Power-Invariant Radio Map

• Power-Invariant Radio Maps are representations that preserve spatial patterns by normalizing RSS or gain measurements to eliminate the effect of unknown transmit power.
• They employ normalization techniques and strategic sensor placement, such as at obstacle vertices, to robustly reconstruct radio fields in obstructed environments.
• This approach enables advanced localization and active sensing by fusing data-driven inference with geometry-aware learning for improved emitter detection.

A power-invariant Radio Map (RM) is a radio-field representation whose spatial pattern is constructed so that unknown or changing transmit power does not alter the map structure. In the cited literature, this is realized in two closely related ways: by normalizing sparse received signal strength (RSS) samples by the maximum observed sample to build a relative RM for non-cooperative NLoS localization, and by normalizing large-scale channel gain with respect to transmit power so that the shape of the gain map does not depend on the absolute base-station power (Wang et al., 2 Sep 2025, Zeng et al., 7 Apr 2026). Power-invariant RMs therefore occupy a junction between localization, radio cartography, sparse reconstruction, geometry-aware learning, and active sensing.

1. Definitions and invariant formulations

Two formulations appear in the recent literature.

Setting	Power-invariant quantity	Invariance statement
Sparse RSS in NLoS localization	$\tilde y_i=\dfrac{y_i}{\max_{j=1,\dots,M}y_j}$	Unknown transmit power is removed by normalization
Large-scale gain cartography	$G'(\mathbf q)=G(\mathbf q)-P_{\rm tx}$	The shape of the gain map does not depend on the absolute transmit power

In the NLoS localization setting, the measurements are raw RSS values $y_i$ at sensor locations $\{\mathbf r_i\}\subset\mathcal S_s$. RadioDiff-Loc constructs power-invariant measurements $\{\tilde y_i\}\in[0,1]$ by normalizing every sampled RSS by the maximum observed sample, and then defines a relative RM through

$$\mathcal R(\mathbf x)\;\approx\;\mathbb E\bigl[\tfrac{r(\mathbf x)}{\max_j r(\mathbf r_j)}\mid\mathrm{layout},\{\tilde y_i\}\bigr].$$

This is explicitly a relative, power-invariant RM rather than an absolute field estimate (Wang et al., 2 Sep 2025).

In the gain-cartography setting, the primary quantity of interest is the large-scale channel gain

$$G(\mathbf q)\;=\;P_{\rm tx}\;-\;L(\mathbf q)\;-\;S(\mathbf q)\;+\;\varepsilon(\mathbf q),$$

or, in linear units,

$$G_{\rm lin}(\mathbf q)\;=\;\frac{P_{\mathrm{rx}(\mathbf q)}}{P_{\mathrm{tx}}}.$$

If $P_{\rm tx}$ is changed by a constant $\Delta P$, then $G'(\mathbf q)=G(\mathbf q)-P_{\rm tx}$0 shifts by $G'(\mathbf q)=G(\mathbf q)-P_{\rm tx}$1 everywhere, whereas the normalized gain

$G'(\mathbf q)=G(\mathbf q)-P_{\rm tx}$2

does not change its spatial pattern. The same normalization is written in dB as

$G'(\mathbf q)=G(\mathbf q)-P_{\rm tx}$3

This establishes power invariance as invariance of map shape, not invariance of absolute received power (Zeng et al., 7 Apr 2026).

2. Construction from sparse unknown-power RSS

RadioDiff-Loc addresses non-cooperative scenarios in which the transmit power $G'(\mathbf q)=G(\mathbf q)-P_{\rm tx}$4 is unknown, so all RSS readings are only proportional to the true field. Its normalization step is

$G'(\mathbf q)=G(\mathbf q)-P_{\rm tx}$5

with $G'(\mathbf q)=G(\mathbf q)-P_{\rm tx}$6 the sparse sampling positions and $G'(\mathbf q)=G(\mathbf q)-P_{\rm tx}$7 the measured RSS. The normalized values $G'(\mathbf q)=G(\mathbf q)-P_{\rm tx}$8 become the conditioning input for constructing a relative RM (Wang et al., 2 Sep 2025).

The conditioning representation is spatially embedded. The model uses a binary layout map $G'(\mathbf q)=G(\mathbf q)-P_{\rm tx}$9, a sparse-measurement tensor $y_i$0,

$y_i$1

and a binary mask

$y_i$2

These are stacked into the three-channel input

$y_i$3

This formulation is specific to unknown-power RSS localization in obstructed environments. A plausible implication is that the normalization step is not merely a preprocessing convenience; it is the mechanism that makes the learned RM transferable across different unknown source powers while preserving the geometry-dependent intensity pattern.

3. Information-optimal sparse sampling at obstacle vertices

RadioDiff-Loc does not treat sparse measurement placement as arbitrary. Its sampling strategy is derived from the physical insight that diffracted electromagnetic energy concentrates near building edges, and, more specifically, that most of the diffracted energy in a NLoS scenario concentrates near sharp edges and corners through the “knife-edge” effect (Wang et al., 2 Sep 2025).

The sampled RSS vector is modeled as

$y_i$4

where $y_i$5 are unknown boundary fields, $y_i$6 is a Green’s-function matrix, and $y_i$7 is noise. The Fisher information matrix is

$y_i$8

Under far-field diffraction each column $y_i$9 of $\{\mathbf r_i\}\subset\mathcal S_s$0 has norm $\{\mathbf r_i\}\subset\mathcal S_s$1, where $\{\mathbf r_i\}\subset\mathcal S_s$2 is the distance from the corner. Hence

$\{\mathbf r_i\}\subset\mathcal S_s$3

which is maximized at $\{\mathbf r_i\}\subset\mathcal S_s$4, namely at the obstacle vertex.

The same conclusion is obtained from mutual information. The mutual information between $\{\mathbf r_i\}\subset\mathcal S_s$5 and $\{\mathbf r_i\}\subset\mathcal S_s$6 is

$\{\mathbf r_i\}\subset\mathcal S_s$7

and it grows fastest where $\{\mathbf r_i\}\subset\mathcal S_s$8 is largest, again at vertices. The stated consequence is direct: placing sparse sensors at obstacle corners yields maximal information for RM reconstruction and localization, even with very few samples. This makes vertex sampling a structural component of the power-invariant RM pipeline rather than a heuristic measurement policy.

4. Conditional diffusion completion and RM-based localization

The completed power-invariant RM in RadioDiff-Loc is generated by a decoupled denoising diffusion model conditioned on the layout map and the sparse normalized RSS map (Wang et al., 2 Sep 2025). Let $\{\mathbf r_i\}\subset\mathcal S_s$9 denote a ground-truth power-invariant RM. The forward kernel is

$\{\tilde y_i\}\in[0,1]$0

with continuous-time form

$\{\tilde y_i\}\in[0,1]$1

where $\{\tilde y_i\}\in[0,1]$2 and $\{\tilde y_i\}\in[0,1]$3.

A U-Net or similar CNN $\{\tilde y_i\}\in[0,1]$4 is trained to estimate the score $\{\tilde y_i\}\in[0,1]$5. In practice the simplified denoising–noise prediction loss is

$\{\tilde y_i\}\in[0,1]$6

where $\{\tilde y_i\}\in[0,1]$7 and $\{\tilde y_i\}\in[0,1]$8.

At inference, the model simulates the reverse DDPM updates

$\{\tilde y_i\}\in[0,1]$9

where $$\mathcal R(\mathbf x)\;\approx\;\mathbb E\bigl[\tfrac{r(\mathbf x)}{\max_j r(\mathbf r_j)}\mid\mathrm{layout},\{\tilde y_i\}\bigr].$$0 and $$\mathcal R(\mathbf x)\;\approx\;\mathbb E\bigl[\tfrac{r(\mathbf x)}{\max_j r(\mathbf r_j)}\mid\mathrm{layout},\{\tilde y_i\}\bigr].$$1. The sample at $$\mathcal R(\mathbf x)\;\approx\;\mathbb E\bigl[\tfrac{r(\mathbf x)}{\max_j r(\mathbf r_j)}\mid\mathrm{layout},\{\tilde y_i\}\bigr].$$2 is the completed, power-invariant RM $$\mathcal R(\mathbf x)\;\approx\;\mathbb E\bigl[\tfrac{r(\mathbf x)}{\max_j r(\mathbf r_j)}\mid\mathrm{layout},\{\tilde y_i\}\bigr].$$3.

Localization is then performed by identifying the brightest point on the generated RM: $$\mathcal R(\mathbf x)\;\approx\;\mathbb E\bigl[\tfrac{r(\mathbf x)}{\max_j r(\mathbf r_j)}\mid\mathrm{layout},\{\tilde y_i\}\bigr].$$4 The paper states that this approximates the MAP estimate under a uniform prior. Repeating the generative sampling yields an ensemble $$\mathcal R(\mathbf x)\;\approx\;\mathbb E\bigl[\tfrac{r(\mathbf x)}{\max_j r(\mathbf r_j)}\mid\mathrm{layout},\{\tilde y_i\}\bigr].$$5 for uncertainty quantification. The same completed RM can also be fused with classical RSS-based localization methods in a “dual-driven” paradigm: trilateration or nonlinear least squares, fingerprinting/MLE, and centroid methods such as Top-$$\mathcal R(\mathbf x)\;\approx\;\mathbb E\bigl[\tfrac{r(\mathbf x)}{\max_j r(\mathbf r_j)}\mid\mathrm{layout},\{\tilde y_i\}\bigr].$$6 weighted centroid or threshold-region centroid.

5. Geometry-aware gain RM reconstruction and uncertainty modeling

GeoUQ-GFNet studies sparse gain radio map reconstruction from a geometry-aware and active sensing perspective. It defines the true gain RM after discretizing $$\mathcal R(\mathbf x)\;\approx\;\mathbb E\bigl[\tfrac{r(\mathbf x)}{\max_j r(\mathbf r_j)}\mid\mathrm{layout},\{\tilde y_i\}\bigr].$$7 into an $$\mathcal R(\mathbf x)\;\approx\;\mathbb E\bigl[\tfrac{r(\mathbf x)}{\max_j r(\mathbf r_j)}\mid\mathrm{layout},\{\tilde y_i\}\bigr].$$8 grid with cell centers $$\mathcal R(\mathbf x)\;\approx\;\mathbb E\bigl[\tfrac{r(\mathbf x)}{\max_j r(\mathbf r_j)}\mid\mathrm{layout},\{\tilde y_i\}\bigr].$$9 as

$$G(\mathbf q)\;=\;P_{\rm tx}\;-\;L(\mathbf q)\;-\;S(\mathbf q)\;+\;\varepsilon(\mathbf q),$$0

and performs reconstruction on the normalized gain so that the learned map is independent of the actual base-station transmit power (Zeng et al., 7 Apr 2026).

The geometry priors are encoded as

$$G(\mathbf q)\;=\;P_{\rm tx}\;-\;L(\mathbf q)\;-\;S(\mathbf q)\;+\;\varepsilon(\mathbf q),$$1

where $$G(\mathbf q)\;=\;P_{\rm tx}\;-\;L(\mathbf q)\;-\;S(\mathbf q)\;+\;\varepsilon(\mathbf q),$$2 is obstacle occupancy, $$G(\mathbf q)\;=\;P_{\rm tx}\;-\;L(\mathbf q)\;-\;S(\mathbf q)\;+\;\varepsilon(\mathbf q),$$3 is building height, $$G(\mathbf q)\;=\;P_{\rm tx}\;-\;L(\mathbf q)\;-\;S(\mathbf q)\;+\;\varepsilon(\mathbf q),$$4 are horizontal offset maps from the transmitter, $$G(\mathbf q)\;=\;P_{\rm tx}\;-\;L(\mathbf q)\;-\;S(\mathbf q)\;+\;\varepsilon(\mathbf q),$$5 is transmitter–grid distance, and $$G(\mathbf q)\;=\;P_{\rm tx}\;-\;L(\mathbf q)\;-\;S(\mathbf q)\;+\;\varepsilon(\mathbf q),$$6 is a line-of-sight–blocked indicator. Sparse observations are

$$G(\mathbf q)\;=\;P_{\rm tx}\;-\;L(\mathbf q)\;-\;S(\mathbf q)\;+\;\varepsilon(\mathbf q),$$7

with sampling mask $$G(\mathbf q)\;=\;P_{\rm tx}\;-\;L(\mathbf q)\;-\;S(\mathbf q)\;+\;\varepsilon(\mathbf q),$$8 and accessibility mask $$G(\mathbf q)\;=\;P_{\rm tx}\;-\;L(\mathbf q)\;-\;S(\mathbf q)\;+\;\varepsilon(\mathbf q),$$9. A dense initialization $$G_{\rm lin}(\mathbf q)\;=\;\frac{P_{\mathrm{rx}(\mathbf q)}}{P_{\mathrm{tx}}}.$$0 is obtained by a fast neighborhood-filling of $$G_{\rm lin}(\mathbf q)\;=\;\frac{P_{\mathrm{rx}(\mathbf q)}}{P_{\mathrm{tx}}}.$$1. The full input is

$$G_{\rm lin}(\mathbf q)\;=\;\frac{P_{\mathrm{rx}(\mathbf q)}}{P_{\mathrm{tx}}}.$$2

The architecture first separates

$$G_{\rm lin}(\mathbf q)\;=\;\frac{P_{\mathrm{rx}(\mathbf q)}}{P_{\mathrm{tx}}}.$$3

and encodes them into $$G_{\rm lin}(\mathbf q)\;=\;\frac{P_{\mathrm{rx}(\mathbf q)}}{P_{\mathrm{tx}}}.$$4. The observation branch is geometry-gated: $$G_{\rm lin}(\mathbf q)\;=\;\frac{P_{\mathrm{rx}(\mathbf q)}}{P_{\mathrm{tx}}}.$$5 The paper states that this explicitly blocks propagation of spurious information across walls or into inaccessible areas.

The encoder uses three cascaded stages with Ghost residual blocks, while the deeper two stages insert Grid-KAN nonlinear enhancers,

$$G_{\rm lin}(\mathbf q)\;=\;\frac{P_{\mathrm{rx}(\mathbf q)}}{P_{\mathrm{tx}}}.$$6

and the bottleneck employs a large-kernel depthwise convolutional context module. The decoder is FPN-style, producing a shared representation $$G_{\rm lin}(\mathbf q)\;=\;\frac{P_{\mathrm{rx}(\mathbf q)}}{P_{\mathrm{tx}}}.$$7 that feeds a residual gain head and an uncertainty head. Reconstruction is

$$G_{\rm lin}(\mathbf q)\;=\;\frac{P_{\mathrm{rx}(\mathbf q)}}{P_{\mathrm{tx}}}.$$8

and the uncertainty output is

$$G_{\rm lin}(\mathbf q)\;=\;\frac{P_{\mathrm{rx}(\mathbf q)}}{P_{\mathrm{tx}}}.$$9

Training uses a weighted sum of four losses over the unobserved accessible set $P_{\rm tx}$0: $P_{\rm tx}$1 with default weights

$P_{\rm tx}$2

The stated losses are masked $P_{\rm tx}$3, gradient consistency, heteroscedastic Gaussian negative log-likelihood, and a small regularizer on $P_{\rm tx}$4.

6. Active sensing, benchmark evidence, and reconstruction performance

GeoUQ-GFNet couples power-invariant gain RM reconstruction with uncertainty-guided measurement selection (Zeng et al., 7 Apr 2026). After each reconstruction step, the candidate region is

$P_{\rm tx}$5

and the next measurements are chosen by

$P_{\rm tx}$6

The mask and observed sparse map are then updated: $P_{\rm tx}$7 The paper states that this policy yields faster reduction in reconstruction error than random sampling under the same budget because the model flags geometrically difficult or sparsely supported regions with high predicted uncertainty.

All experiments use the UrbanRT-RM ray-tracing benchmark, described as seven scene layouts × eight BS deployments, 128×128 patches, 3.5 GHz, Sionna RT. The reported metrics over the accessible region are $P_{\rm tx}$8 and $P_{\rm tx}$9. Under overall reconstruction with random 10% sampling, the reported results are as follows.

Model	$\Delta P$0	$\Delta P$1
GeoUQ-GFNet	7.1025	1.6191
Eff-UNet	7.7557	2.0449
ViT-UNet	7.7895	2.0978
ResNet-UNet	7.8529	2.1139
Nearest	11.9412	3.7052

For active sensing with initial 10% plus 4 rounds of +1% each, at total additional budget 4%, the reported results are:

Model	UQ-guided RMSE	Random RMSE
GeoUQ-GFNet	3.1002	5.9144
ViT-UNet	3.4558	6.2599
Eff-UNet	3.5101	6.2742
ResNet-UNet	3.5543	6.3106
Nearest	9.8791	9.8791

The active sensing curves are summarized in the source as showing that GeoUQ-GFNet’s uncertainty policy steadily drives down RMSE from approximately $\Delta P$2 dB at 10% to approximately $\Delta P$3 dB at +4%, whereas random sampling yields only modest improvement. This places power-invariant RM reconstruction within a broader closed-loop sensing framework rather than a one-shot completion problem.

7. Applications, fusion strategies, and conceptual boundaries

Power-invariant RMs serve two application classes in the cited works. In RadioDiff-Loc, the completed relative RM is used for emitter localization in non-line-of-sight environments and is also compatible with existing RSS-based localization algorithms, enabling a “dual-driven” paradigm that fuses physical knowledge and data-driven inference (Wang et al., 2 Sep 2025). In GeoUQ-GFNet, the normalized gain map is motivated by environment-aware wireless communication, network planning, and radio resource optimization, and the uncertainty map is used to guide active measurement selection under limited sensing budgets (Zeng et al., 7 Apr 2026).

A key conceptual boundary is that power invariance does not recover absolute transmit power. In the gain-map formulation, the only difference between two deployments with powers $\Delta P$4 and $\Delta P$5 is an additive shift in the entire map; no architectural or data-collection changes are required. In the RSS-based formulation, normalization by the maximum observed sample produces a relative map,

$\Delta P$6

so the output is expressly power-invariant rather than absolutely calibrated.

The two works also delimit two methodological regimes. One regime reconstructs a relative RSS field from sparse, strategically placed measurements using a conditional diffusion model and then localizes by an argmax rule. The other reconstructs a dense gain map and a spatial uncertainty map from sparse measurements and structured scene priors, then uses uncertainty for iterative measurement selection. This suggests that “power-invariant RM” is best understood not as a single algorithm, but as a representation principle: remove the transmit-power ambiguity, preserve the spatial pattern, and then apply a reconstruction or inference mechanism suited to the downstream task.