Physically Aligned Normalization (PAN)

• Physically Aligned Normalization (PAN) is a parameter-free, closed-form preprocessing method that normalizes images by suppressing chromatic bias and separating pseudo-reflectance from pseudo-illumination.
• It employs a three-stage process—Gray-World color normalization, log-domain Retinex decomposition, and dynamic range recombination—to align data with physical image formation models.
• Empirical results indicate PAN enhances shadow removal performance with measurable improvements in PSNR and SSIM, outperforming classical color correction techniques.

Physically Aligned Normalization (PAN) is a fully parameter-free, closed-form preprocessing method developed for image shadow removal and ambient-light normalization, particularly under challenging illumination conditions with mixed or colored light sources. Its primary function is to suppress chromatic bias and align input data to the Retinex-inspired physical prior, facilitating accurate disentanglement of illumination and reflectance for downstream networks. PAN was introduced as a core component in the PhaSR (Physically Aligned Shadow Removal) pipeline but is formulated independently for use in diverse imaging tasks where adherence to the physical image formation model is desired (Lee et al., 24 Jan 2026).

1. Motivation and Physical Model

Shadow removal and related intrinsic image tasks require recovering scene reflectance from images subject to spatially and chromatically varying illumination. Real-world illumination commonly involves colored sources (e.g., incandescent bulbs, daylight, or multi-source environments), which induce global chromatic biases that confound standard reflectance-recovery architectures. The physical model for image formation is

$$\mathbf{I}(x) = \mathbf{R}(x) \otimes \mathbf{S}(x),$$

where $\mathbf{I}(x)$ is the observed RGB image at spatial location $x$, $\mathbf{R}(x)$ is the view-dependent reflectance, and $\mathbf{S}(x)$ is the spatially varying illumination. Shadows primarily affect $\mathbf{S}(x)$, while $\mathbf{R}(x)$ encodes intrinsic scene color and material properties.

The primary challenge arises when global illumination chromaticity is not properly “factored out,” resulting in networks focusing on irrelevant global statistics rather than local boundary and detail recovery. PAN addresses this by explicitly normalizing color, separating pseudo-reflectance and pseudo-illumination in logarithmic space, and recombining these to serve as a physically-aligned normalizing transform.

2. PAN Algorithmic Steps

PAN is structured as a three-stage, closed-form preprocessing module that operates directly on the input RGB image:

2.1 Gray-World Color Normalization

Under the Gray-World assumption, the mean intensity of each RGB channel over a neutral scene is expected to be equal [J. Buchsbaum, 1980]. To suppress dominant color casts, PAN scales each pixel of the input image by the ratio of the global mean intensity to its per-channel mean: $$\mathbf{I}_{\mathrm{norm}} = \mathbf{I} \odot \frac{\mathbb{E}[\mathbf{I}]}{\mathbb{E}_c[\mathbf{I}] + \varepsilon},$$ where $\mathbb{E}[\mathbf{I}]$ is the scalar mean across all pixels/channels, $\mathbb{E}_c[\mathbf{I}]$ is the per-channel mean, and $\varepsilon = 10^{-6}$ prevents division by zero.

2.2 Log-Domain Retinex Decomposition

The normalized image is further decomposed in the log-domain—a standard approach in Retinex theory: $$\log \mathbf{I}_{\mathrm{norm}}(x) = \log \mathbf{R}(x) + \log \mathbf{S}(x).$$ Assuming the global illumination chromaticity can be approximated by the mean in log-space, PAN estimates

$$\log \widehat{\mathbf{S}} = \mathbb{E}_{(x,y)} [\log(\mathbf{I}_{\mathrm{norm}}(x) + \varepsilon)],$$

$$\log \widehat{\mathbf{R}}(x) = \log(\mathbf{I}_{\mathrm{norm}}(x) + \varepsilon) - \log \widehat{\mathbf{S}},$$

and exponentiates to recover pseudo-components

$$\widehat{\mathbf{R}}(x) = \exp(\log \widehat{\mathbf{R}}(x)),\quad \widehat{\mathbf{S}} = \exp(\log \widehat{\mathbf{S}}).$$

2.3 Dynamic Range Recombination

To form the output, these pseudo-reflectance and pseudo-illumination are recombined and contrast-stretched: $$\widetilde{\mathbf{I}}(x) = \frac{\widehat{\mathbf{R}}(x) \otimes \widehat{\mathbf{S}} - \min_{u,v}[\widehat{\mathbf{R}}(u)\otimes\widehat{\mathbf{S}}]}{\max_{u,v}[\widehat{\mathbf{R}}(u)\otimes\widehat{\mathbf{S}}] - \min_{u,v}[\widehat{\mathbf{R}}(u)\otimes\widehat{\mathbf{S}}] + \varepsilon}.$$ The resulting $\widetilde{\mathbf{I}}$ is normalized, free from dominant chromatic bias, and exhibits uniform illumination statistics, with local reflectance boundaries preserved.

PAN Pseudocode

def PAN(I, eps=1e-6): global_mean = mean(I) # scalar channel_means = mean(I, axis=(0,1)) # 3-vector I_norm = I * (global_mean / (channel_means + eps)) L = log(I_norm + eps) # H×W×3 logS = mean(L, axis=(0,1)) # 3-vector logR = L - logS # H×W×3 R_hat = exp(logR) S_hat = exp(logS) J = R_hat * S_hat # H×W×3 Jmin = J.min() Jmax = J.max() I_out = (J - Jmin) / (Jmax - Jmin + eps) return I_out

3. Physical Interpretation and Theoretical Rationale

Each stage of PAN maps directly to standard physical and perceptual assumptions about image formation and perception:

• Gray-World normalization removes scene-wide chromatic bias, setting the global scene color to approximate achromaticity under the assumption of unbiased illumination.
• Log-domain Retinex decomposition leverages multiplicative separation of illumination and reflectance and utilizes global log-space averaging to approximate the illumination component, leaving residuals that approximate reflectance.
• Dynamic range recombination and contrast normalization restore the output to a normalized [0,1] range, suitable for network input, while preserving important structure in high-frequency regions, such as reflectance or shadow boundaries.

Together, these transformations align the preprocessed input with the assumption that shadowing exclusively affects the illumination component, not the intrinsic reflectance, facilitating subsequent learning or inference modules.

4. Quantitative Effectiveness

Empirical evaluation across multiple publicly available shadow removal and ambient-light datasets demonstrates the effectiveness of PAN in reducing residual shadow error and improving network performance when used as a plug-in preprocessing step. Notable metrics include:

Dataset/Scenario	Key Metric	Improvement with PAN
ISTD (outdoor)	Residual error	-26.4% (train), -17.3% (test)
Ambient6K (indoor, multi-source)	Residual error	-0.8% (train), -1.3% (test)
WSRD+ (OmniSR)	PSNR, SSIM	+0.22 dB, +0.005
WSRD+ (DenseSR)	PSNR, SSIM	+0.33 dB, +0.013
WSRD+ (PhaSR)	PSNR, SSIM	+0.27 dB, +0.017

A comparison against classical color correction techniques (ACE, white-balance, white-patch, CIELab) shows PAN with PSNR = 28.44 and SSIM = 0.9418 on WSRD+, where all baselines remain below PSNR 27.13 and SSIM 0.92. Ablation studies indicate that eliminating PAN degrades PhaSR performance substantially: for ISTD+, PSNR/SSIM drops from (34.48/0.960) to (33.15/0.952); on WSRD+, from (28.44/0.942) to (28.17/0.925) (Lee et al., 24 Jan 2026).

5. Application Scope and Integration

Although PAN was proposed in the context of PhaSR for general shadow removal, its parameter-free, closed-form nature and general adherence to the physical image formation model make it suitable as a preprocessing module in broader settings requiring color normalization and intrinsic image factorization. Its plug-and-play design enables integration with a variety of architectures, as evidenced by consistent PSNR and SSIM improvements across diverse shadow removal backbones and datasets. The process requires no supervision or training, computes efficiently via global statistics and pointwise operations, and maintains fine spatial structure critical for downstream networks.

6. Relationship to Prior Art and Comparative Analysis

Classical color correction methods, including ACE, simple white-balance, white-patch referencing, and CIELab normalization, operate via local or global channel statistics but do not explicitly separate illumination and reflectance based on a physical prior. PAN’s distinguishing factor is the log-domain decomposition aligned with Retinex theory and the explicit recombination step, which better preserves scene structure and supports physical interpretability.

Empirical comparison corroborates this difference, with PAN consistently surpassing classical methods in standard shadow-removal and ambient-light benchmarks. This suggests that explicit physical prior alignment within preprocessing, without introducing learnable parameters, constitutes a practically and theoretically superior modality for challenging illumination normalization tasks.

7. Limitations and Practical Considerations

PAN requires only global and per-channel means, logarithms, pointwise operations, and contrast normalization, introducing negligible computational overhead and no tunable hyperparameters except for a numerical stability constant $\varepsilon$. While PAN substantially reduces chromatic bias in strongly colored or mixed-source environments, it assumes the Gray-World prior holds and that global averaging sufficiently captures dominant illumination effects. Scenes with significant violation of Gray-World assumptions or highly localized illuminants may present challenging cases. Integration into domain-specific pipelines should consider the underlying scene statistics and whether the approximations in PAN’s closed-form decomposition remain congruent with target physical conditions.

In summary, Physically Aligned Normalization provides a parameter-free, physically-motivated normalization strategy that improves robustness and generalization for shadow-removal and related intrinsic-image tasks under diverse illumination conditions (Lee et al., 24 Jan 2026).