# Optimal Subpattern Assignment Metric

**Libraries:**

Sensor Fusion and Tracking Toolbox /
Track Metrics

## Description

The Optimal Subpattern Assignment Metric block computes the optimal subpattern assignment metric between a set of tracks and known truths. You can enable different types of OSPA metrics by configuring these parameters:

Traditional OSPA — Specify the

**Metric**parameter as`OSPA`

and specify the**Labeling error**parameter, on the**Properties**tab, as`0`

. The traditional OSPA metric, which evaluates instantaneous tracking performance, contains two components:Localization error component — Accounts for state estimation errors between assigned tracks and truths.

Cardinality error component— Accounts for the number of unassigned tracks and truths.

Labeled OSPA — Specify the

**Metric**parameter as`OSPA`

and specify the**Labeling error**parameter, on the**Properties**tab, as a positive scalar. The labeled OSPA (LOSPA) metric, which evaluates instantaneous tracking performance and includes penalties for incorrect assignments, contains three components:Localization error component — Accounts for state estimation errors between assigned tracks and truths.

Cardinality error component— Accounts for the number of unassigned tracks and truths.

Labelling error component — Accounts for the error of incorrect assignments.

OSPA

^{(2)}— Specify the**Metric**parameter as`OSPA(2)`

. The OSPA^{(2)}metric evaluates cumulative tracking performance for a duration of time.

You can output each component individually from the block. For more details on the algorithm, see Algorithm and References.

## Examples

### Extended Object Tracking of Highway Vehicles with Radar and Camera in Simulink

Track highway vehicles around an ego vehicle in Simulink. In this example, you use multiple extended object tracking techniques to track highway vehicles and evaluate their tracking performance. This example closely follows the Extended Object Tracking of Highway Vehicles with Radar and Camera MATLAB® example.

- Since R2021b
- Open Model

## Ports

### Input

**Tracks** — Track list

Simulink^{®} bus containing MATLAB^{®} structure

Track list, specified as a Simulink bus containing a MATLAB structure.

If you specify the **Track bus** parameter on the **Port
Setting** tab to `objectTrack`

, the structure must
use this form:

Field | Description |
---|---|

`NumTracks` | Number of tracks |

`Tracks` | Array of track structures |

Each track structure must contain `TrackID`

and `State`

fields. Additionally, if you specify an NEES-based distance (`posnees`

or `velnees`

) in the **Distance type** parameter, each
structure must contain a `StateCovariance`

field.

Field | Definition |
---|---|

`TrackID` | Unique track identifier used to distinguish multiple tracks, specified as a nonnegative integer. |

`State` | Value of state vector at the update time, specified as an |

`StateCovariance` | Uncertainty covariance matrix, specified as an |

If you specify an NEES-based distance (`posnees`

or
`velnees`

) in the **Distance type** parameter,
then the structure must contain a `StateCovariance`

field.

If you specify the **Track bus** parameter to
`custom`

, then you can use your own track bus format. In
this case, you must define a track extractor function using the **Track
extractor function** parameter. The function must use this
syntax:

tracks = trackExtractorFcn(trackInputFromBus)

`trackInputFromBus`

is the input from the track bus and
`tracks`

must return as an array of structures with
`TrackID`

and `State`

fields. **Truths** — Truth list

Simulink bus containing MATLAB structure

Truth list, specified as a Simulink bus containing a MATLAB structure.

If you specify the **Truth bus** parameter on the **Port
Setting** tab to
`Platform`

, the structure
must use this form:

Field | Description |
---|---|

`NumPlatforms` | Number of truth platforms |

`Platforms` | Array of truth platform structures |

Each platform structure has these fields:

Field | Definition |
---|---|

`PlatformID` | Unique identifier used to distinguish platforms, specified as a nonnegative integer. |

`Position` | Position of the platform, specified as an |

`Velocity` | Velocity of the platform, specified as an |

If you specify the **Truth bus** parameter as
`Actor`

, the structure must
use this form:

Field | Description |
---|---|

`NumActors` | Number of truth actors |

`Actors` | Array of truth actor structures |

Each actor structure has these fields:

Field | Definition |
---|---|

`ActorID` | Unique identifier used to distinguish actors, specified as a nonnegative integer. |

`Position` | Position of the actor, specified as an |

`Velocity` | Velocity of the actor, specified as an |

If you specify the **Truth bus** parameter to
`custom`

, then you can define your own truth bus format. In this
case, you must define a truth extractor function using the **Truth extractor
function** parameter. The function must use this
syntax:

truths = truthExtractorFcn(truthInputFromBus)

`truthInputFromBus`

is the input from the truth bus and
`truths`

must return as an array of structures with
`PlatformID`

, `Position`

, and `Velocity`

fields.**Assignments** — Known assignment

*K*-by-2 matrix of nonnegative integers

Known assignment, specified as a*K*-by-2 matrix of nonnegative
integers. *K* is the number of assignment pairs. The first column
elements are track IDs, and the second column elements are truth IDs. The two IDs in a
row are assigned to each other. If a track or truth is not assigned, specify
`0`

as the other element in the row.

Assignments between tracks and truths must be unique. Redundant or false tracks
should be treated as unassigned tracks by assigning them to the "`0`

"
`TruthID`

.

#### Dependencies

To enable this port, on the **Port Setting** tab, select
**Assignments**.

### Output

**OSPA Metric** — OSPA metric

nonnegative real scalar

OSPA metric, returned as a nonnegative real scalar. Depending on the values of the
**Metric** and **Labeling error** parameters, the
block can output traditional OSPA, labeled OSPA (LOSPA), or
OSPA^{(2)}.

Metric Parameter Value | Labeling error Parameter Value | Metric |
---|---|---|

`OSPA` | `0` | OSPA |

`OSPA` | Positive scalar | LOSPA |

`OSPA(2)` | Not applicable | OSPA^{(2)} |

**Example: **10.1

**Localization Error** — Localization error component

nonnegative real scalar

Localization error component, returned as a nonnegative real scalar.

**Example: **8.5

#### Dependencies

To enable this port, on the **Port Setting** tab, select
**Localization error**.

**Cardinality Error ** — Cardinality error component

nonnegative real scalar

Cardinality error component, returned as a nonnegative real scalar.

**Example: **6

#### Dependencies

To enable this port, on the **Port Setting** tab, select
**Cardinality error**.

**Labeling Error** — Labeling error component

nonnegative real scalar

Labeling error component, returned as a nonnegative real scalar.

**Example: **7.5

#### Dependencies

To enable this port, on the **Port Setting** tab, select
**Labeling error**.

## Parameters

**Properties**

**Metric** — Metric option

`OPSA`

(default) | `OSPA(2)`

Metric option, specified as `OSPA`

or `OSPA(2)`

.

`OSPA`

— Computes the traditional OSPA metric by default, or computes the labeled OSPA metric after additionally specifying the**Labeling error**parameter as a positive value.`OSPA(2)`

— Computes the OSPA^{(2)}metric, which evaluates cumulative tracking performance. Selecting this option enables these parameters for configuring the metric:**Window length****Window sum order (q)****Window weights****Window weight exponent (r)****Custom window weights**

Selecting this option also disables two parameters used to evaluate the labeling error component:

**Assignments****Labeling error**

**Cutoff distance** — Threshold for cutoff distance between track and truth

`30`

(default) | real positive scalar

Threshold for the cutoff distance between track and truth, specified as a real positive scalar. If the computed distance between a track and the assigned truth is higher than the threshold, the actual distance incorporated in the metric is reduced to the threshold.

**Example: **`40`

**Order** — Order of OSPA metric

`2`

(default) | positive integer

Order of the OSPA metric, specified as a positive integer.

**Example: **
`3`

**Distance type** — Distance type

`posnees`

(default) | `velnees`

| `posabserr`

| `velabserr`

| `custom`

Distance type, specified as `posnees`

, `velnees`

,
`posabserr`

, or `velabserr`

. The distance type
specifies the physical quantity used for distance calculations:

`posnees`

– Normalized estimation error squared (NEES) of track position`velnees`

– NEES error of track velocity`posabserr`

– Absolute error of track position`velabserr`

– Absolute error of track velocity`custom`

– Custom distance error

If you select `custom`

, you must also specify a distance function
in the **Custom distance function** parameter.

**Custom distance function** — Custom distance function

function handle

Custom distance function, specified as a function handle. The function must support the following syntax:

d = myCustomFcn(Track,Truth)

`Track`

is a structure of track information, `Truth`

is a structure of truth information, and `d`

is the distance between
the truth and the track. See `objectTrack`

for an example on how
to organize track information.
**Example: **
`@myCustomFcn`

#### Dependencies

To enable this property, set the **Distance type** parameter to
`custom`

.

**Motion model** — Desired platform motion model

`constvel`

(default) | `constacc`

| `constturn`

| `singer`

Desired platform motion model, specified as `constvel`

,
`constacc`

, `constturn`

, or
`singer`

. This property selects the motion model used by the
**Tracks** input port.

The motion models expect the `State`

field of the track structure
to have a column vector containing these values:

`constvel`

— Position is in elements [1 3 5], and velocity is in elements [2 4 6].`constacc`

— Position is in elements [1 4 7], velocity is in elements [2 5 8], and acceleration is in elements [3 6 9].`constturn`

— Position is in elements [1 3 6], velocity is in elements [2 4 7], and yaw rate is in element 5.`singer`

— Position is in elements [1 4 7], velocity is in elements [2 5 8], and acceleration is in elements [3 6 9].

The `StateCovariance`

field of the track structure input
must have position, velocity, and turn-rate covariances in the rows and columns
corresponding to the position, velocity, and turn-rate of the `State`

field of the track structure.

**Labeling error** — Penalty for incorrect assignment

`0`

(default) | nonnegative scalar

Penalty for incorrect assignment of track to truth, specified as a nonnegative
scalar. The function decides if an assignment is correct based on the provided known
assignment input from the **Assignments** input port. If the assignment
is not provided as an input, the last known assignment is assumed to be correct.

**Example: **
`5`

#### Dependencies

To enable this parameter, set the **Metric** parameter to
`OSPA`

.

**Window length** — Sliding window length for OSPA^{(2)} metric

`100`

(default) | positive integer

Sliding window length for the OSPA^{(2)} metric, specified
as a positive integer. The window length defines the number of time steps from a
previous time to the current time used to evaluate the metric. For more details, see
OSPA(2) Metric.

#### Dependencies

To enable this parameter, set the **Metric** parameter to
`OSPA(2)`

.

**Data Types: **`single`

| `double`

**Window sum order (q)** — Order of weighted sum for track and truth history

`2`

(default) | positive scalar

Order of weighted sum for the track and truth history, specified as a positive scalar. For more details, see OSPA(2) Metric.

#### Dependencies

To enable this parameter, set the **Metric** parameter to
`OSPA(2)`

.

**Data Types: **`single`

| `double`

**Window weights** — Options for window weights

`auto`

(default) | `custom`

Options for window weights, specified as `auto`

or
`custom`

.

`auto`

— Automatically generates the window weights using the algorithm given in OSPA(2) Metric.`custom`

— Customizes the window weights using the**Custom window weights**parameter.

#### Dependencies

To enable this parameter, set the **Metric** parameter to
`OSPA(2)`

.

**Data Types: **`single`

| `double`

**Window weight exponent (r)** — Exponent for automatic weight calculation

`1`

(default) | nonnegative scalar

Exponent for automatic weight calculation, specified as a nonnegative scalar. An
exponent value, *r*, of `0`

represents equal weights
in the window. A higher value of *r* assigns more weights on recent
data. For more details, see OSPA(2) Metric.

#### Dependencies

To enable this parameter, set the **Window weights** parameter to
`auto`

.

**Data Types: **`single`

| `double`

**Custom window weights** — Custom weights in time window

*N*-element of vector of nonnegative values

Custom weights in the time window, specified as an *N*-element of
vector of nonnegative values, when *N* is the window length, specified
in the **Window length** parameter.

#### Dependencies

To enable this parameter, set the **Window weights** parameter to
`custom`

.

**Data Types: **`single`

| `double`

**Simulate using** — Type of simulation to run

`Interpreted execution`

(default) | `Code Generation`

Select a simulation type from these options:

`Interpreted execution`

— Simulate the model using the MATLAB interpreter. This option shortens startup time. In`Interpreted execution`

mode, you can debug the source code of the block.`Code generation`

— Simulate the model using generated C code. The first time you run a simulation, Simulink generates C code for the block. The C code is reused for subsequent simulations as long as the model does not change. This option requires additional startup time.

**Port Setting**

**Assignments** — Enable assignment input

`off`

(default) | `on`

Select this parameter to enable the input of known assignments through the
**Assignments** input port.

#### Dependencies

To enable this parameter, set the **Metric** parameter to
`OSPA`

.

**Localization error** — Enable localization error component output

`off`

(default) | `on`

Select this parameter to enable the output of the localization error component
through the **Localization Error** output port.

**Cardinality error** — Enable cardinality error component output

`off`

(default) | `on`

Select this parameter to enable the output of the cardinality error component
through the **Cardinality Error** output port.

**Labeling error** — Enable labeling error component output

`off`

(default) | `on`

Select this parameter to enable the output of the labeling error component through
the **Labeling Error** output port.

#### Dependencies

To enable this parameter, set the **Metric** parameter to
`OSPA`

.

**Track bus** — Track bus selection

`objectTrack`

(default) | `custom`

Track bus selection, specified as `objectTrack`

or
`custom`

. See the description of the **Tracks**
input port for more details about each selection.

**Truth bus** — Truth bus selection

`Platform`

(default) | `Actor`

| `custom`

Truth bus selection, specified as `Platform`

,
`Actor`

, or `custom`

. See the
description of the **Truths** input port for more details about each
selection.

**Track extractor function** — Track extractor function

function handle

Track extractor function, specified as a function handle. The function must support this syntax:

tracks = trackExtractorFcn(trackInputFromBus)

`trackInputFromBus`

is the input from the track bus and
`tracks`

must return as an array of structures with
`TrackID`

and `State`

fields. If you specify an
NEES-based distance (`posnees`

or `velnees`

) in the
**Distance type**parameter, then the structure must contain a

`StateCovariance`

field.**Example: ** `@myCustomFcn`

#### Dependencies

To enable this property, set the **Track bus** parameter to `custom`

.

**Truth extractor function** — Truth extractor function

function handle

Truth extractor function, specified as a function handle. The function must support this syntax:

truths = truthExtractorFcn(truthInputFromBus)

`truthInputFromBus`

is the input from the track bus and
`truths`

must return as an array of structures with
`PlatformID`

, `Position`

, and
`Velocity`

as field names.
**Example: **
`@myCustomFcn`

#### Dependencies

To enable this property, set the **Truth bus** parameter to
`custom`

.

## Algorithms

### OSPA Metric

At time *t*_{k}, a list of truths is:

$$X=[{x}_{1},{x}_{2},\dots ,{x}_{m}]$$

At the same time, a tracker obtains a list of tracks:

$$Y=[{y}_{1},{y}_{2},\dots ,{y}_{n}]$$

The traditional OSPA metric is:

$$OSPA={({d}_{loc}^{p}+{d}_{card}^{p})}^{1/p}$$

Assuming *m* ≤ *n*, the two components,
*d*_{loc} and
*d*_{card} are calculated using
these equations. The *localization error* component
*d*_{loc} is computed as:

$${d}_{loc}={\left\{\frac{1}{n}{\displaystyle \sum _{i=1}^{m}{d}_{c}^{p}\left({x}_{i},{y}_{\pi (i)}\right)}\right\}}^{1/p}$$

where *p* is the order of the OSPA metric,
*d*_{c} is the cutoff-based
distance, and
*y*_{π(i)}
represents the track assigned to truth
*x*_{i}. The cutoff-based
distance *d*_{c} is defined as:

$${d}_{c}(x,y)=\mathrm{min}\left\{{d}_{b}(x,y),c\right\}$$

where *c* is the cutoff distance threshold, and
*d*_{b}(*x*,*y*)
is the distance between truth *x* and track *y* calculated
by the distance function. The cutoff-based distance
*d*_{c} takes the smaller
value of *d*_{b} and
*c*.

The *cardinality error* component
*d*_{card} is:

$${d}_{card}={\left\{\frac{n-m}{n}{c}^{p}\right\}}^{1/p}$$

The labeled OSPA (LOSPA) is:

$$OSPA={({d}_{loc}^{p}+{d}_{card}^{p}+{d}_{lab}^{p})}^{1/p}$$

Here, additionally, the labeling error component
*d*_{lab} is:

$${d}_{lab}={\left\{\frac{1}{n}{\displaystyle \sum _{i=1}^{m}{\alpha}^{p}\gamma \left(L({x}_{i}),L({y}_{\pi (i)})\right)}\right\}}^{1/p}$$

where *α* is the penalty for incorrect assignment in the
labeling error component,
*L*(*x*_{i})
represents the truth ID of *x*_{i},
and
*L*(*y*_{π(i)})
represents the track ID of
*y*_{π(i)}.
The function *γ* = 0 if the IDs of the truth and track pair agree with the
known assignment given by the `assignment`

input, or agree with the
assignment in the last update if the known assignment is not given. Otherwise,
*γ* = 1.

If *m* > *n*, exchange *m* and
*n* in the formulation to obtain the OSPA metric.

### OSPA^{(2)} Metric

Consider a time period of *N* time steps, from time
*t*_{k-N+1}
to time *t*_{k}. During this time
period, you have a list of *m* truth histories:

$$X=[{x}_{1},{x}_{2},\dots ,{x}_{m}]$$

Each truth history *x*_{i}, is
composed of :

$${x}_{i}=[{x}_{i(k-N+1}),\dots ,{x}_{i(k)}]$$

where
*x*_{i(s)}
is the track history for *x*_{i} at
time step *t*_{s}, and
*x*_{i(s)}= ∅
if *x*_{i} does not exist at time
*t*_{s}. For the same time
period, you have a list of *n* track histories:

$$Y=[{y}_{1},{y}_{2},\dots ,{y}_{n}]$$

Each track history *y*_{i} is
composed of :

$${y}_{i}=[{y}_{i(k-N+1}),\dots ,{y}_{i(k)}]$$

where
*y*_{i(s)}
is the track history at time step
*t*_{s}, and
*y*_{i(s)}= ∅
if *y*_{i} does not exist at time
*t*_{s}.

Assuming *m* ≤ *n*, the OPSA^{(2)}
metric is calculated as:

$$OSP{A}^{(2)}={\left[{d}_{loc}^{p}+{d}_{card}^{p}\right]}^{1/p}$$

where the *cardinality error* component
*d*_{card} is:

$${d}_{card}={\left\{\frac{n-m}{n}{c}^{p}\right\}}^{1/p}$$

In this equation, *p* is the order of the OSPA metric,
and *c* is the cutoff distance threshold.

The *localization error* component
*d*_{loc} is computed as:

$${d}_{loc=}{\left\{\frac{1}{n}{\displaystyle \sum _{i=1}^{m}{d}_{q}({x}_{i},{y}_{\pi (i)})}\right\}}^{1/p}$$

where
*y*_{π(i)}
represents the track assigned to truth
*x*_{i}, and
*d*_{q} is the *base
distance* between a truth and a track, accounting for cumulative tracking
errors.

You can obtain *d*_{q} between a truth *x*_{i} and a track *y*_{j} as:

$${d}_{q}({x}_{i},{y}_{j})={\left[{\displaystyle \sum _{\tau =k-N+1}^{k}w(\tau ){d}^{*}{\left({x}_{i}(\tau ),{y}_{j}(\tau )\right)}^{q}}\right]}^{1/q}$$

where *N* is the *window length*,
*w*(*τ*) is the *window weight*
at time step *τ*, and *q* is the *window sum
order*. *d** is defined as:

$${d}^{*}\left({x}_{i}(\tau ),{y}_{j}(\tau )\right)=\{\begin{array}{l}{d}_{c}\left({x}_{i}(\tau ),{y}_{j}(\tau )\right)=\mathrm{min}\left\{{d}_{b}\left({x}_{i}(\tau ),{y}_{j}(\tau )\right),c\right\},\text{\hspace{1em}}if\text{\hspace{0.17em}}{x}_{i}(\tau )\ne \varnothing \text{\hspace{0.17em}}and\text{\hspace{0.17em}}{y}_{i}(\tau )\text{\hspace{0.17em}}\ne \varnothing \text{\hspace{0.17em}}\\ c,\text{\hspace{1em}}if\text{\hspace{0.17em}}{x}_{i}(\tau )=\varnothing \text{\hspace{0.17em}}and\text{\hspace{0.17em}}{y}_{i}(\tau )\text{\hspace{0.17em}}\ne \varnothing ,\text{\hspace{0.17em}}or\text{\hspace{0.17em}}{x}_{i}(\tau )\ne \varnothing \text{\hspace{0.17em}}and\text{\hspace{0.17em}}\text{\hspace{0.17em}}{y}_{i}(\tau )\text{\hspace{0.17em}}=\varnothing \\ 0,\text{\hspace{1em}}if\text{\hspace{0.17em}}{x}_{i}(\tau )={y}_{i}(\tau )=\varnothing \text{\hspace{0.17em}}\end{array}$$

From the equation, the cutoff-based distance
*d*_{c} takes the smaller
value of *d*_{b} and
*c*, where
*d*_{b}(*x*_{i}(*τ*),*y*_{j}(*τ*)
) is the distance between truth
*x*_{i} and track
*y*_{j} at time
*τ*, calculated by the distance function.

If you do not customize the window weights, the object assigns the *window
weights* as:

$$w(\tau )=\frac{{(N-k+\tau )}^{r}}{{\displaystyle \sum _{\tau =k-N+1}^{k}{(N-k+\tau )}^{r}}}$$

where *r* is the *window weight
component*.

If *m* > *n*, exchange *m* and
*n* in the formulation to obtain the OSPA^{(2)}
metric.

## References

[1] Schuhmacher, B., B. -T. Vo, and B.
-N. Vo. "A Consistent Metric for Performance Evaluation of Multi-Object Filters."
*IEEE Transactions on Signal Processing*, Vol, 56, No, 8, pp.
3447–3457, 2008.

[2] Ristic, B., B. -N. Vo, D. Clark,
and B. -T. Vo. "A Metric for Performance Evaluation of Multi-Target Tracking Algorithms."
*IEEE Transactions on Signal Processing*, Vol, 59, No, 7, pp.
3452–3457, 2011.

[3] M. Beard, B. -T. Vo, and B. -N.
Vo. “OSPA (2) : Using the OSPA Metric to Evaluate Multi-Target Tracking Performance.”
*2017 International Conference on Control, Automation and Information
Sciences*, IEEE, 2017, pp. 86–91.

## Extended Capabilities

### C/C++ Code Generation

Generate C and C++ code using Simulink® Coder™.

## Version History

**Introduced in R2021a**

### R2023a: Simulink buses do not show in workspace

As of R2023a, the Simulink buses created by this block no longer show in MATLAB workspace.

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