What is Random Finite Set of Trajectories?
Random Finite Set of Trajectories is a set-valued random variable whose elements are complete finite trajectories, allowing Bayesian multi-object tracking to represent uncertain births, deaths, states, and associations without imposing arbitrary labels.
Quick Facts
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How It Works
Represent a trajectory as one set element
A trajectory can be written as X = (t, x^(1:i)), where t is its start time, i is its length, and the state sequence covers consecutive times from t through t+i-1. A set of trajectories may contain histories with different start times and lengths without assigning them arbitrary array positions.
García-Fernández, Svensson, and Morelande formalize this space and its finite-set integral. Projecting every trajectory alive at time k recovers a current target set, but repeated current-set marginals generally do not reconstruct the joint posterior over histories.
Choose whether the posterior covers alive or all trajectories
A filtering problem may retain only trajectories alive at the current time or all trajectories that have existed up to that time. The first limits state growth; the second preserves dead histories for retrospective questions. A time-window marginal can keep only the interval required by an evaluator or downstream decision.
Prediction extends surviving trajectories and adds newborn ones. The standard measurement likelihood depends on the current-state projection of alive trajectories, while the posterior remains a density over histories. Filtering, fixed-lag smoothing, and full smoothing therefore answer different conditioning questions.
Separate trajectory state from labels and algorithms
A trajectory-set RFS is a state representation, not a single filter. TPHD and TCPHD use Poisson and IID-cluster approximations; TPMBM retains Poisson undetected mass and an MBM of detected trajectory hypotheses. Their approximation and hypothesis-management errors remain different.
Labels can index tracks, but an arbitrary label is not an observable serial number. Evaluate claimed history quality with T-GOSPA or another declared trajectory metric, then report localization, missed and false objects, switches, cardinality, calibration, runtime, memory, and sensitivity to the birth, survival, detection, and clutter models.
Key Characteristics
- Uses complete finite trajectories rather than current object states as set elements
- Represents random trajectory count, start times, lengths, and state sequences
- Avoids arbitrary ordering and does not require artificial physical identities
- Supports posteriors over alive trajectories, all trajectories, or time windows
- Projects naturally to the set of object states at any represented time
- Requires approximations or hypothesis reduction for practical inference
Common Use Cases
- Bayesian multi-object tracking through crossings and occlusions
- Fixed-lag or full-history smoothing of radar and sonar tracks
- Reasoning about whether past and present detections share one history
- Comparing TPHD, TCPHD, TPMBM, and labeled RFS trackers
- Evaluating track continuity with trajectory-set metrics
Example
Loading code...Frequently Asked Questions
How is an RFS of trajectories different from an ordinary RFS?
An ordinary tracking RFS commonly uses current object states as its elements. An RFS of trajectories uses complete histories with a start time and state sequence. Its posterior can therefore answer temporal association and smoothing questions that current-set marginals alone cannot answer.
Does a set of trajectories require labels?
No. Each trajectory is already one complete set element, so histories can be represented without adding arbitrary labels to current states. Implementations may still use internal identifiers for storage, but those identifiers are not automatically observable physical identities.
What is the difference between alive and all trajectory posteriors?
An alive-trajectory posterior contains histories whose objects exist at the current time. An all-trajectory posterior also retains histories that have ended. The latter supports retrospective queries but grows with time unless the implementation uses windows or approximations.
Can current multi-object filtering densities recover trajectory uncertainty?
Not in general. A sequence of marginal posteriors over current sets omits joint dependence across times, including which past and present states belong to the same history. A trajectory posterior or a separately justified smoother is needed for those questions.
How should trajectory-set methods be evaluated?
Use a declared trajectory metric such as T-GOSPA for localization, missed and false objects, and switches. Also report cardinality, uncertainty calibration, retained hypothesis mass where applicable, latency, memory, and sensitivity to model and pruning choices.