to a new state (x+ , `+ ) with probability density fS (x+ |x, `)δ` (`+ ), or dies with probability qS (x, `) = 1 − pS (x, `). Assuming that, conditional on X, the transition of kinematic states x ∈ X are mutually independent, we model the set W of surviving objects at the next time as a conditional LMB RFS distributed according to [21] X

fS (W|X) = ∆(X)∆(W)1L(X) (L(W)) [Φ (W; ·)]

where X+ ∩(X×L) = {x+ ∈ X+ : L(x+ ) ∈ L} is the subset of X+ consisting of surviving objects. Given a multi-target posterior at the current time π distributed according to (3), the predicted multi-target density π + is determined via (1) and (9). The reader is referred to the original works [21], [22] for the predicted multi-target density expression and detailed expositions. Model : The multi-target observation Z =  2) Measurement z1 , . . . , z|Z| is the superposition of detected objects and clutter. Given a multi-target state X, each state (x, `) ∈ X is either detected with probability pD (x, `) and generates an observation z ∈ Z with likelihood g(z|x, `) or missed with probability qD (x, `) = 1−pD (x, `). The multi-target likelihood is given as [21], [22] X X g(Z|X) = πK (Z) 1Θ(L(X)) (θ) [ψZ (·|θ)] (10) θ∈Θ

where ψZ (x, `|θ) = δ0 (θ(`))qD (x, `) + (1 − δ0 (θ(`)))

pD (x, `)g(zθ(`) |x, `) , κ(zθ(`) )

πK (Z) = exphκ,1i [κ]Z is Poisson RFS distribution of clutter with intensity function κ(·), and each association map θ : L → {0 : |Z|} , {0, 1, . . . , |Z|} belongs to the space of positive 1-to-1 association mappings Θ. Note that θ is 1-to-1 when restricting its range to positive integers, i.e., θ(i) = θ(j) > 0 implies that i = j. Given a predicted multi-target density π + distributed according to (3) and multi-target observation Z, the multi-target posterior is determined via (2) and (10). For the multi-target posterior expression and further details of its derivation, refer to the original works [21], [22].

(7)

where

D. GLMB Approximation

Φ(W; x, `) =

X

δ` (`+ )pS (x, `)fS (x+ |x, `)

(x+ ,`+ )∈W

  + 1 − 1L(W) (`) qS (x, `). A new object with state (x+ , `+ ) appears at the next time with probability rB (`+ ) and probability density pB (x+ , `+ ), or does not with probability 1 − rB (`+ ). Modeling target birth as an LMB RFS, the set Y of new objects born at the next time is distributed according to [21], [31] Y

fB (Y) = ∆(Y)wB (L(Y)) [pB ] B+ −L

(8)

L

where wB (L) = 1B+ (L) [1 − rB ] [rB ] . The multi-target state at the next time is the superposition of birth and surviving objects, i.e., X+ = W ∪ Y, and since L ∩ B+ = ∅, labeled birth and surviving objects are mutually independent. Thus, following from FISST the multitarget transition kernel ultimately reduces to the product of birth and survival transition densities f (X+ |X) = fS (X+ ∩ (X × L)|X)fB (X+ − X × L),

Relevant to treatments in this work, consider the case when we have an arbitrary labeled multi-target density on F(X × L) of the form X π(X) = ∆(X) w(c) (L(X))p(c) (X) (11) c∈C

where C is a discrete index set, the weights w(c) (·) satisfy (4), and with n = |X|, Z p(c) ({(x1 , `1 ), . . . , (xn , `n )}) d(x1 , . . . , xn ) = 1. Using [24, Proposition 3], we can approximate the labelconditioned joint densities, of the arbitrary labeled multi-target density π in (11), by the product of its marginals to form an approximate δ-GLMB density of the form ˆ π(X) = ∆(X)

(9)

X (c,I)∈C×F (L)

h iX δI (L(X))w ˆ (c,I) pˆ(c,I) (12)

4

where w ˆ (c,I)

pˆ

(c,I)

=w

(x, `) =

(c)

(I),

(13)

(c) 1I (`)pI−{`} (x, `),

(14)

(c)

p{`1 ,...,`n } (x, `) = Z p(c) ({(x, `), (x1 , `1 ), . . . , (xn , `n )})d(x1 , . . . , xn ). (15) A salient feature of this approximation method is that both the cardinality distribution and PHD of π are preserved. Additionally, note that C can take the form of any discrete index set, including the set of indices for the Cartesian product of a collection of finites subsets of some label space and an association history space, i.e., letting C = F(L) × Ξ is possible. E. Fast Delta-GLMB Implementation As originally formulated, δ-GLMB filter implementation consists of prediction and update stages, each requiring independent truncations of δ-GLMB densities [21], [22]. Alternatively, a substantially more efficient implementation of the δ-GLMB filter [23], [31], hereafter referred to as the fast δ-GLMB implementation, employs a single joint prediction/update stage requiring only one truncation procedure. This work employs the fast δ-GLMB implementation, thus for convenience, we introduce pertinent expressions and conventions for δ-GLMB joint prediction/update and formulation of the δ-GLMB truncation problem originally presented in [23], [31]. We expand on this material in Section III-B to incorporate spawning. Given the δ-GLMB density (3) at the current time, the jointly predicted and updated δ-GLMB density at the next time is given by [23]1

Though (16) is not strictly δ-GLMB, it does take on δ-GLMB form when rewritten as a sum over I+ , ξ, θ+ with weights [23] X I ,ξ,θ I,ξ,I ,θ wZ++ + ∝ w(I,ξ) wZ+ + + . (22) I

Efficient implementation of (16)’s recursion is achieved by (I,ξ,I ,θ ) propagating only the components with significant wZ+ + + through time, i.e., for a given fixed component (I, ξ) from the δ-GLMB density at the current time and a fixed multi-target observation Z+ at the next time, the set of pairs (I+ , θ+ ) ∈ (I,ξ,I ,θ ) F(L)×Θ+ (I+ ) with significant wZ+ + + are retained while the rest are discarded or truncated. The truncation procedure is performed by enumerating Z+ = {z1:|Z+ | }, B+ = {`1:K }, and I = {`K+1:P }. For each pair (I+ , θ+ ) ∈ F(L)×Θ+ (I+ ), define a P dimensional vector γ = (γ1:P ) ∈ {−1 : |Z+ |}P by ( θ+ (`i ), if `i ∈ I+ , γi = (23) −1, otherwise. Noting that γ inherits the positive 1-to-1 property from θ+ and letting Γ denote the set of all positive 1-1 elements of {−1 : |Z+ |}P , we can recover I+ and θ+ such that θ+ : I+ → {0 : |Z+ |} from γ ∈ Γ by I+ = {`i ∈ B+ ∪ I : γi ≥ 0} ,

I,ξ,I+ ,θ+

where I ∈ F(L), ξ ∈ Ξ, I+ ∈ F(L+ ), θ+ ∈ Θ+ , and (I,ξ,I+ ,θ+ )

B −I

B ∩I

= [1 − rB ] + + [rB ] + + h iI−I+ h iI∩I+ (ξ) (ξ) × 1 − p¯S p¯S h iI (ξ,θ ) + × 1Θ+ (I+ ) (θ+ ) ψ¯Z+ + , D E (ξ) p¯S (`) = p(ξ) (·, `), pS (·, `) , D E (ξ,θ ) (ξ) ψ¯Z+ + (`+ ) = p¯+ (·, `+ ), ψZ+ (·, `+ |θ+ ) , wZ+

(17) (18) (19)

(ξ)

p¯+ (x+ , `+ ) = 1B+ (`+ )pB (x+ , `+ ) + 1L (`+ )

 pS (·, `+ )fS (x+ |·, `), p(ξ) (·, `+ ) , (20) × (ξ) p¯S (`+ ) (ξ,θ )

pZ+ + (x+ , `+ ) = 1 In

(ξ) p¯+ (x+ , `+ )ψZ+ (·, `+ |θ+ ) . (ξ,θ ) ψ¯Z+ + (`+ )

the interest of simplifying notation, note that

P (I,ξ)∈F (L)×Ξ

P I,ξ

a(I,ξ) when the definitions I ∈ F (L) and ξ ∈ Ξ are provided.

(21) a(I,ξ) =

(24)

Then, for each j ∈ {−1 : |Z+ |} define   1 − rB (`i ), 1 ≤ i ≤ K, j < 0,    r (` )ψ¯(ξ,j) (` ), 1 ≤ i ≤ K, j ≥ 0, B i i Z+ (25) ηi (j) = (ξ) 1 − p¯S (`i ), K + 1 ≤ i ≤ P, j < 0,    p¯(ξ) (` )ψ¯(ξ,j) (` ), K + 1 ≤ i ≤ P, j ≥ 0, i i S

π Z+ (X+ ) ∝ ∆(X+ ) h i X+ X (I,ξ,I ,θ ) (ξ,θ ) × w(I,ξ) wZ+ + + δI+ (L(X+ )) pZ+ + (16)

θ+ (`i ) = γi .

Z+

(ξ)

assuming that, for D all i ∈ {1 : EP }, p¯S (`i ) ∈ (0, 1) (ξ) (ξ) and p¯D (`i ) , p¯+ (·, `i ), pD (·, `i ) ∈ (0, 1). Ultimately, every Q set of positive 1-to-1 vectors γ that generates a signifP icant i=1 ηi (γi ), where γ is extracted from the component combination denoted by (I, ξ, I+ , θ+ ), specifies a significant (I,ξ,I ,θ ) wZ+ + + [23]. Established methods for obtaining a set of positive 1-to-1 vectors γ include [22], [23] • solving a ranked assignment problem using Murty’s algorithm [32], which finds the N best vectors in nonincreasing order, • and, a more efficient method using Markov Chain Monte Carlo (MCMC) to simulate an unordered set of significant positive 1-to-1 vectors via exploitation of the Gibbs sampler [33], [34]. III. D ELTA -GLMB F ILTER WITH S PAWNING This section presents the formulation of δ-GLMB filtering equations and their implementation when spawning objects are present. In Section III-A, we present a predicted labeled multitarget density which is then jointly updated and approximated forming a posterior δ-GLMB density. Section III-B discusses filter implementation.

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multi-target states

A. Delta-GLMB Prediction and Update with Spawning

X

fT (V|X) = ∆(X)∆(V)1T+ (L(V)) [Φ(V; ·)]

(26)

where, with `β ∈ L{`},T and (x+ , `+ ) ∈ V, " Y X ΦT (V; x, `) = δ`β (`+ )pT (`β )fT (x+ |x, `) `β

(x+ ,`+ )

#   + 1 − 1L(V) (`β ) qT (`β ) .

(1,2)

((1,2),(5,1)) (1,1)

0

1

4 ... time

3

2

Fig. 1: An example of label assignment for birth and spawn tracks. Two tracks are born at time 1 and are assigned labels (1, 1) and (1, 2). At time 5, a track is spawned from track (1, 2) and is assigned label ((1, 2), (5, 1)). Proposition 1. If the multi-target posterior is a δ-GLMB of the form (3), then the predicted labeled multi-target density formed by surviving, birth and spawning processes is given by X π(X+ ) = ∆(X+ ) w(I,ξ) (L(X+ ))p(I,ξ) (X+ ) (29) I,ξ

where, letting XS = X+ ∩ (X × L), XT = X+ ∩ (X × T+ ), and XB = X+ ∩ (X × B+ ) such that XS ∪ XT ∪ XB = X+ , w(I,ξ) (L(X+ )) = w(I,ξ) 1I (L(XS ))1LI,T (L(XT ))1B+ (L(XB )) × (1 − rB ) (I,ξ)

p

B+ −L(XB )

[rB ]L(XB ) ,

(30)

(X+ )

= [pB ]

XB

E YD ΦS (XS ; ·, `)ΦT (XT ; ·, `), p(ξ) (·, `) . (31) `∈I

Proof. Using the Chapman-Kolmogorov equation (1) and (28) with fS (XS |X) from (7) and fT (XT |X) from (26), we have π(X+ ) Z X = fB (XB ) ∆(XS )1L(X) (L(XS )) [ΦS (XS ; ·)] X

(27)

The multi-target state at the next time is now X+ = W ∪ V ∪ Y, i.e., the superposition of surviving, birth and spawn objects. Since the label spaces L, T+ , and B+ are mutually disjoint, it follows that the (labeled) RFSs W, V, and Y are also mutually disjoint. Hence, we can equivalently express X+ as a disjoint union, i.e., X+ = W ] V ] Y. It then follows from the FISST fundamental convolution theorem [14], [21] that the multi-target transition kernel is given by f (X+ |X) = fS (W|X)fT (V|X)fB (Y).

tracks state space

We adhere to the following labeling convention for spawned tracks. If a parent track with label ` at time k spawns Nβ children at time k + 1, then each spawn takes on the label `β = (`, k + 1, i), where i = 1, . . . , Nβ . As a result, spawn labels consist of an ancestral element, i.e., the parent’s label, and a non-ancestral element that distinguishes multiple spawned objects originating simultaneously from the same parent. Letting Lβ,+ = {(k + 1, i), i = 1, . . . , Nβ }, we denote the set of spawn track labels generated from a set I of parent labels as LI,T = I × Lβ,+ . For the purposes of exposition, it is assumed that all targets follow the same static spawning model, though it is trivial to relax this assumption. Hence, given the label space Lk for objects at the current time, let Tk+1 denote the label space for tracks spawned at the next time such that Tk+1 = Lk × ({k + 1} × N). Hereafter, in an effort to simplify notation, we revert to the convention letting the symbol + denote the “next time” index. E.g., the label space Lk at the current time becomes L, and the label space Lk+1 at the next time becomes L+ . Accordingly, we follow the same construction in [21] by letting T+ denote the label space for objects spawned at the next time, then L+ = L ∪ T+ ∪ B+ . Note that L, T+ , and B+ are mutually disjoint, i.e., L ∩ T+ = L ∩ B+ = T+ ∩ B+ = ∅. Hence, we can distinguish surviving, spawn, and birth objects from their labels. Fig. 1, modeled after [22, Fig.1] in the interest of consistency, illustrates label assignment to birth and spawn tracks. Given the current multi-target state X and with `β ∈ L{`},T for a given label `, each state (x, `) ∈ X either spawns an object with state (x+ , `+ ) with probability pT (`β ) and probability density fT (x+ |x, `)δ`β (`+ ), or it does not with probability qT (`β ) = 1 − pT (`β ). We model the set V of objects spawned at the next time as a conditional LMB RFS2 distributed according to

× ∆(XT )1LL(X),T (L(XT )) [ΦT (XT ; ·)] h iX X × ∆(X) w(I,ξ) δI (L(X)) p(ξ) δX = ∆(XS )∆(XT )fB (XB )

XZ

∆(X)w(I,ξ) δI (L(X))

I,ξ

× 1L(X) (L(XS ))1LL(X),T (L(XT )) h iX × ΦS (XS ; ·)ΦT (XT ; ·), p(ξ) δX X X = ∆(XS )∆(XT )fB (XB ) w(I,ξ) δI (J)

(28)

2 It is possible to derive a GLMB based spawning model, but an LMB is presented for compactness

(32)

I,ξ

(33)

I,ξ J∈F (L)

× 1J (L(XS ))1LJ,T (L(XT )) E YD × ΦS (XS ; ·, `)ΦT (XT ; ·, `), p(ξ) (·, `) `∈I

(34)

6

where the last line follows from [21, Lemma 3]. Using fB (XB ) from (8), ∆(X+ ) = ∆(XS )∆(XT )∆(XB ), and noting that the only non-zero inner summand occurs when I = J, we have (29).

C −1 =

R

π(X+ )g(Z+ |X+ )δX+ and λ = πK (Z+ ), we have

π(X+ |Z+ ) = Cπ(X+ )g(Z+ |X+ ), (37) X (I,ξ) w (L(X+ ))1Θ+ (L(X+ )) (θ+ ) = Cλ∆(X+ ) I,ξ,θ+

Implicit in (29) is that, even though the survival and spawn RFSs are mutually disjoint due to their labels (see (28)), they are both conditioned on the same multi-target state X, thereby correlating the probability distributions of a parent object and its spawned objects. Hence, the labeled multi-target density (29) is not δ-GLMB. The labeled multi-target density (29) can be approximated via (12) prior to performing a measurement update, which would essentially eliminate the correlations between parents and their spawn. However, we propose a more prudent approach whereby approximation is performed on the posterior multi-target density, thus preserving more information with potential for improving filter performance, albeit at the cost of increased complexity. Namely, using (29) and (10), we seek to approximate π(X+ |Z+ ) = R

π(X+ )g(Z+ |X+ ) . π(X+ )g(Z+ |X+ )δX+

(35)

Whether an increase in complexity in exchange for improved filter performance (and vice versa) is justifiable is subject to the conditions of a given scenario. Proposition 2. Given the predicted multi-target density (29) and multi-target likelihood (10), the joint updated δ-GLMB density which preserves the cardinality distribution and PHD of (35) is given by

 X+ (X+ ) ψZ+ (·|θ+ ) , (38) X (I,ξ) w (L(X+ ))1Θ+ (L(X+ )) (θ+ ) = Cλ∆(X+ ) ×p

I,ξ,θ+

 XB (I,ξ,θ+ ) × pB ψZ+ (·|θ+ ) pZ+ (XM ), (39) where, letting XM = XS ∪ XT = X+ − XB , E YD (I,ξ,θ ) ΦS (XS ; ·, `)ΦT (XT ; ·, `), p(ξ) (·, `) pZ+ + (XM ) = `∈I

 XM × ψZ+ (·|θ+ ) . We apply the δ-GLMB approximation (12) to (39), however, note that determining the product of the marginals of the XB single-object birth densities encapsulated in pB ψZ+ (·|θ+ ) is redundant. Only the single-object densities encapsulated in (I,ξ,θ ) pZ+ + (XM ) require marginalization. Hence, applying the δ-GLMB approximation from (12) gives ˆ + |Z+ ) π(X X ˆ = Cλ∆(X δI+ (L(X+ ))w(I,ξ) (L(X+ )) +) I,ξ,I+ ,θ+

 XBh (I,ξ,I+ ,θ+ ) iXM ×1Θ+ (L(X+ )) (θ+ ) pB ψZ+ (·|θ+ ) pˆZ+ (I,ξ,I+ ,θ+ )

where I+ ∈ F(L+ ), pˆZ+

X

(I,ξ,I+ ,θ+ )

δI+ (L(X+ ))w ˆZ+

I,ξ,I+ ,θ+

XB h (I,ξ,I+ ,θ+ ) iXM pB ψZ+ (·|θ+ ) pˆZ+ × h i I+ (I,ξ,I ,θ ) p¯Z+ + + (x+ , ·) 

(36)

(40)

is defined in (45), and

Cˆ −1 =λ

ˆ + |Z+ ) = ∆(X+ ) π(X

(I,ξ)

X Z

∆(X+ )δI+ (L(X+ ))

I,ξ,I+ ,θ+

× w(I,ξ) (L(X+ ))1Θ+ (L(X+ )) (θ+ )  XB h (I,ξ,I+ ,θ+ ) iXM × pB ψZ+ (·|θ+ ) δX+ pˆZ+ X X =λ δI+ (L)w(I,ξ) (L)1Θ+ (L) (θ+ )

(41)

I,ξ,I+ ,θ+ L⊆L+ (I,ξ,I+ ,θ+ )

p¯Z+

(x+ , `+ )

(42)

`+ ∈L

=λ Proof. With g(Z+ |X+ ) from (10), π(X+ ) from (29), letting

Y

×

where I ∈ F(L), ξ ∈ Ξ, I+ ∈ L+ , θ+ ∈ Θ+ , and [see (44)-(46) at bottom of the page].

X

h i I+ (I,ξ,I ,θ ) w(I,ξ) (I+ )1Θ+ (I+ ) (θ+ ) p¯Z+ + + (x+ , ·) (43)

I,ξ,I+ ,θ+

i I+ h (I,ξ,I ,θ ) w(I,ξ) (I+ )1Θ+ (I+ ) (θ+ ) p¯Z+ + + (x+ , ·) (I,ξ,I ,θ ) w ˆZ+ + + = h i I+ P (I,ξ,I ,θ ) w(I,ξ) (I+ )1Θ+ (I+ ) (θ+ ) p¯Z+ + + (x+ , ·) I,ξ,I+ ,θ+ Z (I,ξ,I+ ,θ+ ) (I,ξ,θ ) pˆZ+ (x+ , `+ ) = 1I+ (`+ ) pZ+ + ({(x+ , `+ ), (x1,+ , `1,+ ), . . . , (xn,+ , `n,+ )}) d(x1,+ , . . . , xn,+ ), D E

 (I,ξ,I ,θ ) (I,ξ,I ,θ ) p¯Z+ + + (x+ , `+ ) = 1B+ (`+ ) pB (·, `+ ), ψZ+ (·, `+ |θ+ ) + (1 − 1B+ (`+ )) pˆZ+ + + (·, `+ ), 1 .

(44)

(45) (46)

7

P

∆(X+ ) ˆ + |Z+ ) = π(X

I,ξ,I+ ,θ+

 XB h (I,ξ,I+ ,θ+ ) iXM δI+ (L(X+ ))w(I,ξ) (I+ )1Θ+ (I+ ) (θ+ ) pB ψZ+ (·|θ+ ) pˆZ+ P I,ξ,I+ ,θ+

(47)

h iI+ (I,ξ,I ,θ ) w(I,ξ) (I+ )1Θ+ (I+ ) (θ+ ) p¯Z+ + + (x+ , ·)

where (42) follows from [21, Lemma 3], which simplifies in (43) since the only non-zero inner summand occurs when L = (I,ξ,I ,θ ) I+ , and p¯Z+ + + (x+ , `+ ) is defined in (46). Substituting (43) into (40) we have (47) [see top of page] which simplifies to (36) via (44).

recover I+ and θ+ such that θ+ : I+ → {0 : |Z+ |} from γ ∈ Γ by

I+ = {`i ∈ B+ ∪ I ∪ LI,T : γi ≥ 0},

θ+ (`i ) = γi .

(54)

B. Joint Prediction and Update Implementation In this section, we expand on the implementation material discussed in Section II-E. The inclusion of spawn modeling naturally increases the complexity of the δ-GLMB filter, thus making use of the fast δ-GLMB implementation an obvious choice. Marginalization is another source of complexity and its use in the truncation procedure is potentially inefficient, especially in cases where many δ-GLMB components are ultimately discarded. In an effort to offset complexity and minimize inefficiency, we invoke a proposal density for the purpose of generating candidate components (I+ , θ+ ). Definition 1. Given a δ-GLMB density (3) at the current time, ˜ at the next time be of form (16) let the proposal density π where h iI∩I+ h iT ∩I (I,ξ,I ,θ ) (ξ) (ξ) + + B ∩I p¯T w ˜Z+ + + = [rB ] + + p¯S h iI−I+ h iT −I (ξ) (ξ) + + B −I × [1 − rB ] + + 1 − p¯S 1 − p¯T iI h (ξ,θ ) + , (48) × 1Θ+ (I+ ) (θ+ ) ψ˜Z+ + D E (ξ) p¯T (`+ ) = p(ξ) (·, `), pT (`+ ) , (49) D E (ξ,θ ) (ξ) ψ˜Z+ + (`+ ) = p˜+ (·, `+ ), ψZ+ (·, `+ |θ+ ) , (50)

Then, for each j ∈ {−1 : |Z+ |} define

ηi (j) =

  1 − rB (`i ),   (ξ,j)  rB (`i )ψ˜Z (`i ),  +   (ξ) 1 − p¯ (` ),

1 ≤ i ≤ K, j < 0, 1 ≤ i ≤ K, j ≥ 0,

K + 1 ≤ i ≤ L, j < 0, i S (55) (ξ) (ξ,j) ˜  p ¯ (` ) ψ (` ), K + 1 ≤ i ≤ L, j ≥ 0, i i  S Z +   (ξ)   1 − p¯T (`i ), L + 1 ≤ i ≤ P, j < 0,    (ξ) p¯ (` )ψ˜(ξ,j) (` ), L + 1 ≤ i ≤ P, j ≥ 0, i i T Z+

(ξ)

(ξ)

assuming that, for all i ∈D{1 : P }, p¯S (`i ) ∈ E(0, 1), p¯T (`i ) ∈ (ξ) (ξ) (0, 1), and p¯D (`i ) , p˜+ (·, `i ), pD (·, `i ) ∈ (0, 1). Sets of positive 1-to-1 vectors γ that correspond to significant (I,ξ,I ,θ ) w ˜Z+ + + are found using (55) with the Gibbs sampling technique presented in [23]. In turn, the jointly predicted and updated δ-GLMB density is computed by using the sets of γ vectors and their associated (I, ξ, I+ , θ+ ) with (36).

(ξ)

p˜+ (x+ , `+ ) = 1B+ (`+ )pB (x+ , `+ ) (ξ)

(ξ)

+ 1L (`+ )˜ pS + 1T+ (`+ )˜ pT , 

pS (·, `+ )fS (x+ |·, `), p(ξ) (·, `+ ) (ξ) p˜S = (ξ) p¯S (`+ )

 pT (`+ )fT (x+ |·, `), p(ξ) (·, `) (ξ) p˜T = (ξ) p¯T (`+ )

(51) (52) IV. S IMULATION (53)

(ξ)

and p¯S is given in (18). During the truncation procedure, we use the proposal den˜ to find sets of positive 1-to-1 vectors γ that corresity π (I,ξ,I ,θ ) spond to significant w ˜Z+ + + . Expanding on the procedure detailed in Section II-E, this is done by enumerating Z+ = {z1:|Z+ | }, B+ = {`1:K }, I = {`K+1:L }, along with the additional set of spawn labels at the next time LI,T = {`L+1:P }. For each pair (I+ , θ+ ) ∈ F(L) × Θ+ (I+ ) we define a P dimensional vector γ and the set of positive 1-to-1 γ vectors, Γ, following the same definitions from Section II-E. We can

A linear Gaussian example is used to verify the proposed spawn model incorporating δ-GLMB filter. Consider a scenario with multiple trajectories within a [−1000, 1000] m × [−1000, 1000] m region, as illustrated in Fig. 2. Over the 100 s scenario duration, the number of targets varies due to birth, spawning, and death. In total, 6 targets are born and 6 targets are spawned. The first 3 appearing spawn targets, generated from birth tracks, each spawn a target of their own. Three spawn tracks cross at the origin at time k = 45, and at time k = 85 three pairs of birth and spawn tracks cross at (−412, −424), (−150, 515), and (530, −155).

8

1000

of 10 m from a parent state xk and in a direction that is perpendicular to the parent’s bearing θ, i.e.,   10 cos(θ + φ(i) )  10 sin(θ + φ(i) )  (i)  dT =    −p˙x,k −p˙y,k

800 600

y coordinate (m)

400 200 0 -200 -400 -600 -800 -1000 -1000

-800

-600

-400

-200

0

200

400

600

800

1000

x coordinate (m)

Fig. 2: Target trajectories in the xy plane. A circle “#” indicates where a target is born, a square “” indicates where a target is spawned, and a triangle “4” indicates where a target dies.

where φ(1) = −90 deg and φ(2) = 90 deg. The maximum number of δ-GLMB filter components is capped at 1000. Using the Gibbs sampler to randomly generate hypotheses, the probabilities of survival and detection are tempered with values set to p˘S,k = 0.95pS,k and p˘D,k = 0.95pD,k , respectively. This biases the sampler, yielding more track termination and miss detection hypotheses, which expedites the killing off of truly dead tracks while reducing the occurrence of dropped tracks, respectively. For more details on tempering techniques, see [23]. Results are presented for 100 Monte Carlo simulations. The mean and standard deviation of cardinality estimates over time are shown in Fig. 3 and mean Optimal Sub-Pattern Assignment (OSPA) [35] distance is shown in Fig. 4. 10

I Fk = 2 02

 ∆I2 , I2

" Qk =

σν2

∆4 4 I2 ∆3 2 I2

∆3 2 I2 ∆2 I2

6 4 True -GLMB Est Std

2 0

#

10

20

30

40

,

where ∆ = 1 s, σν = 5 m s−2 , and In and 0n denote the n×n identity and zero matrices, respectively. Each target is detected with probability pD,k = 0.88 and T each target generated measurement zk = [zx,k , zy,k ] consists of the target’s position with noise added to each component. Measurements follow the linear Gaussian measurement model gk (zk |xk ) = N (zk ; Hk xk , Rk ) such that

50 Time

60

70

80

90

100

Fig. 3: Cardinality statistics (100 Monte Carlo trials). 100

OSPA (m)(c=100, p=1)



8 Cardinality Statistics

The single-target state describing a target’s planar position T and velocity coordinates is xk = [px,k , py,k , p˙x,k , p˙y,k ] . Each target has a probability of survival pS,k = 0.99 and follows linear Gaussian dynamics with transition density fk|k−1 = N (xk ; Fk xk−1 , Qk ) such that

80 60 40 20 0

Hk = [I2 02 ] ,

Rk = σ2 I2 ,

10

20

30

40

50 Time

60

70

80

90

100

Fig. 4: OSPA distance (100 Monte Carlo trials). where σ = 10 m. Clutter is modeled as a Poisson RFS with an average intensity of λc = 1.65 × 10−5 m−2 yielding an average of 66 clutter returns per scan. Targets can appear either by birth or spawning. The birth (i) (i) model is an LMB RFS with parameters πB = {rB , pB }3i=1 (i) (i) (i) where rB = 0.02 and pB (x) = N (x; mB , PB ) with (1) (2) (3) T T mB = [0, 600, 0, 0] , mB = [519.615, −300, 0, 0] , mB = T 2 [−519.615, −300, 0, 0] , and PB = σB I4 where σB = 10 m. The spawn model is a conditional LMB RFS with (i) (i) (i) parameters πT = {pT , fT (·|x)}2i=1 where pT = 0.0133 (i) (i) (i) (i) and fT (·|x, `) = N (·; F x(`) + dT , QT ) with QT = (i) diag([20, 20, 20, 20]T )2 . Each dT is configured such that a spawn track with zero velocity is generated at a distance

It can be seen that the δ-GLMB filter accurately estimates target cardinality while the overall miss distance converges toward the given measurement error. The most notable behavior involves the delays in convergence within the time intervals k = [10, 20] and k = [50, 75]. This is jointly attributed to the distance spawn targets appear from their parents combined with the given measurement error, process noise, and relatively high occurrence of miss detections. With the initial distance between spawn targets and their parents relatively small, it is not until the two begin to separate that the filter begins to converge on the correct data associations. This process is exacerbated when miss detections occur around the time of parent-spawn separation, further delaying convergence.

9

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