Euler-inspired Decoupling Neural Operator for Efficient Pansharpening

Anqi Zhu; Mengting Ma; Yizhen Jiang; Xiangdong Li; Kai Zheng; Jiaxin Li; Wei Zhang

cs.CVApr 14, 2026

Euler-inspired Decoupling Neural Operator for Efficient Pansharpening

Authors: Anqi Zhu, Mengting Ma, Yizhen Jiang, Xiangdong Li, Kai Zheng, Jiaxin Li, Wei Zhang

Organizations: School of Software Technology, Zhejiang University, Ningbo, China · School of Computer Science and Technology, Chongqing University of Post and Telecommunications, Chongqing, China

Abstract

Pansharpening aims to synthesize high-resolution multispectral (HR-MS) images by fusing the spatial textures of panchromatic (PAN) images with the spectral information of low-resolution multispectral (LR-MS) images. While recent deep learning paradigms, especially diffusion-based operators, have pushed the performance boundaries, they often encounter spectral-spatial blurring and prohibitive computational costs due to their stochastic nature and iterative sampling. In this paper, we propose the Euler-inspired Decoupling Neural Operator (EDNO), a physics-inspired framework that redefines pansharpening as a continuous functional mapping in the frequency domain. Departing from conventional Cartesian feature processing, our EDNO leverages Euler's formula to transform features into a polar coordinate system, enabling a novel explicit-implicit interaction mechanism. Specifically, we develop the Euler Feature Interaction Layer (EFIL), which decouples the fusion task into two specialized modules: 1) Explicit Feature Interaction Module, utilizing a linear weighting scheme to simulate phase rotation for adaptive geometric alignment; and 2) Implicit Feature Interaction Module, employing a feed-forward network to model spectral distributions for superior color consistency. By operating in the frequency domain, EDNO inherently captures global receptive fields while maintaining discretization-invariance. Experimental results on the three datasets demonstrate that EDNO offers a superior efficiency-performance balance compared to heavyweight architectures.

Figures & tables

Figure 1. The motivation and performance of EDNO. (Top left) Conventional pixel-domain interactions are computationally expensive and prone to edge ghosting. (Bottom left) Our method utilizes Fast Fourier Transform (FFT) to interact in Euler space, enabling efficient global modeling of geometric structures and spectral energy. (Right) Performance benchmarking shows that EDNO (red star) dominates the upper-left corner of both PSNR-vs-Params and PSNR-vs-FLOPs charts, demonstrating superior efficiency and reconstruction quality.

Figure 2. Overall Architecture of the proposed Euler-inspired Decoupling Neural Operator (EDNO). The EDNO framework follows a structured three-stage pipeline to achieve resolution-agnostic pansharpening. Starting with the Lifting Stage, a dual-mapping operator $L=(L_{1},L_{2})$ transforms the input PAN and LR-MS images into a unified, high-dimensional continuous-feature manifold $z_{0}(x)$ by combining a feature encoder $E_{\psi}$ with implicit coordinate-based sampling. Subsequently, these latent representations undergo the Iterative Kernel Integral Stage, where $T$ iterations of kernel operations ( $\mathcal{K}$ ) progressively refine cross-modal features to resolve spectral-spatial coupling. Finally, in the Projection and Fusion Stage, the refined features are mapped back to the target output space through a projection operator and integrated with a global residual bypass to ensure maximum spectral fidelity in the reconstructed HR-MS output. The Lifting Detail and Sampling Detail (bottom) further elucidate the coordinate-aware mechanism, where neighboring grid features $a_{l}$ are aggregated via spatial weights $s_{l}$ and subsequently concatenated with fractional coordinate offsets $\delta_{l}$ , to maintain consistency across continuous function spaces ( Wei and Zhang, 2023 ) .

Figure 3. The detailed architecture of the Euler Feature Interaction Layer (EFIL). The input modalities are first projected onto a complex-valued frequency manifold via FFT. Within each EFIL, the Explicit Phase Interaction Module (EFIM) aligns geometric structures through phase-shifting, while the Implicit Magnitude Interaction Module (IFIM) adaptively enhances spectral energy. These decoupled components are progressively refined through $T$ iterations and finally reconstructed into the fused feature $F^{T}$ via IFFT.

			Reduced-Resolution					Full-Resolution
Method	Params (K)	FLOPs (G)	PSNR $\uparrow$	SSIM $\uparrow$	$Q_{4}\uparrow$	SAM $\downarrow$	ERGAS $\downarrow$	$D_{\lambda}\downarrow$	$D_{s}\downarrow$	QNR $\uparrow$
SFIM ( Liu, 2000 )	-	-	32.6334	0.8728	0.5159	0.0597	3.1919	0.0737	0.0899	0.8439
Wavelet ( King and Wang, 2001 )	-	-	32.1992	0.8500	0.4577	0.0638	3.3799	0.0968	0.1020	0.8126
IHS ( Faragallah, 2018 )	-	-	32.8250	0.8775	0.5305	0.0637	3.0585	0.0874	0.1187	0.8053
GPPNN ( Xu et al., 2021 )	119.8	11.2	40.5086	0.9698	0.8009	0.0275	1.2060	0.0670	0.0785	0.8607
CTINN ( Zhou et al., 2022a )	378.2	13.3	41.2019	0.9735	0.8149	0.0246	1.0879	0.0658	0.0776	0.8632

Table 1. Evaluation indexes of different methods on WorldView-2 (WV-2) dataset.

			Reduced-Resolution					Full-Resolution
Method	Params (K)	FLOPs (G)	PSNR $\uparrow$	SSIM $\uparrow$	$Q_{4}\uparrow$	SAM $\downarrow$	ERGAS $\downarrow$	$D_{\lambda}\downarrow$	$D_{s}\downarrow$	QNR $\uparrow$
SFIM ( Liu, 2000 )	-	-	34.7715	0.8572	0.4584	0.0657	4.2073	0.0737	0.0899	0.8439
Wavelet ( King and Wang, 2001 )	-	-	33.9208	0.8197	0.4033	0.0695	4.6445	0.0968	0.1020	0.8126
IHS ( Faragallah, 2018 )	-	-	35.2315	0.8837	0.5217	0.0661	3.9912	0.0874	0.1187	0.8053
GPPNN ( Xu et al., 2021 )	119.8	11.2	43.5980	0.9764	0.8663	0.0326	1.5218	0.0670	0.0785	0.8607
CTINN ( Zhou et al., 2022a )	378.2	13.3	44.3181	0.9788	0.8719	0.0291	1.4050	0.0699	0.1561	0.7841

Table 2. Evaluation indexes of different methods on GaoFen-2 (GF-2) dataset.

Figure 4. Visual comparison between our model and other methods on WorldView-2 (WV-2) example. The top line represent the reconstructed results, and the bottom line represents the corresponding MAE maps.

Figure 5. Visual comparison on WorldView-2 (WV-2) example at full resolution.

$T$	PSNR $\uparrow$	SSIM $\uparrow$	$Q_{4}\uparrow$	ERGAS $\downarrow$	$D_{s}\downarrow$
2	41.1870	0.9724	0.8157	1.0925	0.0771
3	41.4968	0.9738	0.8218	1.0472	0.0775
4	41.6488	0.9747	0.8262	1.0259	0.0768
5	41.4173	0.9733	0.8217	1.0527	0.0790

Table 3. Performance analysis of EDNO with varying iteration numbers

T

of the Euler Feature Interaction Layer.

Config	PSNR $\uparrow$	SSIM $\uparrow$	$Q_{4}\uparrow$	ERGAS $\downarrow$	$D_{s}\downarrow$
(I)	38.7992	0.9552	0.7607	1.4415	0.0810
(II)	38.9401	0.9551	0.7580	1.4080	0.0822
(III)	37.4784	0.9434	0.7145	1.6593	0.0874
(IV)	41.2360	0.9724	0.8150	1.0785	0.0789
ours	41.6488	0.9747	0.8262	1.0259	0.0768

Table 4. Ablation study of different architectural configurations and internal components of EDNO.

Figure 6. Zero-shot scale generalization from $\times 2$ to $\times 10$ resolution.

Figure 7. Qualitative comparison and absolute error maps under $5\%$ input scale perturbation.

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RAFNet: Region-Aware Fusion Network for Pansharpening

Jianing Zhang, Zijian Zhou, Kai Sun

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Pansharpening aims to generate high-resolution multispectral (HRMS) images by fusing low-resolution multispectral (LRMS) and high-resolution panchromatic (PAN) images. Although deep learning has advanced this field, mainstream frequency-based methods relying on standard scaled dot-product attention suffer from quadratic computational complexity and fail to exploit the inherent regional sparsity of remote sensing imagery. Furthermore, existing spatial enhancement strategies typically employ static convolution kernels, which struggle to adapt to the complex frequency and regional variations of PAN and MS images. To address these bottlenecks, we propose a Region-Aware Fusion (RAFNet) Network that synergistically models spatial and frequency information. Specifically, we design a Spatial Adaptive Refinement (SAR) module that leverages the discrete wavelet transform (DWT) for directional frequency separation and K-means clustering for regional partitioning, which enables the dynamic construction of region-specific adaptive convolution kernels, achieving spatially and frequency-adaptive feature enhancement. Moreover, we introduce a Clustered Frequency Aggregation (CFA) module based on a sparse attention mechanism guided by the semantic clusters, which executes a region-aware sparse attention strategy that drastically reduces computational redundancy while ensuring high-quality frequency feature extraction. In addition we integrated these modules into a progressive, multi-level spatial-frequency network architecture to facilitate robust interaction and accurate image reconstruction. Extensive experiments on multiple benchmark datasets demonstrate that the proposed RAFNet significantly outperforms state-of-the-art pansharpening methods in both reduced- and full-resolution assessments. The code is available at https://github.com/PatrickNod/RAFNet.

Abstract

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