Four scenarios + GrBP · Joint PDF derivationsJOINT PDF FACTORIZATION
Joint PDF
factorization
From a generative model to local factors.
Point and extended objects.
Known and unknown object counts.
GrBP: Precluster, then associate whole groups.
S1Point · known
S2Point · unknown
S3Extended · known
S4Extended · unknown
The four scenarios and a GrBP derivationJOINT PDF FACTORIZATION
Two modeling choices shape the graph.
Object model: one measurement or several per scan.
Object count: fixed, or represented by potential objects and existence variables.
Common techniqueJOINT PDF FACTORIZATION
Prior × allocation × measurements
- Read the counts from the association.
- Insert the Poisson count laws.
- Cancel allocation factorials.
- Collect constants; marginalize auxiliaries.
collects object states; collects associations and any existence variables needed for the allocation.
Notation and model boundaryJOINT PDF FACTORIZATION
What is observed, and what is inferred?
Observed: the ordered measurement vector and its count.
Inferred: states, associations, and existence when applicable.
Clutter parameters and measurement models are supplied.
For fixed data and clutter model, is constant in the inferred states and assignments.
S1 · Point / known countJOINT PDF FACTORIZATION
Each target makes at most one claim.
existing point targets; measurements.
A nonzero claim must be unique. Several targets may have the null claim.
S1 · LikelihoodJOINT PDF FACTORIZATION
Divide by the clutter explanation.
Claimed measurements use the target likelihood.
Unclaimed measurements use .
Multiplying and dividing by clutter densities leaves a ratio for each detection.
S1 · Count cancellationJOINT PDF FACTORIZATION
A permutation removes the clutter factorial.
Given the detected targets, all consistent assignments to distinct measurement slots are uniform.
The clutter Poisson factorial cancels the allocation factorial.
One factor remains for each detected target.
S1 · Local factorsJOINT PDF FACTORIZATION
Detection ratios and missed detections
Each target contributes its prior and one local factor .
The consistency factor couples their association choices.
S1 · Factor graphJOINT PDF FACTORIZATION
One chain per target; one shared constraint.
Prior → state → local likelihood → association.
forbids duplicate nonzero claims.
Original Scenario 1 factor graph from the restored article.S2 · Point / unknown countJOINT PDF FACTORIZATION
Augment each state with existence.
Legacy potential objects carry .
An existing object can be detected or missed. A nonexistent object cannot claim a measurement.
S2 · Count cancellationJOINT PDF FACTORIZATION
Split unclaimed points into new objects and clutter.
Each measurement has a new potential object .
On the valid support, new-object and clutter counts follow directly from existence and association.
S2 · New-object factorJOINT PDF FACTORIZATION
A claimed point cannot also start a new object.
contains points already claimed by legacy objects.
A valid new-object branch needs both the birth intensity and state density .
S2 · Factorized posteriorJOINT PDF FACTORIZATION
Legacy chains plus new-object factors
The same enforces unique legacy claims.
Each new-object factor depends on the entire legacy association vector.
S2 · Factor graphJOINT PDF FACTORIZATION
New candidates share the association decisions.
Legacy chains remain on the left.
New candidates add factors.
The colored edges prevent a legacy claim and a new-object claim from explaining the same point.
Original Scenario 2 graph.S3 · Extended / known countJOINT PDF FACTORIZATION
Let every measurement name its source.
The extended state is .
Several measurements can share one object. Each measurement has exactly one source; means clutter.
S3 · Count cancellationJOINT PDF FACTORIZATION
The allocation binomials telescope.
Each object selects its cell from the remaining measurements.
The allocation contributes a factorial for each cell, cancelling the Poisson factorials.
S3 · Local factorsJOINT PDF FACTORIZATION
One unary factor, then one edge per point
The zero-count exponential belongs to the object’s unary factor.
An assigned point contributes a measurement-rate likelihood ratio. Every other edge is neutral.
S3 · Factorized posteriorJOINT PDF FACTORIZATION
Put all edges inside the object product.
Each object contributes its unary and all pairwise factors.
All objects meet at the measurement-oriented association variables.
S3 · Factor graphJOINT PDF FACTORIZATION
Many measurements may connect to the same object.
Each object–measurement pair has a factor .
Variable selects the source of measurement .
The source choice itself gives each measurement one owner.
Original Scenario 3 graph.S4 · Extended / unknown countJOINT PDF FACTORIZATION
Anchor each new object to its first point.
legacy potential objects and new potential objects.
New candidate is anchored at measurement and can explain later measurements.
S4 · Measurement countsJOINT PDF FACTORIZATION
A newly detected object has a nonzero count.
Newly detected objects use a zero-truncated Poisson count.
Existing legacy objects use an ordinary Poisson count; nonexistent ones generate zero points.
S4 · Auxiliary constructionJOINT PDF FACTORIZATION
Expand first, then remove the auxiliaries.
- Introduce newly detected states and their associations.
- Map them onto measurement-anchored candidates.
- Integrate copied states; sum auxiliary labels and mappings.
S4 · MarginalizationJOINT PDF FACTORIZATION
Relabeling cancels the new-object factorial.
Permuting the auxiliary labels of newly detected objects gives the same canonical assignment.
The sum contributes .
S4 · Conditional-prior resultJOINT PDF FACTORIZATION
The count model is complete; values come next.
: existing new candidates. : existing legacy objects.
S4 · Unary potentialsJOINT PDF FACTORIZATION
Separate legacy prediction and new-object evidence.
includes the legacy existence mass and its conditional state density.
The new-object unary carries the detected-object intensity and the truncated-count normalization.
S4 · Legacy measurement edgesJOINT PDF FACTORIZATION
An absent object cannot own a point.
Write the rate-weighted likelihood ratio as .
Assignment to an existing object activates that ratio; an absent object forbids the assignment.
S4 · New-object anchorJOINT PDF FACTORIZATION
The anchor enforces existence in both directions.
If candidate exists, measurement must belong to it.
If it does not exist, the anchor cannot name it.
S4 · Later measurementsJOINT PDF FACTORIZATION
An extended new object needs more than its anchor.
For every , candidate can explain measurement through .
Earlier measurements cannot refer to a later candidate.
S4 · Final factorizationJOINT PDF FACTORIZATION
Two families of local products
Legacy objects connect to every measurement. New candidates connect to their anchor and all later measurements.
S4 · Factor graphJOINT PDF FACTORIZATION
Prediction, birth, and measurement edges
: legacy prediction.
: new-object unary.
The anchor and later edges encode the canonical labeling.
Red annotations identify BP messages on this graph.
Original Scenario 4 graph. Its h / ḡ labels follow the source figure.Derivation for GrBP / Fixed partitionJOINT PDF FACTORIZATION
Derivation for GrBP
GrBP uses grouped measurements: external clustering fixes one partition.
Data association then decides which target explains each whole group.
Introduce one newborn candidate per group.
01 · OBSERVERaw detections
↓02 · PRECLUSTERFixed partition
↓03 · ASSOCIATEGroups
→ target choices
Derivation for GrBP / A.1JOINT PDF FACTORIZATION
Points, groups, and targets are different counts.
raw detections form disjoint groups.
legacy components carry predicted beliefs.
Every component has kinematics, extent, and existence; is shorthand.
Derivation for GrBP / A · AssociationJOINT PDF FACTORIZATION
Two variables record the same claim.
: legacy target claims group .
: group comes from legacy target .
leaves two alternatives: newborn or clutter.
Derivation for GrBP / A · Count identitiesJOINT PDF FACTORIZATION
Read the counts from the assignments.
Detected legacy targets, existing newborns, and clutter partition the groups on every feasible hypothesis.
Derivation for GrBP / A.2JOINT PDF FACTORIZATION
Separate the prior, allocation, and likelihood.
Condition throughout on the fixed partition . Its notation is suppressed in the equations.
Derivation for GrBP / A.3JOINT PDF FACTORIZATION
Clutter groups have their own Poisson mean.
: mean number of clutter groups.
: mean number of detected newborns.
Generally : a clutter group can contain several detections.
Derivation for GrBP / A.3 → A.4JOINT PDF FACTORIZATION
The allocation factorials cancel the count factorials.
Choose which unclaimed groups are newborns.
Assign detected legacy targets to distinct group slots.
Multiply by the two Poisson count laws; the remaining denominator is .
Derivation for GrBP / A.4JOINT PDF FACTORIZATION
Keep the existence branches and newborn densities.
collects newborn state densities; is normalized. The nonempty legacy count mass enters the likelihood next.
Derivation for GrBP / A.5 · Group modelsJOINT PDF FACTORIZATION
A detected newborn uses a nonempty-group likelihood.
A legacy group uses the extended-target Poisson likelihood.
A newborn is detected by definition, so condition on a nonzero count.
Keep one exponential per target, not one per point.
Derivation for GrBP / A.5JOINT PDF FACTORIZATION
Multiply the group likelihoods by source.
Conditioned on a feasible association, the approximate model treats groups as independent.
Derivation for GrBP / A.5 → A.6JOINT PDF FACTORIZATION
Divide each target group by its clutter explanation.
The product of all group clutter densities is a baseline for the entire scan.
Redistributing leaves one for every target-generated group.
Derivation for GrBP / A.6JOINT PDF FACTORIZATION
Separate the constant and define the group ratio.
is fixed for the observed grouped scan and fixed model parameters.
abbreviates the per-point likelihood ratios inside group .
Derivation for GrBP / A.7JOINT PDF FACTORIZATION
A legacy target has four local cases.
An existing target can claim a group or generate no detections.
A nonexistent target can only make a null claim.
The predicted prior remains a separate factor.
Derivation for GrBP / A.8JOINT PDF FACTORIZATION
An unclaimed group can create one newborn.
: newborn and clutter compete.
: a legacy target already owns the group, so the newborn cannot exist.
The absent candidate carries dummy density .
Derivation for GrBP / A.9JOINT PDF FACTORIZATION
The grouped joint PDF becomes a product of local factors.
Predicted priors × legacy factors × newborn factors × group-association consistency.
Derivation for GrBP / Model boundaryJOINT PDF FACTORIZATION
Association cannot repair a fixed partition.
S4: infer assignments of individual detections; several points can choose the same target.
GrBP: infer whole-group assignments after external clustering.
One group per target.
One source per group.
Target splits and cross-target merges are outside this model.
The number of groups is not the inferred number of targets.
Derivation for GrBP / Factor graphJOINT PDF FACTORIZATION
Factor graph for GrBP
Legacy chains and group-indexed newborn chains meet at .
Circles: variables · Boxes: factors · Blue: prediction between scans. The central factor collects all pairwise association-consistency factors.SummaryJOINT PDF FACTORIZATION
From raw measurements to fixed groups
Unknown object count: introduce potential objects and existence.
Multiple points per object: use measurement-oriented assignments and Poisson counts.
GrBP: precluster once, then associate whole groups.
| Case | Association | Coupling |
|---|
| S1 | Target-oriented | Global exclusivity |
| S2 | Target-oriented | Exclusivity + new objects |
| S3 | Measurement-oriented | Object–point factors |
| S4 | Measurement-oriented | Legacy + anchor + later edges |
| GrBP | Group-level | Exclusive target–group factors |
Before implementationJOINT PDF FACTORIZATION
Check the tiny cases first.
- Keep ordered-vector and finite-set conventions consistent.
- Retain every state-dependent factor.
- Verify counts and normalization by enumeration.
- Keep in new-object factors.
- Enforce the anchor condition and include every later edge.
- Compare approximate BP marginals against exact inference on small scans.
ReferencesJOINT PDF FACTORIZATION
The page, the scans, and the primary papers
R1 · Point-object models
Meyer et al., “Message passing algorithms for scalable multitarget tracking,” Proceedings of the IEEE, 2018.
R2 · Extended-object model
Granström, Baum & Reuter, “Extended Object Tracking: Introduction, Overview and Applications.”
R3 · Extended objects with uncertain count
Meyer & Williams, “Scalable Detection and Tracking of Geometric Extended Objects,” 2021.
The companion page contains the full derivations and original handwritten scans for S1–S4, plus the derivation for GrBP.