NHPP Measurement Model

This post offers a gentle introduction to the Non-Homogeneous Poisson Process (NHPP) measurement model for multi-object tracking. NHPP models are commonly used in applications such as radar and sonar, where sensors detect a variable number of measurements from both moving objects and background clutter.

To see why NHPP is a useful model, watch the following example. It shows how NHPP-generated measurements model the data from three moving objects. While this is simulated, it captures the key characteristics of radar-like measurement data: variable number of detections, spatial spread, and background clutter. Multi-object tracking algorithms typically operate on this kind of data to track moving objects—for example, tracking the three objects shown in the left panel.

So how does the NHPP model represent such a scene?

In fact, the full set of measurements can be viewed as arising from four sources: three components associated with the three moving objects, and one component representing background clutter. The following video visualises this decomposition, with each measurement coloured according to its source.

In this example, each subset of coloured points is itself generated by an independent NHPP. In fact, this forms the starting point for modelling multi-object measurements using NHPP. Specifically, we model the measurements from each object, as well as the clutter, as independent NHPPs. Then, thanks to the NHPP superposition property, the combined set of measurements (formed by taking all points from these independent NHPPs) is itself an NHPP. The resulting model retains an elegant mathematical form, with statistical properties fully derivable from those of the individual NHPP components.

We have not yet provided a formal definition of the NHPP. Without loss of generality, and to make things concrete, let us now take a closer look at one NHPP corresponding to a single object (denoted by the label $k$ and with states $X_k$ that includes positions, velcoity, etc.).

In fact, any NHPP — including the one we are considering here — can be fully characterised by two components:

1. A rate parameter $\Lambda_k$ that determines the average number of points (or measurements) generated by the NHPP. The actual number of points follows a Poisson distribution with this mean rate, $\mathrm{Pois}(\Lambda_k)$.
2. A spatial distribution $\ell(Y_j \mid X_k)$ that determines how the points (or measurements) $Y_j$ from object $k$ are distributed in space. Given the number of points, their spatial locations are independently sampled according to this distribution.

In the following example, we zoom in on this single object and visualise both of these aspects. You will see that the number of measurements fluctuates from frame to frame, reflecting the Poisson variability, while their spatial distribution follows the object’s 2D Gaussian shape model.

In this example, the rate is set to $\Lambda_k = 12$. The spatial distribution $\ell(Y_j \mid X_k)$ is a 2D Gaussian distribution aligned with the object’s direction of motion, with principal axes of length 5.0 and 1.5. We can clearly see how the NHPP model is useful for characterising both the object’s shape and its measurement rate.

As for the clutter in the previous NHPP realisation, the rate is set to $\Lambda_0 = 15$, and the spatial distribution $\ell_0(Y_j)$ is a uniform distribution over the region $[-40, 40] \times [-40, 40]$. The following video shows a realisation of measurements generated solely from this clutter process.

Now the overall measurement process, formed by the superposition of these NHPPs, once again gives rise to the black dots we saw in the first video.

By the NHPP superposition property, this combined NHPP has an overall rate of $\Lambda_0 + \sum_{k=1}^K \Lambda_k$, where $K$ is the total number of objects ($K = 3$ in our example). The corresponding spatial distribution is:

This allows us to express the overall measurement likelihood as:

\begin{equation} \label{eq:nhpp lieklihood} p(Y, M \mid X_{1:K}) =\frac{1}{M!} \exp\left( - \Lambda_0 - \sum_{k=1}^K \Lambda_k \right) \prod_{j=1}^{M} \left( \Lambda_0 \, \ell_0(Y_j) + \sum_{k=1}^K \Lambda_k \, \ell(Y_j \mid X_k) \right), \end{equation}

where $Y = [Y_1,Y_2,…,Y_M]$ denotes all measurements observed in the frame, and $M$ is the total number of measurements.

This form makes it clear that while the overall likelihood is tractable, it does not indicate which measurements came from which object or from clutter. Hence, instead of the coloured points shown in this video, it is the unlabelled measurements, as shown again in the video video below, that are used in practice for tracking. Resolving this ambiguity, known as the data association problem, is a key challenge in multi-object tracking.

This NHPP model can be further extended to incorporate object existence and detectability, and the tracking problem is further complicated by the fact that the number of objects $K$ is unknown and may vary over time.

Moreover, the difficulty of tracking increases when many objects are present in close proximity, or when there is a large amount of clutter. For example, the following video showcases a challenging tracking scenario with 50–100 objects under heavy clutter, where even the human eye struggles to distinguish and track the individual objects.

If you are interested in an efficient and reliable multi-object tracking algorithm capable of handling such challenging scenarios, please see PiVoT.