What is Probability Hypothesis Density Filter?
Probability Hypothesis Density Filter is a Random Finite Set filtering approximation that propagates the first-order moment of a multi-object posterior so its integral gives expected object count and its peaks indicate likely states.
Quick Facts
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How It Works
Propagate the first moment of an RFS posterior
Mahler's first-order multitarget moment paper derives the PHD recursion as a tractable approximation to multi-target Bayes filtering. Prediction combines birth intensity with surviving and spawned intensity transported through the motion model.
The intensity has count units. A component weight can represent expected target mass and need not be bounded by one, while the total integral need not equal one. Normalizing it like an ordinary probability density destroys cardinality information. Birth, survival, and spawning parameters therefore affect both where targets are expected and how many are expected.
Update collectively with detections and clutter
The update retains a missed-detection term and adds one intensity contribution per measurement. Each measurement contribution is normalized by its clutter intensity plus the total predicted target-generated measurement intensity, which couples all predicted states competing to explain that observation.
PHD filtering avoids an explicit one-to-one assignment output, but it does not remove measurement-model assumptions. The standard point-target model usually permits at most one detection per object and treats clutter independently. Extended targets, merged detections, unknown detection rates, and structured clutter require different models or robust extensions.
Choose an implementation and evaluate extraction separately
Vo and Ma's GM-PHD paper gives a closed-form Gaussian-mixture implementation for linear Gaussian models; SMC-PHD instead represents intensity with weighted particles. Mixture pruning, merging, capping, particle proposals, resampling, and state extraction are implementation decisions, not invisible details.
The Stone Soup GM-PHD tutorial demonstrates birth intensity, clutter density, prediction, update, and mixture reduction. Report cardinality error, OSPA or GOSPA, localization, false and missed objects, extraction sensitivity, component count, runtime, and memory. A Trajectory PHD Filter moves the intensity to trajectory space; it does not make a state-only PHD estimate carry identity.
Key Characteristics
- Propagates the first-order moment of a multi-object posterior
- Uses intensity whose integral equals expected object count
- Models target birth, survival, spawning, detection, and clutter
- Updates all predicted target mass against the measurement set
- Supports Gaussian-mixture and sequential Monte Carlo implementations
- Does not preserve full cardinality uncertainty or identity by itself
Common Use Cases
- Radar or sonar tracking with an unknown target count
- Pedestrian and vehicle tracking with births and disappearances
- Dynamic occupancy mapping from noisy sensor detections
- Cell or particle tracking in microscopy sequences
- Baselines for CPHD, multi-Bernoulli, GLMB, and PMBM filters
Example
Loading code...Frequently Asked Questions
What does the intensity of a PHD filter mean?
The integral of PHD intensity over any state-space region is the expected number of objects in that region. It is not a normalized probability density for one object, and individual intensity values or mixture weights need not lie between zero and one.
Does a PHD filter perform data association?
It updates a collective intensity directly from the measurement set rather than outputting explicit track-to-measurement assignments. Association uncertainty is still present in the multi-object likelihood and denominator, and extraction or labeling may introduce additional matching decisions.
What is the difference between PHD and CPHD filters?
PHD propagates only the first-order intensity, so it represents expected cardinality but not the full count distribution. CPHD also propagates a cardinality distribution, usually improving object-count estimates at additional computational and modeling cost.
How do GM-PHD and SMC-PHD differ?
GM-PHD represents intensity as Gaussian components and has closed-form recursion under linear Gaussian assumptions. SMC-PHD uses weighted particles and supports broader nonlinear or non-Gaussian models, but requires proposal, resampling, clustering, and higher compute choices.
Does a PHD filter preserve track identities?
Not by itself. Peaks describe likely current states, but the unlabeled first moment does not encode persistent object identity. Track extraction heuristics may connect peaks over time; labeled RFS or trajectory-set filters model identity and trajectory uncertainty more explicitly.