GM-PHD Filter

Gaussian Mixture Probability Hypothesis Density Filter

A Gaussian Mixture Probability Hypothesis Density (GM-PHD) Filter handles multi-object tracking in a low signal-to-noise environment by recursively propagating an unnormalized Gaussian Mixture Model, which when integrated over a volume corresponds with the expected number of objects in that volume.

GaussianFilters.GaussianMixture — Type.
GaussianMixture(N::Int64, w::Vector{Number}, μ::Vector{AbstractVector},
    Σ::Vector{AbstractMatrix})
GaussianMixture(w::Vector{Number}, μ::Vector{AbstractVector},
    Σ::Vector{AbstractMatrix})

Arguments:
N: Number of models
w: Weights
μ: Means of the model
Σ: Covariances of the model
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Building a GM-PHD Filter

Building a GM-PHD Filter requires defining a birth intensity model of type GaussianMixture for where objects can be globally born, a spawn intensity model of type Spawn for how new objects can spawn off old objects, a Vector of possible linear dynamics models of type Dynamics, a linear measurement model of type Measurement, a survival probability, a detection probability, and a clutter modeling function.

GaussianFilters.Spawn — Type.
Spawn(β::GaussianMixture, dyn::Vector{Dynamics})

Arguments:
β: Gaussian Mixture model determining the
spawning intesity of the target
dyn: Dynamics
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GaussianFilters.Dynamics — Type.
Dynamics(A::AbstractMatrix, Q::AbstractMatrix, d::AbstractVector)
Dynamics(A::AbstractMatrix, Q::AbstractMatrix)

Construct linear dynamics model with; transition matrix A, process noise matrix Q and offset vector d

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GaussianFilters.Measurement — Type.
Measurement(C::AbstractMatrix, R::AbstractMatrix)

Construct measurement model with observation matrix C and sensor noise matrix R

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Once all of these are defined, a GM-PHD filter can be constructed with PHDFilter.

GaussianFilters.PHDFilter — Type.
PHDFilter(γ::GaussianMixture, spawn::Spawn, dyn::Vector{Dynamics},
    meas::Measurement, Ps::Float64, Pd::Float64, κ::Function)
PHDFilter(γ::GaussianMixture, spawn::Spawn, dyn::Dynamics,
    meas::Measurement, Ps::Float64, Pd::Float64, κ::Function)

Sets up a PHD Filter

Arguments:
γ: Birth intensity
spawn: Spawning intensity
dyn: Vector of possible Dynamics models
meas: Measurements
Ps: Survival probability
Pd: Detection probability
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Running a GM-PHD Filter

Currently, the GM-PHD Filter must be run step-by-step. Similar to the Kalman-Class filters, this can be done with a single call to update, which wraps functions to predict the next state, perform a measurement update with measure, and prune the resulting mixture model of low-probability and sufficiently close mixtures.

GaussianFilters.update — Function.
update(filter::AbstractFilter, b0::GaussianBelief, u::AbstractVector,
    y::AbstractVector)

Uses AbstractFilter filter to update gaussian belief b0, given control vector u and measurement vector y.

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update(phd::PHDFilter, b0::GaussianMixture, Z::Vector{AbstractVector},
	T::Real, U::Real, J_max::Integer)

Perform an update step using a GMPHD Filter.

Arguments:

  • phd::PHDFilter PHD filter to step through. [PHD Filter]
  • b0::GaussianMixture Prior state distribution. [Gaussian Mixture]
  • Z::Matrix Matrix of measurements. [Measurements]
  • T::Real Truncation threshold. Drop distributions with weight less than T
  • U::Real Merging threshold. Merge if (μ1-μ2)^T * Σ^-1 * (μ1-μ2) < U
  • J_max::Integer Maximum number of features

Returns:

  • bn::GaussianMixture Describe first return value. [Gaussian Mixture]

References:

  1. Vo, B. N., & Ma, W. K. (2006). The Gaussian mixture probability hypothesis

density filter. IEEE Transactions on signal processing, 54(11), 4091-4104.

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GaussianFilters.predict — Function.
predict(m::LinearDynamicsModel, x::AbstractVector{<:Number}, u::AbstractVector{<:Number})
predict(m::LinearDynamicsModel, x::AbstractVector{<:Number}, u::AbstractVector{<:Number}, rng::AbstractRNG)

Uses the linear dynamics model to propagate the state x one step forward in time with control input u. If rng is given, it adds process noise.

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predict(m::NonLinearDynamicsModel, x::AbstractVector{<:Number}, u::AbstractVector{<:Number})
predict(m::NonLinearDynamicsModel, x::AbstractVector{<:Number}, u::AbstractVector{<:Number}, rng::AbstractRNG)

Uses the non linear dynamics model to propagate the state x one step forward in time with control input u. If rng is given, it adds process noise.

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predict(filter::KalmanFilter, b0::GaussianBelief, u::AbstractVector)

Uses Kalman filter to run prediction step on gaussian belief b0, given control vector u.

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predict(filter::ExtendedKalmanFilter, b0::GaussianBelief, u::AbstractVector)

Uses Extended Kalman filter to run prediction step on gaussian belief b0, given control vector u.

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predict(filter::UnscentedKalmanFilter, b0::GaussianBelief, u::AbstractVector)

Uses Unscented Kalman filter to run prediction step on gaussian belief b0, given control vector u.

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predict(phd::PHDFilter, b0::GaussianMixture)

Make a prediction on next state based on PHD dynamics

Arguments:

  • phd::PHDFilter PHD filter to step through. [PHD Filter]
  • b0::GaussianMixture Prior state distribution. [Gaussian Mixture]

Returns:

  • bp::GaussianMixture Predicted next state distribution. [Gaussian Mixture]

References:

  1. Vo, B. N., & Ma, W. K. (2006). The Gaussian mixture probability hypothesis

density filter. IEEE Transactions on signal processing, 54(11), 4091-4104.

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GaussianFilters.measure — Function.
measure(m::LinearObservationModel, x::AbstractVector{<:Number}, u::AbstractVector{<:Number})
measure(m::LinearObservationModel, x::AbstractVector{T}, u::AbstractVector{T}, rng::AbstractRNG) where T<:Number

Returns an observation of state x according to the linear observation model m, with control inputs u. If rng is passed, adds additive Gaussian noise to the observation.

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measure(m::LinearObservationModel, x::AbstractVector{<:Number}, u::AbstractVector{<:Number})
measure(m::LinearObservationModel, x::AbstractVector{T}, u::AbstractVector{T}, rng::AbstractRNG) where T<:Number

Returns an observation of state x according to the non linear observation model m, with control inputs u. If rng is passed, adds additive Gaussian noise to the observation.

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measure(filter::KalmanFilter, bp::GaussianBelief, y::AbstractVector;
    u::AbstractVector = [false])

Uses Kalman filter to run measurement update on predicted gaussian belief bp, given measurement vector y. If u is specified and filter.o.D has been declared, then matrix D will be factored into the y predictions

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measure(filter::ExtendedKalmanFilter, bp::GaussianBelief, y::AbstractVector;
    u::AbstractVector = [false])

Uses Extended Kalman filter to run measurement update on predicted gaussian belief bp, given measurement vector y. If u is specified and filter.o.D has been declared, then matrix D will be factored into the y predictions.

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measure(filter::UnscentedKalmanFilter, bp::GaussianBelief, y::AbstractVector;
    u::AbstractVector = [false])

Uses Unscented Kalman filter to run measurement update on predicted gaussian belief bp, given measurement vector y. If u is specified and filter.o.D has been declared, then matrix D will be factored into the y predictions.

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measure(phd::PHDFilter, bp::GaussianMixture, Z::Vector{AbstractVector})

Perform a measurement update on a predicted PHD next state.

Arguments:

  • phd::PHDFilter PHD filter to step through. [PHD Filter]
  • bp::GaussianMixture Prior state distribution. [Gaussian Mixture]
  • Z::Vector{AbstractVector} Array of measurements. [Measurements]

Returns:

  • bm::GaussianMixture Describe first return value. [Gaussian Mixture]

References:

  1. Vo, B. N., & Ma, W. K. (2006). The Gaussian mixture probability hypothesis

density filter. IEEE Transactions on signal processing, 54(11), 4091-4104.

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GaussianFilters.prune — Function.
prune(b::GaussianMixture, T::Real, U::Real, J_max::Integer)

Prune the posterior Gaussian-Mixture next PHD state

Arguments:

  • b::GaussianMixture Posterior state distribution. [Gaussian Mixture]
  • T::Real Truncation threshold. Drop distributions with weight less than T
  • U::Real Merging threshold. Merge if (μ1-μ2)^T * Σ^-1 * (μ1-μ2) < U
  • J_max::Integer Maximum number of features
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Target locations can be extracted from a GaussianMixture state using multiple_target_state_extraction.

GaussianFilters.multiple_target_state_extraction — Function.
multiple_target_state_extraction(b::GaussianMixture, th::Real)

Extracts targets whose weights (x.w) are above threshold.

Arguments: b::GaussianMixture: Set of Gaussian Mixtures th::Real: Threshold on weights. Above this threshold, state estimate is extracted

Returns: X: Multi-Target State Estimate

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Examples

Full implementation examples can be found in the notebooks folder of the repo:

GM-PHD Object Surveillance Example

GM-PHD Aircraft Carrier Example