Radio Map Estimation (RME) Overview

• Radio Map Estimation (RME) is the process of reconstructing spatial radio metrics from sparse measurements, often integrating side-information like maps or transmitter details.
• Methodologies span from classical interpolation and grid-based deep completion to grid-free and Bayesian approaches, addressing various sensing scenarios.
• Empirical studies and benchmark evaluations reveal trade-offs between model complexity, training data needs, and estimation accuracy in practical deployments.

Radio Map Estimation (RME), also termed spectrum cartography in parts of the literature, is the inference problem of reconstructing a spatial field of radio metrics—most commonly received signal strength, received power, power spectral density, or path loss—over a geographic region from sparse measurements, often together with side information such as building maps, transmitter descriptors, or environmental priors. Contemporary RME spans continuous-space interpolation, grid-based map completion, multi-frequency tensor recovery, spatio-temporal-spectral estimation, cooperative and distributed sensing, and Bayesian posterior inference; the field now includes both classical interpolation/model-based estimators and learned generators, transformers, unrolled optimization networks, and diffusion models (Teganya et al., 2020, Viet et al., 2024, Ha et al., 8 Aug 2025).

1. Problem formulations and mathematical structure

At its most basic, RME seeks a map $p(\mathbf{x})$ over a region $\mathcal{A}\subset\mathbb{R}^d$ from irregular measurements $\{(\mathbf{x}_m,y[m])\}_{m=1}^M$, with observation model $y[m]=p(\mathbf{x}_m)+n_m$ in continuous formulations (Viet et al., 2024). In grid-aware formulations, the target area is discretized into an $H\times W$ or $R_1\times R_2$ lattice, and the problem becomes reconstruction of a matrix or tensor from sparse observed entries. Examples include a single-band map $\mathbf{P}\in\mathbb{R}^{R_1\times R_2}$, a multi-frequency tensor $\mathcal{D}\in\mathbb{R}^{H\times W\times K}$, and a 4D spatio-temporal-spectral tensor $$\mathbf{\Phi}\in\mathbb{R}^{N_x\times N_y\times N_t\times N_f}$$ (Zhang et al., 2024, Wang et al., 7 Feb 2025, Yang et al., 15 Feb 2026).

The estimated quantity varies across subliteratures. Some works focus on received signal power or RSS, others on PSD, and others on path loss or large-scale fading. The RMDirectionalBerlin benchmark defines the target as path loss for a fixed transmitter over receiver locations on a $256\times256$ grid, with values clipped to $\mathcal{A}\subset\mathbb{R}^d$0 dB and linearly rescaled to $\mathcal{A}\subset\mathbb{R}^d$1 (Jaensch et al., 2024). Other works define the radio map as received power $\mathcal{A}\subset\mathbb{R}^d$2 over multiple bands, or as aggregate PSD $\mathcal{A}\subset\mathbb{R}^d$3 (Guo et al., 21 Nov 2025, Yang et al., 15 Feb 2026).

A major axis of variation is what side information is assumed. Transmitter-aware methods use transmitter locations, power, or beam descriptors together with map geometry. Cooperative methods instead infer the full field from sparse user-side RSS and a geographic map, explicitly removing the need for transmitter locations, number, or transmit powers at inference time (Zhang et al., 2024). Other works assume building maps, depth maps, aerial images, transmitter-location maps, or k-nearest-neighbor graph structure over visible measurements (Jaensch et al., 2024, Guo et al., 21 Nov 2025, Yang et al., 15 Feb 2026). Bayesian formulations shift the target further: instead of only estimating $\mathcal{A}\subset\mathbb{R}^d$4, they aim to infer the posterior $\mathcal{A}\subset\mathbb{R}^d$5 of the entire map given measurements, so that arbitrary functionals $\mathcal{A}\subset\mathbb{R}^d$6 can be estimated through $\mathcal{A}\subset\mathbb{R}^d$7 rather than by plug-in evaluation $\mathcal{A}\subset\mathbb{R}^d$8 (Ha et al., 8 Aug 2025).

Two practical distinctions recur throughout the literature. The first is grid-agnostic versus grid-aware estimation. The real-data study on UAV and 4G measurements shows that grid-aware formulations smooth out small-scale fading and are generally easier, whereas grid-agnostic formulations seek to predict raw held-out measurements directly (Shrestha et al., 2023). The second is whether the target is static or dynamic. FM-RME explicitly generalizes RME from static spatial estimation to joint estimation over space, time, and frequency, motivated by dynamic transmitters and changing spectrum environments (Yang et al., 15 Feb 2026).

2. Structural priors and theoretical analyses

Theoretical work has clarified that radio maps are neither arbitrary images nor trivially smooth surfaces. For free-space power maps, one line of analysis models

$\mathcal{A}\subset\mathbb{R}^d$9

and shows that the relevant function class is rich enough that exact finite-sample recovery cannot be expected in general, while still possessing strong regularity: derivatives are bounded, pointwise variation is controlled, and the spatial Fourier spectrum decays exponentially (Romero et al., 2023). A central scalar complexity measure is the proximity coefficient

$\{(\mathbf{x}_m,y[m])\}_{m=1}^M$0

which quantifies how nearby transmitters increase RME difficulty. This analysis also derives reconstruction-error bounds for nearest-neighbor, linear, and sinc interpolation, and empirically corroborates on ray-tracing data that larger transmitter distance tends to reduce NMSE (Romero et al., 2023).

A second theoretical direction concerns low-dimensional recovery under coarse or partial observations. Quantized RME is formulated as recovery of a spatio-spectral tensor from heavily quantized sensor reports, using either a block-term tensor decomposition or a deep generative model under a Gaussian quantizer and maximum-likelihood estimation. Recoverability guarantees are derived under partial sampling, quantization, noisy measurements or dithering, and model mismatch, thereby extending prior full-resolution theory to a more realistic communication setting (Timilsina et al., 2023). Related low-rank work models multi-frequency radio maps as a low-rank tensor plus sparse residual and unrolls an ADMM-style solver into a trainable network, explicitly preserving the meaning of low-rank shrinkage, sparse shrinkage, data consistency, and multiplier updates (Wang et al., 7 Feb 2025).

Recent theory has also turned to generative models. Diffusion-based RME is recast as a non-linear matrix completion problem, and theoretical lower bounds are derived for ultra-low sampling regimes. In that view, the minimum achievable estimation error is governed not only by sparsity but by the discrepancy between the deployment distribution and the true underlying radio propagation law learned at training time. The same framework derives a critical sampling-rate threshold above which further measurements yield limited gains, and proposes empirical surrogates based on observable quantities such as obstacle layout and local power variation (Liu et al., 24 Jun 2026). This suggests that, for diffusion priors, adding sensors cannot compensate indefinitely for deployment-distribution mismatch.

A further theoretical development addresses sampling-distribution shift itself. Under UAV-style trajectory-based sensing, the observation set is no longer i.i.d. random over the map but generated sequentially along feasible paths. The trajectory-aware analysis models the field as a Gaussian random field and the sparse observation set as a random subset $\{(\mathbf{x}_m,y[m])\}_{m=1}^M$1, then decomposes performance loss into an intrinsic difficulty term and a distribution-mismatch term. Using pairwise-distance statistics and a log-determinant mutual-information expansion, it argues that trajectory-based sampling yields more short-distance pairs, greater redundancy, and lower effective information than random sampling of the same cardinality (Qiu et al., 27 May 2026).

3. Data, benchmarks, and empirical evaluation protocols

RME has historically depended heavily on simulated data, but benchmark diversity has expanded. The open dataset RMDirectionalBerlin contains 74,515 radio maps on 424 city maps from Berlin, generated by 3D ray tracing with directional Tx antennas, vegetation, and aerial imagery, over $\{(\mathbf{x}_m,y[m])\}_{m=1}^M$2 regions at $\{(\mathbf{x}_m,y[m])\}_{m=1}^M$3 m resolution (Jaensch et al., 2024). RadioMapSeer remains a canonical benchmark for single-band urban RME, and several later works use derived cooperative or trajectory-based versions of it (Zhang et al., 2024, Qiu et al., 27 May 2026). BRAT-LabW provides a multiband benchmark with 340 radiomaps at five frequencies $\{(\mathbf{x}_m,y[m])\}_{m=1}^M$4 GHz, while the BART-Lab Radiomap Dataset is used for multi-frequency tensor completion and unrolled reconstruction (Guo et al., 21 Nov 2025, Wang et al., 7 Feb 2025).

The field now includes real measurement datasets as well. The real-world validation study collects 18 USRP-based UAV measurement sets and a separate real 4G dataset, and uses them to compare classical interpolators and DNNs under four metrics. A main conclusion is that estimation error is reasonably small even with few measurements, and that deep neural networks require large training datasets to exhibit a significant advantage over traditional methods (Shrestha et al., 2023). STORM is also validated on one ray-tracing dataset and two real-measurement datasets, including UAV-USRP data and 4G cellular measurements (Viet et al., 2024).

The following resources are repeatedly used in the literature.

Dataset/resource	Characteristics	Representative use
RMDirectionalBerlin	3D Berlin geometry, vegetation, directive Tx antennas, aerial imagery, $\{(\mathbf{x}_m,y[m])\}_{m=1}^M$5 maps	Input-feature and architecture studies (Jaensch et al., 2024)
RadioMapSeer	Single-band urban benchmark, $\{(\mathbf{x}_m,y[m])\}_{m=1}^M$6, multiple cities	CNN, GAN, cooperative, distributed, and shift studies (Zhang et al., 2022)
BRAT-LabW	Multiband $\{(\mathbf{x}_m,y[m])\}_{m=1}^M$7 benchmark at five frequencies	Knowledge-guided multiband RME (Guo et al., 21 Nov 2025)
USRP / 4G real datasets	Large real-measurement RME datasets	Empirical validation and estimator comparison (Shrestha et al., 2023)
SpectrumNet	Open-source 3D radio-map repository	Trajectory-based sampling-shift study (Qiu et al., 27 May 2026)

Evaluation protocols differ according to formulation. Grid-aware CNN papers usually report RMSE, NMSE, MSE, PSNR, and sometimes outage error on held-out map pixels or entries (Teganya et al., 2020, Wang et al., 7 Feb 2025). Grid-free estimators such as STORM use patch-based RMSE over arbitrary query locations rather than raster completion error (Viet et al., 2024). The RMDirectionalBerlin benchmark reports RMSE in grayscale and NMSE after conversion to dB, with the conversion $\{(\mathbf{x}_m,y[m])\}_{m=1}^M$8 due to the $\{(\mathbf{x}_m,y[m])\}_{m=1}^M$9 dB dynamic range (Jaensch et al., 2024). Trajectory-aware work evaluates under matched and mismatched train/test sampling distributions, making the sampling process itself part of the benchmark definition (Qiu et al., 27 May 2026).

4. Methodological families

Methodologically, RME spans several distinct but increasingly overlapping families.

Classical interpolation and model-based methods include kriging, kernel interpolation, kernel ridge regression, K-nearest neighbors, radial basis interpolation, and model-based path-loss fitting. They remain relevant because they are simple, interpretable, and often competitive when training data are scarce. The real-world study concludes that deep neural networks necessitate large volumes of training data to exhibit a significant advantage over more traditional methods, and simple Kriging appears especially versatile in practice (Shrestha et al., 2023). At the other end of the model-based spectrum, ray tracing offers high-fidelity propagation simulation but requires detailed environment and transmitter information and is computationally expensive (Zhang et al., 2024).

Grid-based deep completion models treat sparse measurements as partially observed images or tensors. Deep Completion Autoencoders recast RME as tensor completion with masks, using a fully convolutional encoder-decoder that learns spatial propagation structure from past environments rather than only interpolating the current measurements (Teganya et al., 2020). Subsequent U-Net-style models, including RadioUNet-derived architectures, use map geometry, transmitter-location images, or other auxiliary channels to infer dense path-loss or power fields (Jaensch et al., 2024, Zhang et al., 2022). GAN-based models strengthen this line by adding an adversarial realism prior. RME-GAN combines a log-distance path-loss interpolation prior with a conditional GAN in a two-phase training scheme, first extracting global propagation patterns and then learning local shadowing and obstacle effects (Zhang et al., 2022). GAN-CRME removes transmitter metadata altogether and reconstructs the map from sparse user-side RSS and a geographic map, using a conditional GAN with a UNet generator and a CNN discriminator (Zhang et al., 2024).

A second family imposes explicit low-dimensional or optimization-grounded structure. DULRTC-RME models multi-frequency maps as a low-rank background plus sparse foreground and unrolls an ADMM/tensor-completion algorithm into a 10-stage trainable network, thereby positioning itself between purely model-based completion and black-box CNNs (Wang et al., 7 Feb 2025). Quantized SC adopts either block-term tensor decomposition or a deep generative model for emitter-specific spatial loss fields, and estimates the map by maximum likelihood under ordinal quantized observations (Timilsina et al., 2023). These methods are explicitly motivated by interpretability, data efficiency, or recoverability theory rather than only empirical accuracy.

A third family is grid-free or continuous-coordinate estimation. STORM formulates RME as set-to-point interpolation from irregular measurements, using translation-invariant query-centered coordinates and a canonical rotation to obtain a gridless transformer with full spatial resolution, translation equivariance, rotation equivariance, and about 100k parameters in the default configuration (Viet et al., 2024). CGFormer is also grid-free, but replaces STORM’s target-specific preprocessing with a lightweight spatial semantic embedding and cross-attention between target queries and sparse measurements. It explicitly incorporates building semantics and a measurement mask, and the paper reports an average RMSE reduction of 12.9% over the second-best DCAE baseline while also noting an abstract-level claim of “up to 6%,” which the text does not fully reconcile (Nan et al., 22 Mar 2026).

Knowledge-guided and foundation-style models introduce stronger architectural priors. RadioKMoE combines a KAN stage for coarse global propagation prediction with a MoE-based TransUNet-style refinement stage, together with building maps, transmitter-location maps, and a physics-inspired depth map (Guo et al., 21 Nov 2025). FM-RME extends the scope further to joint spatial, temporal, and spectral RME, using a geometry-aware feature extraction module that encodes translation and rotation symmetries and a masked self-supervised pre-training strategy over diverse 4D radio-map datasets (Yang et al., 15 Feb 2026).

5. Cooperative, distributed, active, and Bayesian RME

A notable shift in the literature is from centralized, transmitter-aware map reconstruction toward sensing architectures in which measurements are decentralized, opportunistic, or uncertainty-aware.

Cooperative RME addresses the setting where the network operator does not know the active transmitters’ locations, number, or transmit powers. GAN-CRME formalizes this regime by defining a sparse observation matrix $y[m]=p(\mathbf{x}_m)+n_m$0 from mobile-user RSS uploads and learning a direct mapping $y[m]=p(\mathbf{x}_m)+n_m$1. The paper argues that the missing transmitter state is latent in the observed RSS field and reports that the method can outperform RadioUNet once the number of RSS samples exceeds 300, despite not using transmitter information, and can perform coarse error-correction when the geographic map is flawed (Zhang et al., 2024).

Distributed and federated RME addresses privacy, communication, and heterogeneity across clients. PI-DRME assumes transmitter locations are known but landscaping information is unavailable, and splits the model into a globally shared pathloss-oriented autoencoder module and a client-specific shadowing-oriented autoencoder module. Only the common encoder is aggregated across clients; the common decoder and the individual autoencoder remain local. A dense pathloss template is obtained from log-distance path-loss fitting, and the shared loss uses gradient similarity to this template together with a parameter regularizer. The paper reports that PI-DRME achieves the best RMSE across three sampling cases and that vanilla FL can perform worse than standalone training under strong heterogeneity (Yang et al., 1 Feb 2025).

Active sensing adds measurement-selection into the RME loop. STORM extends its transformer to select the next measurement location based on current observations and candidate locations, motivated by minimization of drive tests. The candidate-quality branch is trained jointly with the estimator, and the paper reports that choosing the next measurement according to STORM significantly improves RMSE relative to random selection on both ray-tracing and USRP datasets (Viet et al., 2024).

Bayesian RME changes the inferential target from a single estimate to a posterior over maps. In this formulation, the objective is to determine $y[m]=p(\mathbf{x}_m)+n_m$2, from which standard Bayesian estimators and arbitrary functional estimates can be computed. The paper emphasizes that $y[m]=p(\mathbf{x}_m)+n_m$3 in general, so posterior inference is preferable whenever downstream quantities are nonlinear in the map, such as capacity, BER, outage, or coverage area. A conditional diffusion model is proposed to sample from an approximate posterior, and the resulting Bayesian estimator is shown empirically to outperform non-Bayesian plug-in estimation for nonlinear map functionals (Ha et al., 8 Aug 2025).

6. Empirical trends, controversies, and open problems

Several empirical trends now recur across the literature. First, dense-map recovery from sparse measurements is feasible in practice. The real-world measurement study reports reasonably small errors even with few measurements and thereby establishes the viability of RME beyond simulation (Shrestha et al., 2023). Second, the most effective prior depends strongly on regime. For transmitter-aware single-band urban benchmarks with abundant aligned geometry, U-Net-like map-completion methods remain strong (Teganya et al., 2020, Jaensch et al., 2024). Under multi-band or extreme sparsity, structured priors such as low-rank tensor models, KAN+MoE coarse-to-fine pipelines, and diffusion or generative priors become more attractive (Wang et al., 7 Feb 2025, Guo et al., 21 Nov 2025, Liu et al., 24 Jun 2026).

Third, the common assumption that deep learning uniformly dominates classical methods is not supported by the strongest real-data evidence. The empirical validation paper concludes that deep neural networks necessitate large volumes of training data to exhibit a significant advantage over more traditional methods, and that combining both types of schemes yields a hybrid estimator with the best performance in most situations (Shrestha et al., 2023). This finding coexists with benchmark-specific results in which learned models clearly outperform strong baselines: RadioKMoE achieves the best performance on both BRAT-LabW and RadioMapSeer at all sampling rates, DULRTC-RME exceeds HaLRTC, RBF, FISTA-Net, and RadioUNet at 10% observed entries with PSNR 25.81, RMSE 0.0620, and outage error 0.1261, and STORM outperforms all compared methods on one ray-tracing and two real-measurement datasets while using about 100k parameters rather than the millions reported for several CNN baselines (Guo et al., 21 Nov 2025, Wang et al., 7 Feb 2025, Viet et al., 2024).

Fourth, data realism and deployment mismatch remain central unresolved issues. Much of the literature still trains and tests on simulated or ray-traced datasets, even when those datasets are high fidelity (Jaensch et al., 2024, Liu et al., 24 Jun 2026). Trajectory-aware work shows that a model trained with random sampling can fail badly under continuous-path observations: on SpectrumNet, RMSE rises from 0.0391 under Random$y[m]=p(\mathbf{x}_m)+n_m$4Random testing to 0.2632 under Random$y[m]=p(\mathbf{x}_m)+n_m$5ST-TBS, while ST-TBS training reduces the trajectory-test RMSE to 0.0571 (Qiu et al., 27 May 2026). This suggests that benchmark design must align the training mask distribution with actual sensing trajectories rather than only measuring interpolation capacity under i.i.d. sparse masks.

Finally, several limitations recur across otherwise disparate approaches. Many papers omit runtime tables, parameter counts, or training hyperparameters; several note notation corruption or incomplete optimization detail (Guo et al., 21 Nov 2025, Yang et al., 1 Feb 2025). Side-information requirements vary sharply: some methods need transmitter locations, building layouts, or depth maps, while others explicitly aim to avoid them (Zhang et al., 2024, Guo et al., 21 Nov 2025). Extensions to true 3D propagation, heterogeneous sensor quality, temporal dynamics, multi-frequency fusion, and robust cross-domain deployment remain open in multiple subfields (Zhang et al., 2024, Yang et al., 15 Feb 2026). A plausible implication is that future progress in RME will depend less on isolated gains under one benchmark and more on principled integration of sensing policy, structural priors, uncertainty, and deployment-specific observation models.