Brain Inspired Probabilistic Occupancy Grid Mapping with Hyperdimensional Computing

Initially introduced in [9], traditional methods represent the most extensively documented and comprehensively understood approaches for OGM. These methods are characterized by their use of dense arrays of individually parameterized statistical distributions. Moreover, these methods are known for high levels of accuracy with a corresponding high level of determinism. The most well-known traditional method, described in [9], estimates voxel occupancy through a geometric sensor model with a conditional Bayesian update. Building on this approach, [5] extends beyond the traditional approach by introducing a forward sensor model that generates statistical maps with the highest probability of generating the prior observations.

The major limitation of traditional methods is the significant computational and space complexity associated with the learning of dense statistical distributions with time and space complexities of $O(n^{3})$ and $O(n^{2})$, respectively. More recent advancements, such as [10, 11], utilize Bayesian Hilbert Maps (BHMs). BHMs offer reduced runtime complexity for traditional methods by eliminating the need to retain previously observed points and eliminating regularization parameter optimization. Although Fast-BHM [8] managed to lower the runtime complexity of methods developed by Elfes and Thrun [9, 5] to sub-quadratic levels, runtime complexity continues to pose a significant challenge.

Neural methods, also known as neural implicit representations, leverage deep learning and back-propagation to map perceptions into implicit representations [12]. Various approaches are being explored in this domain, including Neural Radiance Fields (NeRF) [13], occupancy networks [14], as well as traditional convolutional and recurrent neural networks [15, 16]. While neural methods have shown remarkable capabilities, their implicit operational characteristics pose challenges in terms of validation for safety critical applications and knowledge-transferability across diverse domains [7].

Quasi-neural methods emerge at the intersection of traditional methods and neural methods, aiming to harness the increased computational efficiency from neural methods while retaining the accuracy and interpretability of traditional methods. One notable quasi-neural method is [7] where a shallow convolutional neural network (CNN) and a sparse kernel are combined to update a three-dimensional semantic occupancy map by post processing previously labelled semantic point clouds. A major limitation of this work lies in the shallow CNN architecture, which necessitates retraining with domain shifts. Taking inspiration from recent works in cognitive science [19] and connectionist models [20], VSA-OGM is a foundational approach to quasi-neural OGM based on biologically-inspired hyperdimensional mathematics.

Regardless of the perception capabilities of any given robotic system, its perspective of the world should be projected to a stable reference frame for accurate map building across individual or multiple agents. This projection is a non-trivial task with a rich collection of literature on the topic [32]. This process is commonly achieved through a linear transformation calculated based on inertial measurement units, and localization methods to ensure differentiability between successive data samples [33]. To remove any additional noise from these calculations in our experiments, all data generated within simulated environments are natively located on the global reference frame. Moreover, all real-world datasets are preprocessed to the global reference frame as in [8].

Robotic systems sample multiple point clouds $\{\lambda_{0},\lambda_{1},...,\lambda_{t}\}$, with the coordinates projected to the global reference frame at time step $0$. For each $t$, $\lambda_{t}$ consists of a point vector $\gamma$ and an attribute vector $\psi$ of labels. All experiments in this work limit the environmental dimensionality of $\gamma$ to $2$ but this can be generalized to support higher dimensional problem spaces. We limit the dimensionality of $\psi$ to $1$, where binary values such that $\psi\in\{0,1\}$ specifies if a voxel is occupied $\psi=1$ or empty $\psi=0$.

Multiple properties of the VSA system must be specified before any point clouds can be processed. The most critical parameter is the individual vector dimensionality $d$, which has a direct impact on the noise resilience and information capacity of the hypersphere [19]. We explore the impact of this parameter in Section IV-B1. Length scale $l$ should be adjusted according to the desired map resolution. We explore the impact of this parameter in Section IV-B2. Encoding n-dimensional spatial information requires the specification of an axis vector $\phi_{axis}$ per dimension. In the case of our 2D problem, we define axis vectors $\phi_{axis}^{x}$ and $\phi_{axis}^{y}$.

Inspired by recent understanding of chunking where mammalian brains separate large scale spatial phenomenon into different portions of memory [34], we create an abstraction of this theory to separate the global space into a matrix of hypercubes. Specified by $\delta$, we split each dimension into $\delta$ evenly space chunks. To further discretize the environment, each chunk is also separated on a class-by-class basis. As such, for a given experiment with $\delta$ quadrants per dimension and $2$ classes, there are $\delta^{n}*2$ memory vectors, $\omega_{i}$, with $d$ dimensionality:

We explore the impact of this approach in Section IV-B3 where we evaluate multiple parameter values and determine the required VSA dimensionality.

For each time step $t$, a point cloud $\lambda_{t}$ contains $j$ points and labels in $\gamma$ and $\psi$, respectively. Across all $j$, $\gamma_{i}$ contains an $n$-dimensional matrix of coordinates and $\psi_{i}$ contains a vector of attributes. All results in this work assume $\gamma_{i}$ is a 2D vector across the $x-y$ plane and $\psi_{i}$ contains one binary attribute specifying occupancy.

Each point $\gamma_{i}$ in $\lambda_{t}$ is encoded into a set of location vectors $\{\phi_{0},\phi_{1},...,\phi_{j}\}$ by fractionally binding along each individual dimension of $\gamma_{i}$ and using circular convolution to bind the individual dimensions together as a single vector $\phi_{i}$ with a variable length scale $l$ by expanding Equation 6:

This process is repeated for all $j$ points in $\gamma$ resulting in a set of $j$ location vectors representing each point in $\lambda_{t}$ encoded into hyperdimensional space.

Given the set of location vectors $\{\phi_{1},\phi_{2},...,\phi_{j}\}$ and their corresponding labels $\{\psi_{1},\psi_{2},...,\psi_{j}\}$ from $\lambda_{t}$, we follow Algorithm 1 to add the $j$ points to the memory $\beta$, corresponding to the correct quadrant and class. The process begins by calculating the centers of each quadrant: $\mu$. Looping over each point, we determine the corresponding quadrant center with the minimum Euclidean distance to the given point and add the point vector to that quadrant’s memory representing the point’s label $\psi_{i}$.

With all labeled points fully realized in each quadrant memory, we can query individual quadrant memories for specific information about the environment. For occupancy grid mapping, we are primarily concerned with two main types: (1) what portions of the map are occupied, and (2) what parts of the map are unoccupied or empty. Given our fully specified quadrant memory vectors $\{\omega_{1},\omega_{2},...,\omega_{\delta^{n}*2}\}$, we can make inferences about occupancy and emptiness through another series of deterministic vector operations and a novel application of Shannon entropy [23].

Throughout the continuous operation of any SSP-based VSA system, successive bundling will slowly increase the magnitude of the memory vectors. Without correction, this will drastically reduce information capacity as the unit-modulus property is violated [19]. Therefore, a normalization operation transforming the memory vector back to unit length is required before querying any information:

With the quasi-probabilistic property of our VSA framework, we calculate the probability of a coordinate $\gamma_{i}$, located within a quadrant $\beta$, with a class label $\psi_{i}\in\{0,1\}$ as:

given a location vector $\phi_{loc}$ representing $\gamma_{i}$ in hyperdimensional space as described in Equation 10. Depending on the specific application and the amount of available memory, the computational complexity of calculating $\phi_{loc}$ in Equation 12 can be minimized by precalculating a matrix of location vectors and storing them in memory. The class of a region is calculated by expanding Equation 12 across a matrix of $\phi_{loc}$.

Even with the implementation of vector normalization as described in Equation 11, the inherent noise within VSAs makes quantitatively decoding the information non-trivial. This leads many hyperdimensional computing systems to use some form of clean-up memory or associative memory to transform the noisy representations into more clearly defined and linearly separable representations [28].

With our focus of maximizing the interpretability and minimizing the stochasticity of our system, we decided to take a radically different approach. Shannon’s theory of information has revolutionized modern communication systems and allows us to quantify information within noisy signals [23]. Recently, many works in material sciences [35] and radar imaging [36] use this well studied approach to extract feature rich information from noisy or otherwise incomprehensible data.

Taking inspiration from these works, we employ an entropy-based mechanism to extract labeled spatial information from the probability heat maps calculated by applying Equation 12 across the entire map. To avoid violating the axioms of probability assumed in information theory [37], we convert the quasi-probability values from our VSA system to true probability values with a loose interpretation of the Born rule [38] where each true probability is equal to the squared quasi-probability. Using disk filters with radii $r(\psi=1)$ and $r(\psi=0)$ for both classes, we compute the local Shannon entropy $\mathcal{S}$ within a radius of any heat map voxel at $(x,y)$ representing the $\mathcal{S}(\psi|x,y)$. This process is repeated across all voxels resulting in global class-wise entropies that increase class separability and reduce background noise. Globally, we subtract the entropy values of occupancy probability from the entropy values of empty probability resulting in a final entropy matrix $\mathcal{S}(\psi)$. This process is highlighted visually in Figure 2 and programatically in Algorithm 2. The specification of $r(\psi=1)$ and $r(\psi=0)$ and the implications thereof are explored in our ablation study in Section IV-B4.

With $\mathcal{S}(\psi)$, we classify each point $(x,y)$ as:

where $\rho$ is the decision threshold. The sensitivity of our method to the proper specification of $\rho$ is highlighted with the area under the receiver operating characteristic curve [39] in Section IV-A.

Our hyperdimensional occupancy grid mapping system supports single agent mapping and multi-agent mapping through memory fusion. Given a set of $n$ independently operating agents $\{\Gamma_{1},...,\Gamma_{n}\}$, their quadrant memory vectors can be merged if and only if three conditions are satisfied. First, each agent $\Gamma_{i}$ must be operating on the same axis basis vectors for all dimensions of their operating environment. In this work focusing on two-dimensional occupancy grid mapping, all agents must have the same axis basis vectors $\phi_{axis}^{x}$ and $\phi_{axis}^{y}$. Secondly, each agent should be learning the same number of classes with each class having the same $\psi$ value. Lastly, each agent must use the same value of $\delta$ to separate their environment into $\delta$ distinct chunks along the global reference frame.

Assuming all three conditions are met, their memories can be merged into a cohesive set of quadrant memories representing the collective knowledge amongst the group. The process begins by initializing an empty set $\Omega$ that will contain the unique memory vectors within all agents. Now looping over each agent, the following steps are followed. First, check if all quadrant memories within the agent are represented within $\Omega$. Second, if all memories exist within $\Omega$, proceed to the third step. Otherwise, empty memory vectors are initialized within $\Omega$ corresponding to the unique memory vectors within the current agent. Third, use Equation 5, to bundle each memory vector within the current agent with the global memory vector within $\Omega$. Lastly, repeat this process until all agent memories have been accounted for in $\Omega$. This approach is also highlighted programmatically in Algorithm 3.

We created a synthetic LiDAR dataset with a re-implementation of the simulator introduced in BHM [11]. This dataset uses a single agent equipped with a single LiDAR sensor having a $360^{\circ}$ field of view, a view distance of $20$ meters, and $50$ individual beams. The total area of the environment is $100\mbox{m}^{2}$.

The dataset contains $4500$ individual points with a class distribution of $60\%$ occupied and $40\%$ empty. We train VSA-OGM and BHM [10] on $90\%$ of the dataset with $10\%$ reserved for validation where the final AUC score is determined.

This dataset is particularly difficult due to the sparse features. Many applications of probabilistic occupancy grid mapping avoid this issue by augmenting the training data. This is accomplished by interpolating a linear line with random points between the agent’s location and the LiDAR points [10]. We chose to avoid this preprocessing step to highlight how theoretical methods and VSA-OGM respond to data sparsity.

With a vector dimensionality of $32000$, a length scale of $2$, occupied and empty disk filter radii of $4$ and $10$ voxels, and $2$ quadrants per dimension, VSA-OGM achieves an AUC score of $0.959$ versus BHM with a score of $0.978$. We were unable to find a working GPU-compatible implementation of BHM.

Despite adjusting the parameters for BHM, we were unable to build an occupancy grid map that successfully interpolates between sparse points to broadly classify empty regions with no training or testing points. This is shown in Figure 3 where the majority of the map is labelled as unknown, highlighted by the teal color. On the other hand, VSA-OGM was able to successfully interpolate between the empty points and label the majority of the area as empty. Our increased interpolation performance also comes with significantly reduced inference latency at approximately 240x (CPU) and 2800x (GPU) reductions versus BHM (CPU) while maintaining comparable accuracy on the training and validation sets. Furthermore, our approach reduces the total memory size of the learned representation by 800x. These results are summarized in Table III. Furthermore, the ground truth representation and trained occupancy grid maps are visualized in Figure 3.

We created a second dataset from our re-implemented simulation environment from [11]. Rather than being focused on a single agent, this experiment simulates the accumulation of data from two independent agents operating within the same environment. As such, each will have a unique perspective of the environment that should help improve the collective world model.

Both agents are equipped with a single LiDAR sensor having a $360^{\circ}$ field of view, a view distance of $20$ meters, and $50$ beams. The total area of the environment is $100\mbox{m}^{2}$. The data sequences from both agents contain $18350$ points across $367$ individual point clouds. That leads to a total of $36700$ points across $734$ point clouds. The class distribution for agent 1 is $48.9\%$ occupied and $51.1\%$ empty with agent 1 having $53.1\%$ and $46.9\%$, respectively.

The same configuration parameters are used from Section IV-A1 except that agent 1 has $r(\psi=1)=5$ voxels and $r(\psi=0)=6$ voxels versus agent 2 with $r(\psi=1)=4$ voxels and $r(\psi=0)=10$ voxels. Therefore, both agents are equipped with identically configured hyperdimensional mapping systems with the only difference being the disk radii. The radius values were selected through cross validation on the validation set. Both agent’s are trained simultaneously on $90\%$ of their training data along with fusion occurring at every time step. The AUC score is calculated across both agent’s individually with agent 1 achieving $0.975$ and agent 2 with $0.976$. Furthermore, the final fused map achieves a score of $0.974$, suggesting that our fusion methodology results in negligible information losses while combining the unique perspectives of each agent into an accurate representation of the entire environment. The individual training results amongst both agents are shown in Figure 4 with the fused results highlighted in Figure 5 and quantitative results in Table IV.

The Intel Campus dataset [25] is a commonly used, real-word dataset for benchmarking semantic mapping algorithms. It contains $383118$ data points with $910$ point clouds collected over an area $40$ meters by $36$ meters. The data has a class distribution of $42\%$ occupied and $58\%$ empty. Using a vector dimensionality of $200000$, a length scale of $0.2$ meters, occupied and empty disk filter radii of $1$ voxels, and $2$ quadrants per dimension, our hyperdimensional mapping system achieves an AUC score of $0.95$ with an average per point cloud processing time of $125$ms versus Bayesian Hilbert Maps (BHM) [10] with a score of $0.96$ and per frame processing time of $22425$ms. Furthermore, Fast Bayesian Hilbert Maps (Fast-BHM) [8], assuming independence between model weights, achieved an AUC score of $0.95$ with a per frame processing time of $463$ms. These results are summarized in Table V with the ground truth representation and trained occupancy grid maps visualized in Figure 6. With an inference latency of $125$ms, our method achieves comparable performance to the baseline methods while reducing inference latency by approximately $180$x and $4$x respectively. Our method reduces the runtime of Fast-BHM [8] by approximately 4 times while maintaining the covariance between voxels. We also see a 1000x memory reduction compared to BHM [10]. Compared to Fast-BHM [8] we do not see a memory improvement because they assume independence between points which itself drastically reduces the model’s memory requirement.

Based on the multi-map fusion technique described in [8], we split the Intel dataset [25] into $4$ quadrants across the $x-y$ plane. The final splits of the global map are highlighted in Figure 7.

As summarized in Table VI, quadrant 1 (upper-right) contains $95301$ points with a class distribution of $38.83\%$ occupied and $61.16\%$ empty. Quadrant 2 (upper left) contains $99142$ points with a class distribution of $43.35\%$ occupied and $56.64\%$ empty. Quadrant 3 (lower-left) contains $120135$ points with a class distribution of $42.29\%$ occupied and $57.70\%$ empty. Quadrant 4 (lower right) contains $68540$ points with a class distribution of $42.05\%$ occupied and $57.94\%$ empty.

Similar to Section 2, each of the four agents are equipped with identically configured hyperdimensional mapping systems. Each system is initialized with a vector dimensionality of $200000$, a length scale of $0.2$ meters, disk radii of $1$ voxel, and $2$ quadrants per dimension. Individually, the final AUC values on the $10\%$ validation set are 0.887, 0.927, 0.930, and 0.916. Furthermore, the global fused AUC was 0.915 compared to Fast-BHM [8] with a final AUC of $0.936$. Therefore our approach achieves comparable performance to Fast-BHM in multi-agent fusion. These values are summarized in Table VII with the training data and occupancy grid maps visualized in Figure 7.

Introduced in [12], Evidential Lidar Occupancy Grid Mapping (EviLOG) uses a multi-layer convolutional neural network architecture with $4.7$ million trainable parameters. They demonstrate the ability of their system to perform localized occupancy grid mapping. They introduce a fully-labelled synthetic dataset using a VLP32C lidar sensor mounted on the roof of the moving vehicle.

The EviLOG dataset is separated into three sets for training, testing, and validation with each containing $10000$, $100$, and $1000$ point clouds (PCs), respectively. Each PC is bounded within a $56.32$ by $81.92$ meter rectangular region. Furthermore, each PC has a corresponding ground truth occupancy grid map with a voxel resolution of $0.16$ meters.

Training EviLOG for 100 epochs with a batch size of 2000 took approximately $30$ hours. This comes in stark contrast to our approach where no pre-training is required and represents a major limitation of neural methods where the training process induces domain shift sensitivity. Furthermore, our method has a per frame inference time of 98ms versus EviLOG at 153ms. Therefore without pretraining, VSA-OGM is approximately 1.5x faster than EviLOG.

With a vector dimensionality of 32000, length scale of 0.2, an axis resolution of 0.1, 1 quadrant per dimension, and disk radii of $r(\psi=1)=2$ and $r(\psi=0)=4$, our method achieves a mean AUC score of 0.977, recall of 0.94, precision of 0.92, and F1-Score of 0.93 across the testing set. This results in a final model size of 0.25MB versus 18.9MB for EviLOG representing a 75x memory reduction with VSA-OGM.

EviLOG [12] calculates performance based on the KL-divergence across the entire domain, including areas with no labelled points so we are unable to compare via quantitative analysis as in Section 3. However, qualitative (visual) analysis provides a clear story of the differences between our quasi-neural approach and their neural approach.

As shown in Figure 8, the input point clouds are sparser versus any of the prior experiments. EviLOG’s neural network begins with a PointPillars architecture [40] before passing the latent representation series of convolution layers. This architecture allows EviLOG to better interpolate between points that are outside of the observable domain. This is clearly apparent when comparing the output of their network versus the global entropy map $\mathcal{S}(\psi)$ from our VSA method in Figure 8. While our approach accurately interpolates between the points with a higher density in the vicinity of the moving vehicle, it fails to fully model the distant environment where points are extremely sparse. Their neural network was trained on a dataset from the same domain to extrapolate the unseen structure from sparse, unlabelled data points. This comes in contrast to our method where we make no prior assumptions about the observable space but requires a point labelling heuristic. This is a limitation of our quasi-neural method, however, prior experiments show that our method would be able to fully learn this environment provided adequate time to traverse the domain and sample more points.

Vector dimensionality $d$ plays a critical role in the design of VSA based systems. Intuitively, increasing this value should have a positive correlation with model accuracy and noise resilience. Despite the ever increasing power of the cloud, this parameter should be carefully considered as it will have a direct impact upon operational characteristics at the edge.

The ablation dataset consists of $5000$ randomly generated occupied points located along a circle with a radius of $4$ meters. Furthermore, we include an additional $5000$ empty points uniformly distributed within the center of the circle. Consisting of $10000$ points, this allows us to evaluate the qualitative and quantitative impacts of vector dimensionality.

As shown in Figure 9, we have unique columns for every discrete dimensionality value along with a row for each output of our VSA pipeline. This begins with the $Pr(\psi=0)$ and $Pr(\psi=1)$ as well as the global combined quasi-probability $Pr(\psi)$ calculated as $Pr(\psi=1)-Pr(\psi=0)$. The length scale for each test is fixed at $0.2$ meters. We also include the class-wise and global entropy values calculated with $r(\psi=1)=r(\psi=0)=2$.

We conduct a search over a set of vector dimension sizes ranging from $2^{2}$ to $2^{15}$. Qualitatively, it is clear that increasing the dimensionality has an improvement on the visual quality of the decoded heat maps. Quantitatively, performance improves as dimensionality increases. At the lower end with $d=2^{2}$, our classifier achieves an AUC score of 0.511, suggesting that this dimensionality results in a classifier that is better than random guessing but far from being considered high quality. Increasing through the powers of 2 until $2^{15}$, we see final AUC values between $0.511$ and $0.975$. Beyond $d=2^{12}$, the AUC score remains constant within the margin of error.

This suggests that $d=2^{12}$ is adequate for high classification performance on this dataset. However, if a given application does not require this level of classification performance or spatial fidelity, users can reduce the vector dimensionality to achieve their desired performance characteristics. The major takeaways from this experiment are (1) our VSA method has the ability to accurately perform occupancy grid mapping with (2) predictable variations as dimensionality is modified. This confirms the stability of our method despite changes in the operating configuration.

The length scale parameter $l$ is used during the fractional binding process, as shown in Equation 8, to adjust the width of the kernel around a given point. This process can be extended into more than one dimension as in Equation 10. Depending upon the specific application, the proper specification of $l$ is vital as it directly impacts the resolution and noise resilience of the decoded probability and entropy maps. As shown in Figure 10, increasing the length scale has a positive correlation with noise resilience but also corresponds to a decrease in resolution. On the other hand, decreasing the length scale will increase resolution but also increase noise sensitivity.

While our hyperdimensional occupancy grid mapping approach could be fully realized in a single, global memory vector $\omega_{global}$, this would be impractical for real-time distributed systems with power, computational, and memory constraints. These issues arise as a result of the ever increasing vector dimensionality required to accurately model larger and more intricate environments.

Recent research efforts hypothesize the existence of a chunking mechanism within the mammalian brain that break larger spatial phenomenon into smaller, more easily comprehensible chunks or quadrants [34]. Our implementation of chunking comes with the specification of $\delta$ that quantifies the number of discrete regions along each axis. Therefore, increasing $\delta$ will result in an exponential increase in the number of hypercubes in the $n$-dimensional space. Operating in 2D, $n=2$, with two distinct class labels $\psi\in\{0,1\}$ results in a total number of quadrant memories calculated as $\delta^{n}*2$.

The proper specification of $\delta$ is critical as too small of a value results in fewer quadrant memories with each having increased dimensionality. Conversely, a $\delta$ value that is too high will fail to properly account for the probabilistic relationships of nearby points in adjacent quadrants.

As shown in Figure 11, we evaluate the specification of $\delta$ in two scenarios with vector dimensionalities of $d=32000$ and $d=256$. Each experiment uses the same dataset as the prior ablation sections with an area of 8 meters by 8 meters with an axis resolution of 0.2 meters. We specify the length scale and disk radii as 0.2 meters and 2 voxels, respectively.

Beginning with $d=32000$ across a discrete range of $\delta$ values $\{1,2,4,6\}$, all experiments achieve a final AUC score greater than 0.98. This suggests that for this specific problem, a vector dimensionality of 32000 is adequate to accurately model and classify the entire observed space. On the other hand, a qualitative assessment of the decoded probability and entropy maps suggests there is a remarkable improvement in visual quality and overall clarity as $\delta$ increases. This discrepancy should be taken into account depending upon specific applications that simply require a high level of classification performance or also require a high level of visual fidelity.

To show how the proper specification of $\delta$ can be used to model spaces with orders of magnitude less dimensionality, we conduct the same experiment across the same range of $\delta$ values with the vector dimensionality being reduced to $256$. As shown in Figure 11, the qualitative performance of this configuration transforms from completely inaccurate and imperceivable to accurate and perceivably modeling the environment. An important thing to notice is the slight addition of artifacts along the quadrant boundaries, this is due to the discretization of the environment where the crossover of additional information in adjacent quadrants is not reflected.

The visual interpretation of increasing performance is reflected in the final AUC scores with a range of $0.655$ to $0.986$. The major takeaway from this study is that the specification of $\delta$ should be considered in tandem with the specification of vector dimensionality based upon the complexity and goals of the mapping problem along with the available compute hardware.

To study the specification of the local entropy disk filter radii, we transition to a more complex, real world dataset from [25]. Depicting the interior of the Intel Campus in Seattle, this dataset has multiple corridors, rooms, and main pathways. Without any preprocessing, this results in the main pathways being sampled multiple times versus the rooms and corridors providing much less information. This is shown in Figure 12, where the probability of the individual classes along the main corridor are clearly visible with respect to the baseline noise. This comes in stark contrast to the reduced clarity of the decoded information within the corridors and rooms.

We employ localized Shannon entropy [23] to illuminate information encoded within sparser areas and increase qualitative performance and class separability. This operation requires the specification of disk radii for both classes. Specified as $r(\psi=0)$ and $r(\psi=1)$, these values have a drastic impact on the decoded map quality.

Figure 12 visually displays the localized entropy maps for both classes. Specifically in the case of $\psi=1$, the entropy map clearly illuminates information in the less sampled portions of the environment. A similar effect occurs when $\psi=0$ however the effect is less pronounced. We evaluate all combinations of disk radii in $\{1,2,3,4\}$.

At lower radii values, we see the decoded entropy maps are quite sharp along the edges, this is in contrast to higher values where the maps have a blurred effect. At the center of the figure are the combined global entropy maps with the different combinations of disk radii. Compared to the original probability maps, we see radii in the range of 1 to 2 provide the most clarity.

Quantitatively, the final AUC scores reflect our qualitative analysis as $r(\psi=1)=r(\psi=0)=1$ results in a final score of 0.955 versus $r(\psi=1)=r(\psi=0)=4$ with 0.732. These results are also summarized in Table VIII. This suggests our addition of localized Shannon entropy is not overly sensitive to parameter specification and provides predictable performance variations with parameter shifts.