Active Sampling and Gaussian Reconstruction for
Radio Frequency Radiance Field

We use boldface letters to represent random variables and random vectors. $\vec{(.)}$ denotes a vector, while capital letters represent matrices. $(\cdot)^{T}$ denotes the transpose operation. $\mathbb{E}$ is the expectation operator.

Our model focuses on a short time slot, within which the given scene remains static. We model each position in the room as an omnidirectional virtual signal source. Specifically, for time slot $t$, the virtual signal at position $p_{i}$, is denoted as $\boldsymbol{x}_{t,p_{i}}$. Furthermore, the received signal power at any position is modeled as a sum of $R$ different rays from different angles. The received signal from each ray is modeled as a linear combination of the $N$ virtual signal source sampled on the ray, as shown in Fig. 1.

Therefore, for time slot $t$ the received signal can be expressed as:

The notation $RX(r,n)$ refers to the position of the $n-$th sample on the $r-$th ray for the receiver placed at $p_{RX}$. Specifically, there are $R\times N$ virtual signal sources that contribute to the received signal at $p_{RX}$ and their corresponding positions are given by:

For notation simplicity, we rewrite the received signal model for time slot $t$ (1) into vector form:

For a given time slot $t$, we model the RF Radiance Field as a Gaussian random process, with the virtual signal sources—components of the RF Radiance Field—modeled as Gaussian random variables. A Gaussian process is characterized by its mean and covariance (also referred to as the kernel):

• •

  Mean: For any position $p_{i}$, the virtual signal is modeled as a zero-mean Gaussian random variable.

  $\displaystyle\mathbb{E}\left[\boldsymbol{x}_{t,p_{i}}\right]=0$

  (5)

• •

  Covariance: The covariance between each pair of virtual signal sources is given by:

  $\displaystyle Cov(\boldsymbol{x}_{t,p_{i}},\boldsymbol{x}_{t,p_{j}})=\alpha^{2}\exp{\left(-\frac{1}{2l^{2}}||p_{i}-p_{j}||^{2}\right)}$

  (6)

The value of $a^{2}$ represents the variance of the virtual signal sources, which are selected based on the true transmission power of the device within the room. The length-scale parameter $l$, controls the rate of variation of the signal; smaller values of $l$ indicate a sharper change in the function. In a later section, we will demonstrate how to estimate this parameter. Note that (6) represents the Radial Basis Function (RBF) kernel. For information on other types of kernel functions and guidance on selecting the appropriate kernel, we refer the reader to [13].

With the above definitions, $\vec{\boldsymbol{x}}_{t,p_{RX}}$, the collection of virtual signal sources contributing to $\boldsymbol{y}_{t,p_{RX}}$ is a Gaussian random vector:

Here, $\Sigma_{p_{RX}}$ is the covariance matrix where the $(i,j)-$th component represents the covariance between the $i-th$ and $j-th$ components of $\vec{\boldsymbol{x}}_{t,p_{RX}}$.

Also, $\boldsymbol{y}_{t,p_{RX}}$, a linear combination of Gaussian random variables. (4) is a Gaussian random variable with the following distribution:

Next, we describe how to predict the received signal power $\boldsymbol{y}_{t,p_{Target}}$ at a given target position $p_{Target}$ using $M$ observations $\boldsymbol{y}_{t,p_{RX_{1}}},\boldsymbol{y}_{t,p_{RX_{2}}},...,\boldsymbol{y}_{t,p_{RX_{M}}}$, within the same time slot $t$. Since the observation is a Gaussian random variable, the collection of $M$ observations can be represented as a Gaussian random vector:

The matrix $A$ is the linear combination matrix, given by:

where $I_{M}$ is the $M\times M$ identity matrix and $\otimes$ represents the Kronecker product.

Next, by stacking the the received signal power $\boldsymbol{y}_{t,p_{Target}}$ for the target position $p_{Target}$ into (9), we obtain a new Gaussian random vector:

The mean and covariance matrix can be computed using (8). We denote this distribution as:

Note that from (8) and (9), $\vec{\mu}_{t,Past}$ is a zero vector and $\mu_{t,Target}$ is a zero scalar, respectively.

With (26), the predicted value $\boldsymbol{y}_{t,p_{Target}}$ for the target position $p_{Target}$ can be calculated using the conditional probability formula:

• •

  Prediction Mean:

  	$\displaystyle\text{mean}(\hat{\boldsymbol{y}}_{t,p_{Target}})=$
  	$\displaystyle\Sigma_{t,Target,Past}\Sigma^{-1}_{t,Past}(\vec{y}_{t,p_{RX_{1:M}}}-\vec{\mu}_{t,Past})$		(28)

• •

  Prediction Variance:

  	$\displaystyle\text{var}(\hat{\boldsymbol{y}}_{t,p_{Target}})=$
  	$\displaystyle\Sigma_{t,Target}-\Sigma_{t,Target,Past}\Sigma^{-1}_{t,Past}\Sigma_{t,Past,Target}$		(29)

Notice that in (29), the prediction variance at the selected target position depends only on the target position and the locations of all previously sampled observations.

From the conventional Gaussian prediction, it is shown that if the kernel function is given, we can obtain both the prediction mean and the prediction variance for any given target position. The computational bottleneck arises from the inverse operation of the matrix $\Sigma_{t,Past}$ in (28) and (29). The size of $\Sigma_{t,Past}$ for $M$ observations is $(MRN)\times(MRN)$. As a result, the computational complexity is $\mathcal{O}((MRN)^{3})$. This complexity is particularly high in large-scale radiance fields which entail a large number of observations.

Moreover, in a given scene, certain areas may consist of empty space, where virtual signal sources are expected to be highly correlated. In contrast, other areas may be densely populated with objects such as tables, chairs, and other furniture. In these crowded regions, the correlation between virtual signal sources will differ from that in the empty spaces. As a result, the uncertainty model should be adapted based on the characteristics of the observations. Specifically, we assign low uncertainty to smoother areas and higher uncertainty to regions with sharper variations.

To address the two issues mentioned above, we propose a local kernel estimation strategy that tackles both the computational complexity and the correlation locality problems. For a selected target position $p_{Target}$, we use only the past observations sampled close to $p_{Target}$ to perform the local kernel estimation and predict $\boldsymbol{y}_{t,p_{Target}}$. Specifically, observation $\boldsymbol{y}_{t,p_{RX_{i}}}$ will be used for the local kernel estimation for position $p_{Target}$, if the position of the observation, $p_{RX_{i}}$, is sufficiently close $p_{Target}$, that is:

Here, $L$ is the locality parameter, which can be set based on the complexity of the scene structure or learned from the RF Radiance Fields of other scenes. In this work, we present the concept of local kernel estimation, leaving the determination of the optimal value for $L$ for future work.

Once the observations for the selected target are obtained, we need to estimate the kernel’s length-scale parameter $l$, as shown in (6). However, since we are now performing local kernel estimation, this length-scale parameter $l$ should depend on the position of the target. Therefore, we modify (6) as follows:

Next, we perform maximum likelihood estimation to estimate the local length-scale parameter $l_{t,p_{Target}}$ in (31):

Now that we have estimated the local kernel from the local samples near the selected target, we can apply (28) and (29) to obtain the prediction mean and variance of $\boldsymbol{y}_{t,p_{Target}}$.

Since only $M_{local}$ observations are used for the prediction, the computation complexity is reduced from $\mathcal{O}((MRN)^{3})$ to $\mathcal{O}((M_{loacl}RN)^{3})$. If the observations are uniformly distributed within the room, then the ratio $\frac{M_{local}}{M}$ is approximately$\frac{2\pi L^{2}}{\text{size of the scene}}$, where $L$ is the locality parameter.

To illustrate the concept of local kernel estimation, we use simulated data from the NeRF2 pre-trained model, which models a room measuring $10m\times 6m$ room, , as shown in Fig.4. The room contains 200 randomly distributed observations. We set the locality parameter $L=1m$ for the local kernel estimation. As shown in Fig 2, only 7 observations are used to predict the received signal at the selected target.

Fig. 6 and Fig. 6 shows the reconstruction of the RF Radiance Field with and without our proposed local kernel estimation. The results demonstrate that our method effectively captures the sharp details of the RF Radiance Field, leading to a more accurate reconstruction.

As shown in the previous section, we can predict the mean and variance of the RF Radiance Field at any selected target position based on a given set of observations. However, collecting these observations can be expensive, and some may provide limited or negligible information. In such cases, adding these samples to the reconstruction may have little effect on the model’s accuracy. To address this, we propose an active sampling strategy for collecting more informative observations.

Our approach begins by making an initial prediction of the radiance field using a small number of randomly selected observations. We then adaptively choose the next observation position based on the highest predicted variance, as given by (29). Fig. 4 shows an example of the prediction variance of the received signal power across different positions. After a few initial observations, only certain areas of the scene exhibit high uncertainty (i.e., large variance). Thus, further observations are focused on these high-variance regions to improve the model’s accuracy most efficiently. The proposed method for static RF Radiance Field is summarize in Alg. 1. In this work, instead of only selecting the largest variance position each time, we divide the scene into smaller sections and select the position with the highest variance within each section as the next observation point. This approach helps avoid repeatedly computing the matrix inverse in (28) and (29) by adding only one new observation at a time. Note that some sections may be skipped if the predicted variances of all positions within a section are below a certain threshold. By combining Local Kernel Estimation with Active Sampling, we learn the sharpness of different regions of the RF Radiance Field and focus more observations on the areas with higher sharpness. Since smoother regions can be predicted with fewer samples, active sampling allows us to reduce the overall number of samples.

Now, we explore the potential of our proposed method in a quasi-dynamic environment. We focus on the scenario where the RF Radiance Field of the scene has already been fully measured for time slot $t$, and at time slot $t+1$, some parts of the RF Radiance Field have changed due to environmental dynamics. The received signal power at a target position $P_{Target}$ can be written as:

where $\boldsymbol{e}_{t,,p_{Target}}$ denotes the difference of the value between time slot $t$ and $t+1$ for position $p_{Target}$.

In the previous static case, our proposed Local Kernel Estimation and Active Sampling methods learned the sharpness of the RF Radiance Field, focusing more on the sharp areas. However, when transitioning from time slot $t$ to $t+1$, the sharp areas of the RF Radiance Field might remain the same. If we continue to focus on these areas of sharpness, we may end up observing unchanged regions. Therefore, instead of learning the structure of the new RF Radiance Field, we apply our method to learn the difference between the RF Radiance Fields at time slots $t$ and time slot $t+1$. As shown in (33), we can obtain $\boldsymbol{y}_{t+1,p_{Target}}$ by learning the difference $\boldsymbol{e}_{t,,p_{Target}}$. By focusing on the differences between the two time slots, our method prioritizes regions with significant changes, which is exactly what is needed to adapt to the evolving RF Radiance Field.

Here, we present the results of the experiment for both static and quasi-dynamic RF Radiance Fields.