Physics-informed Neural Operator for Pansharpening
Authors: Xinyang Liu
Junming Hou
Chenxu Wu
Xiaofeng Cong
Zihao Chen
Shangqi Deng
Junling Li
Liang-Jian Deng
Bo Liu
Venue: NeurIPS 2025 poster ()
Keywords: Pan-sharpening; Neural Operator; Multispectral and hyperspectral image fusion
Primary area: applications
TL;DR: This paper introduces a physics-informed neural network that recover high-res hyperspectral data by modeling sensor optics, reducing distortions and improving accuracy.

Abstract: Over the past decades, pansharpening has contributed greatly to numerous remote sensing applications, with methods evolving from theoretically grounded models to deep learning approaches and their hybrids. Though promising, existing methods rarely address pansharpening through the lens of underlying physical imaging processes. In this work, we revisit the spectral imaging mechanism and propose a novel physics‐informed neural operator framework for pansharpening, termed PINO, which faithfully models the end‐to‐end electro‐optical sensor process. Specifically, PINO operates as: (1) First, a spatial-spectral encoder pair is introduced to aggregate multi-granularity high-resolution panchromatic (PAN) and low-resolution multispectral (LRMS) features.
(2) Subsequently, an iterative neural integral process utilizes these fused spatial-spectral characteristics to learn a continuous radiance field $L_i(x, y, \lambda)$ over spatial coordinates and wavelength, effectively emulating band-wise spectral integration. (3) Finally, the learned radiance field is modulated by the sensor’s spectral responsivity  $R_b(\lambda)$ to produce physically consistent spatial–spectral fusion products. This physics-grounded fusion paradigm offers a principled solution for reconstructing high-resolution multispectral and hyperspectral images in accordance with sensor imaging physics, effectively harnessing the unique advantages of spectral data to better uncover real-world characteristics. Experiments on multiple benchmark datasets show that our method surpasses state-of-the-art fusion algorithms, achieving reduced spectral aberrations and finer spatial textures. Furthermore, extension to hyperspectral (HS) data demonstrates its generalizability and universality. The code will be available upon potential acceptance.

Physics-informer Neural Operator for Pansharpening
Xinyang Liu Junming Hou'; Chenxu Wu', Xiaofeng Cong?, Zihao Chen Shangqi Deng', Junling Li', Liang-Jian Deng", Bo Liu
School of Information Science and Engineering, Southcast University
2 School of Engineering Mathematics and Technology. University of Bristol
School of Cyber Science and Engineering, Southeast University
4 School of Mathematical Sciences, University of Electronic Science and Technology of China
5 Institute of Artificial Intelligence and Robotics, Xi' an Jiaotong University
Faculty of Computer Science and Control Engineering, Shenzhen University of Advanced Technology codex 1xy@gmaill c junming houdseu. edu. 1 liangjian ian. dengduestc. edu, en, li..uobsuatt-sz. edu. cn
Abstract
Over the past decades, pansharpening has contributed greatly numerous remote sensing applications, with methods evolving from theoretically grounded models to deep learning approaches and their hybrids. Though promising, existing methods rarely address pansharpening through the lens of underlying physical imaging pro- cesses. In this work, we revisit the spectral imaging mechanism and propose novel physics-informed neural operator framework for pansharpening, termed PINO, which faithfully models the end-to-end electro-optical sensor process. Specifically, PINO operates as: (1) First, a spatial-spectral encoder is introduced to aggregate multi- guuti-granullartty high-resolution panchromatic (PAN) and lowrresouution multi- spectral (LRMS) features. (2) Subsequently, an iterative neural integral process utilizes these fused spatial-spectral characteristics to learn a continuous radiance field (a y, X) over spatial coordinates and wavelength, effectively emulating band-wise spectral integration. (3) Finally, the learned radiance field modu- lated by the sensor's i spectral responsivity (A) to produce the desired fusion fusion paradigm offers a principled solution for pansharpening in accordance with sensor imaging physics. Experiments on multiple benchmark datasets show that our method surpasses state-of-the-e fu- sion algorithms, achieving reduced spectral aberrations and finer spatial textures. Furthermore, extension to hyperspectral (HS) data demonstrates its generalizability and universality. The code is available at https://ittab.com///iiion//////
products. This
1 Introduction
Remote sensing satellites have become indispensable tools for Earth observation and monitoring, which span diverse civilian and military applications, such as precision agriculture, environmental monitoring, and mineral exploration [56] 67 50 49). Unfortunately, existing optical satellite sensors struggle to directly acquire high-resolution multispectral (HRMS) observations. Instead, they typically acquire paired rew-lsolutionn (LR) MS and high-resolution panchromatic (PAN) images of the same scene. Pansharpening techniques aim to bridge the resolution gap by fusing complementary PAN and MS modalities to produce HRMS images, benefiting downstream remote sensing tasks.
*Co-first authors
I Corresponding Author
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Over the past decades, the pansharpening community has witnessed a proliferation of methodologies, spanning from traditional model-driven algorithms to deep learning approaches and their hybrid variants. Despite notable advancements, most existing methods inadequately model the fundamental electro-optical process inherent to optical sensor systems, such as continuous radiance formation, optical filtering and sensor responsivity (59 33 79]. Traditional approaches, including component substitution and multiresolution analysis, typically operate on tensorized band data, applying heuris- tic transformations to align spatial details with spectral content. Deep learning approaches learn direct mappings from LRMS to HRMS. Both paradigms exhibit critical shortcomings: model-driven methods rely on simplified physical assumptions that often violate sensor- -ensor--pecfiiccec radiometric con- straints, while data-driver approaches implicitly capture sensor characteristics through training data without explicit physical constraints. This disconnect from the underlying sensor imaging physics leads to critical challenges, including sensor-specific biases, spectral aberrations and spatial artifacts, restricting the applicability in real-world scenes.
In this work, we propose a physics-informan infoomeed neural operator framework, named PINO, which unifies continuous radiance field modeling with learnable spectral responsivity, significantly boosting spatial- spectral fidelity and enhancing generalization across heterogeneous sensors and varying imaging conditions. The key procedures of PINO are follows: First, we employ spatial-spectral encoder to aggregate multi granularity high-resolution PAN and LRMS features. Second, we exploit the fused spatial-spectral characteristies learn continuous radiance field over spatial coordinates and wavelengths, enabling the coexistence of sub-pixel spatial details and sub-band spectral nuances within the implicit neural representation. This process is implemented using an iterative kernel integral operator, effectively emulating band-wise spectral integration. Finally, we modulate the learned radiance field with the sensor's spectral responsivity, in accordance with the camera forward model, to generate high-resolution fusion products. We summarize the main contributions as follows:
We propose
presically-groundde pansharpening framework that establishes new paradigm bridging neural representations with physical sensor models, providing a princi- pled solution applicable to multispectral and hyperspectral image fusion tasks.
We employ an iterative kernel integral operator that leverages multi- graniiiraanlarity spa- tial-spectral features to learn a continuous radiance field over spatial coordinates and wavelength, enabling sub-pixel and sub-band fusion of different modalities while effectively emulating band-wis wise spectral integration.
We introduce a learnable band wise spectral responsivity modulation simulate the sensor's spectral properties, allowing the simultaneous optimization of response functions.
Experiments on multiple remote sensing benchmark datasets reveal that PINO consistently outperforms state-of tf-tte--rtt methods in fusion capability, generaizalibity and adaptability.
2
Related Works
Deep Learning-based MSI and HSI Fusion In recent years, fueled by the availability of large- scale datasets, deep learning-basee methods have achieved impressive progress in MS and HS image sharpening. significantly outperforming traditional model-driven approaches
[6 67
24
Typical techniques range from convolutional neural networks (CNNs) [78, 9, 25|
27
19 global
transformers 82 23
4 73,, to recently emerging generative diffusion models [55
In addition,
growing interest has been directed toward hybrid approaches that integrate traditional fusion principles
with deep learning architectures to enhance model interpretability 775 promising, these methods are largely grounded in mathematical models and rarely explore potential solutions from the physical imaging process. Concurrently, recent methods SSMNet and FBS-PS begin to incorporate sensor-related physics (c.g. band-separable properties) into learning- based fusion. However, they still fall short of modeling the full electro-optical imaging process.
84
63
35
51
While
42 30
Implicit Neural Representation Implicit neural representations (INRs) have drawn significant attention for their ability to encode continuous signals with high fidelity in a parametric form. This approach has been revolutionary in the domain of 3D computer vision, as exemplified by NeRF [574] which demonstrated the potential of encoding intricate 3D structures with just 2D pose images. Recent advancements in INR architecture have led to the development of continuous image functions, which leverage the power of multi-layer perceptrons (MI.ss) for tasks such as super-resolution (SR). For instance, LIIF 77 introduced a local implicit image function for SR, where the network performs
2

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localized learning across the spatial domain to improve high-resolution reconstruction. Similarly, UltraSR [77] and DIINN [58] employ deep neural networks with residual connections and dual- interactive architectures, respectively, to further enhance decoding capabilities and spatial detail recovery. Furthermore, SSIF learns a continuous spatial-spectral representation for spectral image super-resolution [53]. and SeeF performs physically based spectral rendering with neural radiance field (36., illustrating how INR can extend beyond purely spatial signals. This line of work has naturally extended to the pansharpening and MHIIF domain [54] 39 [3), where INRs have shown promise in enhancing the representation spectral data. However, the current methods mainly use spatial coordinates or/and simple band index to encode the continuous signal of spectral data.
Image Super-resolution Neural Operator Neural operators (NOs) have recently been introduced to learn mappings between dnfiite-dimensione function spaces [32 31. In contrast to classical neural networks, which typically operate on finite-dimensional Fuclidean spaces or discrete sets, neural operators optimize network training within function spaces, offering superior nonlinear As resolution-invariant architecture, neural operators generalize effectively across different discretizations, rendering them highly suitable for tasks involving continuous or resolution- varying data. Wei et al. introduce SRNO, which integrates the function space mapping of neural operators with the efficient non-local modeling of Galerkin- style attention, offering a novel approach to continuous super- repoetrrsooution [71]. DiffFNO designs a Weighted Fourier Neural Operator with novel mode rebalancing mechanism address the loss of segh-freueency details traditional counterparts caused by mode truncation [41]. HiNOTE introduces hierarchical neural operator framework that effectively broadens the scope of neural operators to scientific data [46]. Very recently, He et al. apply neural operators to the hyperspectral pan-sharpening, developing spectral-spatial continuous function mapping to solve the challenging scale generalization problem (20).
fitting and invariance to discretization 45
Methodology
In this section, first introduce the preliminary of the spectral sensor imaging mechanism and then derive the proposed physics-informed neural operator framework. Subsequently, we elaborate on the implementation details of continuous radiance field learning, band-wise spectral responsivity modulation, optimization strategies and objectives within the proposed PINO.
3.1 Preliminary Spectral Imaging Sensor Model
A multispectral or hyperspectral sensor records the scene's emitted or reflected radiance by focusing light through its optics and spectral filters onto detector array. Along this optical path, the radi- ance spatial distribution undergoes ra- diometric, spatial and spectral transfor- mations: lenses and dispersive elements shape the beam, filters select wave- length bands, detectors convert photons into electronic signal, and A/D converter quantizes that signal into dig- ital numbers. Modeling these steps is essential to recover both high-resolution spatial detail and faithful spectra in subsequent fusion.
Imaging Process
At-Sensor Radiance
Imaging Optics
Scanner
Detectorg
A/D
Spectral Observation
Figure 1: The key components and processes in an electro- optical pppettallimaginn system.
As schematically depicted in Figure [ this imaging pipeline operates as: (i) the scene radiance is collected and formed on the focal plane; (ii) it passes through band-selecting filters or prisms; (iii) detectors measure the filtered irradiance; (iv) the resulting analog signal is digitized. Let L, represent spectral irradiance on detector located on the optical axis, it is related to the at-sensor radiance Li by the camera equation 59]: L, v, A) (To(A) (,(,,
(W-m-2 ......mmm pm-
4N2
where is the opties f-umbber, given by the ratio of the optical focal length divided by the aperture stop diameter. For simplicity, we assume unit geometric magnification between the scene and the
3

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Wavelength Range
Up-anoiing
+
+
ww
Merge
Wavelength Sampling
Dyy,
Rg(2)
A2582
Encoder
UnFald
LRMS
Sampling
Iterative Kernel Integral N- h
h2 h2a
Lift
PAN
Loc. [x,y]
14 Kl(sas). (e af sarda mayer
Le M.A)
Riemman Sum Discretization Approximation
Concatenate
hist
Σ LiR.AAAAAAAAAA L. RAAAM
Embedding
GT
HRMS
Figure 2: Overview of the proposed PINO. features three key ingredients: (i) Spatial-spectral encoder: Aggregating multi-granularity spatial-spectral features; (ii) Iterative kernel integral operator: Learning a continuous radiance field over spatial coordinates and wavelength; (iii) Radiance Field Modulation: Modulating the learned continuous radiance field with the spectral responsivity to generate the desired HRMS images.
image plane, and use the same (æ, y) coordinate system for both the scene and image. The optical transmittance (A) (excluding filters) remains high (>90%) and spectrally uniform in reflective systems, causing minimal spectral distortion.
We then introduced multispectral filters wavelength dispersion elements, such as prisms, split the incoming energy into distinct wavelength bands. Denote the combined filter transmittance and detector sensitivity by the spectral responsivity R&,A), the signal intensity Itb measured by the sensor in band b is:
- L, A) (W-m-3.
[,,(0,)) =
2
where the integral is over the sensor's spectral range Ap. In practice, we discretize this integral via Riemann summation sampling wavelengths {6 E As:
1, Σ Li (,)
(3)
where (Ay) represents weighted spectral responsivity that incorporates both the original spectral responsivity and the wavelength sampling interval. Finally, the analogue signal [1,( y) is converted digital values via A/D conversion. In general, this model reveals critical insight: The sensor output is weighted integral of continuous radiance, motivating us to learn this mapping directly.
3.2 Physics-informed Neural Operator for Radiance Formation
To approximate the integral of continuous radiance, we first employ a transformer- based encoder to extract features from the input MS image MS R l''xs and PAN image IPAN E H shown in Figure 2 Specifically, we use the bidirectional dilation transformer (BDT) architecture [2), which leverages dilated spatial self-attention and grouped spectral self-attention in a bidirectional hierarchy, expanding the spatial receptive field while capturing local spectral correlations. This allows the model aggregate both global context and fine details across the multi-resolution inputs. The encoder thus outputs an intermediate feature map F that jointly encodes information from the upsampled MS image and the PAN image. More details about the encoder can be found in the Appendix A.3.3

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Given the fused feature map F, we could query and unfold the feature to obtain each point-wise feature in the input resolution, which de facto standard for SR-related INRs 39) Meanwhile, we use periodic Fourier encoding
[52
for both sampled spatial coordinates and wavelengths:
(1( p sin
0o 2
cos( (.) .... sin ,r-;), cos ())
0₂ = Ciin g ,-, t t= 0, -
where v denotes the input, T is the total number of grid scales and g Cmax, predefined constants and present the minimum and maximum grid scale, p could parameterized by a MLP or neural operator.
Cminn Cmaz are
Cuin
Consequently, weight each neighboring feature by the fraction of its overlap with the query coordinates (2r, y) in the targeted-resolution to obtain an interpolated pixel- wise feature that captures local image content. As iullustrated in Figure [ concatenate these features with periodic fourier encodings of the same spatial coordinates, yielding the final fetched feature F. Based on F, we approximate the continuous radiance field L( y, A) by leveraging an iterative kernel integral operator with Galerkin-type integral form [5]
1 F, h 12
N- NN,
with each hidden feature h" (x, y. A), 1.2,.... .N. defined as:
g n- "0,'(x,)) = 1 Jo K((0, v), (u, h2 (u, v) dudv,
(6)
n (x,, A) = J 1 (IT,)))(A)ddX
(7)
S. K( v). (u, h" v) dudv) H[))dX
+ hh (z. 1e (2,
(8)
after iterations, the hidden feature A. is projected to obtain the radiance field L(x, y, A). To make the computation tractable, we discretize the outer integral over À in Eq 8 approximating via finite summation, which we dubbed as wavelength modulation.
3.3 Band-wise Spectral Responsivity Modulation
While we have obtained the estimated radiance, as described in Section 3.1 we still need to integrate the radiance field through each band's responsivity to produce the recovered sensor measurements. Specifically, for each spectral band b, we parameterize its spectral responsivity function R2 (A) as a learnable MLP layer that maps wavelengths to responsivity values. Note that this learnable responsivity function shared across spatial coordinates and image contents, and the output of this MIP will be applied to perform band- wise modulation for estimated radiance, such that there are 8 MLP layers in total. To ensure physical validity, we enforce non-negativity and smoothness constraints on the learned responsivity functions by adding sigmoid activation function:
R (A,, 0,) (0.
where 0g is learnable and parameterized by a MLP, A, E (A In summary, the learnable R2, (Ay, 0.) lies within the range of [0. 1] and could be regarded as relative responsivity function (59].
Finally, we could predict the signal intensity and the pansharpened pixel value at the corresponding spatial coordinate and band:
L,((,,)) = Σ L,(2,V ELo (A,,,, 0b),
(10)
Pub(I,u) = 1,,(2, v) + P.M. pms (x,
where P.M (z. y) is the pixel value from the upsampled MS image at the corresponding spatial coordinates and band. The residual value could help the neural network to align the scale, as the physical dimension is not specified in the implicit neural representations.
5

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3.4 Optimization Strategies and Objectives
Since learning implicit representations relies on sampling spatial coordinates, direct optimization can lead to the degradation or loss of spatial and semantic information from the input MS and PAN images. To mitigate this issue and build a robust foundation for spatial-spectral feature extraction, we adopt two-stage training strategy. In the first stage, the encoder is trained independently using reconstruction losses, specifically I.1 and SSIM, defined as:
L = L1 + 0x,1 Lusim:
(11)
where aal is a trade-off hyperparameter that balances two loss terms, L measures the mean of absolute error, Leeim measures structural similarity.
Then the encoder is fine-tuned in conjunction with PINO, which learns the continuous radiance field and spectral responsivity. Since the second-stage -tage training involves uniform pixel sampling. local structural consistency is disrupted during the process. To address this, we incorporate the histogram loss Chist 442 to align the statistical distributions of the predicted and reference pixel samples. Given the sampled HRMS pixels V and corresponding reference pixels Va. the loss for the second stage defined as follows:
L.2 L1 + 01,2
1
(12)
Chist
D D
where 02 is introduced to balances the loss terms, D2 and Dwa are the estimated ID cumulative histogram vectors of the sampled pixels from the i-th fused spectral band and its corresponding reference sampled pixels, respectively.
Experiments
Datasets, Metrics and Implementaion Details. We assess the effectiveness of our method using data collected from the WorldView-3 (WV3), GaoFen-2 (GF2), and World View-2 (WV2) satellites, which publicly available through the PanCollection dataset [10]. Owing to the absence of ground-truth references, the training datasets are simulated using observations from the original satellite imagery in accordance with Wald's protocol. In particular, WV2 samples are used for cross-sensor evaluation to further validate the model's generalizability and transferability. Reduced resolution evaluation conducted using five established metrics: PSNR, SAM (0). ERGAS (68), Q2n and SCC (8I). Full resolution performance is assessed through three no-reference indicators: Dx. and D., and HQNR 22] All experiments are conducted an NVIDIA GeForce GTX 4090 GPU, using the PyTorch framework. More implementation and training details can be found in Appendix A.3.4
Baselines. We compare model against several state-of-the-e art DL-based methods to validate its superiority, including FusionNet (9. GPPNN (75), Fourmer 831, HFIN [62], and HOIF [85),
PanMamba (21), I.ormer 23 ADWM 28 For comprehensive evaluation, also include three
classical algorithms: BTTH BDSD-PC 65
LRTCFPan 74]
4.1 Comparison With SOTA Methods
Evaluation on Reduced Resolution Scene. We first quantitatively assess the similarity between the fused images and the ground truth reduced resolution scenarios. As shown in Table our model surpasses state-of-the-ard pan-sharpening methods across all evaluation metrics for both the WV3 and GF2 datasets. Figure displays the RGB visualizations alongside their corresponding Mean Absolute Error (MAE) residues against the ground truth for both benchmarks. Our model's outputs exhibit minimal aberrations, as evidenced by its sparse MAE residue maps. These quantitative and qualitative findings together corroborate the superior fusion capability of our method.
Evaluation on Full Resolution Scene. We then evaluate our model on full resolution data assess real-world applicability. As summarized in Table it achieves the optimal outcomes on the majority of metrics for both the WV3 and GF2 datasets, mirroring the positive trends seen at reduced resolution. In particular, it delivers the highest HQNR values, surpassing both traditional and deep-learning baselines, which reflects superior spectral fidelity and spatial detail. These full resolution results confirm our method's 's robust generalization in practical settings.

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Table I: Quantitative results for reduced and full resolution WV3 and GF2 samples, comparing
several representative state-of-the-an methods. Bold: Best; Underline: Second best.
Reduced Resolution
Hull Resolution
Method
PSNR(=
SAM(# std)
ERGAS(= sid)
Q2"(+ std)
SCCC+
DyC+
D,0+ sid)
HQNR(= std)
BDSDAPC 65
32.96942.784
1
1.696:1.617
0829:500097
0.90
0.063-00024
0.073+00.36
0870:60005
BT-H 9
33.080+2.880 4,920+1445 125
4.579+1.496
0.832+0.094
0.925=0..02
0.057-0.023
0.081+0.027
0.867+0.054
LRTCFPan
33.61342.839 4.737
4.31511.422
0.84610.091
0.92710.023
0.018
0.053:10.026
0.93110.031
FusionNet
38.04242.592
3.52510.698
2.46710.645
0.9011000900
0.98110.007
0.024=0.009
0.036:0.014
0.941:0.020
WV3 Darsect
GPPNN
05
38.57112.776
3.05510.610
2.30610.557
0,911100087
0.98110.0066
0.028
0.038.10.016
0.935.1.0022
Fourmer
38.268::2.777
682
24119:0.665
0.911+000900
0.984:88005
0.022-0.010
0.035::0.004
0.944-60013
HFIN
38.534#22786
3.088=06335
2.306+0.557
0.912+00.89
0.984+0.006
0.025-0.008
0.043+0.017
0.934+0.024
HOIr
K5
38.35242.855
3.18610.643
2.38510.668
0,9131100066
0.96210.007
0.029
0.039100010
0.92410.026
PunMamba
39.0124:2888
2.91410.592
2.184:0.521
0.920:0.085
0.95618.005
0.018-0.007
0.031:00011
0.95-+0.015
[.oomee
39,075+2.844
2.89900584
2.165+0.500
0.919+00.86
0.986+8.005
0.037-0.022
0.036+0.002
0.929+0.027
ADWM 23
29.170+22878
2,914+0.589
2114+00531
0.919:00.06
0.96++0.005
0.024-0.010
0.029:0.015
0.948+0.021
PINO (ours)
39.383=22889
2.845:0.586
2.097:0.496
0,922::0.085
0.988=00.004
0.014-0.005
0.032:+0.003
0.954=00006
BDSD-PC 65
35.180::2.317
1.68110.360
1.66750444
0.892::0.035
0.952:0.016
0.076=0.030
0.155::0.028
0.781
BT-H 6
36.051:12.236
1.69910.360
1.52810.109
0.918110:255
0.95710.0155
0.060-00025
0, 1.0011
0.17110.031
LRTCHPn Y4
37.59992.3331 1.298+0.312
1.272+0,343
0,955-+0.030
D 0.964-00012
0.033-0.027
0.0M0=0.014
0.88--0.025
FusionNet
39.639:2..200
0,97410:211 212
0.958::0.222
0,964:10.009
0.981
0.040010.013
0.101:0.0113
0.863-10.188
GP2 Datoset
GPPNN
42.42661.8000 0.797+0.161
0.711+0.130
0.979++0.006
0.900--B.002
0.023-0.012
0.067+00.09
0.912+00014
Fourmer
3
40.670++1.003 0:976::0.209
0,885 185
0.9701100011
0.967
0.047=00.039
0.038:10.010
0.917:10.025
HFIN
12.189:1..722 0.843110.14 148
0.735 126
0.977:10.011
0.99010.0022
0.027=0.020
0.062:10.009
0.912:10.018
HOIF
40.962+1.801 01943+0.205
0.641-0.122
0.974++0.009
0.029-0.015
0.051+0.011
0.922+-0.11
PanMamba
121107 0.74320.156
0.684:0.129
0.962++0.08
0.901
0.023=0.011
0.057=0.012
0.921=00.155
[.oomee
25
44 19196+1.800
0.648+0.190
0.578+0.112
0.985+00007
0.993+0.0002
0.021-0.010
0.050+0.008
0.930+0.013
ADWM
18771
0.67210.130
0.597
0,965:100006
0.903
0.022
0.052.10.011
0.928:100044
PINO (ours)
44.705:1.819
0.61510.129
0,544:0.107
0,987 10.006
0.994:00001
0,016 0.009
0.018
0.967:10009
Ideal value
0
0
1
WV5 Dutaset
GF2 Datuset
FisiooNic
GEPNN
Foumner
HINN
HOIF
PazMasria
L.Ferner
ADW
PINO
Figure 3: The visual results (top) and mean absolute error maps (bottom) of all compared DL-based methods on a
sample from WV3 and GF2 sensors, respectively.
Generalizing to New Satellite Data. We further evaluate adaptability by deploying the WV3-trained model on unseen WV2 samples without any fine-tuning. As reported in Table 2 our method outperforms all deep-learning baselines across all reduced and full resolution metrics, demonstrating its robust cross-satellite generalization and adaptability.
Evaluation on Hyperspectral Datasets. We further demonstrate our model's versatility by applying it to multispectral and hyperspectral image fusion (MHIF) task, which shares degradation principles with multispectral pansharpening. We benchmark against leading MHIF methods over the widely used CAVE dataset. As shown in Table 3. our approach outperforms all basclines across all metrics.
4.2 Ablation Studies
We first investigate the contribution of different core ingredients by performing ablation studies on the WV3 dataset with reduced resolution setting. To demonstrate the adaptability of our proposed
7

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Table 2: Quantitative results for reduced and full resolution WV2 samples, comparing several state-of-the-an deep learning methods. Bold: Best; Underline: Second best.
Redicee Resolation
Full Resolution
Method
PSNR(= std)
SAMC# std)
ERGASC# std)
(2" std)
SCC std)
DaC+ std)
D std)
HQNRC= std)
FussinNNtt
0
28.734+2.400
6.426-0.860
5.136+00.55
0.796-0.074
0,875+00013
0.052+00.229
0.056-0.015
0.894-0.019
GPPNN
65
28.384+1.876
6823=08882
5.208=00:73
0:791:00005
0911+0.011
0.11::0.064
0.055=0.012
0.822:0.063
Fourmer
3
28.924#2.418 6.224=0:459
4.915=0.348
0,521=00078
0.89120.017
0.038:0.027
0.085=0.017
0.879=0.017
HFIN
30.09312.208
5.467=0.681
4.411=0.447
0,835:0.083
0.91810.008
0.059.0.045
0.03920.013
0.905=0.054
HOIF
85
29.917#2.242 5.490=0.655
4,517=00:47
0.834:0.085
0.908+0.0000
0.068=0.044
0.043=0.058
0.894=0.088
PanMamba
21
29,372+2.654 6.471=0.943
0.790=0.399
0.817:0.076
0.898:0.018
0.056=00.027
0.040=0.011
0,907=0.05
LFormer
28
30.077+2.338 5.613=0.594
4.411:0.399
0.838.0.081
0,91610.010
0.056:10.038
0.038=0.010
0.908=0.040
ADWM
2
30,27012.227 5.483
4.343.10.460
0,84110.083
0.92110.010
0.19210.310
0.073100076
0..30
PINO (ours)
31.509:2.323 4.892-0.545
3.750-00.59
0.871
0.025 0.941
0.034:0.002
0,942:0.00 0.018
Ideal value
1
0
Table 3: The average and standard deviation calculated for all the compared approaches on 11 CAVE examples simulating scaling factor of 4 and 8. Bold: Best; Underline: Second best.
CAVE x4
CAVE x3
Method
PSNRC1
SAMCL sd)
ERGASCL sdd)
SSIMLL sid)
PSNRCL sid)
SAMEL sd)
ERGASEL
SSIMLL
Bicubic
34.326-1.822
4.45
7.2055:4900 902
0.94220.029
29.96223.544
5.890=2.322
5.563=33081
0.86
CSTF-FUS 53
14.38-5.300 302
7.8444.562
0.86++0.073
38 48.44344002
7.010=2660
2.083+1007
0.960+0:027
LITR
15
35.85153
6.990:2.554
5.82242.79
0.95410.028
37.92243.594
5.373 11960
2.441 #10050
0,97220.018
LTMR
36.543-3.300
6,711=2 193
5.241+2.419
0.962+0.020
38.413+3.552
5.041=1.704
2.244 +0.973
0.974+0.017
TR-TenSR
35.608
12.295-4.683
5.715+2.899
0.944+0.026
36.787436688
12.865444979
2.667+1.300
0.943+0-030
Pusformer
26
49:983-80097
85203-0881
200
0.992+0.011
48 38.363+55.0 108
2,665-00768
01865-00844
0,994+0.004
DHIF
hD
51.072:4.165 6:008:10660
1.222:00.97
0.997.0.002
46.4611-44 893
2.5051007887
0.83610.662
0.095
PSRT
11
50.467-6.187
2.19310.610
2.05713.713
47.857.7.5522 0.99610.003
2.73110.801
1.521333023
0,990100.005
STT
57
51.376-44.77 1.79
2.161:00605
1.137:0.996
0.996:00.003
48. 995
2.299:0.653
1.185=22.277
0.996 003
DSPNet
61
51.182._3.921
2.148100612
1.133,0,820
0.99710.001
5.3.33
2.722.007877
0,811100.11
0.095
QIS
86
02.218-4.213
1.977-0.599 599
1,025-0080 806
0.997+0.001
441 +-4.988
2.545-0.772
0,733-06625
0.996+
DCT
8
49.973 3.584
2.511.0.800 802
1,250L00.884
0.99610..02
46 142.339988
3.460L00.988
0,967.10.656
0.000
MIMO
50.855_3.444
2:285-10772
1.17710.717
0,99610.0011
47.976614.305
0.955
0.864100614
0.994
PINO (ours)
52.362 3.432
1.857.0.556
0.946.00.60
0.99810.001
50.221114.374
2. 10666
0.631 10.478
1.996.00000
Ideal
PINO, we further apply it to different encoders and compare the performance gain. Moreover, we also reveal the importance of different optimization strategies and objectives. We also refer readers to Appendix AA.6 for qualitative visualizations of the learned radiance maps and sensor responsivity, and A.2 for theoretical analysis and physical-consistengg" details.
to Appendix|A I.
Ablation Study of Core Design. We investigate the integration of neural operators and ra- diance field estimation in the context of pan-sharpening. The radiance field plays a critical role in accurately representing the spectral information within produced MS images. To as- sess its contribution, we conduct ablation studies that explore different design configurations. Specifically, we consider the inclusion or exclusion of the neural operator as well as the impact of incorporat- ing or omitting the radiance field estimation process. We also examine the role of spectral responsivity modula- tion, which refines the output align with sensor-specific characteristics. Let "w/o no", "w/o sr" and "w/o wm" " represent the results: "without neural operator", "without spectral responsivity" and "without wavelength modula- tion", respectively. As shown in Figure 4 we report three representative metrics "PNSR", "SAM" and "ERGAS" on the reduced resolution WV3 examples. We can find that the wavelength modulation is critical for the accurate radiance field estimation, without it, the pansharpening performance of PINO degrade largely. The neural operator and spectral responsivity are also essential for improving the performance. Overall, the results demonstrate that integrating these components improves fusion accuracy, resulting in better spectral integrity and spatial detail.
PSNRA
30
17
Oirra
20:303
2.07
2.145
ERGASI
Figure 4: Ablation study of core design.
8

---

PSNR
stage
one stoge
wio has
1O0 53tm
WO same
We
Gues
Ours
Figure 5: Ablation results for optimization strategies and objectives. "one stage", "w/o ssim" and "w/o hist" represent the results of "only training encoder with PINO one stage", "without ssim loss" and "without histogram loss", respectively.
Optimization Strategies and Objectives. We investigate different optimization strategies, com- paring single-stage versus ttogstage training processes. In the single-stage training, the model is optimized as a whole, while the two-stage approach, the encoder is first pretrained independently to extract ppatial-spectrac features. Then the encoder is fine-tuned in conjunction with PINO, which learns the continuous radiance field. We also explore the use of different loss functions, including SSIM loss and histogram loss, ensure that both spatial and spectral integrity are preserved during the fusion process. Figure compprrss the impact of different optimization strategies on fusion quality, where method yields the optimal results, highlighting the efficacy of the selected techniques.
ERGAS
40.0
2.999
2.200
2.1GG
2,150
39.5
3002
00-084
075
19.188
39.
30 30,006
2.95
3,800
0888
2.150
2.130
20
2.90
845
2.125
2.85
2.100
2.075
38.5
2.80
SSMNer
SSMNNd+
LFormer
LFormer
hDT
BUT
SSMNe
SoMPNut
LForeer
LPormmr
hoT
BUT+
SSMINe
SoMNet+
LFormer
LPormer's
HOT
BUT
Figure 6: Ablation results for different encoder architectures with and without PINO. X means original encoder architecture while X+ denotes corresponding encoder with PINO.
Encoder Architecture. We explore the influence of different encoder architectures the per- formance of PINO. The encoder responsible for aggregating multi-granularing features from the input high-resolution PAN and LRMS images. We conduct comparative analysis of various coder architectures, such as SSMNet, LFormer and BDT, focusing on evaluating the performance improvement when the encoder is combined with PINO compared the encoder alone. The results in Figure 6 highhighh the advantages of PINO-enhanced encoder architectures, which offer improved feature fusion and support more accurate pansharpening.
5 Limitations and Conclusion
Unlike conventional INRs developed for super-resolution tasks, in this paper, we mainly focus on the spectral imaging process and incorporate its physical sensor characteristics. We do not explore the use of arbitrary resolution settings for training and inference, which remains uncommon in current practices of the pansharpening community. Nevertheless, investigating this capability could greatly reduce the computation and parameters by unifying the redundant models and training for each targeted resolution. Moreover, cross-sensor or cross-dataset training is not explicitly addressed in current design. The computational complexity and model size of our framework could also

---

be further optimized to enhance efficiency and scalability. In the wavelength modulation, we use uniform spectral sampling. which simple and empirically effective but potentially suboptimal under strong wavelength variability; sensor- aussor-vwrr/adaptiie schemes (e.g., denser samples where spectral responsivity or radiance changes rapidly) could better capture fine spectral detail.
We present Shyyically-groundde pansharpening framework that establishes a new paradigm bridging neural representations with physical sensor models, providing a principled solution applicable to multispectral and hyperspectral fusion tasks. At its core, our method first learns a continuous radiance field (I, y. A) over spatial coordinates and wavelength using multi- gaaliigraaularity spatial- spectral features, effectively emulating band-wir -wise spectral integration. By modulating this radiance field with the sensor's spectral responsivity (A), can generate figh-fideitty fusion outputs. Extensive experiments on multiple benchmark datasets demonstrate that our approach outperforms Hate-offthheaar fusion algorithms in both quantitative metries and qualitative comparisons.
Acknowledgements
This work is funded by National Natural Science Foundation of China (Grant No. 62203309 and 12271083), and Guangdong Basic and Applied Basic Research Foundation (Grant No. 2024A151 15501 1333), and the Project of the Department of Science and Technology of Sichuan Province (Grant No. 2025YFNH0001).
References
[1] Aiazzi, Luciano Alparone, S Baronti, R Carlå, Andrea Garzelli, and L. Santurri. Full-scale assessment of pansharpening methods and data products. In Image and Signal Processing for Remote Sensing XX, volume 9244, page 924402. SPIE, 2014.
[2] Alberto Arienzo, Gemine Vivone, Andrea Garzelli, Luciano Alparone, and Jocelyn Chanussot. Full-resolution quality assessment of pansharpening: Theoretical and hands-on approaches. IEEE Geoscience and Remote Sensing Magazine, 10(3): (000::68--11111 2022.
[3] Kamyar Nzizzadenesheli, Nikola Kovachki, Zongyi Li, Miguel Liu-Schiaffin, Jean Kossaifi, and Anima Anandkumar. Neural operators for accelerating scientific simulations and design. Nature Reviews Physics, 6(5):320-328, 2024.
[4] Jonathan T Barron, Ben Mildenhall, Dor Verbin, Pratul P Srinivasan, and Peter Hedman. Zip-nerf: Anti-aliased grid-based neural radiance fields. In Proceedings of the IEEE/CVF International Conference Computer Vision, pages 19697-1970S 2023.
[5] Shuhao Cao. Choose transformer: Fourier or galerkin. Advances in neural information processing systems, 34:24924-2444 24940, 2021.
[6] Zihan Cao, Shiqi Cao, Liang-Jian Deng, Xiao Wu, Junming Hou, and Gemine Vivone. Diffusion model with disentangled modulations for sharpening multispectral and hyperspectral images. Information Fusion, 104: 104:102158, 2024.
[7] Yinbo Chen, Sifei Liu, and Xiaolong Wang. Learning continuous image representation with local implicit image function. In Proceedings of the IEEE/CVF conference computer vision and pattern recognition, pages 8628-8638, 2021.
[8] Shengyang Dai, Mei Han, Wei Xu, Ying Wu, Yihong Gong, and Aggelos K Katsaggelos. Softcuts: soft edge smoothness prior for color image super-resolution. IEEE transactions image processing, 1865):96--911, 2009.
[9] Liang-Jian Deng, Gemine Vivone, Cheng Jin, and Jocelyn Chanussot. Detail injection- deep convolutional neural networks for pansharpening. IEEE Transactions Geoscience and Remote Sensing, 5995))6995-7000, 2020.
[10] Liang-Jian Deng, Gemine Vivone, Mercedes Paoletti, Giuseppe Scarpa, Jiang He, Yongjun Zhang, Jocelyn Chanussot, and Antonio Plaza. Machine learning in pansharpening: A bench- mark, from shallow to deep networks. IEEE Geoscience and Remote Sensing Magazine, 10(3):279-315, 2022.
10

---

[11] Shang- Qi Deng, Liang-Jian Jian Deng, Xiao Wu, Ran Ran, Danfeng Hong, and Gemine Vivone. Psrt: Pyramid shuffle-and-reshuiee transformer for multispectral and hyperspectral image fusion. IEEE Transactions on Geoscience and Remote Sensing, 61:1-15, 2023.
[12] Shangqi Deng, Liang-Jian Deng. Xiao Wu, Ran Ran, and Rui Wen. Bidirectional dilation transformer for multispectral and hyperspectral image fusion. In Proceedings of the Thirty- Second International Joint Conference on Artificial Intelligence, pages 3633-3641, 2023.
[13] Shangqi Deng, Jun Ma, Liang-Jian Deng, and Ping Wei. Otias: Octree implicit adaptive sampling for multispectral and hyperspectral image fusion. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 39, pages 2708 27162 2025.
[14] Renwei Dian and Shutao Li. Hyperspectral image super-resolution via subspace-basedd low tensor multi-rank regularization. IEEE Transactions on Image Processing, 28(10):5135-5) 2019.
[15] Renwei Dian, Shutao Li, and Leyuan Fang. Learning a low tensor-train rank representation for hyperspectral image super-resolution. IEEE transactions on neural networks and learning systems, 30(9):2672- 2683, 2019.
[16] Renwei Dian, Shutao Li, Bin Sun, and Anjing Guo. Recent advances and new guidelines on hyperspectral and multispectral image fusion. Information Fusion, 69:40-51, 2021.
[17] Jian Fang. Jingxiang Yang, Abdolraheem Khader, and Liang Xiao. Mimo-sst: Multi-input multi-output spatial-spectral transformer for hyperspectral and multispectral image fusion. IEEE Transactions on Geoscience and Remote Sensing, 62:1-20, 2024.
[18] Andrea Garzelli and Filippo Nencini. Hypercomplex quality assessment of multi/hyperspectral images. IEEE Geoscience and Remote Sensing Letters, 6(4):662-665, 2009.
[19] Anjing Guo, Renwei Dian, and Shutao Li. A deep framework for hyperspectral image fusion between different satellites. IEEE Transactions on Puttern Analysis and Machine Intelligence, 4537):7939-7954, 2022.
[20] Lin He, Zhou Fang, Jun Li, Hanghui Ye, and Antonio Plaza. Arbitrary-resolution hyperspectral pansharpening neural operators. IEEE Transactions on Geoscience and Remote Sensing, 63:1- 19, 2025.
[21] Xuanhua He, Ke Cao, Jie Zhang, Keyu Yan, Yingying Wang, Rui Li, Chengjun Xie, Dan- feng Hong, and Man Zhou. Pan-mamba: Effective pan-sharpening with state space model. Information Fusion, 115:102779, 2025.
[22] Alain Hore and Djemel Ziou. Image quality metrics: Psnr vs. ssim. In 2010 20th international conference on pattern recognition, pages 2366-2369. IEEE, 2010.
[23] Junming Hou, Zihan Cao, Naishan Zheng, Xuan Li, Xiaoyu Chen, Xinyang Liu, Xiaofeng Cong, Danfeng Hong, and Man Zhou. Linearly-evolved transformer for pan-sharpening. In Proceedings of the 32nd ACM International Conference on Multimedia, pages 1486-1494, 2024.
[24] Junming Hou, Xiaoyu Chen, Ran Ran, Xiaofeng Cong, Xinyang Liu, Jian Wei You, and Liang- Jian Deng. Binarized neural network for multi-spectral image fusion. In Proceedings of the Computer Vision and Pattern Recognition Conference, pages 2236-2245, 2025.
[25] Junming Hou, Xinyang Liu, Chenxu Wu, Xiaofeng Cong, Chentong Huang, Liang-Jian Deng, and Jian Wei You. Bidomain uncertainty gated recursive network for pan-sharpening. Informa- fion Fusion, 118: 102:00,,38, 2025.
[26] Jin-Fan Hu, Ting-Zhu Huang, Liang-Jian Deng, Hong-Xia Dou, Danfeng Hong, and Gem- ine Vivone. Fusformer: A transformer-basee fusion network for hyperspectral image super- resolution. IEEE Geoscience and Remote Sensing Letters, 19:1-5,202.
11

---

[27] Jin-Fan Hu, Ting-Zhu Huang, Liang- -lang-Jian Deng, Tai-Xiang Jiang, Gemine Vivone, and Jocelyn Chanussot. Hyperspectral image super-resolution via deep spatiospectral attention convolutional neural networks. IEEE Transactions Neural Networks and Learning Systems, (3(1)))7251- 7265, 2021.
[28] Jie Huang, Haorui Chen, Jiaxuan Ren, Siran Peng, and Liangjian Deng. A general adaptive dual-level weighting mechanism for remote sensing pansharpening. IEEE/CVF Computer Vision and Pattern Recognition Conference (CVPR), 2025.
[29] Tao Huang, Weisheng Dong, Jinjian Wu, Leida Li, Xin Li, and Guangming Shi. Deep hyper- spectral image fusion network with iterative spatio-spectral regularization. IEEE Transactions on Computational Imaging. 8:201-214, 2022.
[30] Hyun-Ho Kim and Munchurl Kim. Fbs-ps: Fully band-separable pan-sharpening considering the physical characteristics of electro-optical sensors. IEEE Transactions on Geoscience and Remote Sensing, 2024.
[31] Diederik P Kingma. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014.
[32] Nikola Kovachki, Zongyi Li, Burigede Liu, Kamyar Azizzadenesheli. Kaushik Bhattacharya, Andrew Stuart, and Anima Anandkumar. Neural operator: Learning maps between function spaces with applications to pdes. Journal of Machine Learning Research, 24(89):1-9) 2023.
[33] Hoyoung Lee, Soo Hyun Park, Sang Ha Noh, Jongguk Lim, and Moon Kim. Development of a portable 3ced camera system for multispectral imaging of biological samples. Sensors, 2014.
14(11):20222-200
[34] Fan Li, Zhiyi Wang, and Guoliang He. Ap shadow net: A remote sensing shadow removal network based atmospheric transport and poisson's 's equation. Entropy, 24(9): 2022.
[35] Jiaxin Li, Ke Zheng, Lianru Gao, Li Ni, Min Huang, and Jocelyn Chanussot. Model-informed multi-stage unsupervised network for hyperspectral image super-resolution. JEEE Transactions on Geoscience and Remote Sensing, 2024.
[36] Ru Li, Jia Liu, Guanghui Liu, Shengping Zhang, Bing Zeng. and Shuaicheng Liu. Spectralnerf: Physically based spectral rendering with neural radiance field. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 38, pages 3154-3162, 2024.
[37] Shutao Li, Renwei Dian, Leyuan Fang, and José M Bioucas-Dias. Fusing hyperspectral and multispectral images via coupled sparse tensor factorization. IEEE Transactions Image Processing. 27(8):4118 4130, 2018.
[38] Zongyi Li, Nikola Borislavov Kovachki, Kamyar Arizzadenesheli, Burigede liu, Kaushik Bhattacharya, Andrew Stuart, and Anima Anandkumar. Fourier neural operator for parametric partial differential equations. In International Conference on Learning Representations, 2021.
[39] YuJie Liang, Zihan Cao, Shangqi Deng, Hong -ia Dou, and Liang-Jian Deng. Fourier-enhanced implicit neural fusion network for multispectral and hyperspectral image fusion. Advances in Neural Information Processing Systems, 37:63441 17774441-33465, 2024.
[40] Shuaiqi Liu, Tong Liu, Lele Gao, Hailiang Li, Qi Hu, Jie Zhao, and Chong Wang. Convolutional neural network and guided filtering for sar image denoising. Remote Sensing, 11(6):702, 2019.
[41] Xiaoyi Liu and Hao Tang. Difffno: Diffusion fourier neural operator. arXiv preprint 241.:92111. 2024.
[42] Xinyang Liu, Junming Hou, Xiaofeng Cong, Hao Shen, Zhuochen Lou, Liang-Jian Deng, and Jian Wei You. Rethinking pan-sharpening via spectral-band modulation. IEEE Transactions on Geoscience and Remote Sensing, 62:1-16, 2023.
[43] Yuanye Liu, Renwei Dian, and Shutao Li. Low-rank transformer for high-resolution hyper- spectral computational imaging. International Journal of Computer Vision, 1339 2025.
12

---

[44] Simone Lolli, Luciano Alparone, Andrea Garzelli, and Gemine Vivone. Haze correction for contrast-based based multispectral pansharpening. IEEE Geoscience and Remote Sensing Letters, 14(12):2255-23) -259,, 2017.
[45] Lu Lu, Pengzhan Jin, Guofei Pang, Zhongqiang Zhang, and George Em Karniadakis. Learning nonlinear operators via deeponet based on the universal approximation theorem of operators. Nature machine intelligence, 3(3):218-223) 2021.
[46] Xihaier Luo, Xiaoning Qian, and Byung-Jun Yoon. Hierarchical neural operator transformer with learnable frequency-aware loss prior for arbitrary-scale super-resolution. In Proceedings of the s1 International Conference on Machine Learning, volume 235 of Proceedings of Machine Learning Research, pages 33466-33483 PMLR, 21-27 Jul 2024.
[47] Qing Ma, Junjun Jiang, Xianming Liu, and Jayy Ma. Learning a 3d-cnn and transformer prior for hyperspectral image super-resolution. Information Fusion, 100:101907, 2023.
[48] Qing Ma, Junjun Jiang, Xianming Liu, and Jiayi Ma. Reciprocal transformer for hyperspectral and multispectral image fusion. Information Fusion, 104: 104:102148, 2024.
[49] Xiaowen Ma, Rongrong Lian, Zhenkai Wu, Renxiang Guan, Tingfeng Hong, Mengjiao Zhao, Mengting Ma. Jiangtao Nie, Zhenhong Du, Siyang Song, and Wei Zhang. novel scene coupling semantic mask network for remote sensing image segmentation. ISPRS Journal of Photogrammetry and Remote Sensing, 221 221:44663. 63, 2025.
[50] Xiaowen Ma, Rongrong Lian, Zhenkai Wu, Hongbo Guo, Fan Yang, Mengting Ma. Sensen Wu, Zhenhong Du, Wei Zhang, and Siyang Song. Logcan++: Adaptive local-global class-aware network for semantic segmentation remote sensing images. IEEE Transactions on Geoscience and Remote Sensing, pages 2025.
[51] Xiaowen Ma, Zhenliang Ni, and Xinghao Chen. Tinyvim: Frequency decoupling for tiny hybrid vision mamba, 2024.
[52] Gengchen Mai, Krzysztof Janowicz, Bo Yan, Rui Zhu, Ling Cai, and Ni Lao. Multi- scale representation learning for spatial feature distributions using grid cells. arXiv preprint arXiv:2003.00824, 2020.
[53] Gengehen Mai, Ni Lao, Weiwei Sun, Yuchi Ma, Jiaming Song, Chenlin Meng, Hongxu Ma. Jinmeng Rao, Ziyuan Li, and Stefano Ermon. Ssif: Learning continuous image representation for spatial-spectral super-resolution. arXiv preprint arXiv:2310.00413, 2023.
[54] Ge Meng, Jingjia Huang, Yingying Wang, Zhenqi Fu, Xinghao Ding, and Yue Huang. Progres- sive high-frequency reconstruction for pan-sharpening with implicit neural representation. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 38, pages 4197, 2024.
[55] Qingyan Meng, Wenxu Shi, Sijia Li, and Linlin Zhang. Pandiff: A novel pansharpening method based on denoising diffusion probabilistic model. IEEE Transactions on Geoscience and Remote Sensing, 61:1-177 2023.
[56] Xiangchao Meng, Huanfeng Shen, Huifang Li, Liangpei Zhang, and Randi Fu, Review of the pansharpening methods for remote sensing images based the idea of meta-analysis: Practical discussion and challenges. Information Fusion, 16:1022-13 2019.
[57] Ben Mildenhall, Pratul P Srinivasan, Matthew Tancik, Jonathan T Barron, Ravi Ramamoor- thi, and Ren Ng. Nerf: Representing scenes as neural radiance fields for view synthesis. Communications of the ACM, 65(1):99-106, 2021.
[58] Quan H Nguyen and William J Beksi. Single image super-resolution via dual interactive implicit neural network. In Proceedings of the IEEE/CV Winter Conference on Applications of Computer Vision, pages 4936-4945, 2023.
[59] Robert A Schowengerdt. Remote sensing: models and methods for image processing. elsevier, 2006.
13

---

[60] Philip N Slater. Remote sensing: optics and optical systems. Reading, 1980.
[61] Yucheng Sun, Han Xu, Yong Ma, Minghui Wu, Xiaoguang Mei, Jun Huang, and Jiayi Ma. Dual spatial-spectral pyramid network with transformer for hyperspectral image fusion. IEEE Transactions on Geoscience and Remote Sensing, 61:1-16, 2023.
[62] Jiangtong Tan, Jie Huang, Naishan Zheng, Man Zhou, Keyu Yan, Danfeng Hong, and Feng Zhao. Revisiting panchromatic and multi- multi-spectral image fusion. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 25922 25931, 2024.
information integration from hierarchical perspective for
[63] Xin Tian, Kun Li, Wei Zhang. Zhongyuan Wang, and Jiayi Ma. Interpretable model-driven deep network for hyperspectral, multispectral, and panchromatic image fusion. IEEE Transactions on Neural Networks and Learning Systems, 2023.
[64] Alasdair Tran, Alexander Mathews, Lexing Xie, and Cheng Soon Ong. Factorized fourier neural operators. In The Eleventh International Conference on Learning Representations, 2023.
[65] Gemine Vivone. Robust band-dependent spatial-detail approaches for panchromatic sharpening.
IEEE transactions on Geoscience and Remote Sensing, 57(9))6421-6433, 2019.
[66] Gemine Vivone, Luciano Alparone, Jocelyn Chanussot, Mauro Dalla Mura, Andrea Garzelli, Giorgio A Licciardi, Rocco Restaino, and Lucien Wald. A critical comparison among pansharp- ening algorithms. IEEE Transactions on Geoscience and Remote Sensing, 53(5):2565-2555 2014.
[67] Gemine Vivone, Liang-Jian Deng, Shangqi Deng, Danfeng Hong, Menghui Jiang, Chenyu Li, Wei Li, Huanfeng Shen, Xiao Wu, Jin-Liang Xiao, et al. Deep learning in remote sensing image fusion:
Methods, protocols, data, and future perspectives. IEEE Geoscience and Remote Sensing Magazine, 2024.
[68] Lucien Wald. Data fusion: definitions and architectures: fusion of images of different spatial resolutions. Presses des MINES, 2002.
[69] Lucien Wald, Thierry Ranchin, and Mare Mangolini. Fusion of satellite images of different spatial resolutions: Assessing the quality of resulting images. Photogrammetric engineering and remote sensing, 63(6))691-699, 1997.
[70] Zhou Wang, Alan C Bovik, Hamid R Sheikh, and Eero Simoncelli. Image quality assessment: from error visibility structural similarity. IEEE transactions on image processing, (3()):600- 612, 2004.
[71] Min Wei and Xuesong Zhang. Super-resolution neural operator. In Proceedings of the IEEE/CVF Conference Computer Vision and Pattern Recognition, pages 18247-18256, 2023.
[72] ChenYang Yang Weng, Yulin Zhan, Xingfa Gu, Jian Yang, Yan Liu, Hong Guo, Zilong Lian, Shiyuan Zhang, Zhangjie Wang, and Xuechun Zhao. The spatially seamless spatiotemporal fusion model based on generative adversarial networks. IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing, 17:1 17:12760-12771, 2024.
[73] Xiao Wu, Zi-Han Cao, Ting-Zhi Zhu Huang. Liang-Jian Deng, Jocelyn Chanussot, and Gemine Vivone. Fully-comnected transformer for multi-source image fusion. IEEE Transactions on Pattern Analysis and Machine Intelligence, 47(3):2071-200 2025.
[74] Zhong-Cheng Wu, Ting-Zhu Huang, Liang-Jian Deng, Jie Huang, Jocelyn Chanussot, and Gemine Vivone. Lrtefpan: Low-rank tensor completion based framework for pansharpening. IEEE Transactions on Image Processing, 32: 12:1640-1655, 2023.
[75] Shuang Xu, Jiangshe Zhang, Zixiang Zhao, Kai Sun, Junmin Liu, and Chunxia Zhang. Deep gradient projection networks for pan-sharpening. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 1366-1375 2021.
[76] Ting Xu, Ting-Zhu Huang, Liang-Jian Deng, and Naoto Yokoya. An iterative regularization method based on tensor subspace representation for hyperspectral image super-resolution, IEFE Transactions Geoscience and Remote Sensing, 60: 2022.
14

---

[77] Xingqian Xu, Zhangyang Wang, and Humphrey Shi. Ultrasr: Spatial encoding is a miss- ing key for implicit image tunction-based cibitrarysscale super-resolution. arXiv preprint arXiv:2 /2716, 2021.
[78] Junfeng Yang, Xueyang Fu, Yuwen Hu, Yue Huang, Xinghao Ding, and John Paisley. Pannet: deep network architecture for pan- ran-sharpening. In Proceedings of the IEEE international conference computer vision, pages 1449-5457, 2017.
[79] Fumihito Yasuma, Tomoo Mitsunaga, Daisuke Iso, and Shree K Nayar. Generalized assorted pixel camera: postcapture control of resolution, dynamic range, and spectrum. IEEE transactions on image processing, 19(9):2211-25533 2010.
[80] Roberta H Yuhas, Alexander FH Goetz, and Joe W Boardman. Discrimination among semi-arid landscape endmembers using the spectral angle mapper (sam) algorithm. In JPL, Summaries of the Third Annual JPL Airborne Geoscience Workshop. Volume 1: AVIRIS Workshop, 1992.
[81] Jie Zhou, Daniel Civco, and John A Silander. wavelet transform method merge landsat tm and spot panchromatic data. International journal of remote sensing, 19(4):743-757, 1998.
[82] Man Zhou, Jie Huang, Yanchi Fang, Xueyang Fu, and Aiping Liu. Pan-sharpening with customized transformer and invertible neural network. In Proceedings of the AAAI conference on artificial intelligence, volume 36, pages 3553-356 053-3561, 2022.
[83] Man Zhou, Jie Huang, Chun-Le Guo, and Chongyi Li. Fourmer: An efficient global modeling paradigm for image restoration. In International conference on machine learning, pages 42589-42601. PMLR, 2023.
[84] Man Zhou, Jie Huang, Naishan Zheng, and Chongyi Li. Learned image reasoning prior penetrates deep unfolding network for panchromatic and multi-spectral image fusion. In Proceedings of the IEEF/CVE International Conference on Computer Vision, pages 12398- 12407, 2023.
[85] Man Zhou, Naishan Zheng, Xuanhua He, Danfeng Hong, and Jocelyn Chanussot. Probing synergistic high-order interaction for multi-modal image fusion. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2024.
[86] Chunyu Zhu, Shangqi Deng, Yingjie Zhou, Liang-Jian Deng, and Qiong Wu. Qis-gan: A lightweight adversarial network with quadtree implicit sampling for multispectral and hy- perspectral image fusion. IEEE Transactions on Geoscience and Remole Sensing, 61:1-15, 2023.
[87] Feida Zhu, Zhetong Liang, Xixi Jia, Lei Zhang, and Yizhou Yu. A benchmark for edge- preserving image smoothing. IEEE Transactions on Image Processing, 28(7):3556-355 3570. 2019.
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A Appendix/ Supplementary Material
This supplementary material provides additional insights into the background, methodologies, and experimental details outlined in our paper. It includes details about theoretical analysis and physical consistency of proposed physical modeling, datasets, metries, the employed bidirectional transformer encoder, implementation and training. Furthermore, extensive qualitative visualizations are presented across standard pansharpening benchmarks and multispectral and hyperspectral image fusion, along with analyses of the learned radiance field and spectral responsivity, to substantiate the interpretability and physical grounding of our model. The provided information aims to enhance the reader's understanding of the intricacies involved in our research and its practical applications.
A.1 Theoretical Analysis of Physical Modeling
We begin by revisiting and summarizing the electro electro-optical image formation model used throughout the paper and detailing how each component is instantiated within PINO, providing the context for the subsequent discussion on boundary behavior and physical consistency.
1) Sensor physics modeling. (i) Irradiance formation follows the camera equation (Eq. (1) in the main paper).
π (A( y,,
L, y, A) = 4N2
(13)
which is realized within spatial-spectral encoder. (ii) Band-wise spectral integration (Eq. (2) in the main paper) is
L, = S L,(2, A) Ro(A) dA,
(14)
and in our PINO this integral is approximated via an iterative kemel integral operator (Eq. (8) in the main paper), while R&(A is a learnable, modulated function (Eq. (9) in the main paper). (iii) Discretization (Eq. (3) in the main paper) reflects sensor quantization:
I, y) = > E(((E.VAA)) R( Adj =
Σ L. R(A,))
(15)
The above formulas explicitly encodes key electro-optical process, and the proposed learning frame- work is built on this theoretical base.
2) Neural operator as physics-informed solver. The iterative Galerkin-style kernel integral operator acts as a functional approximator for the spectral integral in Eq. (2):
h (x, y. A) 1 fa K((x, ()) (u, u)) (", v) du du II()),
(16)
where IC captures ppatial-radiame coupling and II(A) provides implicit spectral basis functions (e.g.. Fourier encoding); the wavelength discretization above mirrors sensor quantization. This iterative operator lets the model learn a continuous radiance field aligned with addiative-imaging physics.
3) Spectral responsivity as differentiable physics. Parameterizing each band's responsivity as:
R (Ag 0,) = ((())))
(A5 0p) E [0, 1].
(17)
which guarantees the non-negativity that aligns with the bounded nature of real sensor spectral responses; the differentiable sigmoid (with gradient d'(x) (1 - a (z))) avoids non- non-physical discontinuities, II(A) is a spectral encoding (e.g., Fourier features), and the learnable mapping A (-) (e.g., MLP) adapts to sensor- sensor-specific bands.
A.2 Boundary Conditions of the Radiance Field and Physical Consistency
Building on the above modeling, we next state the boundary assumptions for the continuous radiance field and clarify how physical consistency is enforced in the learned sensor model.
We assume homogeneous Neumann boundary conditions for the radiance field in Eq. (1) of the main paper, i.e.,
(L y, A)
0,
(18)
In
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which reflects the smooth transition commonly observed at image boundaries in natural and remote- sensing scenes. Such boundary smoothness standard modeling choice in image processing and remote sensing 8 34 72
Although we do not explicitly encode boundary conditions into the network or loss, they can be implicitly learned from data due to the inductive bias of convolutional operations and padding. Prior work reports that networks trained for restoration remote-sensing tasks learn smooth boundary transitions even without explicit constraints 88 40 Visualizations in Appendix A.6 further show that our learned radiance fields exhibit boundary smoothness closely matching the ground truth.
While the boundary behavior of L is handled implicitly, we place an explicit physics prior on the sensor's spectral responsivity ensure global physical consistency. Because responsivity is sensor-specific and independent of scene content, we enforce non-negativity and smoothness by parameterizing each band with sigmoid activation. This guarantees learned spectral responses bounded, smooth, and sensor-consistent, leading to radiance maps that preserve fine spatial structures and realistic wavelength--dependent variations; see Appendix [A.6] Together with wavelength modu- lation and the neural operator, this yields a PINO that is both physically consistent and spectrally faithful.
A.3 Datasets, Metrics and Implementation Details
A.3.1 Datasets
We evaluate the pan-sharpening performance of our framework over several public benchmark datasets, including WorldView-3 (WV3), GaoFen-2 (GF2) and View2 (WV2). MS im- ages within the WV3 and WV2 datasets contain eight spectral bands: coastal 397-454/400-450 nm, blue 045-5174450-555 626-696/630-666 690 nm, red-edge 698-749/705-77 705 nm, near-IRI (165-899/770-888 nm, and nea 857-1039/860-10000 1040 nm. The corresponding PAN images observed from the same scene are single- channel images. The GF2 dataset pairs a single-band PAN image with four MS channels covering blue 450-520 nm, green 520-590 nm, red 620-690 nm and NIR 770-890 nm. Notably, due to the absence of ground-truth (GT) images, we generate reduced-resolltion MS-PAN training and testing pairs following Wald's protocol [69].
nm, green 507-586/510 5866510-501000 nm, yellow 1000-22/585-4460 625 nm, red
To evaluate the versatility of our model, we further conducted experiments on the multispectral and hyperspectral image fusion (MHIF) task by using the CAVE dataset. The CAVE dataset comprises 32 Hyperspectral Images (HSIs) with 31 spectral bands spanning from 400 nm to 700 nm at 10 nm intervals. We randomly selected 20 images for training and used the remaining 11 for testing (same as 39 13). The RGB visualization of the test set is shown in F
In experiments, we uniformly sample and normalize the wavelength with min-max scaling according to the specific spectral range of each band:
X Amin
Ā 2 Amax
1,
(19)
Amin
where Amin and Amaz represent the minimum and maximum value of spectral wavelength for the corresponding band. For example, Amii 397, Amaz 454, X U(\mans Amaz) for the WV3 coastal band spans 397-454 nm, and the sampled wavelength À is normalized to [-1, 1].
A.3.2 Metrics
Following standard evaluation protocols in the pansharpening community, we assess the quality of fused images using both reference-based and no-reference metrics. For reduucd-rrsolltionn qual- ity assessment, we adopt the Peak Signal-to-Noise
Ratio (PSNR) 22 Spectral Angle Mapper (SAM) 80), Relative Dimensionless Global Error in Synthesis (ERGAS) 68), Spatial Correlation Coefficient (SCC) [81). Structural Similarity (SSIM) [70] and the generalized Q2" index [18), where n corresponds to the number of spectral bands (e.g., Q8 for 8-band data and Q4 for 4-band data). For full-resolution no-reference evaluation, we utilize three widely accepted indicators: the Hybrid Quality with No Reference (HQNR) (I. Spectral Distortion Index (Dx), and Spatial Distortion Index (D,) (66). The formal definitions of these metrics are outlined below.
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Figure 7: The RGB visualization of CAVE testing samples.
(1) Peak Signal-to-Nont Ratio (PSNR): PSNR evaluates the pixel-wise fidelity of each reconstructed band by comparing it to the ground truth. We compute the average PSNR over all bands as
B PSNR M PSN Io),
(20)
where To, 1o E Hx are the b-th bands of the reference and fused images, and
max(Ib)
(I. fb) = 20 log //MSE(L,.. (I,,
(21)
where MSE represents Mean Square Error, and max represents the maximum value.
(2) Spectral Angle Mapper (SAM): SAM evaluates the angular difference between spectral vectors of the fused image and ground truth (GT), with an ideal value of It is defined as:
1 SAM
B
Σ
arecos
|rol2
Fb 1%b
|
(22)
where B the number of spectral bands, rs and is denote the b-th spectral vector from the GT and fused image, respectively.
(3) Relative Dimensionless Global Error in Synthesis (ERGAS); ERGAS measures the global radiometric distortion, ideally approaching zero. is computed as:
B
RMSE(T,, ib)
ERGAS = 100 8
(23)
B
H4b
1=1
where s is the spatial resolution ratio between PAN and MS images, is the mean of the b-th GT band, and RMSE denotes the root mean square error.
(4) Spatial Correlation Coefficient (SCC): SCC quantifies the similarity of spatial details between the fused image and GT using high-pass filter and correlation coefficient (CC). The specific calculation
of SCC includes two steps: 1) Using high-pass filter to extract the high frequencies of images. 2) Calculating the CC between the high frequencies obtain the SCC. The commonly used Laplacian filter has the following form:
L=
8
-1
(24)
-1
The CC is another widely used spectral indicator which is defined as follows:
CC = Σx- Et
F)
(25)
-)2
where ray and fiy are pixel values from GT and fused images, and 7 are their respective means, w and h are the width and height of the image.
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(5) Quality Index (Q2"): Q2" extends the universal image quality index (UIQI) metric to multi- spectral hyper-spectral images and is computed as:
Q2" [ov((r,
20, OF or oF of 0
2 [ r |
(26)
where and are multiband pixel vectors from the GT and fused image, respectively, Cov(:,)), and & denotes covariance, standard deviation and mean, respectively.
(6) Spectral Distortion Index (Dx): This metric assesses spectral consistency among bands in the fused image compared to the low-resolution input by using inter-band Q values. Mathematically, can be expressed as:
B Σ
D
Σ B(B 1 j=1
1
- 15)1
(27)
where (): is the Quality Index between bands, f; and y: denote the i-th band of the fused and low-resolution MS images, respectively. q is typically set to
(7) Spatial Distortion Index (D.): D& measures spatial fidelity relative to the PAN image, which is complementary Dx. It defined as follows:
:
B D3 Σ II((..)) Qw...))) -I
(28)
where p the original PAN image, P represents its downsampled version, Q(--)) denotes the Quality Index and q is typically set to I.
(8) Hybrid Quality with No Reference (HQNR): HQNR provides an global no-reference quality estimate by combining spectral and spatial distortions (Dx and D,). Specifically, is defined as:
HQNR = (1 (1- (1-D 11-110 D.)
(29)
where K, a, and B are hyperparameters (typically, 1).
(9) Structural Similarity (SSIM): SSIM captures structural similarity by combining luminance and structural contrast functions:
1 (2p. + C2) - C2) (4 A +1) +
SSIM
(30)
B
where Mr. A are the mean intensities of I, and î, o? are their variances, 0g is their covariance, C₁ and C2 are fixed constants.
A.3.3 Encoder Details: Bidirectional Transformer
The encoder is designed to jointly capture the spatial detail of the high-resolution PAN input and the spectral content of the coarser MS input. To this end, we adopt bidirectional dilation transformer architecture with parallel spatial and spectral branches [12]. In the spatial branch, the PAN image is concatenated with the MS image after upsampling (bicubic interpolation to PAN resolution), then passed through an initial convolutional block that increases the channel dimensionality. This is followed by a cascade of Dilated Spatial Self-Attention (D-Spa) modules. Analogously, the spectral branch processes the original low-resolution MS image with its convolutional block, followed by Grouped Spectral Self-Attentic (G-Spe) modules. Each branch thus produces a hierarchy feature maps at progressively deeper levels. By operating in a bidirectional, multi-scale fashion, these two branches extract complementary representations of the two inputs that can later be fused.
Specifically, we implement dral-branch design: (i) spatial branch applies 3 D-Spa layers with dilation rate and 8 attention heads, and (ii) a spectral branch uses same 3 G-Spe layers and attention heads with 8 non-overlapping spatial groups (i.e. window size is 8 Two branches operate in parallel and are fused to generate unified latent representation for the following radiance estimation. Each D-Spa module implements spatial attention over dilated neighborhood. Concretely,
19

---

the input feature map is linearly projected (via Ixl convolutions or fully-connee connected layers) into query, key, and value tensors for each attention head. Instead of attending to a contiguous local window, D-Spa applies fixed "hollow" window defined by dilation rate. This dilation enlarges each token's receptive field without introducing additional parameters or sliding computations. the spectral branch, each G-Spe module performs channel self-attention in grouped manner over the spatial domain. The input feature map is first partitioned into a fixed number of spatial groups (i.e. non-overlapping patches). After processing all groups independently, their outputs are concatenated to reform the full feature map. This grouped design retains the dense channel connectivity and content-aa aware nature of spectral attention while focusing each operation a local region, enabling the model to learn fine-grate une-grained spectral correlations within every area of the image.
Overall, the multi-level outputs from the D-Spa and G-Spe branches form a hierarchical representation, which fused through feature integration and progressive upsampling to produce unified high- resolution prectral-spatial representation. The enriched spectral-spatial feature map is then fed to inform the radiance estimation.
A.3.4 Implementation and Training Details
The implementation of our proposed PINO uses PyTorch 2.1.0 and Python 3.10 on Ubuntu 20.04.6 OS, with training conducted on an NVIDIA RTX 4090 GPU, A two-stage training strategy is adopted, with optimization performed using the Adam algorithm (with parameters 81 0.9 and B3 = 0.999) 3T. In the first stage, the encoder trained independently using a learning rate of 0.0002 to minimize composite loss function consisting of the MAE loss and the structural similarity index loss (L1 + Ct.1 Lasim. where æg = 0.1). The encoder is trained for 100, 600, and 2000 epochs on the WV3, GF2, and CAVE datasets, respectively. In the second stage, the entire framework is tuned with a learning rate halved from the first stage, optimizing a loss composed of MAE and histogram loss (In + 022 Thist, where 0s2 0.01). This stage is conducted over 200 epochs for WV3 and GF2, and 2000 epochs for CAVE.
GF2 Dataset
Dataset
Dataser
FusionNet
GPPNN
Fourner
HFIN
HOIF
PacMamba
LF orrner
ADWM
PINO
Figure& 8: The visual results (odd rows) and the corresponding mean absolute error (MAE) maps (even rows) of all compared DL-based methods on restuutiresolttionn samples from the GF2, WV3 and WV2 sensors, respectively.
20

---

GF2 Detaset
Dutases
W Dataset
GPPNN
Figure 9: The visual results (odd rows) and the corresponding HQNR maps (even rows) of all compared DL-based methods on full-resolution samples from the GF2, WV3 and WV2 sensors, respectively.
PSAX
3UE-NcI
DSS
MUNIC
PIN
Figure 10: The visual results (odd rows) and the corresponding mean absolute error (MAE) maps (even rows) of all compared DL-based methods on two CAVEX4 testing samples.
21

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A.4 More Visualization Results on Pansharpening Benchmarks
To further illustrate the visual effectiveness our proposed PINO framework, we provide additional pansharpening examples reduced- and fell-resolution samples from the WV3, GF2 and WV2 datasets. As shown in Figure and we compare PINO with all DL-based methods presented in the main paper using RGB visualization and corresponding error maps.
On reduced-resolution samples (Figure (). our method exhibits clearer spatial textures and more faithful spectral appearance, especially in vegetation and urban structures. More importantly, the associated MAE maps reveal lower error distributions across most spatial regions, suggesting superior spectral-spatic consistency. On full-resolution benchmarks (Figure 9). PINO produces significantly sharper images with better preservation of high-frequency details, while achieving lower distortion in HQNR maps compared to competing baselines. These results confirm the robust performance of our method not only under supervised conditions but also in realistic, fabel-free scenarios.
A.5
Visualization Results Cave Dataset
We further evaluate the proposed framework on the CAVE hyperspectral dataset to verify its perfor- mance hitiiseetrralhyypsspeeete image fusion task. As shown in Figure 10, we visualize the fused results and corresponding error maps for two representative CAVE testing samples.
Figure 11: The visualized high-resolution PAN images (odd rows) and estimated radiance field (even rows) on 5 peduced-resolutic samples and 5 full-resolution samples from the WV3 sensor, respectively. Please zoom in for more details.
Our method demonstrates high fidelity in reconstructing both spatial structures and spectral pro- files, with MAE maps revealing minimal residuals across regions of varying complexity. Unlike competing methods, PINO effectively retains subtle spectral variations while preserving spatial continuity, particularly in edge-rich and texture-sensitive areas such as printed text, fabrics, and natural surfaces. These observations validate the versatility of our physics-informe design when extended to multispectral- trapppecctrra-yyypsseeeerall image fusion task.
A.6 Visualization of Learned Radiance Field and Spectral Responsivity
To qualitatively assess the spectral fidelity and physical interpretability of the proposed PINO framework, we visualize both the learned radiance field and the corresponding spectral responsivity functions. These visualizations aim demonstrate how effectively the model captures the intrinsic characteristics of the spectral imaging process.
22

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Figure 12: The visualized high-resolution PAN images (odd rows) and estimated radiance field (even rows) 5 reduced-resolution samples and 5 full-resolution samples from the GF2 sensor, respectively. Please zoom in for more details.
Figure 13: The visualized high- nigh-resolution PAN images (odd rows) and estimated radiance field
(even rows) on 5 reduced-resolution samples and 5 full-resolution samples from the WV2 sensor, respectively. Please zoom in for more details.
23

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Figure 14: The visualized high- eighrresolution RGB images (first row) and estimated radiance field (second row) on 5 test samples from the CAVE dataset, respectively. Please zoom for more details.
Learned Spectral Resporsiviy Curve
Learned Spectral Resporsivily Curve
Spectral Rergoaare
0.300
Blue
Green
Spectral Possensivas
0.20
Red
Green
0.25
NIR
Yellow
0.15
0.10
0.005
0.05
0.00
0.00
-10
-0.75
-0.50
-0.25
0.000
0.25
0.50
0.75
1.00
Normalized Wavelength
Normalized Wavelen.tth
Figure 15: The learned spectral responsivity curves across various spectral bands and different datasets. Left: five bands from the WV3 dataset; Right: four bands from the GF2 dataset.
and 14 present visualizations of the learned radiance field on both reduced- and full-resolution test samples from the WV3, GF2, WV2 datasets, and test samples from CAVE x4 dataset. The radiance field is rendered based on the indexed spatial coordinates and wavelengths (randomly sampled), Compared to the input PAN image, the learned radiance map preserves fine spatial details while modulating them with wavelength-depeenden structures. This highlights the capacity of PINO to simultaneously resolve high-frequency textures and continuous spectral variation.
Figures
Complementing this, Figure illlustttts the learned spectral responsivity curves on the WV3 and GF2 datasets (for better visualization, we select five bands for WV3). Each curve represents the model's 's learned responsivity over the wavelength domain for specific sensor band. Notably, the curves exhibit band-specific shapes similar to real sensor characteristics. This indicates that the model successfully internalizes the spectral selectivity imposed by sensor filters, despite learning these responsivity functions purely from data, without explicit supervision.
Together, these results validate the model's capacity approximate the underlying physical formation of sensor measurements through implicit neural modeling. The radiance field captures the latent spectral-spatial structure of the scene, while the learned responsivity functions align closely with real- world sensor profiles. Such physically grounded representations offer both enbanced interpretability and robust generalization across varying sensor domains.
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Reviews:
[{"review_id": "H0qGWogjzG", "reviewer": "Reviewer_3YGA", "summary": "The PINO paper proposes a physics-based neural operator framework for pansharpening of remote sensing images. By simulating the physical imaging process, this framework provides a new solution that can significantly improve the quality and generalization ability of fused images. Although there are some challenges in terms of computational complexity and model complexity, PINO's outstanding performance on multiple benchmark datasets demonstrates its potential and value in the field of pansharpening. Future research can further optimize computational efficiency, explore cross-sensor generalization capabilities, and expand its application in other image fusion tasks.", "questions": "1. The encoder details are not detailed enough. The BDT architecture does not specify key parameters such as the number of layers/heads, and the appendix only mentions \"Bidirectional Dilated Transformer\".\n2. For the boundary conditions of the radiation field, Formula 1 does not discuss the boundary constraints of the scene radiation $L_i^s$, which may affect physical consistency.\n3. In the implementation of wavelength modulation, the discretization does not specify the sampling strategy (uniform/adaptive?).", "limitations": "Same weaknesses and problems as before.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "# Strengths\n1. The continuous radiation field modeling $L_i(x,y,\\lambda)$ is combined with the learnable spectral response modulation $R_b(\\lambda)$ to explicitly simulate the physical process of sensor imaging, surpassing the traditional heuristic fusion method. The learned spectral response curve is consistent with the real sensor characteristics, enhancing the credibility of the model.\n\n2. Continuous radiation field modeling is achieved through Galerkin-type integration, preserving sub-pixel spatial details and sub-band spectral features. Pre-train the encoder (L1+SSIM) and then fine-tune the whole (L1+histogram loss) to alleviate the information degradation problem in implicit representation training.\n\n3. Multi-granularity feature aggregation (expanded spatial attention + grouped spectral attention) effectively fuses the HR details of PAN and the spectral information of LRMS.\n\n# Weaknesses\n1. The high computational cost caused by iterative neural integration and radiation field modulation makes training time-consuming (2000 rounds for CAVE), and the authors did not evaluate the inference speed.", "was_revised": true, "final_justification": "The author has solved my questions in detail, so I will increase my score.", "rebuttal": "Dear Reviewer,\n\nWe sincerely appreciate your thorough review and insightful suggestions, which are invaluable for enhancing the quality of our paper. We are especially encouraged by the comments: “this framework provides a new solution, surpassing traditional heuristic fusion methods, with outstanding performance.” \n\nWe will carefully revise the paper in accordance with your feedback. Below, we provide point-by-point responses to your questions (Q) and identified weaknesses (W).\n\n**W1. Number of training epochs and inference speed:**\nWhile we agree that the introduction of iterative neural integration and radiance field modulation does result in increased training time in some cases, we would like to clarify that our two-stage training approach reduces training time in certain scenarios.\n\nTo be specific, we summarize the training epochs and inference speed as follows:\n\n(1) **WV3 dataset**: The encoder is trained for 100 epochs, followed by 200 epochs in the second stage. In comparison, the baseline methods such as LFormer requires 800 epochs, and Fourmer, ADWM and PanMamba require 500 epochs [R1-R4]. \nThus, the total number of training epochs of our method is reduced compared to the baseline methods.\n\nFor a fair comparison, we evaluate the inference speed of our method and these recent baselines on the WV3 dataset with a resolution of 256×256:\n\nOurs: 0.064 s;   Fourmer: 0.263 s;   LFormer: 0.233 s;   ADWM: 0.022 s;   PanMamba: 0.034 s\n\nIt is clear that our method delivers competitive inference speed while achieving state-of-the-art performance.\n\n(2) **GF2 dataset**: The encoder is trained for 600 epochs, followed by 200 epochs in the second stage. In comparison, the baseline methods such as LFormer requires 800 epochs, and Fourmer, ADWM and PanMamba require 500 epochs [R1-R4]. \nThus, the total training epochs of our approach are competitive with these baselines.\n\n(3) **CAVE dataset**: While it does indeed require 2000 epochs, it is important to note that baseline methods such as BDT and PSRT also require 2000 epochs [R5, R6], and QIS-GAN requires 10,000 training epochs [R7]. Thus, our method does not significantly increase the training time compared to these representative approaches.\n\nAdditionally, the number of training epochs used in our experiments follows the original BDT method, which we do not adjust manually. Our primary focus in this work is to propose a physics-informed modeling approach for pansharpening that outperforms existing methods, rather than optimizing computational cost or complexity. The number of training epochs is, to some extent, a hyperparameter and is dependent on factors such as dataset size, learning rate, batch size, and pixel sampling per image. In future work, we could further optimize these hyper-parameters to reduce computational overhead.\n\n**Q1. Encoder architectural details:**\nOur encoder is based on the Bidirectional Dilation Transformer introduced in [R5] and we have presented details about the encoder input, up-sampling operation, core modules, feature embedding process and its design principle in the Appendix A.1.3. Specifically, we implement a dual-branch design: \n(i) a spatial branch applies $3$ Dilated Spatial Self-Attention (D-Spa) layers with dilation rate $2$ and $8$ attention heads, \nand (ii) a spectral branch uses same $3$ Grouped Spectral Self-Attention (G-Spe) layers and $8$ attention heads with 8$\\times$8 non-overlapping spatial groups (i.e. window size is 8$\\times$8). \n\nThese details will be added to the appendix, and full code/configuration will be released upon acceptance for reproducibility.\n\n**Q2. Boundary conditions of the radiance field $\\( L_i^s(x, y, \\lambda) \\) $ and physical consistency:** Thank you for your insightful comment on the boundary conditions of the radiation field in Formula 1. Although we did not explicitly discuss this in the original manuscript, we appreciate the reviewer bringing this up, and we are now addressing it. We will also add further details in the supplementary materials.\n\n**1. Boundary Conditions of the Radiation Field:**\nWe assume Homogeneous Neumann Boundary Conditions for the radiation field, which implies that the gradient of the field at the boundary is zero:\n$\\left. \\frac{\\partial L_s^i(x, y, \\lambda)}{\\partial n} \\right|_{\\partial \\Omega} = 0$\n\nThis assumption reflects the smooth transition at the boundary, which is commonly observed in natural and remote sensing images. It is a standard practice in image processing and remote sensing, as real-world scenes typically exhibit smooth transitions at boundaries [R8-R10].\n\n**2. Implicit Learning of Boundary Conditions and Explicit Spectral Responsivity Constraints.**\nAlthough we did not explicitly incorporate boundary conditions into the network architecture or loss function, the network could implicitly learn these boundary conditions. \nIn supervised learning, networks can learn smooth transitions at the boundaries through the spatial smoothness enforced by the convolutional operations, especially with the use of padding operations. \nThis behavior has been well-documented in the literature. \nFor instance, networks trained on image restoration or remote sensing tasks have been shown to learn boundary smoothness even without explicit constraints [R11, R12].\n\nFurthermore, as illustrated in the Appendix, our visual analysis of the learned radiance fields demonstrates that the model indeed predicts smooth transitions at the image boundaries, which highly similar to the ground truth images.\nIn addition, while the radiation field’s boundary conditions are implicitly learned by the network, we focus more on ensuring the **physical consistency of spectral responsivity**. Since spectral responsivity is **sensor-specific** and independent of the image content, we enforce **smoothness and non-negativity** constraints on the learned responsivity functions using a sigmoid activation function (Please refer to the Eq. 9 in the main paper).\n\nThis function ensures that the spectral responsivity remains positive and smooth across the spectrum, aligning with physical sensor properties. The responsivity function is modeled through content-independent learnable parameters, providing the network with the flexibility to learn sensor-specific behaviors.\nUnlike the implicit boundary learning in the radiation field, the explicit constraint on the spectral responsivity is crucial because it governs sensor-specific characteristics that are independent of image content. This approach guarantees that the model’s learned spectral response adheres to realistic sensor constraints, ensuring the overall physical consistency of the method. As shown in Appendix A.4, our model learns radiance maps that preserve fine-grained structures and wavelength-dependent variations, as well as sensor responsivity curves that resemble real-world spectral responses, further supporting the physical consistency of our model.\n\n**Q3. Discretization sampling strategy in wavelength modulation:**\nIn our current implementation, we adopt a *uniform sampling* strategy across the wavelength domain, as we do not make dataset-specific assumptions about spectral distribution and uniform sampling yields effective performance and results in our experiments. \nTo enhance the clarity of our paper, we will emphasise this in our revised paper.\nThe *adaptive sampling* strategies, for example, based on the second derivative of the sensor's spectral response curve to focus on regions with high spectral variability, could potentially enhance spectral representation. We consider this a valuable future direction.\n\n[R1] Zhou M, Huang J, Guo C L, et al. Fourmer: An efficient global modeling paradigm for image restoration[C]//International conference on machine learning. PMLR, 2023: 42589-42601.\n\n[R2] Hou J, Cao Z, Zheng N, et al. Linearly-evolved transformer for pan-sharpening[C]//Proceedings of the 32nd ACM international conference on multimedia. 2024: 1486-1494.\n\n[R3] Huang J, Chen H, Ren J, et al. A General Adaptive Dual-level Weighting Mechanism for Remote Sensing Pansharpening[C]//Proceedings of the Computer Vision and Pattern Recognition Conference. 2025: 7447-7456.\n\n[R4] He X, Cao K, Zhang J, et al. Pan-mamba: Effective pan-sharpening with state space model[J]. Information Fusion, 2025, 115: 102779.\n\n[R5] Deng S, Deng L J, Wu X, et al. Bidirectional dilation transformer for multispectral and hyperspectral image fusion[C]//Proceedings of the Thirty-Second International Joint Conference on Artificial Intelligence. 2023: 3633-3641.\n\n[R6] Deng S Q, Deng L J, Wu X, et al. PSRT: Pyramid shuffle-and-reshuffle transformer for multispectral and hyperspectral image fusion[J]. IEEE Transactions on Geoscience and Remote Sensing, 2023, 61: 1-15.\n\n[R7] Zhu C, Deng S, Zhou Y, et al. QIS-GAN: A lightweight adversarial network with quadtree implicit sampling for multispectral and hyperspectral image fusion[J]. IEEE Transactions on Geoscience and Remote Sensing, 2023, 61: 1-15.\n\n[R8] Dai S, Han M, Xu W, et al. Softcuts: a soft edge smoothness prior for color image super-resolution[J]. IEEE transactions on image processing, 2009, 18(5): 969-981.\n\n[R9] Li F, Wang Z, He G. AP shadow net: A remote sensing shadow removal network based on atmospheric transport and Poisson’s equation[J]. Entropy, 2022, 24(9): 1301.\n\n[R10] Weng C Y, Zhan Y, Gu X, et al. The Spatially Seamless Spatiotemporal Fusion Model Based on Generative Adversarial Networks[J]. IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing, 2024, 17: 12760-12771.\n\n[R11] Liu S, Liu T, Gao L, et al. Convolutional neural network and guided filtering for SAR image denoising[J]. Remote Sensing, 2019, 11(6): 702.\n\n[R12] Zhu F, Liang Z, Jia X, et al. A benchmark for edge-preserving image smoothing[J]. IEEE Transactions on Image Processing, 2019, 28(7): 3556-3570."}, {"review_id": "38eGtsM3I0", "reviewer": "Reviewer_8nfB", "summary": "This paper proposed a physics-informed neural operator framework, named PINO, which unified continuous radiance field modeling with learnable spectral responsivity, significantly boosting spatial spectral fidelity and enhancing generalization across heterogeneous sensors and varying imaging conditions.\nThe main contributions as follows:\n1)\tThis paper proposed a physically-grounded pansharpening framework that established a new paradigm bridging neural representations with physical sensor model.\n2)\tIt employed an iterative kernel integral operator that leverages multi-granularity spatial–spectral features to learn a continuous radiance field over spatial coordinates and wavelength, enabling sub-pixel and sub-band fusion of different modalities while effectively emulating band-wise spectral integration.\n3)\tIntroduced a learnable band-wise spectral responsivity modulation to simulate the sensor’s spectral properties, allowing the simultaneous optimization of response functions.\n4)\tExperiments on multiple remote sensing benchmark datasets reveal that PINO consistently outperforms state-of-the-art methods in fusion capability, generaizalibity and adaptability.", "questions": "1.\tWhy implement analog imaging process operators in imaging products? From a physical principle perspective, why is this useful?\n2.\tLack of more detailed experimental analysis.\n3.\tWill the approach used in this paper result in a significant increase in computational load and a significant decrease in training speed?", "limitations": "yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths : \nThe idea is novel, and the theoretical reasoning is correct.\nWeaknesses:\n1.Readers are not clear about why physical information neural operators are useful.\n2. The experimental data set is small and insufficient.", "was_revised": true, "final_justification": "Thanks for the responses.. I'll maintain my original rating.", "rebuttal": "Dear Reviewer,\n\nWe sincerely appreciate your detailed feedback and instructive comments, which are invaluable for enhancing the quality of our paper. We are especially encouraged by the comment: “The idea is novel, and the theoretical reasoning is correct.” \n\nWe will carefully revise the paper in accordance with your comments. Below, we provide point-by-point responses to your questions (Q) and identified weaknesses (W).\n\n**W1 & Q1. Why implement analog imaging process operators in imaging products? From a physical principle perspective, why is this useful?** \n\nTo clarify, our goal is not to explicitly implement an analog imaging process, but rather to design learning components inspired by the underlying *physical principles and mathematical models* that govern the sensor imaging process. Specifically, pansharpening requires fusing low-spatial high-spectral MS images with high-spatial low-spectral PAN (or RGB) images to reconstruct high-resolution multispectral images. This inherently involves the inverse of a physical image formation process that is both spatially and spectrally entangled. By analyzing the mathematical formulation behind this process (rather than relying purely on end-to-end black-box learning), we can guide network design to more effectively leverage complementary modalities and enhance interpretability.\n\nThis paradigm is conceptually aligned with advances in other areas such as NeRF, 3D Gaussian Splatting, and Physics-Informed Neural Networks [R1-R3], where physical models inform the training pipeline or network architectures. While prior work in pansharpening has introduced model components inspired by data properties [R4, R5], our approach uniquely formalizes the imaging process through a learnable wavelength-modulated radiance field and band-specific sensor response modeling. These design choices not only yield superior performance but also enhance interpretability. As shown in Appendix A.4, our model learns radiance maps that preserve fine-grained structures and wavelength-dependent variations, as well as sensor responsivity curves that resemble real-world spectral responses, further supporting the value of this physically-informed modeling approach.\n\n**We kindly refer you to our response to Q1 from Reviewer qiJf, where we provide a detailed explanation of the sensor imaging physics and how it is explicitly implemented in our proposed method.**\n\n**W2 & Q2. Experimental data and more detailed experimental analysis:** We respectfully clarify that our experimental setup follows widely accepted protocols in the pansharpening community [R6-R8]. We evaluate our model on three standard satellite datasets: WV3, WV2, and GF2, with training samples simulated from original satellite imagery using Wald’s protocol [R9]. We report results under both reduced (256×256) and full (512×512) resolution settings, and importantly, we also evaluate cross-dataset generalization on GF2, which is not seen during training. Beyond pansharpening, we validate spectral learning capacity on hyperspectral datasets (CAVE ×4 and CAVE ×8), demonstrating the model’s ability to generalize to diverse spectral settings.\n\nTo complement quantitative metrics, we provide detailed qualitative analysis in the original paper and appendix, including RGB outputs, HQNR maps, and comparison visualisations across benchmarks. The results show that PINO effectively retains subtle spectral variations while preserving spatial continuity, especially in edge-rich or texture-sensitive regions. Furthermore, we visualize internal representations such as the learned radiance field and spectral responsivity, which confirm the model’s ability to reconstruct fine spatial structures while learning band-specific spectral selectivity imposed by the sensor. These analyses support both the performance and interpretability of our approach.\n\n**Q3. Computational load and training speed:** \n\nWhile we agree that the introduction of iterative neural integration and radiance field modulation does result in increased training time in some cases, we would like to clarify that our two-stage training approach reduces training time in certain scenarios.  \n\nTo be specific, we summarize the training epochs and inference speed (computational load) as follows:\n\n(1) **WV3 dataset**: The encoder is trained for 100 epochs, followed by 200 epochs in the second stage. In comparison, the baseline methods such as LFormer requires 800 epochs, and Fourmer, ADWM and PanMamba require 500 epochs [R10-R13]. \nThus, the total number of training epochs of our method is reduced compared to the baseline methods.\n\nFor a fair comparison, we evaluate the inference speed of our method and several recent baselines on the WV3 dataset with a resolution of 256×256:\n\nOurs: 0.064 s;   Fourmer: 0.263 s;   LFormer: 0.233 s;   ADWM: 0.022 s;   PanMamba: 0.034 s\n\nIt is clear that our method delivers competitive inference speed while achieving state-of-the-art performance.\n\n(2) **GF2 dataset**: The encoder is trained for 600 epochs, followed by 200 epochs in the second stage. In comparison, the baseline methods such as LFormer requires 800 epochs, and Fourmer, ADWM and PanMamba require 500 epochs [R10-R13]. \nThus, the total training epochs of our approach are competitive with these baselines.\n\n(3) **CAVE dataset**: While it does indeed require 2000 epochs, it is important to note that baseline methods such as BDT and PSRT also require 2000 epochs [R14, R15], and QIS-GAN requires 10,000 training epochs [R16]. Thus, our method does not significantly increase the training time compared to these representative approaches.\n\n[R1] Kerbl B, Kopanas G, Leimkühler T, et al. 3D Gaussian splatting for real-time radiance field rendering[J]. ACM Trans. Graph., 2023, 42(4): 139:1-139:14.\n\n[R2] Mildenhall B, Srinivasan P P, Tancik M, et al. Nerf: Representing scenes as neural radiance fields for view synthesis[J]. Communications of the ACM, 2021, 65(1): 99-106.\n\n[R3] Raissi M, Perdikaris P, Karniadakis G E. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations[J]. Journal of Computational physics, 2019, 378: 686-707.\n\n[R4] Kim H H, Kim M. FBS-PS: Fully Band-Separable PAN-Sharpening Considering the Physical Characteristics of Electro-Optical Sensors[J]. IEEE Transactions on Geoscience and Remote Sensing, 2024.\n\n[R5] Liu X, Hou J, Cong X, et al. Rethinking pan-sharpening via spectral-band modulation[J]. IEEE Transactions on Geoscience and Remote Sensing, 2023, 62: 1-16.\n\n[R6] Hou J, Cao Q, Ran R, et al. Bidomain modeling paradigm for pansharpening[C]//Proceedings of the 31st ACM international conference on multimedia. 2023: 347-357.\n\n[R7] Vivone G, Deng L J, Deng S, et al. Deep learning in remote sensing image fusion: Methods, protocols, data, and future perspectives[J]. IEEE Geoscience and Remote Sensing Magazine, 2024.\n\n[R8] Zhong Y, Wu X, Cao Z, et al. Ssdiff: Spatial-spectral integrated diffusion model for remote sensing pansharpening[J]. Advances in Neural Information Processing Systems, 2024, 37: 77962-77986.\n\n[R9] Wald L, Ranchin T, Mangolini M. Fusion of satellite images of different spatial resolutions: Assessing the quality of resulting images[J]. Photogrammetric engineering and remote sensing, 1997, 63(6): 691-699.\n\n[R10] Zhou M, Huang J, Guo C L, et al. Fourmer: An efficient global modeling paradigm for image restoration[C]//International conference on machine learning. PMLR, 2023: 42589-42601.\n\n[R11] Hou J, Cao Z, Zheng N, et al. Linearly-evolved transformer for pan-sharpening[C]//Proceedings of the 32nd ACM international conference on multimedia. 2024: 1486-1494.\n\n[R12] Huang J, Chen H, Ren J, et al. A General Adaptive Dual-level Weighting Mechanism for Remote Sensing Pansharpening[C]//Proceedings of the Computer Vision and Pattern Recognition Conference. 2025: 7447-7456.\n\n[R13] He X, Cao K, Zhang J, et al. Pan-mamba: Effective pan-sharpening with state space model[J]. Information Fusion, 2025, 115: 102779.\n\n[R14] Deng S, Deng L J, Wu X, et al. Bidirectional dilation transformer for multispectral and hyperspectral image fusion[C]//Proceedings of the Thirty-Second International Joint Conference on Artificial Intelligence. 2023: 3633-3641.\n\n[R15] Deng S Q, Deng L J, Wu X, et al. PSRT: Pyramid shuffle-and-reshuffle transformer for multispectral and hyperspectral image fusion[J]. IEEE Transactions on Geoscience and Remote Sensing, 2023, 61: 1-15.\n\n[R16] Zhu C, Deng S, Zhou Y, et al. QIS-GAN: A lightweight adversarial network with quadtree implicit sampling for multispectral and hyperspectral image fusion[J]. IEEE Transactions on Geoscience and Remote Sensing, 2023, 61: 1-15."}, {"review_id": "PvjswfbBqJ", "reviewer": "Reviewer_qiJf", "summary": "This paper presents an interesting model called PINO for pansharpening, which combines deep learning with the underlying physics of spectral imaging. The approach is implemented using implicit neural representations and neural operators, offering a novel perspective on the problem. Given that physics-driven methods are still relatively rare in remote sensing (RS) research, this work could inspire more explorations in this direction. Overall, the paper is well-structured, and the experimental results are compelling.", "questions": "See the Weaknesses.", "limitations": "Yes.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 2, "originality": 3, "strengths_and_weaknesses": "# Strengths\n1. The paper introduces an interesting method for pansharpening that integrates physical imaging using implicit neural representation and neural operators. The idea is innovative compared with existing pansharpening works. \n2. The experiment results are significantly improved over the existing methods, while the ablation studies show the effectiveness of the core designs.\n3. The motivation and presentation of the paper are clear.\n\n# Weaknesses \n1. While combining pansharpening with physical imaging is reasonable, the paper needs deeper theoretical analysis of the proposed model, particularly regarding its physical modeling components.\n2. The physical interpretability of the proposed model remains unclear. Additional experiments or in-depth analysis would help demonstrate whether it properly captures physical imaging properties.\n3. The discussion should include more relevant physically-driven pansharpening works, such as those presented in DOI: 10.1109/TGRS.2024.3523865, for better contextualization.", "was_revised": true, "final_justification": "Thanks for the responses. My concerns are addressed. I'll maintain my original rating.", "rebuttal": "Dear Reviewer,\n\nWe sincerely appreciate your detailed feedback and insightful comments, which are invaluable for enhancing the quality of our paper. We are especially encouraged by the comments: “the idea is innovative compared with existing pansharpening works, the experiment results are significantly improved, the motivation and presentation of the paper are clear.” \n\nWe will carefully revise the paper in accordance with your comments. Below, we provide point-by-point responses to the weaknesses (W).\n\n**W1. Theoretical analysis of physical modeling:** \nOur approach is grounded in a detailed formulation of the physical imaging process and corresponding mathematical modeling, which serve as the theoretical foundation of our approach. To clarify this further, we revisit and summarize the key components of our method as follows:\n\n**1. Sensor Physics Modeling:** \n\n(1) Radiance formation follows the camera equation (Eq. 1 in our main paper): $L_i(x, y, \\lambda)=\\frac{\\pi \\tau_o(\\lambda)}{4 N^2} L_i^{s}(x, y, \\lambda)$, which is implicitly realized through a spatial-spectral encoder within our PINO.  \n\n(2) Spectral integration (Eq. 2 in our main paper) models the sensor's band-wise response: $I_{i, b}(x, y) = \\int_{\\Lambda_b} L_i(x, y, \\lambda)R_b(\\lambda)\\$. In our PINO, the spectral integration is approximated via an iterative kernel integral (*i.e.*, neural operator) in Eq. 8, while the sensor responsivity $R_b(\\lambda)$ is implemented as a learnable function, modulated using Eq. 9.\n\n(3) Discretization (Eq. 3 in our main paper) enables the practical implementation of the integration, reflecting the quantization of real sensor outputs: $I_{i,b}(x,y)\\approx\\sum_j L_i(x,y,\\lambda_j)R_b(\\lambda_j)\\Delta\\lambda_j=\\sum_j L_i(x,y,\\lambda_j)R^{\\prime}_b(\\lambda_j)$.\n\nThe above formulas explicitly encodes key electro-optical process into the learning framework, ensuring that PINO is physically consistent and spectrally faithful.\n\n**2. Neural Operator as Physics-Informed Solver:**\n\nThe iterative Galerkin-style kernel integral operator (Eq. 8 in our main paper) is designed as a functional approximator for the integral in Eq. 2 of the mian paper: $h^{n+1}(x, y, \\lambda) = \\int_{\\Lambda} \\left( \\int_{\\Omega_x} \\int_{\\Omega_y} \\mathcal{K} \\left( (x, y), (u, v) \\right) h^{n}(u, v) \\, \\mathrm{d}u \\, \\mathrm{d}v \\right) \\Pi(\\lambda) \\mathrm{d}\\lambda'$, where $\\mathcal{K}$ captures spatial-radiance coupling;  $\\Pi(\\lambda)$ implements implicit spectral basis functions via Fourier encoding; discretization (*i.e.*, wavelength modulation) mirrors sensor quantization. This iterative integral allows the model to learn a continuous radiance field that aligns with the physical principles of radiative imaging.\n\n**3. Spectral Responsivity as Differentiable Physics:**\n\nTo account for sensor-specific spectral filtering, we parameterize each band’s responsivity $R^\\prime_b(\\lambda_j, \\theta_b)$ as: $R^\\prime_b(\\lambda_j, \\theta_b) = \\mathrm{Sigmoid}(\\theta_b(\\Pi(\\lambda_j)))$. In this formulation, $R^\\prime_b(\\lambda_j, \\theta_b)\\in [0, 1]$ guarantees the non-negativity that aligns with the bounded nature of real sensor spectral responses; the differentiable sigmoid function (with gradient $\\sigma'(x) = \\sigma(x) (1 - \\sigma(x))$) enables smoothness, avoiding non-physical spectral discontinuities; the learnable MLP mapping $\\theta_b(\\cdot)$ allows the model to adaptively fit sensor-specific band responses from data.\n\nWe believe that this deeper theoretical insight further clarifies how the neural operator, wavelength modulation, and learnable spectral responsivity collectively contribute to the accuracy and effectiveness of our pansharpening framework. The extra content about more detailed theoretical analysis will be included in the revised version of the Appendix for better clarity.\n\n**W2. Physical interpretability:**\n\n1. We would like to emphasize that the proposed model is physically interpretable by design, as it explicitly incorporates a wavelength-modulated radiance field and a learnable spectral responsivity function. This modeling aligns closely with the physical principles of sensor imaging physics, as detailed in our preceding analysis.\n\n2. Moreover, we visualize the learned radiance maps and sensor responsivity curves in Appendix A.4. The radiance maps preserve fine-grained structures and exhibit wavelength-dependent variations, while the learned responsivity curves resemble real-world sensor spectral responses. These results support the model’s ability to capture and reproduce key physical characteristics of the imaging process, providing further evidence of interpretability. \n\nTo further improve clarity, we will add a related description of these analyses in the main paper as per your instructive suggestion.\n\n**W3. More discussion about relevant physically-driven pansharpening works:** \n \nWe thank the reviewer for pointing this out and for suggesting relevant references. In the revised version, we will update the Related Work section to better contextualize our approach by including recent physically-driven pansharpening methods, such as the SSMNet and FBS-PS [R1, R2], both of which incorporate spectral-band separability based on sensor characteristics to inform network design. While these prior works has introduced model components inspired by data properties or spectral decomposition, our method uniquely formalizes the imaging process via a learnable wavelength-modulated radiance field and band-specific sensor response modeling, thereby offering a principled physically-informed framework.\n\n[R1] Kim H H, Kim M. FBS-PS: Fully Band-Separable PAN-Sharpening Considering the Physical Characteristics of Electro-Optical Sensors[J]. IEEE Transactions on Geoscience and Remote Sensing, 2024.\n\n[R2] Liu X, Hou J, Cong X, et al. Rethinking pan-sharpening via spectral-band modulation[J]. IEEE Transactions on Geoscience and Remote Sensing, 2023, 62: 1-16."}, {"review_id": "begxLFjjsB", "reviewer": "Reviewer_eGPc", "summary": "This paper proposes a novel physics-informed neural operator framework for pansharpening, termed PINO. A spatial-spectral encoder is introduced to aggregate multi-granularity high-resolution panchromatic (PAN) and low-resolution multi-spectral (LRMS) features. An iterative neural integral process utilizes these fused spatial-spectral characteristics to learn a continuous radiance field. The learned radiance field is modulated by the sensor’s spectral responsivity to produce the desired fusion products.", "questions": "See the weakness.", "limitations": "N/A", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths \n1, this paper proposed a novel physics-informed neural operator framework for pansharpening, termed PINO.\n2, A spatial-spectral encoder is introduced to aggregate multi-granularity high-resolution panchromatic (PAN) and low-resolution multi-spectral (LRMS) features.\n3, An iterative neural integral process utilizes these fused spatial-spectral characteristics to learn a continuous radiance field. \nWeaknesses\n1, It would be better to describe section 3.1 in the background section.\n2, It seems that the contribution of this paper comes from the utilization of PIOO for Pansharpening. Theoretical analysis is encouraged to investigate.", "was_revised": true, "final_justification": "The paper investigates the physical imaging process and corresponding mathematical modeling, which serve as the theoretical foundation of the proposed approach.", "rebuttal": "Dear Reviewer,\n\nWe thank you for your careful comments on our paper and sincerely appreciate your recognition of our work as “a novel physics-informed neural operator framework.”\n\nWe will carefully revise the paper in accordance with your comments. Below, we provide point-by-point responses to the weaknesses (W).\n\n**W1. Reorganization of Sec. 3.1:**\n\nThank you for this instructive suggestion, we will move Sec. 3.1 to a new “Background and Motivation” section for better clarity.\n\n**W2. Contributions and Theoretical analysis:** We would like to clarify that the primary contribution of our paper lies in the theoretical analysis of the physical imaging process and mathematical modeling, which leads to a physically principled solution applicable to both multispectral and hyperspectral image fusion tasks. \n\n**Rather than simply introducing a neural operator as a black-box component, our method leverages the iterative kernel integral operator as a functional approximator for the spectral integral in Eq. 2, allowing the learning of a continuous radiance field modulated by spatial coordinates and wavelength**. This design is grounded in the theoretical analysis of the physical sensor imaging process, as detailed in Section 3.2. It enables sub-pixel and sub-band fusion of different modalities, which is essential for the pansharpening task. Although the ablation studies (please see Sec 4.2 in the main paper) demonstrate that the neural operator is critical to the performance improvement, the wavelength modulation brings more benefits to our final result.\n\nAdditionally, our framework incorporates learnable band-wise spectral responsivity modulation to simulate the sensor’s spectral properties. This component is also crucial to the improvement in performance. We provide visualizations of the learned radiance field and spectral responsivity in the appendix A.4, which further support the interpretability and effectiveness of the proposed method and its components. We believe these contributions offer a solid theoretical foundation for advancing the field of pansharpening. The mentioned contributions are summarized in our main paper, and also presented in the review summary.\n\nTo further clarify the role of the iterative kernel integral operator, we provide the following theoretical analysis of our approach:\n \nOur approach is grounded in a detailed formulation of the physical imaging process and corresponding mathematical modeling, which serve as the theoretical foundation of our approach. To clarify this further, we revisit and summarize the key components of our method as follows:\n\n**1. Sensor Physics Modeling:** \n\n(1) Radiance formation follows the camera equation (Eq. 1 in our main paper): $L_i(x, y, \\lambda)=\\frac{\\pi \\tau_o(\\lambda)}{4 N^2} L_i^{s}(x, y, \\lambda)$, which is implicitly realized through a spatial-spectral encoder within our PINO.  \n\n(2) Spectral integration (Eq. 2 in our main paper) models the sensor's band-wise response: $I_{i, b}(x, y) = \\int_{\\Lambda_b} L_i(x, y, \\lambda)R_b(\\lambda)\\$. In our PINO, the spectral integration is approximated via an iterative kernel integral (*i.e.*, neural operator) in Eq. 8, while the sensor responsivity $R_b(\\lambda)$ is implemented as a learnable function, modulated using Eq. 9.\n\n(3) Discretization (Eq. 3 in our main paper) enables the practical implementation of the integration, reflecting the quantization of real sensor outputs: $I_{i,b}(x,y)\\approx\\sum_j L_i(x,y,\\lambda_j)R_b(\\lambda_j)\\Delta\\lambda_j=\\sum_j L_i(x,y,\\lambda_j)R^{\\prime}_b(\\lambda_j)$.\n\nThis formulation explicitly encodes key electro-optical processes into the learning framework, ensuring that PINO is physically consistent and spectrally faithful.\n\n**2. Neural Operator as Physics-Informed Solver:**\n\nThe iterative Galerkin-style kernel integral operator (Eq.8 in our main paper) is designed as a functional approximator for the integral in Eq. 2: $h^{n+1}(x, y, \\lambda) = \\int_{\\Lambda} \\left( \\int_{\\Omega_x} \\int_{\\Omega_y} \\mathcal{K} \\left( (x, y), (u, v) \\right) h^{n}(u, v) \\, \\mathrm{d}u \\, \\mathrm{d}v \\right) \\Pi(\\lambda) \\mathrm{d}\\lambda'$, where $\\mathcal{K}$ captures spatial-radiance coupling;  $\\Pi(\\lambda)$ implements implicit spectral basis functions via Fourier encoding; discretization (*i.e.*, wavelength modulation) mirrors sensor quantization. This iterative integral allows the model to learn a continuous radiance field that aligns with the physical principles of radiative imaging.\n\n**3. Spectral Responsivity as Differentiable Physics:**\n\nTo account for sensor-specific spectral filtering, we parameterize each band’s responsivity $R^\\prime_b(\\lambda_j, \\theta_b)$ as: $R^\\prime_b(\\lambda_j, \\theta_b) = \\mathrm{Sigmoid}(\\theta_b(\\Pi(\\lambda_j)))$. In this formulation, $R^\\prime_b(\\lambda_j, \\theta_b)\\in [0, 1]$ guarantees the non-negativity that aligns with the bounded nature of real sensor spectral responses; the differentiable sigmoid function (with gradient $\\sigma'(x) = \\sigma(x) (1 - \\sigma(x))$) enables smoothness, avoiding non-physical spectral discontinuities; the learnable MLP mapping $\\theta_b(\\cdot)$ allows the model to adaptively fit sensor-specific band responses from data.\n\nWe believe that this deeper theoretical insight further clarifies how the neural operator, wavelength modulation, and learnable spectral responsivity collectively contribute to the accuracy and effectiveness of our pansharpening framework. The extra content about more detailed theoretical analysis will be included in the revised version of the Appendix for better clarity."}]

Author rebuttal:

Meta-review: The paper proposes a physics-informed neural operator framework for pansharpening that explicitly models the sensor imaging process, combining a spatial-spectral encoder, neural integral operator, and learnable spectral responsivity. Reviewers praised the novelty, strong empirical performance, and improved interpretability compared to prior methods. Concerns were raised regarding computational cost, theoretical depth of the physical modeling, and limited dataset scale, but these were addressed satisfactorily during rebuttal with clarifications on efficiency, boundary conditions, and extended analysis. Reviewers subsequently increased or confirmed their positive ratings. Overall, the contribution is technically solid, impactful for both remote sensing and physics-informed learning, and merits acceptance.
Decision: Accept (poster)