FourierMamba: Fourier Learning Integration with State Space Models for Image Deraining
Authors: Dong Li
Yidi Liu
Xueyang Fu
Jie Huang
Senyan Xu
Qi Zhu
Zheng-Jun Zha
Venue: ICML 2025 poster (2025)
Keywords: Fourier
deraining
Mamba
Primary area: applications->computer_vision

Abstract: Image deraining aims to remove rain streaks from rainy images and restore clear backgrounds. Currently, some research that employs the Fourier transform has proved to be effective for image deraining, due to it acting as an effective frequency prior for capturing rain streaks. However, despite there exists dependency of low frequency and high frequency in images, these Fourier-based methods rarely exploit the correlation of different frequencies for conjuncting their learning procedures, limiting the full utilization of frequency information for image deraining. Alternatively, the recently emerged Mamba technique depicts its effectiveness and efficiency for modeling correlation in various domains (e.g., spatial, temporal), and we argue that introducing Mamba into its unexplored Fourier spaces to correlate different frequencies would help improve image deraining. This motivates us to propose a new framework termed FourierMamba, which performs image deraining with Mamba in the Fourier space. Owing to the unique arrangement of frequency orders in Fourier space, the core of FourierMamba lies in the scanning encoding of different frequencies, where the low-high frequency order formats exhibit differently in the spatial dimension  (unarranged in axis) and channel dimension (arranged in axis). Therefore, we design FourierMamba that correlates Fourier space information in the spatial and channel dimensions with distinct designs. Specifically, in the spatial dimension Fourier space, we introduce the zigzag coding to scan the frequencies to rearrange the orders from low to high frequencies,  thereby orderly correlating the connections between frequencies; in the channel dimension  Fourier space with arranged orders of frequencies in axis, we can directly use Mamba to perform frequency correlation and improve the channel information representation. Extensive experiments reveal that our method outperforms state-of-the-art methods both qualitatively and quantitatively.

Mauriar Fourier Learning Integration with State Space Models for Image Deraining
Dong Li
Yidi Liu"
Xueyang Fu'
Jie Huang'
Senyan Xu
Qi Zhu'
Zheng-Jun Zha'
Abstract
orders of frequencies in axis, we can directly use Mamba to perform frequency correlation and im- prove the channel information representation. Ex- tensive experiments reveal that our method outper- forms state-of-the-art methods both qualitatively and quantitatively.
Image deraining aims to remove rain streaks from rainy images and restore clear backgrounds. Cur- rently, some research that employs the Fourier transform has proved to be effective for image de- raining, due it acting as effective frequency prior for capturing rain streaks. However, despite there exists dependency of low frequency and high frequency in images, these Fourier-based methods rarely exploit the correlation of different frequencies for conjuncting their learning proce- dures, limiting the full utilization of frequency information for image deraining. Alternatively, the recently emerged Mamba technique depicts its effectiveness and efficiency for modeling correla- tion in various domains (e.g., spatial, temporal), and we argue that introducing Mamba into its unexplored Fourier spaces to correlate different frequencies would help improve image deraining. This motivates us to propose a new framework termed FourierMamba, which performs image deraining with Mamba in the Fourier space. Ow- ing to the unique arrangement of frequency or- ders in Fourier space, the core of FourierMamba lies in the scanning encoding of different fre- quencies, where the low-high frequency order formats exhibit differently in the spatial dimen- sion (unarranged in axis) and channel dimension (arranged in axis). Therefore, we design Fourier- Mamba that correlates Fourier space information in the spatial and channel dimensions with distinct designs. Specifically, in the spatial dimension Fourier space, we introduce the zigzag coding to scan the frequencies to rearrange the orders from low high frequencies, thereby orderly correlat- ing the connections between frequencies; in the channel dimension Fourier space with arranged
1. Introduction
Images taken in rainy conditions exhibit significant degra- dation in detail and contrast due to rain in the air, leading to unpleasant visual results and the loss of frequency infor- mation. This issue can severely impact the performance of outdoor computer vision systems, such as autonomous driving and video surveillance (Wang al., 2022a). To mit- igate the effects of rain, many image deraining methods (Fu et al., 201 I; Xiao et al., 2022) have emerged in recent years, aiming to remove rain streaks and restore clear backgrounds in images.
The advent of deep learning has spurred this field forward, with several learning-based deraining methods achieving remarkable success (Fu et al., 2017b; Yang et al., 2017; Zhang & Patel, 2018). Among them, some studies utilize the Fourier transform for deraining in the frequency do- main (Zhou et al., 2023; Guo et al., 2022), proving effective. The key insights inspiring the use of the Fourier transform for image deraining are twofold: 1) The Fourier transform can separate image degradation and content components to some extent, serving as a prior for image deraining, as shown in Figure 1; 2) The Fourier domain possesses global properties, where each pixel in Fourier space is involved with all spatial pixels. Thus, it makes sense to explore the task of rain removal using the Fourier transform. However, despite the existence of low frequency and high frequency dependencies in images, previous Fourier-based methods rarely utilize the correlation of different frequencies com- bine their learning process. As shown in Figure 1, the commonly used 1 x convolutions cannot correlate differ- ent frequencies, limiting the full utilization of frequency information in the image. Therefore, we seek to exploit the beneficial properties of the Fourier transform while explor- ing correlating different frequencies.
Eqqual contribution 'University of Science and Technology of China, Hefei, China. Correspondence to: Zheng-Jun Zha <zhazj@uste
Chaz.j@ustc.ccu.cn..... edu.en>.
Proceedings of the 42nd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Mambi:: Fourier Learning Integration with State Space Models for Image Deraining
Raieyy Enagge
WT
IFFT
Claan tunge
(b)
(d)
Figure 1. Obsorvation and comparison of different frequeney modeling methods. (a) Observation of the amplitude spectrum exchange. The degradation is mainly in amplitude components, so the Fourier transform helps to disentangle the image content and rain. (b) The commonly used x convolution cannot model the relationship between different frequencies. (c) Previous scanning in Fourier space will fail to establish the ordered dependence between frequencies. (d) Our proposed method achieves ordered frequency dependence from low high (or vice versa), thus fully utilizing frequency information.
Recently, an improved structured state-space sequence model (S4) with a selective scanning mechanism, Mamba, gives us hope. The selective methodology of Mamba can explicitly build the correlation among image patches pix- els. Recent studies have witnessed the effectiveness and efficiency of Mamba in various domains such as spatial and temporal. Therefore, we believe that introducing Mamba into its unexplored Fourier space to correlate different fre- quencies will improve image deraining.
employ the zigzag coding in its originally used space; in- stead, we implement it in a circling-like manner that matches the symmetric frequency orders in Fourier space. In this way, this method orderly correlates the connections between frequencies, as shown in Figure 1. In the channel dimen- sion of the Fourier space, the frequency order is arranged along the axis, following the order of low in the middle to high on both sides. Therefore, we can directly use Mamba for frequency correlation, thus improving channel informa- tion representation and enhancing global properties on the channels.
In this paper, we propose novel framework named Fouri- erMamba, which performs image deraining using mamba in the Fourier domain. Following the "spatial interaction + channel evolution" rule that has also been validated on Mamba (Guo et al., 2024; Behrouz et al., 2024), we design the Mamba framework in the Fourier domain from both spatial and channel dimensions. Considering the unique arrangement of frequency orders in the Fourier domain, the core of FourierMamba lies in the scanning encoding of different frequencies, where the low-high frequency der formats unarranged in the spatial axis and arranged in the channel axis. Therefore, our proposed FourierMambaa correlates Fourier space information in spatial and channel dimensions with distinct designs.
In summary, our contributions are as follows: (1) We propose a novel framework FourierMamba that combines Fourier priors and State Space Model for correlating dif- ferent frequencies in the Fourier space to enhance image deraining. (2) To rearrange the order from low to high fre- quency in the spatial dimension Fourier space, we propose a scanning method based on zigzag coding to orderly correlate different frequencies. (3) Based the channel-dimense Fourier transform, we utilize Mamba to scan on the channels and correlate different frequencies to improve channel infor- mation representation. Extensive experiments demonstrate that the proposed FourierMamba surpasses statc-of-the- methods both qualitatively and quantitatively.
Specifically, in the spatial dimension of the Fourier space, low-high frequencies follow a concentric circular arrange- ment with lower frequencies near the center and higher frequencies around the periphery. If previous scanning method (Liu et al., 2024) is used directly, the orderliness between frequencies will be destroyed, as shown in Figure We note that the zigzag coding in the JPEG compression field can place lower-frequency coefficients at the forefront of the array, while higher-frequency coefficients are posi- tioned at the end. Hence, we introduce the zigzag coding to scan the frequency in the spatial dimension, rearranging the order from low to high frequency. Due to the symmetry of the frequency orders in the Fourier space, we do not directly
2. Related Works
Fourier transform. Recently, the Fourier Transform has demonstrated its effectiveness in global modeling (Chi et al., 2019; 2020). This transformation converts signals into a domain characterized by global statistical properties, facil- itating advancements across various fields (Huang al., 2022; Lee et al., 2018; Li et al., 2023; Pratt et al., 2017; Xu et al., 2021; 1; Yang & Soatto, 2020). Due its efficacy in global modeling, the Fourier Transform has been introduced into low-level vision tasks (Fuoli et al., 2021; Mao et al., 2023). As an early attempt, (Fuoli et al., 2021) proposes
2

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Mambi:: Fourier Learning Integration with State Space Models for Image Deraining
High Freq
HSS
Progrssive Z grag
Bilate
Zig
High Freq
Low Freq
High Freg
C4FFT
Symmetric Spectrum Halving(SSH)
High Freq
Low Freq
Figure 2. Our proposed Fourier space scanning method in the spatial dimension (top) and channel dimension (bottom). For simplicity, only one direction shown for each scanning method,
each method also performs a scan opposite to that shown.
a Fourier Transform-basee loss to optimize global high- frequency information for efficient image super-resolution. DeepRFT (Mao et al., 2023) is proposed for image deblur- ring, employing global receptive field to capture both low and nigh--frequent characteristics of various blurs, cept similarly applied in image inpainting (Suvorov et al., 2022). FECNet (Huang et al., 2022) demonstrates that the amplitude of Fourier features decouples global luminance components, thereby proving effective for image enhance- ment. (Yu et al., 2022) observes a similar phenomenon in image dehazing, where the amplitude reflects global haze- introduce progres- sive scanning strategy in the Fourier domain, enhancing the global modeling capability while addressing the directional sensitivity issues of visual Mamba.
frequency-domain processing branches with the Mamba branch. (Zhen et al., 2024) introduced a wavelet transform branch, yet the scanning in the wavelet domain fails to fully extract global frequency domain information. This paper proposes novel Mamba restoration network based Fourier transform, aiming to comprehensively exploit the frequency domain information of images. Please see the Appendix for more related works.
3. Methodology
related information. In contrast,
3.1. Preliminary
Fourier transform. Fourier transform is widely used technique for analyzing the frequency content of an im- age. For images with multiple color channels, the Fourier transform is applied to each channel separately. Given an image X E R Hx xC the Fourier transform F converts to Fourier space as the complex component F(x), which expressed as:
State Space Models. State Space Models (SSMs) have received lot of attention recently due to their global mod- eling capabilities well linear complexity, with (Gu et al., 2022) initially introducing the base design of SSM models, and (Mehta et al., 2022) further enhancing their performance through gating Moits.More recently, the per- formance of Mamba (Gu & Dao, 2023), proposed based on selective scan mechanism and efficient hardware de- sign, has seen significant enhancement. It stands as an efficient alternative to Transformers, finding applications in various domains including image classification (Zhu al., al., 2024),
H- W-1
F (x) (u,v)
Σ 4=0 w0=0
(h,w)) w)e 52n (
VHW
where u and v indicate the coordinates of the Fourier space. F-1 (x)) defines the inverse Fourier transform accordingly. Both the Fourier transform and its inverse procedure be efficiently implemented using FFT/IFFT algorithms (Frigo & Johnson, 1998). The amplitude component A (xr) (u, v) and phase component P (x) (u, u) are expressed as:
2024)(Liu et al., 2024), object detection and remote sensing(Zhao et al., 2024).In the field of image restoration, (Guo et al., 2024) (Shi et al., 2024) initially introduced general restoration framework based on the Mamba module but did not fully exploit the frequency do- main information of images. (Sun et al., 2024) introduces network combining Transformer and Mamba capture long-range dependencies related to rain. (Yamashita & Ike- hara, 2024) achieves effective deraining by parallelizing
( (u, v) = /R² (x) (u, v) + +I (x) (u, v),
/ (x) (u, v) R (u,
(2)
P (xx (u, v) = arctan
where R(x) (u, and T(x) (u, u) represent the real and
3

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Mambia: Fourier Learning Integration with State Space Models for Image Deraining
Stute-Space
Plattened Sequence
GAP
Spatial Interaction SSM
ID:2D-SSM
1D/2D State-Space Model
PCE-SSM
Fourier Channel Evolation SSM
Multiplicttion
C-FPT
Fourier Channe Evelution SSM
Channel Mamba
Channel Mamba
Input Projection
C-FFT
FRSSB
(S1-88M)
ICE-SSM
1465i
Output WoouProjjan
Degraded Image
Image
Fourier Spatial Interaction SSM
Mamba Block
Spatial
LayerNorm
DWConn
2D-5SM
Conv X
SILU
ID/2D-SSM
Frequency
SILU
SILL
Layer
Spatial Mamba
Frequency Mamba
Prequency Mamba
20-SSM
Spatial Branch
Fourler Branch
ID/2D-SSM
IFFT
A,B exp(aA) Ah.-1 8x,
Ch. + DŽr
1D-SSM
Figure The overall architecture of the FourierMamba. Our FourierMamba consists of multiscale hierarchical design Fourier Residual State-Space Blocks(FRSSBB. The core modules of FRSSB are Fourier Spatial Interaction SSM(FSI-SSM) and Fourier Channel Evolution SSM(FCF-SSM).
imaginary parts respectively. The Fourier transform and its inverse procedure are applied independently to each channel of the feature maps.
are expressed as:
Alv(0 C -
Chaanee-ddimesion Fourier transform. We introduce the channel-dimension Fourier transform (C-FFT) by in- dividually applying the Fourier transform along the chan- nel dimension for each spatial position. For each position (h H w R RW-1) within X H denoted as 2(h, w,O
C = -
(z),
C 1))(2)
(y(0: C- 1)) ()) R (3(0 C. (2)
= arctan
1) and abbreviated as y(0 C 1),
Fourier transform F-) converts Fourier space as the complex component F (y), which is expressed as:
These operations can also be applied for the global vector de- rived by the pooling operation. In this way, A(2) and P(z) signify the magnitude and directional changes in the magni- tude of various channel frequencies, respectively. Both of these metries encapsulate global statistics related channel information.
C-1 y(c)e -2
Fly(0: C - 1)) = с Σ
(3)
State Space Models. State Space Models (SSMs) serve as the cornerstone for transforming one-dimensional inputs into outputs through latent states, utilizing a framework of linear ordinary differential equations. Mathematically, SSMs can be formulated as follows, representing linear
Similarly, the amplitude component A(y(0: C=1))(z) and phase component ((y(0:

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Mambi;: Fourier Learning Integration with State Space Models for Image Deraining
ordinary differential equations (ODEs):
places the less significant high-freyuency coefficients to- wards the end, thereby facilitating more effective compres- sion. Inspired by compression algorithms, we introduce a method that adopts the zigzag coding approach to scan the magnitude and phase spectra.
1/ (1) - Ah(() + Bx((), v(t) - Ch(() + Dx()),
(4)
where, h((1) E R. denotes the hidden state vector, where N
represents the size of the state. The parameters RN*N
Additionally, due to the symmetry of the two-dimensional Fourier transform, scanning the entire spectrum would dis- rupt the symmetry in the Fourier space, potentially leading to the collapse of network optimization. Therefore, we scan half of the spectrum and then deduce the other half based on the central symmetry of the amplitude and the anti-central symmetry of the phase.
B E RRV,V and C E R. are associated with the state size N, while D E R² represents the skip connection.
Discrete versions of these models, such as Mamba(Gu & Dao, 2023), include a discretization step via the zero-order hold (ZOH) method. This enables the models to adaptively sean and adjust to the input data using a selective scanning mechanism. This mechanism provides a global receptive field with linear complexity, which is advantageous for im- age restoration tasks.
Specifically, we design two scanning strategies, as illus- trated in the Figure 2. The first scanning method employs a dual zigzag pattern named bilateral zigzag, starting from the vertex of the highest frequency on one side of the spec- trum, progressing in zigzag pattern toward the center's s low frequencies; similarly, it then zigzags to the opposite side's 's highest frequency. This scanning approach not only models the association between high and low frequencies but also takes into account the periodicity of the Fourier spectrum. Due the periodic nature of the Fourier trans- form, the high- -igh-freqenccy ends on either side should, in fact, be contiguous. The second method builds upon the low- to to-h frequency sequence established by zigzag scanning and conducts a scan from low to high frequencies, which is named progressive zigzag. This method is motivated by the tendency of neural networks to initially learn frequueqecc information when extracting image characteristics. Follow- ing the previous method (Liu et al., 2024; Guo et al., 2024), we reverse the above two scanning methods as additional scanning directions.
3.2. Scanning in Fourier Space
Despite the unique characteristics of the selective sean mech- anism (S6), it processes input data causally. Given the non- causal nature of visual data, directly applying this strategy to patches and flat images fails to estimate relations with un- scanned patches, leading to a "directional sensitivity" issue constrained by the acceptance domain. Numerous meth- ods have attempted to tackle this problem in the spatial domain (Liu et al., 2024; Guo et al., 2024). However, for image restoration, the Fourier space and its associated priors are crucial. Hence, we explore addressing the "directional sensitivity" issue within this domain. Specifically, we cus- tomize Fourier scanning strategies from both spatial and channel dimensions.
For the spatial dimension, each pixel point in the Fourier space contains global information, with its frequencies dis- tributed in concentric circles. Scanning methods based on spatial arrangements (Liu et al., 2024) disrupt the high-low frequency relationships in the frequency domain, thus hin- dering the modeling of image degradation information.
For the channel dimension Fourier space, since is a one- dimensional sequence arranged in order of low to high fre- quencies, we directly scan it one-dimensionally. Similarly, due to the symmetry of the Fourier transform, we scan only half and derive the other half. Through Fourier space scanning in both spatial and channel dimensions, we can correlate the connections between frequencies in an orderly manner, thereby making full use of frequency information to improve rain removal.
Therefore, we aim to devise a scanning method in the Fourier space to progressively model the frequency charac- teristics of images. An intuitive approach is to caleulate the Euclidean distance from each point in the spectrum to the center point. On the shifted Fourier spectrum, the smaller the distance to the center point, the lower the frequency. The flaw of this intuitive approach is that for images of differ- ent sizes, it requires recalculating the Euclidean distance from each point to the center point. The additional computa- tional overhead introduced by this flaw makes this approach impractical in the field of image restoration.
3.3. FourierMamba
3. OVERALL FRAMEWORK
In Figure 3, we illustrate our proposed FourierMamba. Given a rainy image I E #xxx33 FourierMamba first uses 3 x convolution layers to generate shallow features with dimensions of H W C, where H and W represent height and width, and C denotes the number of channels. Subsequently, we employ multi-scale U-Net architecture to obtain deep features. This stage consists of stack of
In JPEG compression, zigzag coding is commonly used among the Discrete Cosine Transform (DCT) coefficients of JPEG, where it prioritizes the muegy--ooncentrann low- frequency coefficients at the beginning of the array, and
5

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Mambi:: Fourier Learning Integration with State Space Models for Image Deraining
Input
GT
Classical
Bilateral
Progressive
Ours
Figure The error map between GT and restored images using various scanning methods in Fourier space. The two scanning methods we propose can achieve smaller errors than using classical scanning method (Liu et al., 2024). And the combination of the two scanning methods is better than either one.
Fourier Residual State-Space Groups, each containing sev- eral Fourier Residual State-Space Blocks (FRSSB). The FRSSB incorporates two core designs: the Fourier Spa- tial Interaction SSM block and the Fourier Channel Evolu- tion SSM block.
activates the features using the SiLU function. The other sub-branch performs spatial Mamba on features after 1 convolution. Specifically, spatial Mamba adopts the same operation sequence the above frequency branch but the scanning in SSM uses the two-dimensional selective scan- ning module shown in Figure 3, which follows previous work (Liu et al., 2024; Guo et al., 2024). Finally, the out- puts of the two sub-branches are multiplied element-wise to obtain the output F..
3.3.2. FOURIER SPATIAL INTERACTION SSM
The structure of the Fourier Spatial Interaction State Space Model (FSI-SSM) is shown in Figure 3. We first apply LayerNorm transform the input features F., into Fi. To facilitate the interaction between spatial and frequency in- formation, FSI-SSM employs both Fourier branch and spatial branch to collaboratively process Fi..
F2 = SpaScan(Conv(F)) @ SiLU(Fi).
(7)
where Conv is 1 x convolution and SpaScan is the spa- tial Mamba mentioned above. Subsequently, we employ a residual connection to add the spatial output to Fin- The spatial branch captures global features in the spatial domain which complement the frequency correlations captured by the Fourier branch in the frequency domain, thereby bene- fiting the performance of image deraining. Hence, we con- catenate the outputs of the spatial and frequency branches and use a x 1 convolution for the fusion of spatial and frequency information.
F2 is transformed into the Fourier spec- trum through the Fast Fourier Transform, subsequently de- composed into the amplitude spectrum A(Fi) and phase spectrum (.F). The amplitude spectrum and phase spec- trum are then processed separately using the progressive frequency scanning method illustrated in Figure 2 to obtain (Fi) and (Fi).
Fourier Branch:
A'(Fi) (A(F)), P'(Fi) (ourScaa(P(FF))
(5)
3.3.3. FOURIER CHANNEL EVOLUTION SSM
Previous work (Guo al., 2024) claims that selecting key channels avoid channel redundancy in SSM. Since each channel contains the information of all channels after the channel ddannel-dimension Fourier transform (C-FFT), we perform channel interaction in the Fourier domain to efficiently cor- relate different frequencies of channels. As depicted Fig- ure 3, our proposed Fourier Channel Evolution SSM (FCE- SSM) consists of three sequential parts: applying the Fourier transform along the channel dimension obtain channel- wise Fourier domain features, scanning its amplitude and phase, then restoring to the spatial domain. Specifically, as- suming the input features are denoted as F, E HR XW.CC. we first perform global average pooling on it.
where FourScan is the sequence transformation using the Fourier space scan described in Sec. 3.2. Following series of works (Liu et al., 2024; Guo et al., 2024; Zhen et al., 2024), the sequence transformation employs the following operation sequence: DW + SiLU + SSM + LayerNore We then perform an inverse Fourier trans- form on the processed spectrum and multiply it with the output of SILU.
Fy = (F (F (A'(F)) o SILU(FI.
(6)
where Fj is the output of the fourier branch, and @ is the Hadamard product.
H- W.
Spatial Branch In the spatial domain, we feed the input features F, into two parallel sub-branches. One sub-branch
Fg H.W,
Σ hm0
Σ Fg(h,w)
(8)
w=U
6

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Mambia: Fourier Learning Integration with State Space Models for Image Deraining
Input
PreNet
MPRNet
MAXIM
Restormer
MambalR
VmambaIR
FreqMamba
Ours
Figure 5. Qualitative comparison Rainl00H (Yang et al., 2017). Zoom in for better visualization.
where Fg Ixxxx corresponds to the center point of the amplitude spectrum of F, (see supplementary material), which effectively encapsulates the global information of the feature. Then, we use the channel-dimensioonal Fourier trans- form shown in Equ. 3 on Fg to obtain (z). Based on this, we use Equ. 4 for (z) to obtain its ampli- tude component A(F))())
where À is the balancing weight. In particular, is set to 0.02 empirically.
4. Experiment
4.1. Experimental Settings
(2) and phase component (z), Since the amplitude spectrum and phase spectrum have ob- vious information meaning, we choose to perform Mamba scanning these components.
Datasets. For training, we employ the widely used Rain 13k dataset (Chen et al., 2021). It contains 13,712 image pairs in the training set, and we evaluate the results on Rain 100H (Yang et al., 2017), Rain100L. (Yang et al., 2017), Test2800 (Fu et al., 2017b), and Test1 (Zhang Patel, 2018).
(=F))(=)' ChaScan (A(F)))))(())())))
(9)
(2) ChaScan(P(F,)())),
Evaluation Metrics. Following previous work (Zamir et al., 2021; 2022), we adopt two commonly used quantitative metrics for evaluations: Peak Signal-to- to-l-tt-Noise Ratio (PSNR) (Huynh-Thae Ghanbari, 2008) and Structural Similarity Index (SSIM) (Wang et al.. 2004).
where ChaScan is one-dimensional sequence transfor- mation that uses the following sequence of operations: DWCone - SiLU + SSMM + Layer Its scan- ning method is shown in Figure 2. After the Mamba cor- relates different frequencies in the channel dimension, we perform an inverse Fourier transform on and multiply the result with the channel features after SiLU activation.
Implementation Details. Our model is implemented within the PyTorch framework and executed on an NVIDIA A100 GPU. The number of blocks per layer has an impact on both the model's 's parameter count and its deraining performance. After balancing the weights, we configure the blocks per layer as [2, 3, 3, 4, 3, 3, 2], which allows us to achieve com- mendable performance with reasonable number of parame- ters. We adopt the progressive training strategy. Specifically, we set the total number of iterations to 80,000 and image sizes to [160, 256, 320. 384], with the corresponding batch sizes of [8, 4, 2, 1]. We utilize the Adam optimizer with default parameters. The initial learning rate is established at -1 followed by a gradual decay to x e- using a cosine annealing schedule.
Fa - (A(F ((F,))))) O SiLU(F)), (10)
where F, E RIXIX., is the channel feature after corre- lating different frequencies. Finally, we multiply it with the spatial feature in a form of attention to get the output F. E IR X..
= Fr.
(11)
3.3.4. OPTIMIZATION
We impose constraints in both the spatial and frequency domains. In the spatial domain, we utilize the L1 loss between the final output Yout and the ground truth Ygt. In the frequency domain, we apply the loss based on the Fourier transform. The overall loss function is formulated follows:
4.2. Comparison with State-of-the-art Methods
Comparison on Benchmark Datasets. first verify the effectiveness of FourierMamba through training models on mixture of synthetic datasets. We compare method
Liotal = Yout Ygell, + ([ooue)) (((gt)lli (12)

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Mambia: Fourier Learning Integration with State Space Models for Image Deraining
Table Quantitative comparison (PSNR/SSIM) for Image Deraining five henchmark datasets. The highest and second-highess
performances are marked in bold and underlined. "-" indicates the result is not available.
Method
Venue
PNSK
Rain10011
(Yang et 2017)
SSIM
FNISR
Rain100L. (Yang 2017)
SSIM 7
PNSR
Test2500
(Fu 2017b)
SSIM
PNSR
Test1200
(2hug Pasel, 2018)
SSIM
Param(M)
GPilgs
DeriaaNe (Fu al., 2017b)
TIP
14.92
01.592
27.03
0.584
24,51
0.861
23.58
0.835
0,058
1.453
UMRL (Yaarrs Pard. 2019)
C9
0.910
RESCAN (Li et 2018)
BCCV118 18
26.36
0.736
29.80
0.881
31.29
0.944
30.51
0.882
1.04
20,361
PreNet (Ren et 2019)
CVPR 19
26.77
0.858
32. 44
0,950
31,75
0.916
31.36
0.911
0.17
73.021
MSPEN (Jing 2020)
CVPR'20
28.66
0.850
32,82
0.930
32.39
0.916
13:22
604.70
SPAIR (Purohit 2021)
ICCV'21
36.53
33.34
0.366
33004
MPRNe: (Zamir at cl., 2021)
CVPR
30.41
0.800
36,
0.965
33.64
0.938
32.91
0.16
3.64
141.28
Restomer iZamir
2022)
22
31,46
0.901
38.99
34,15
0.444
33.19
0.926
24.53
174.7
Fourmer (Zhoou
2023)
ICML'23
30.76
37:47
33,05
0.921
IR-SDE [Luo at 20254)
ICML/23
31,03
0.904
38.30
0.580
30.42
0.891
135.3
119.1
ManihalR (Gio al., 2014)
24
30,62
0.891
38.78
0.977
33.58
0.927
32.56
0.23
31.51
80.64
VMambaIR et 2024)
arsis
ProqMamba (Zben et al., 2024)
22
31,24
0.912
39:18
0.981
34.25
0.951
33.36
0.911
14.52
36.49
31,79
0.913
39.73
0.986
34.23
0.949
34.76
0.938
17.62
22,56
Input
PreNet
MPRNet
Restormer
MambaIR
FreqMamba VmambalR
Ours
Figure 6. Qualitative comparison of real-world rainy images from Internet-Data (Wang et al., 2019).
with these deraining methods: DerainNet (Fu et al., 2017a),
zooming into the red boxed area in Figure 5, our method removes more rain streak residues while better restoring texture details.
UMRL (Yasarla & Patel, 2019), RESCAN (Li et al., 2018),
PreNet (Ren et al., 2019), MSPEN (Jiang et al., 2020).
SPAIR (Purohit et al., 2021). MPRNet (Zamir et al., 2021),
Real-world Deraining Transferred from Synthetic Datasets.
Restormer(Zamir et al., 2022), Fourmer (Zhou et al., 2023).
IR-SDE (Luo al., 2023b), MambalR (Guo et al., 2024)
VMambalR al., 2024) and FreqMamba(Zhen al.,
To verify the generalization of the proposed method in real- world scenarios, examine the real-world deraining capabilities. We evaluate the model trained on the synthetic dataset on the real-world dataset Inrernet-Data (Wang et al., 2019) without ground truth. As shown in Figure 6, FourierMamba able to re- move these complex rains and restore the clean background. In contrast, other deraining methods do not handle the effect of rain cleanly. More generalization results in real-world scenarios can be found in the Appendix.
2024). Table 1 reports the performance evaluation on four datasets. It can be seen that our method achieves the best performance on most datasets, which emphasizes the effec- tiveness of FourierMamba in improving deraining perfor- mance.
the model trained on Rain 13k to
To demonstrate the enhanced fidelity and detail levels ex- hibited by the images generated by our proposed Fouri- erMamba, we compare the visual quality of challenging degraded images from the Rain 1 100H dataset in Figure 5. Our method achieves excellent results when faced with com- plex or extremely severe rain streaks. Compared to previous methods, our FourierMamba achieves impeccable perfor- mance in both global and local restoration. For instance, by
Training On Real-world Rainy Datasets.
To further explore the potential of the proposed method, we use the real-world dataset SPA (Wang et al., 2019) to train FourierMamba. In Table 2, our method compared
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Mambi;: Fourier Learning Integration with State Space Models for Image Deraining
Table 2. Quantitative comparison of training and testing the real-world dataset SPA-Data.
Method
RESCAN
PReNet
SPDNet
DualGCN
Restormer
DRSformer
Ours
PSNR
38.11
40.16
43.20
44.18
47.98
48.54
49.18
SSIM
0.9707
0.9816
0.9871
0.9902
0.9921
0.9924
0.9931
Table 3. Ablation studies of key designs in the proposed method.
Table 4. Ablation of different scanning methods in Fourier space.
w/a ESL-SSM
w/o FCF.SSM
w/o SDF
w/o CDF
Ours
Classic(Liu al.. 2024)
Bilaterul
Progressive
Ours
PSNR SSIM
39.05
39,08
38.25
38,72
39.73
PSNR SSIM
38.82 0.9817
39.31
39.28
39.73
0,9835
0.9836
0,9810
0.9827
0.9856
0.9844
0,9843
0.9856
5. Conclusion
In this paper, we propose novel image deraining frame- work, FourierMamba, which utilizes mamba to correlate frequencies in the Fourier space, thus fully exploiting fre- queney information. Specifically, we design the mamba framework by integrating the unique arrangement of fre- quency orderings within the Fourier domain across spatial and channel dimensions. In the spatial dimension, we devise two zigzag-based methods to scan frequencies, systemat- ically correlating them. In the channel dimension. due to the ordered arrangement of frequencies along the axis, we directly apply mamba for frequency correlation. This work introduces a novel strategy to address the underutilization of frequency information in image deraining that affects per- formance. Extensive experiments multiple benchmarks validate the effectiveness of the proposed method.
with these methods RESCAN (Li et al., 2018), PReNet (Ren et al., 2019), SPDNet (Yi et al., 2021), DualGCN (Fu et al., 2021), Restormer (Zamir et al., 2022), and DRS- former (Chen et al., 2023a) with the same experimental settings. Surprisingly, we observe that FourierMamba ac- quires sigmificant real-world rain removal capabilities. This shows that our method can effectively learn the precipitation model of real rain.
4.3. Ablation Studies
We perform ablations on the key designs and scanning meth- ods of the framework on the Rain 100L.
Fourier Spatial Interaction SSM (FSI-SSM) and Fourier Channel Evolution SSM (FCE-SSM). We replace the mamba scan in FSI-SSM and FCE-SSM with 1 x 1 convolu- tion, called w/o FSI-SSM and w/o FCE-SSM respectively. It can be seen from Table 3 that since 1 X convolution cannot model the dependence of different frequencies, its perfor- mance is worse than the mamba scan in the Fourier domain in both the spatial dimension and the channel dimension.
Impact Statement
Due to uncontrollable weather conditions, image acquisi- tion systems inevitably suffer interference from rain. Images captured during rainy conditions experience significant de- cline in the quality of object details and contrast due rain present in the air. Images tainted by rain can also severely impact the performance of outdoor computer vision sys- tems, including autonomous driving and video surveillance. Therefore, image deraining itself holds significant research and application value. Our proposed FourierMamba com- bines the priors of Fourier space and the correlation mod- eling capability of Mamba, enabling the network tackle more complex image deraining tasks. However, from soci- etal perspective, negative consequences might also follow. For instance, over-reliance on image deraining technology could introduce deviations from actual image textures, af- feeting effective judgment in autonomous driving and video surveillance. In these cases, it is necessary combine expert knowledge to make rational decisions.
Fourier prior. We do not use Fourier transform in the spatial dimension and channel dimension respectively, but directly perform mamba scanning, which are called without spatial dimension Fourier (w/o SDF) and without channel dimension Fourier (w/o CDF) respectively. It can be seen from Table 3 that after losing the Fourier prior in the spatial dimension and channel dimension, the performance drops significantly. This proves the effectiveness of Fourier prior for removing rain from images. The Fourier prior also improves the visual effect; please refer to the Appendix.
Scanning method in Fourier space. We compare several scanning methods of the spatial dimension Fourier space, with the same amount of calculation. Table 4 illustrates that the performance of the two scanning methods pro- posed is better than the classic two-dimensional scanning method (Liu et al., 2024). And thanks to complementar- ity. the combination of the two methods can also further improve performance. The visual comparison in Figure 4 supports this.
A Acknowledgement
This work is supported by the National Natural Science Foundation of China (NSFC) under Grants 62225207, 62436008, 62422609 and 62276243.

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Mambia: Fourier Learning Integration with State Space Models for Image Deraining
References
Dong, H., Pan, J., Xiang, L., Hu, Z., Zhang, X., Wang, F., and Yang, M.-H. Multi-scale boosted dehazing net- work with dense feature fusion. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp. 2157-2167, 2020.
Ancuti, C. O., Ancuti, C., Sbert, M., and Timofte, R. Dense- haze: A benchmark for image dehazing with dense-haze and baze-free images. In 2019 IEEE international confer- ence on image processing (ICIP), pp. 1014-1018. IEEE, 2019.
Frigo, M. and Johnson, S. G. Fftw: An adaptive software ar- chitecture for the fit. In Proceedings of the 1998 IEEE In- ternational Conference Processing, ICASSP 98 (Cat. No. 98CH36181), volume 3, pp. 1381 1381-1384, IEEE, 1998.
Ancuti, C. O., Ancuti, C., and Timofte, R. Nh-haze: An image dehazing benchmark with non-homogeneous hazy and haze-free images. Proceedings of the IEEE/CVF conference on computer vision and pattern recognition workshops. pp. 444-444, 2020.
Acoustics, Speech and Signal
Fu, X., Huang. J., Ding, X., Liao, Y., and Paisley, Clearing the skies: deep network architecture for single-image rain removal. IEEE Transactions on Image Processing, 26(6):2944 2956, June 2017a. ISSN 1941-0042. doi: 10. 1 1109/tip.201..269 URL http: / / dx www.oooo.gg..ooooo 10. 1119/199TPPP 2017 2691802.
Behrouz, A., Santacatterina, M., and Zabih, R. Mam- bamixer: Efficient selective state space models with dual token and channel selection. arXiv preprint arXiv:2403. 19888, 2024.
Bozcan, I., Le Fevre, J., Pham, H. X., and Kayacan, E. Gridnet: Image-agnostic conditional anomaly detection for indoor surveillance. IEEE Robotics and Automation 2021.
Fu, X., Huang. J., Zeng, D., Huang, Y., Ding, X., and Paisley, Removing rain from single images via a deep detail network. In Proceedings of the IEEE conference on computer vision and pallern recognition, pp. 3855-3863, 2017b.
Letters,
Cai, B., Xu, X., Jia, K., Qing, C., and Tao, D. Dehazenet: An end-to-end system for single image haze removal. IEEE transuctions image processing, 25(11):5187- 5198, 2016.
Fu, X., Qi. Q. Zha, Z.-., Zhu, Y., and Ding, X. Rain streak removal via dual graph convolutional network. In Pruceedings of the AAAI Conference on Artificial Intelli- gence, volume 35, pp. 1352-1360, 2021.
Chen, L., Lu, X., Zhang, J., Chu, X., and Chen, C. Hinet: Half instance normalization network for image restora- tion. In Proceedings of the IEEF/CVF Conference Computer Vision and Pattern Recognition, pp. 182-192, 2021.
Fu, X., Xiao, J., Zhu, Y., Liu, A., Wu, F., and Zha, Z.-J. Continual image deraining with hypergraph convolutional networks. IEEE Transactions on Pattern Analysis and Machine Intelligence, 45(8):9534-958 9551,, 2023. doi: 10. 1100/TPAM.2003332417555
Chen, T., Tan, 7., Gong, T., Chu, Q., Wu, Y., Liu, B., Ye, J., and Yu, N. Mim-istd: Mamba-in-mamble for efficient infrared small target detection, 2024.
Fu, Y.-.,, Kang, L.-W., Lin, C.-W., and Hsu, C.-T. Single- frame-base basee rain removal via image decomposition. In 2011 I IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 1453-1456. IEEE, 2011.
Chen, X., Li, H., Li, M., and Pan, J. Learning a sparse transformer network for effective image deraining. In Proceedings of the IEEE/CVF Conference on Computer Vision and Puttern Recognition, pp. 5896-5905, 2023a.
Fuoli, D., Van Gool, L., and Timofte, R. Fourier space losses for efficient perceptual image super-resolution. In Proceedings of the IEEE/CVF International Conference Computer Vision, pp. 2360-2369, 2021.
Chen, X., Pan, J., Dong, J., and Tang, J. Towards unified deep image deraining: A survey and a new benchmark. arXiv preprint arXiv:2310.035355 2023b.
Gao, N., Jiang, X., Zhang, X., and Deng, Y. Efficient frequency- image deraining with contrastive regu- larization.
Chi, L., Tian, G., Mu, Y., Xie, L., and Tian, Q. Fast non- local neural networks with spectral residual learning. In Proceedings of the 27th ACM International Conference on Multimedia, pp. 2142-211 22511 2019.
Gu, A. and Dao. T. Mamba: Linear-time sequence modeling with selective state spaces, 2023.
Chi, L., Jiang, B., and Mu, Y. Fast fourier convolution. Advances in Neural Information Processing Systems, 33: 4479 4488, 2020.
Gu, A., Goel, K., and Ré, C. Efficiently modeling long se- quences with structured state spaces. In The International Conference on Learning Representations (ICLR), 2022.
10

---

Mambia: Fourier Learning Integration with State Space Models for Image Deraining
Gu, S., Meng, D., Zuo, W., and Zhang, L. Joint convo- lutional analysis and synthesis sparse representation for single image layer separation. In Proceedings of the IEEE international conference on computer vision, pp. 1708-1716, 2017.
Li, X., Wu, J., Lin, Z., Liu, H., and Zha, H. Recurrent squeeze-and-excitatinn context aggregation net for single image deraining. In European Conference on Computer Vision, pp. 262-277. Springer, 2018.
Li, Y., Tan, R. T., Guo, X., Lu, J., and Brown, M. S. Rain streak removal using layer priors. In Proceedings of the IEEE conference on computer vision and pattern recogni- tion, pp. 2736-2744, 2016.
Guo, C., Li, C.,, Guo, J., Loy, C. C., Hou, J., Kwong, S., and Cong, Zero-reference deep curve estimation for low-light image enhancement. In Proceedings of the IEEE/CVF conference computer vision and pattern recognition, pp. 1780-1789 2020.
Li, Y. Fang, C., Yang, J., Wang, Z., Lu, X., and Yang. M.-H. Universal style transfer via feature transforms. Advances in neural information processing systems, 30, 2017.
Guo, H., Li, J., Dai, T., Ouyang, Z., Ren, X., and Xia, S.-T. Mambair: A simple baseline for image restoration with state-space model. arXiv preprint arXiv:2402.15648, 2024.
Liu, L.., Liu, B., Huang, H., and Bovik, A. C. No-reference image quality assessment based on spatial and spectral entropies. Signal processing: Image communication, 29 (8):856-863, 2014.
Guo, X., Fu. X., Zhou, M., Huang, Z., Peng, J., and Zha, Z.- J. Exploring fourier prior for single image rain removal. 2022.
Liu, Y., Tian, Y., Zhao, Y., Yu, H., Xie, L.., Wang, Y., Ye, Q.. and Liu, Y. Vmamba: Visual state space model. arXiv preprint arXiv:2401.10166 2024.
In IJCAI, pp.
He, K., Sun, J.. and Tang, X. Single image haze removal using dark channel prior. IEEE transactions on pattern analysis and machine intelligence, 33(12):2341-2353, 2010.
Luo, Z., Gustafsson, F. K., Zhao, 7., Sjölund, J., and Schön, T. B. Image restoration with mean-reverting stochastic differential equations. International Conference on Ma- chine Learning, 2023a.
Hsu, W. Y. and Chang, W.- Wavelet upproximation-awart residual network for single image deraining. IEEE trans- actions on pattern analysis and machine intelligence, 2023.
Luo, Z., Gustafsson, F. K., Zhao, Z., Sjölund, J., and Schön, Enabling large-size realistic image restoration with latent-space diffusion models. In Pro- ceedings of the IEEE/CVF Conference on Computer Vi- sion and Pattern Recognition Workshops, pp. 1680-1691, 2023b.
T. B. Refusion:
Huang, J., Liu, Y., Zhao, F., Yan, K., Zhang, J., Huang, Y., Zhou, M., and Xiong, Z. Deep fourier-based exposure correction network with spatial-frequency interaction. In Eurupean Conference on Computer Vision, pp. 163-180. Springer, 2022.
Mao, X., Liu, Y., Liu, F., Li, Q., Shen, W., and Wang, Y. Intriguing findings of frequency selection for image deblurring. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 37, pp. 1905-1913, 2023.
Jang, K., Wang, Z., Yi, P... Chen, C., Huang, B., Luo, Y., Ma, J., and Jiang, J. Multi-scale progressive fusion network for single image deraining. In IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), June 2020.
Mehta, H., Gupta, A., Cutkosky, A., and Neyshabur, B. Long range language modeling via gated state spaces. arXiv preprint arXiv:2206. / 13947, 2022.
Mittal, A., Moorthy, A. K., and Bovik, A. C. No-reference image quality assessment in the spatial domain. IEEE Transactions on image processing, 21(12)4695-4708. 2012a.
Jiang, L., Dai, B., Wu, W. and Loy, C. C. Focal frequency loss for image reconstruction and synthesis. In Proceed- ings of the IEEF/CVF international conference com- puter vision, pp. 13919-13929) 2021.
Mittal, A., Soundararajan, R., and Bovik, A. C. Making a "completely blind" image quality analyzer. IEEE Signal processing letters, 20(3):209-11,, 2012b.
Lec, J.H.., Heo, M., Kim, K.-R., and Kim, C.-S. Single- image depth estimation based on fourier domain analysis. In Proceedings of the IEEE conference on computer vi- sion and pattern recognition, pp. 330-339, 2018.
Pratt, H., Williams, B., Coenen, F., and Zheng, Y. Fenn: Fourier convolutional neural networks. In Machine Learn- ing and Knowledge Discovery in Databases: European Conference, ECML PKDD 2017, Skopje, Macedonia, September 18-22, 2017, Proceedings, Part 17, pp. 786- 798. Springer, 2017.
Li, C., Guo, C.-L., Zhou, M., Liang, Z., Zhou, S., Feng, R., and Loy, C. C. Embedding fourier for uitra-high- definition lightight image enhancement, arXiv preprint arXiv:2302. / 1831, 2023.

---

Mambiar Fourier Learning Integration with State Space Models for Image Deraining
Purohit, K., Suin, M., Rajagopalan, A., and Boddeti, V. N. Spatially-adaptive image restoration using distortion- guided networks. In Proceedings of the IEEE/CVF Inter- national Conference on Computer Vision, pp. 9909-2319, 2021.
Wang, T., Yang, X., Xu, K., Chen, S., Zhang, Q. and Lau, R. W. Spatial attentive single-image deraining with high quality real rain dataset. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp. 12270-12279, 2019.
Quan, R., Yu, X., Liang, Y., and Yang. Removing rain- drops and rain streaks in one go. In Proceedings of the IEEE/CVF conference computer vision and pattern recognition, pp. 9147-9156, 2021.
Wang, Z., Cun, X., Bao, J., Zhou, W., Liu, J., and Li, H. Uformer: A general u-shaped transformer for image restoration. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp. 17683- 17693, 2022b.
Ran, W., Yang, Y., and Lu, H. Single image rain removal boosting via directional gradient. In 2020 IEEE Interna- tional Conference on Multimedia and Expo (ICME), pp. 1-6. IEEE, 2020.
Wei, C., Wang, W., Yang, W., and Liu, J. Deep retinex decomposition for low-light enhancement, arXiv preprint arXiv: 045.00 2018.
Ran, W., Ma, P., He, Z., Ren, H., and Lu, H. Harnessing joint rain-/detail-avve -warre representations to eliminate in- tricate rains. In International Conference on Learning Representations, 2024.
Wu, H., Qu, Y., Lin, S., Zhou, J., Qiao, R., Zhang, Z., Xie, Y., and Ma, L. Contrastive learning for compact single image dehazing. In Proceedings of the IEEF/CVF conference on computer vision and pattern recognition, pp. 10551-10560, 2021.
Ren, D., Zuo, W., Hu, Q.. Zhu, P., and Meng. D. Progressive image deraining networks: A better and simpler baseline. In IEEE Conference Computer Vision and Pattern Recognition, 2019.
Wu, W., Weng, J., Zhang, P., Wang, X., Yang, W., and Jiang, J. Uretinex-net: Retinex- based deep unfolding network for low-light image enhancement. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp. 5901-5910, 2022.
Shi, Y., Xia, B., Jin, X., Wang,
X., Zhao, T., Xia, X., Xiao, X., and Yang, W. Vmambair: Visual state space model for image restoration. arXiv preprint arXiv:2403.I 11423, 2024.
Xiao, J., Fu, X., Liu, A., Wu, F., and Zha, Z.-J. Image de-raining transformer. IEFE Transactions on Pattern Analysis and Machine Intelligence, 2022.
Sun, S., Ren, W., Zhou, J., Gan, J., Wang, R., and Cao, X. A hybrid transformer- mambaaa network for single image deraining. arXiv preprint 0eXii:244990041000 2024.
Xu, Q., Zhang, R., Zhang, Y., Wang, Y., and Tian, Q. A fourier-based framework for domain generalization. In Proceedings of the IEEF/CVF conference computer vision and pattern recognition, pp. 14383-14392, 2021.
Suvorov, R., Logacheva, E., Mashikhin, A., Remizova, A., Ashukha, A., Silvestrov, A., Kong, N., Goka, H., Park, K., and Lempitsky, V. Resolution-robust large mask in- painting with fourier convolutions. In Proceedings of the IEEE/CVF winter conference on applications of computer vision, pp. 2149-2159, 2022.
Xu, X., Wang, R., Fu, C.-W., and Jia, J. Snr-aware low-tight image enhancement. In Proceedings of the IEEE/CVF conference computer vision and pattern recognition, pp. 17714-17724, 2022.
Yamashita, S. and Ikehara, M. Image deraining with frequency-enhanced state space model. arXiv preprint arXiv:2405. 16470, 2024.
Valanarasu, J. M. J., Yasarla, R., and Patel, V. M. Tran- sweather: Transformer- based restoration of images de- graded by adverse weather conditions. In Proceedings of the IEEF/CVF Conference on Computer Vision and Pattern Recognition, pp. 2353-2363, 2022.
Yang, W., Tan, R. T., Feng. J., Liu, J., Guo, Z., and Yan, S. Deep joint rain detection and removal from single im- age. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 1357-1366, 2017.
Wang, C., Pan, J., and Wu, X.M Online-updated high- order collaborative networks for single image deraining. In Proceedings of the AAAI Conference on Artificial In- telligence, volume 36, pp. 2406-2413, 2022a.
Yang, and Soatto, S. Fda: Fourier domain adaptation for semantic segmentation. In Proceedings of the IEEF/CVF conference on computer vision and pattern recognition, pp.
Wang, H., Xie, Q. Zhao, Q. and Meng, D. A model-driven deep neural network for single image rain removal. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp. 3103-3112, 2020.
2020.
Yasarla, R. and Patel, V. M. Uncertainty guided multi-scale residual learning-using a cycle spinning cnn for single
12

---

Mambi:: Fourier Learning Integration with State Space Models for Image Deraining
image de-raining- In The IEEE Conference on Computer Vision and Pattern Recognition (CVPR), June 2019.
Yi, Q.. Li, J., Dai, Q. Fang, F., Zhang, G., and Zeng, T. Structure-prest prreerrinngg deraining with residue channel prior guidance. In Proceedings of the IEEE/CVF international conference on compuler vision, pp. 4238 4247, 2021.
Yu, H., Zheng, N., Zhou, M., Huang, J., Xiao, Z., and Zhao, F. Frequency and spatial dual guidance for image dehazing. In European Conference on Computer Vision, pp. 181-198. Springer, 2022.
Zamir, S. W., Arora, A., Khan, S., Hayat, M., Khan, F. S., Yang, M.-H., and Shao, L. Multi-stage progressive image restoration. In CVPR, 2021.
Zamir, S. W., Arora, A., Khan, S., Hayat, M., Khan, F. S., and Yang, M.-H. Restormer: Efficient transformer for high-resolution image restoration. In CVPR, 2022.
Zhang, H. and Patel, V. M. Density-aware single image de-raining using multi-stream dense network. In Pro- ceedings of the IEEE conference on computer vision and pattern recognition, pp. 695-704, 2018.
Zhang, Y., Zhang. J., and Guo, X. Kindling the darkness: A practical low-light image enhancer. In Proceedings of the 27th ACM international conference on multimedia, pp. 1632-1640, 2019.
Zhang, Y., Guo, X., Ma, J., Liu, W., and Zhang, J. Beyond brightening low-light images. International Journal of Computer Vision, 1013-1073, 2021.
Zhao, S., Chen, H., Zhang, X., Xiao, P., Bai, L., and Ouyang, W. Rs-mamba for large remote sensing image dense prediction, 2024.
Zhen, Z., Hu, Y., and Feng, Z. Freqmamba: Viewing mamba from frequency perspective for image deraining, 2024.
Zhou, M., Huang, J., Guo, C.-L., and Li, C. Fourmer: An efficient global modeling paradigm for image restoration. In International Conference on Machine Learning, pp. 22589-42601. PMLR, 2023.
Zhu, I.,, Liao, B., Zhang, Q., Wang, X., Liu, W., and Wang, X. Vision mamba: Efficient visual representation learning with bidirectional state space model, 2024.
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A. Appendix
A.1. Limitation
In this work, we introduce FourierMamba and extensively validate its efficacy for image deraining through experiments. Our experiments primarily leverage the widely used U-shaped architecture. We plan to further validate the effectiveness of combining Fourier priors with Mamba on more architectures, such isotropic and multi-stage architecture.
Furthermore, given the proven priors of Fourier transform for capturing rain streaks, choose to first validate Mambaa on image deraining. Our work could also offer novel insights for other low-level vision fields, though it may necessitate integrating priors tailored to the distinct differences between various low-level tasks. Given the universal need across various low-level tasks for Fourier priors and the importance of correlating frequencies, the performing improvements can be positively anticipated. We will explore applications in other low-level tasks in our future work.
A.2. More related works
Image deraining. Traditional image deraining methods focus on separating rain components by utilizing meticulously designed priors, such as Gaussian Mixture Models (Li et al., 2016), Sparse Representation Learning (Gu et al., 2017; Fu al., 2011), and Directional Gradient Priors (Ran et al., 2020). Although these methods are insightful, they often struggle cope with complex precipitation patterns and the diverse real-world scenarios. The advent of deep learning has heralded era for image deraining. (Fu et al., 2017b) introduces pioneering deep residual networks for image deraining. The initiation of CNNs marked significant advancement, facilitating more nuanced and adaptive processing of rain streaks across a vast array of images (Yang et al., 2017; Zhang& & Patel, 2018). With the evolution of transformers, the development of architectures that incorporate attention mechanisms (Valanarasu et al., 2022; Wang et al., 2022b) has further refined the capacity to recognize and eliminate rain components, addressing previous shortcomings in model generalization and detail preservation. COIC (Ran et al., 2024) presents a Context-based Instance-level Modulation mechanism integrated with rain-/detail-awaad aware contrastive learning to enhance CNN and Transformer models for improved image deraining on mixed datasets. (Hsu& & Chang, 2023) proposes wavelet approximation-aval residual network, which efficiently removes rain from low-frequency structures and high-frequency details at each level separately. In this work, we propose novel baseline with block based on Fourier and Mamba to enhance deraining performance.
A.3. Inference time of the model
In this section, compare the inference time of the proposed method with several state-of-the-art methods. The comparison results of the model inference time using 512 x 512 images on NVIDIA RTX 4090 GPU are shown in Table 5. It can be seen that the inference time of our model is comparable to that of other methods.
Table 5. Runtime comparison hetween our method and other approaches.
Method
MambalR
VmambalR
FreqMamba
Restormer
Ours
Runtime (s)
0.534s
0.423s
1.837s
0.253s
0.523s
A.4. Results on test100
In this section, we add some performance comparisons with other methods on Test100 as shown in Table 6. All methods are trained on 13k and then tested on Test100. It can be seen that our method still achieves excellent deraining performance.
Table 6. Performance comparison on Test100. PSNR (t) and SSIM (f) are reported.
Metric PReNet MPRNet Restormer MambalR VmambalR FreqMamba
Ours
PSNR
24.81
30.27
32.00
31.82
31.84
31.89
32.07
SSIM
0.851
0.897
0.923
0.922
0.918
0.921
0.925
A.5. Ablation studies and computational overhead
To further demonstrate the effectiveness of Mamba, we present the impact of computational overhead in the first ablation study. For the ablation of FSI-SSM, we compress our model by reducing the number of channels and blocks, achieving a
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Mambia: Fourier Learning Integration with State Space Models for Image Deraining
computational cost similar that of the "wo FSI-SSM" variant. The comparison is shown in Table 7. As observed, the model with FSI-SSM still achieves better performance. For the ablation of FCE-SSM, the computational overhead of the variant without FCE-SSM (w/o FCE-SSM) in Table 3 is similar to that of the model with FCE-SSM. The "woo FCE-SSM" variant stacks several x 1 convolutions with residual connections to match the parameter count of Mamba. The specific computational overhead and performance are shown in Table 8. It evident that, with a similar parameter count, our method outperforms the "w/o FCE-SSM" variant.
Table 7. The computational overhead of the ablation study on FSI-SSM.
Method
PSNR
SSIM
Flops(G)
(arams(M)
w/o FSI-SSM
39.05
0.9835
14.42
10.82
Ours
39.37
0.9845
14.64
10.12
Table 8.
The computational overhead of the ablation study on FCE-SSM
Method
PSNR
SSIM
Flops(G)
Params(M)
w/o FCE-SSM
39.08
0.9836
21.08
17.81
Ours
39.73
0.9856
22.56
17.62
A.6. Reasons for using channel-dimensional Fourier
To address the limitation of Fourier transform not accounting for channel evolution, we introduce channel-dimension Fourier transform. A pivotal motivation is due to different channels often displaying varying properties of degradation information, which also determine the global information of the image when conjunct different channels. comparable deduction can be drawn from style transfer research, where the Gram matrix signifies global style information (Li et al., 2017). This inspires us to employ Fourier transform on the channel dimension enrich the representation of global information.
A.7. The Relationship between Global Average Pooling and Fourier Transform
We believe that the global average pooling equals A(0, 0) the amplitude. In the Appendix, we further verify this. Typically, the Spatail Fourier transform is expressed as:
F-I--1
F (u,)
Σ
Σ
x (h, wh, )2- (u+t (2
(13)
VHW
A=0 w=0
The center point of the amplitude spectrum means that u and u are 0. The formula is as follows:
H-1W-1
Σ Σ x (h,
F (z) (0,0) = VHW
(14)
It can be seen that the above formula is essentially to find the average value of the entire feature map. Therefore, global average pooling (GAP) is equivalent to taking the center point of the amplitude spectrum.
A.8. Performance on other low-level vision tasks
To further demonstrate the effectiveness of our approach, we investigate the performance of our model on other low- level vision tasks. Following FreqMamba (Zhen et al., 2024), we evaluate our method on low-light enhancement and image dehazing. We use the I.O--VI (Wei et al., 2018) and LOL. performance of our method on low-light enhancement, and the Dense-Haze (Ancuti et al., 2019) and NH-HAZE (Ancuti et al., 2020) datasets are used to evaluate the performance of our method on real-world image dehazing. The results for low-light cnhancement are shown in the Table 9. The comparison results for image dehazing are presented in the Table 10. can be seen that our method also demonstrates significant potential for other image restoration tasks.
(Wei et al., 2018) datasets to evaluate the
A.9. Differences between the proposed method and FreqMamba
Our method focuses on customized design based on the characteristics of Fourier space, combining Fourier priors with state space models and exploring the potential of introducing Mamba directly in the Fourier domain. In contrast, FreqMamba
15

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Mambia:
Fourier Learning Integration with State Space Models for Image Deraining
Table 9. Comparison of methods on L.OL.VVI and LOI.-V2-Syn datasets.
LOL-VI
LOL-V2-Syu
Method
PSNR f
SSIM f
PSNR
SSIM
RetinexNet (Wei al., 2018)
18.38
0.7756
19.92
0.8847
KinD (Zhang al., 2019)
20.38
0.8248
22.62
0.9041
ZeroDCE (Guo et al., 2020)
16.8
0.5573
17.53
0.6072
KinD++ (Zhang et al., 2021)
21.3
0.8226
21.17
0.8814
URetinex-Net (Wu et al., 2022)
21.33
0.8348
22.89
0.895
FECNet (Huang et al., 2022)
22.24
0.8372
22.57
0.8938
Awwre (Xu ct al., 2022)
23.38
0.8441
24.12
0,9222
FreqMamba (Zhen et al., 2024)
23.57
0.8453
24.46
0,9355
Ours
23.78
0.8467
24.75
0,9452
Table 10. Comparison of methods on Dense- Haze and NH-HAZE datasets.
Dense-Haze
NH-HAZE
Method
PSNR f
SSIM
PSNR
SSIM
DCP (He et al., 2010)
10.06
0,3856
10.57
0.5196
DehazeNet (Cai et al., 2016)
13.84
0.4252
16.62
0.5238
GridNet (Bozcan et al., 2021)
13.31
0.3681
13.80
0.5370
MSBDN (Dong al., 2020)
15.37
0.4858
19.23
0.7056
AECR-Net (Wu al., 2021)
15.80
0,4660
19.88
0.7173
Mrmq (Zhen et al., 2024)
17.35
0.5827
19.93
0.7372
Ours
18.91
0.6763
20.03
0.7508
operates in the Fourier space using only x convolutions, which fails to fully utilize the rich frequency information inherent to the Fourier domain. Specifically, Frema applies Mamba scanning in a wavelet-tramsfonet transformed domain lacks the notable advantages of the Fourier domain, such as the Fourier transform's ability to decouple degradations and its global representation properties. Additionally, after wavelet decomposition, Freqa divides the image into multiple patches and performs spatial scanning within each patch. This design limits FreqMamba's ability to effectively model frequency correlations.
domain. However, the
In contrast, our method performs Mamba scanning directly in the Fourier domain, fully leveraging the global characteristics of the Fourier transform. This allows our approach to better capture rain streaks, which often exhibit high apparent repetitiveness. Consequently, from a visual perspective, our method demonstrates significantly better performance in removing rain streaks. As shown in Figure 7, we show the feature maps and restoration results of Freq and our method. It can be seen that our method can better capture the rain lines and thus remove the rain lines more cleanly.
Input
Fieq fea
Ours fea
Frag result
Ours result
GT
Figure 7. Feature maps and restoration results of Froq and our method.
A.10. Comparison with other methods such as DRSformer and FADformer
In this section, we compare method with RCDNet (Wang et al., 2020000MPPM.. (Zamir et al., 2021), SPDNet (Yi et al., (021).DuaIGCN (Fu et al., 2021), (Fu et al., Uformer (Wang et al., 2022b), (Xiao et al., (0022).Restommee (Za- mir et al., DRSoo.err (Chen et al., 2023a) and FADformer (Gao et al.). as shown in Table 11. To ensure fairness, we adopt the same experimental setup as the other methods, performing independent training and testing on each dataset, including Rain200L/H (Yang et al., 2017), DID-Data (Zhang& & Patel, 2018), DDN-DaDt Data (Fu et al., 2017b), and SPA (Wang al., 2019). The results demonstrate that our method achieves superior performance on the majority of the datasets.
16

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Mambia: Fourier Learning Integration with State Space Models for Image Deraining
Table I1. Performance comparison of methods across various datasets.
Rain200L
Rain200H
DID-Data
DDN-Data
SPA-Data
Average
Method
PSNR 1
SSIM
PSNR
SSIM
PSNR
SSIM
PSNR
SSIM
PSNR
SSIM
PSNR
SSIM
RCDNet
39.17
0.9885
30.24
0.9048
34,08
0.9532
33.04
0,9472
43.36
0.9831
35.97
0.9554
MPRNet
39.47
0.9825
30.67
0.911
33.99
0.959
33.1
0.9347
43.64
0,9844
36.17
0.9543
SPDNet
40.5
0.9875
31.28
0,9207
34.57
0,956
33.15
0,9457
43.2
0,9871
36.54
0.9594
DualGCN
40.73
0.9886
31.15
0.9125
34.37
0.962
33.01
0,9489
44.18
0.9902
36.68
0.9604
HCN
41.31
0.9892
31.34
0.9248
34.7
0.9613
33.42
0,9512
45.03
0.9907
37.16
0.9634
Uformer
40.2
0.986
30.8
0.9105
35,02
0.9621
33.95
0.9545
46.13
0.9913
37.22
0.9609
IDT
40.74
0.9884
32.1
0.9344
34,89
0.9623
33.84
0,9549
47.35
0,993
37.78
0.9666
Restormer
40.99
0.989
32,0
0,9329
35.29
0.9641
34,20
0.9571
47.98
0,9921
38.09
0.9670
DRSformer
41.23
0.9894
32.17
0,9326
35,35
0.9646
34.35
0,9588
48.54
0,9924
38.32
0.9676
FADformer
41.80
0.9906
32.48
0.9359
35.48
0.9657
34.42
0,9602
49.21
0.9934
38.67
0.9691
Ours
42.27
0.9908
32.71
0.9395
35.49
0.9659
35.58
0.9599
49.18
0.9931
39.05
0.9698
A.11. Difference between mamba and convolution in processing Fourier frequencies
First, Mamba utilizes sequence modeling to integrate information across all frequency bands, effectively leveraging the complementary relationships between different bands. In contrast, convolution, as a local operation, struggles holistically model global features across all frequency bands when processing frequency information in the Fourier domain. This limitation significantly constrains its capacity in the Fourier space. Second, Mamba's 's sequence modeling is orderly, which can help the network establish an orderly dependency relationship between different frequencies. This characteristic is critical for modeling image degradation information. Conversely, convolution insufficient capturing the dependencies between high and low frequencies in the Fourier domain, thereby weakening its ability to accurately represent degradation features. In summary, based on these two advantages, Mamba achieves better coordination of high-frrquency and cowffrequency information in the Fourier domain during the image restoration process.
We process the Fourier frequencies using both Mamba and convolution separately, and then visualize their features, as shown in Figure 8. It can be seen our method (i.e., Mamba) not only captures rain streaks effectively but also extracts structural information from the background with high accuracy.
Input
Convolution
Mamba
GT
Figure 8. Feature visualization comparison of firquency-domain convolution and mamba on Rain 100..
A. More visual deaining comparison on synthetic datasets
In this section, we provide more visual deraining comparisons on synthetic datasets to further demonstrate the effectiveness of our method. Specifically, we perform visual comparisons on several datasets Table 1. Figure 9 shows more visualization results on Rain 100H. As with the results in the main text, it shows that our method can better remove rain effects and prevent artifacts, which is attributed to the progressive frequency correlation. Figures 10, 11, and show the visualization results
17

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MaubiarMamba: Fourier Learning Integration with State Space Models for Image Deraining
Input
PreNet
MPRNet
MAXIM
Restormer
Ours
GT
Figure 9. More qualitative comparison on Rain100H (Yang et al., 2017).
Table 12. Quantitative comparison of testing on the real-world dataset SPA-Data.
Method PSNR
PReNet
RESCAN
HiNet
MSPFN
Restormer
MPRNet
Ours
31.33
31.56
33.89
34.03
34.18
34.54
35.27
SSIM
0,9501
0.9423
0.9500
0.9471
0.9493
0.9548
0.9575
on the simulation datasets Rain100L, Test2800, and Test respectively.
A.13. More real-world detraining results by using synthetic data
In this section, we provide more real-world rain removal cases to verify the generalization ability of the model trained on the synthetic dataset (rain13k). The quantitative comparisons directly tested on SPA-Data are shown in Table 12. The visualization results are shown in Figure 13. Our method is superior to other methods in rain removal and detail recovery. To further demonstrate its generalization ability the real world, we also tested it on a real-world dataset RE-RAIN -Chhe also
al., 2023b), as shown in Figure 14. FourierMamba can obtain the most visually pleasing results. In addition, tested our method directly on RainDS-Real (Quan al., 2021), and the quantitative results shown in the table 13, Figure 15 shows the visualization results RainDS-Rea Real. It can be seen that our method effectively remove real rain.
A.14. More real-world visual deraining results by training real-world rainy images
Training and testing real-world rainy images can verify the representation ability of the model in the real world. In Section 4.2, we show the visualization results of training and testing using SPA-Data, as shown in Figure 16. It can be seen that our method excels at removing rain and recovering details to obtain pleasing visual results. In addition, we also train and test MourierMamba on RainDS-Ree Real (Quan et al., 2021) to further verify its effectiveness in real-world scenes. As shown in Table 14, our method can still achieve excellent performance.
the real-world dataset SPA-Data to train FourierMamba and report quantitative results. In this section,
A.15. Metrics that can better reflect human perceptions
In this section, we use some metrics that better reflect human perception to evaluate our method. We use the more widely used perceptual metries BRISQUE (Mittal et al., 2012b), NIQE (Mittal et al., 2012a), SSEQ (Liu et al., 2014), as shown in Table 15. can be seen that our method can also achieve excellent performance on perceptual metries.
Table 13. Quantitative results of testing on the real-world dataset RainDS Real (Quan et al., 2021).
Method
PReNet
RESCAN
Restormer
HINet
MSPEN
MPRNet
Ours
PSNR
24.15
24.29
24.54
24.71
24.76
25.07
25.12
SSIM
0.711
0.717
0.727
0.9731
0.729
0.736
0.738
18

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Mambi:: Fourier Learning Integration with State Space Models for Image Deraining
Input
PreNet
MPRNet
MAXIM
Restormer
Ours
Figure 10. Qualitative comparison Rain 100L (Yang al., 2017). Zoom in for better visualization.
Input
MPRNet
MAXIM
Restormer
Ours
GT
Figure 11. Qualitative comparison Test2800 (Fu et al., 2017b). Zoom in for better visualization.
19

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Mambia: Fourier Learning Integration with State Space Models for Image Deraining
Input
PreNet
MPRNet
MAXIM
Restormer
Ours
GT
Figure 12. Qualitative comparison on Test (Zhang & Patel, 2018). Zoom in for better visualization.
Input
PreNet
MPRNet
Restormer
Ours
GT
Ks
cks
Ks
CKS
CKS
cks
Figure 13, Qualitative comparison real-world rainy images from SPA-Data(Wan) Wang et al., 2019).
20

---

Mambi:: Fourier Learning Integration with State Space Models for Image Deraining
Input
PreNet
MPRNet
Restormer
Ours
Figure 14. Qualitative comparison real-world rainy images from RE-RAIN (Chen al., 2023b).
Input
PreNet
MPRNet
Restormer
Ours
GT
Figure 15, Qualitative results of real world rainy images from RainDS Reaal. (Quan al., 2021).
21

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Mamri::Mamba: Fourier Learning Integration with State Space Models for Image Deraining
Input
DualGCN
Ne
Restormer
DRSformer
Ours
GT
Figure 16. Qualitative results of training and testing on SPA-Data. (Wang et al., 2019).
Table 14. Quantitative comparison of training and testing the real-world dataset RainDS-
Method
PReNet
MSPEN
RCDNet
MPRNet
SwinIR
Restormer
Ours
PSNR
26.43
26.45
26.71
27.51
27.53
27.57
27.69
SSIM
0.7294
0.7270
0.7180
0.7355
0.7425
0.7438
0.7482
A.16. More about the Optimization
In the main body, we describe that apply the LI loss based on the Fourier transform. Here, we introduce the loss function in the frequency domain in further detail. we first use the Fourier transform convert Your and Ygt into the Fourier space. Then, the -nrm of the amplitude difference and phase difference between Yout and Ygt are calculated and summed to produce the total frequency loss following:
[( F(Yr:) = [[A((o))) A(Y + -IP(())))-((())))) -P(Y) P()
(15)
A.17. Ablation study on different frequency domain loss functions
We use three additional frequency domain loss functions: Phase Consistency Loss (PCL), Frequency Distribution Loss (PDL), and Focal Frequency Loss (FFL) (Jiang et al., 2021) for comparison with the L1 frequency domain loss we use. PCL defined as the mean squared error of the phase difference between two images in the frequency domain, expressed as:
LPCL
Σ ((,,, p(y
(16)
HW
FDL represents the difference in frequency domain amplitude distributions between two images, expressed as:
LEDL
(A(You)(u,u) A(Y vu,))
(17)
HW
FFL focuses on frequency components that are difficult to synthesize by down-weighting the easier ones, expressed as:
w(u, v) = (FYoo) F(Y)t)
1
(18)
CEF
w(u (Y ou,
HW
where (u,) v) represents the Fourier Transform, w(u, v) is the weight for the spatial frequency at(u, v), and a is the scaling factor for flexibility (a practice).
22

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Mambi:: Fourier Learning Integration with State Space Models for Image Deraining
Table 15. Performance comparison of different methods various datasets. Metries include BRISQUE, NIQE, and SSRQ.
Dataset
Rain100L.
Rain10011
Test2000
Test1200
Test100
Method
BRISQUE
SSBQ
BRISQUE
NIQE
BRISQUE
NIQE:
SSEQ
BRISQUE
SSBQU
BRISQUE.
NIQE
MPRNet
17.791
6.816
12.702
16.287
6.973
13,860
15.782
6.25]
9.470
23.434
5,742
12.653
23.526
6.903
12.767
MAXIM
11.90
6.402
9.688
14,622
6.929
8.034
15.272
6.114
8.760
25.026
5.573
14.760
22.433
6.770
12,615
MambalR
15.662
6.553
11.527
10.350
6.104
8.719
13.246
6.165
8.332
20.743
5.570
10.877
17.856
5.969
8,390
VmambJIR
16.073
6.651
11.061
11,686
5,713
8.302
13,465
6.114
8,306
20.851
5.553
10.610
17.805
5.751
8,548
PregMansaa
15.151
19,942
5,439
5,898
7.247
Ours
12.178
6.261
10.008
10,607
6.009
3.826
12,895
5.258
8.286
21.467
5,538
10.839
17.738
5.958
7,102
We conduct ablation comparison experiments these loss functions, shown in Table 16. It can be seen that the performance obtained by these four loss functions is similar. The focus of this work is on the design of the network architecture, so we follow existing methods (Zhou et al., 2023; Zhen et al., 2024) to use the L1 norm in the frequency domain. We will explore more frequency-domain loss functions in future work.
Table 16. Comparison results different frequency-domain loss functions.
PCL
FDL
FFL
Ours
PSNR
39.67
39.69
39.75
39.73
SSIM
0.9848
0.9852
0.9859
0.9856
A.18. More visualizations for ablation studies
In Section 4.3, we perform ablation studies on the key designs and scanning methods of the proposed method. To further verify the effectiveness of the proposed method, we provide visualizations of above ablation studies. Specifically, we subtract the restored images obtained from each ablation study from the ground truth to show the effeet of each design. Figure 17 shows the visualization of the ablation study in Table 3. It can be seen that all designs have significant effect rain removal. Figure 18 shows the visualization of the ablation study of various scanning methods in Table 4. Both Figures 4 and Figures 18 illustrate that orderly correlation of different frequencies can promote rain removal.
Input
W/o FSI SSM
W/o FCE SSM
W/o SDF
W/o CDF
Ours
Figure 17. Visualization of ablation studies of various key designs of the proposed method.
23

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Mambi:: Fourier Learning Integration with State Space Models for Image Deraining
Input
GT
Classical
Bilateral
Progressive
Ours
Figure 18. Visualization of ablation studies of different scanning methods in Fourier space.
24


Reviews:
[{"review_id": "OIzcmu5lPO", "reviewer": "Reviewer_qXRb", "summary": "This paper introduces FourierMamba, a novel framework for image deraining that leverages the Mamba technique within the Fourier space to effectively correlate different frequency components. Unlike existing Fourier-based methods that fail to fully exploit the dependencies between low and high frequencies, FourierMamba employs a unique scanning mechanism to encode frequencies in both spatial and channel dimensions. In the spatial dimension, it uses zigzag coding to rearrange frequencies from low to high, ensuring orderly correlation. In the channel dimension, it directly applies Mamba to enhance frequency correlation and channel representation. Extensive experiments demonstrate that FourierMamba outperforms state-of-the-art methods in both qualitative and quantitative evaluations, offering a significant advancement in image deraining by better utilizing frequency information.", "questions": "", "limitations": "", "rating": 5, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**R1**: The Necessity of Channel-Dimensional Fourier Transform\n\nIn Appendix A.6 of the paper, titled \"Reasons for Using Channel-Dimensional Fourier,\" we note that different channels typically exhibit distinct degradation characteristics, which collectively determine the global information of an image when integrated across channels. This observation draws inspiration from style transfer research, such as the use of the Gram matrix to represent global style information [1]. Building on this insight, we introduce the Fourier transform in the channel dimension to enhance the representation of global information by capturing frequency dependencies across channels. Directly scanning the sequence (e.g., using Mamba) fails to effectively leverage the global properties inherent in the Fourier domain. In contrast, the channel-dimensional Fourier transform enables the decoupling of degradation information (e.g., rain streaks) from background content, thereby improving Mamba’s capacity to model frequency correlations. The effectiveness of this design is substantiated by ablation studies presented in Table 3 and Figure 17, which provide quantitative metrics and visualizations, respectively, demonstrating its impact.\n\n**R2**: Ensuring Normal Spatial Features After Inverse Transformation from Fourier Space Scanning\n\nYou raised the concern that the paper should discuss how normal spatial features are preserved through inverse transformation following Fourier space scanning. We acknowledge that the current discussion on this aspect is insufficiently detailed and will address this in the revised manuscript by supplementing relevant content. The key to FourierMamba lies in its scanning strategy, which is designed to preserve the symmetry and global properties of the Fourier domain. Specifically, in the spatial dimension, we employ zigzag coding for scanning and process only half of the spectrum, leveraging the symmetry of the Fourier transform (amplitude centro-symmetry and phase anti-centro-symmetry) to deduce the other half (see Section 3.2). This approach ensures the integrity of the Fourier domain information. In the channel dimension, Mamba scanning is performed on a one-dimensional Fourier spectrum, adhering to similar symmetry principles. The inverse transformation relies on the standard Inverse Fast Fourier Transform (IFFT) algorithm, which theoretically guarantees perfect reconstruction from the Fourier domain to the spatial domain, provided that the scanning operation does not introduce irreversible information loss. Our experimental results (e.g., Figures 5 and 9) demonstrate that the inverse-transformed images retain normal spatial features, such as textures and details, owing to the orderly nature of the scanning design and the preservation of symmetry. We will enhance Sections 3.2 and 3.3 with these clarifications and may include a mathematical derivation in the appendix to further elucidate this process.\n\n**R3**: Formatting Issues\n\nWe will address and correct spelling errors in the revised manuscript.\n\n[1] Li, Y., Fang, C., Yang, J., Wang, Z., Lu, X., and Yang, M.-H.Universal style transfer via feature transforms. Advances in neural information processing systems, 30, 2017.", "claims_and_evidence": "The author clearly express the claims made in the manuscript.", "methods_and_evaluation_criteria": "To address the insufficient utilization of correlations among different frequencies, this paper introduces Mamba combined with the Fourier transform to model the dependencies between frequencies, thereby enhancing the representation of frequency information. Experiments conducted on multiple datasets demonstrate that the proposed method achieves effective frequency correlation, showcasing its potential to break through the limitations of existing modeling frameworks.", "theoretical_claims": "The effectiveness of frequency correlation is extensively validated and analyzed on a wide range of rainy datasets.", "experimental_designs_or_analyses": "The experimental design is well-justified, and the proposed method demonstrates competitive performance on both synthetic and real-world datasets.", "relation_to_broader_scientific_literature": "Previous work has demonstrated the importance of frequency in image restoration tasks such as image deraining. This paper emphasizes the significance of establishing correlations between different frequencies and proposes a customized Fourier-based method for image deraining.", "essential_references_not_discussed": "No", "other_strengths_and_weaknesses": "Strengths:\n1.The paper introduces a novel combination of Fourier learning and Mamba, which is well-motivated and effectively improves image deraining. It explores further possibilities of integrating Mamba with the Fourier transform.\n2.The paper proposes a scanning method based on zigzag coding to systematically correlate different frequencies. This method introduces zigzag coding in the Fourier space to rearrange frequency orders, thereby orderly connecting the relationships between frequencies. The zigzag scanning strategy is well-motivated and technically sound.\n3.Compared to other Mamba models for image restoration tasks, Fourier Mamba demonstrates higher efficiency and better performance, showcasing the enhancement of Fourier learning on Mamba's modeling capabilities.\n4.The proposed model achieves a balance between accuracy and reasonable computational cost and model size.\n5.The paper is well-organized and relatively easy to follow, with experimental results validating the effectiveness of the proposed modules and scanning methods.\n\nWeaknesses:\n1.The authors should clarify why the Fourier transform is necessary in the channel dimension instead of directly scanning the sequence.\n2. There should be a proper discussion on how to ensure that normal spatial features can be obtained by inverse transformation after Fourier space scanning?\n3.There are minor formatting issues, such as the need for a space before \"More\" in line 130.", "other_comments_or_suggestions": "See the weakness.", "questions_for_authors": "See the weakness.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "6VVuhKZKRm", "reviewer": "Reviewer_q47J", "summary": "This paper proposes FourierMamba that addresses the problem of single image deraining by introducing a scanning encoding mechanism that correlates different frequencies in both spatial and channel dimensions. Specifically, it employs zigzag coding in the spatial dimension to reorganize frequency orders and improve their connectivity, while utilizing tailored designs in the channel dimension. This approach enables more effective frequency information utilization, leading to improved image deraining performance.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**R1**: Details of Zigzag Scanning Implementation and Its Impact on Inference Speed\n\nSection 3.2 briefly outlines the motivation and design of zigzag scanning, inspired by JPEG’s zigzag encoding, to order frequencies in the Fourier domain from low to high. For the spatial Fourier spectrum, it is divided into symmetric halves, with zigzag path scanning applied to one half, arranging frequencies from center (low) to edges (high), followed by Mamba sequence modeling. The other half is derived via Fourier symmetry, ensuring orderly frequency correlation without compromising symmetry, unlike full-spectrum scanning.\n\nAs a preprocessing step, zigzag scanning’s computational cost, mainly from frequency rearrangement and Mamba modeling, is minimal. Experiments show its additional time is negligible, involving a one-time index reorder—reusable as a dictionary for same-resolution images—while Mamba’s linear complexity ensures efficiency. FourierMamba’s inference time matches MambaIR and FreqMamba, indicating no significant burden. Revised Section 3.2 will detail the implementation and confirm its minimal impact on inference speed.\n\n**R2**: Comparison with FreqMamba in Frequency-Domain Correlation Strategies\n\nWe further clarify the differences and similarities between FourierMamba and FreqMamba in frequency-domain strategies. While FreqMamba employs Mamba in the frequency domain, it primarily conducts spatial scanning in the wavelet domain, underutilizing the global properties of the Fourier domain. In contrast, FourierMamba applies Mamba directly in the Fourier domain, using zigzag encoding for orderly frequency correlation, enhancing rain streak capture. FreqMamba’s wavelet-domain scanning limits its global frequency modeling capacity.\n\nQuantitative comparisons with FreqMamba on Rain13k and SPA datasets, shown in Tables 11 and 16, reveal our method’s improvements of 0.54 dB and 0.71 dB, respectively. Feature map visualizations in Figure 7 demonstrate that our approach more effectively targets rain degradation, yielding superior deraining results.\n\n**R3**: Diversity of assessment indicators\n\nWe have augmented the evaluation with no-reference metric results for the unpaired Internet-Data test set from the SPA dataset, as shown in Table 1 above.\n\n**R4**: Effectiveness and Generalizability of Mamba in Processing Fourier Frequencies\n\nMamba, a state-space model, excels with global modeling and linear complexity, ideal for sequential data. In the Fourier domain, where frequency information is inherently global and sequential, Mamba efficiently captures inter-frequency dependencies. Traditional convolutional or Transformer architectures, however, are less efficient or computationally costly for global frequency processing, while Mamba’s linear global attention effectively combines their strengths.\n\nRegarding generalizability, supplementary results demonstrate FourierMamba’s strong performance in low-light enhancement and dehazing on datasets like LOL-V1 and Dense-Haze, indicating robust task adaptability. For deblurring, where frequency loss is key, its correlation modeling shows promise, to be explored further in future work.\n\nCompared to wavelet transforms, which excel in local frequency and multi-scale analysis, Fourier transforms better support global frequency representation and degradation decoupling, particularly for deraining, where rain streaks are distinctly separable in the frequency domain. Thus, we prioritize Fourier transforms, but future work will explore wavelet integration. These points will be detailed in Sections 2 and 5 of the revised manuscript to fully address Mamba’s effectiveness and the method’s generalizability.\n\n**R5**: Related Works Omissions\n\nWe appreciate your feedback and will include discussions of [1] and [2] in the revised manuscript. While [1] explores cross-modal fusion with Mamba for image restoration, our work focuses on frequency-domain modeling. Similarly, [2] employs linear models for weather degradation, which aligns partially with our approach. We will clarify these connections in Section 2 to better position our contributions.\n\n| Method      | BRIQUE ↓ | NIQE ↓ | SSEQ ↓ |\n|-------------|----------|--------|--------|\n| Rainy Input | 28.517   | 5.095  | 28.280 |\n| MPRNet      | 34.733   | 5.144  | 33.765 |\n| Restormer   | 32.288   | 4.851  | 31.789 |\n| IDT         | 27.042   | 4.536  | 28.314 |\n| DRSformer   | 26.080   | 4.531  | 27.954 |\n| FADformer   | 25.959   | 4.760  | 26.667 |\n| Freqmamba   | 26.172   | 4.890  | 27.387 |\n| Ours        | 25.827   | 4.682  | 26.423 |\n\n[1] Cross-Modality Fusion Mamba for All-in-One Extreme Weather-Degraded Image Restoration, 2025. [2] Restoring images in adverse weather conditions via histogram transformer, 2024.", "claims_and_evidence": "Yes.", "methods_and_evaluation_criteria": "Yes.", "theoretical_claims": "N/A, there is no theoretical claims in the paper.", "experimental_designs_or_analyses": "The experimental analyses are sound because the authors have conducted extensive experiments on five datasets and shown the superiority of their method.", "relation_to_broader_scientific_literature": "The related works section overlooks several recent studies, including [1-2], which explore the concepts of pixel-level alignment and the linear model.\n\n[1] Cross-Modality Fusion Mamba for All-in-One Extreme Weather-Degraded Image Restoration, 2025. \n[2] Restoring images in adverse weather conditions via histogram transformer, 2024.", "essential_references_not_discussed": "N/A.", "other_strengths_and_weaknesses": "Strengths:\n\n1. The authors have conducted extensive experiments on over five commonly used deraining datasets, demonstrating the proposed method's superiority in both complexity and deraining performance, evaluated quantitatively and qualitatively.  \n\n2. The incorporation of zigzag coding in Fourier space and the concept of scanning encoding for different frequencies introduce a novel approach to deraining.  \n\n3. The paper is well-written and easy to follow, with numerous visual comparisons and feature visualizations effectively illustrating the deraining results of the proposed method.\n\nWeaknesses:\n\n1. To improve the implementation and evaluation parts, it is helpful to provide more details on the zigzag scanning implementation and its impact on inference speed. \n\n2. To expanded comparisons and additional experiments, it is suggested to discuss the comparisons and differences between the proposed method and FreqMamba in Fourier correlation strategies. It is helpful to incorporate alternative perceptual metrics beyond PSNR and SSIM to align with human perception on the real-world rainy dataset such as SPA-Data.  \n\n3. It is recommended to justify why Mamba is more effective than other architectures for processing Fourier frequencies. The authors are also suggested to investigate generalization to other tasks (e.g., deblurring) and discuss the applicability of wavelet transformation.", "other_comments_or_suggestions": "N/A", "questions_for_authors": "Please refer to the weaknesses.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "NBjG3MDT4P", "reviewer": "Reviewer_X9L3", "summary": "The paper applies Mamba to dual-domain spatial dimensions and channels for image deraining. For spatial-dimensional Fourier processing, the authors introduce a new scanning method that better models the correlations between different frequencies. The constructed network is evaluated on multiple image restoration tasks beyond deraining and achieves promising performance.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**R1**: Response to “Insufficient Novelty”\n\nThe distinctions between our method and FreqMamba are detailed in Appendix A.9. \n\nHere, we further clarify the novelty:\n\n**Spatial-Domain Zigzag Scanning**: Unlike FreqMamba’s simple wavelet-domain scanning, FourierMamba’s zigzag scanning, inspired by JPEG encoding, orders low-to-high frequencies in the Fourier domain, enhancing correlation modeling while preserving symmetry. This integrates signal processing knowledge into Mamba’s efficient structure, synergistically improving deraining outcomes.\n\n**Channel-Domain Fourier Scanning**: Discussed in Appendix A.6, this design leverages varying degradation characteristics across channels, which collectively define global image information. By introducing Fourier transforms in the channel dimension, we capture inter-channel frequency dependencies, enhancing global representation while decoupling degradation (e.g., rain streaks) from background content using amplitude and phase spectra. Joint modeling with Mamba further strengthens channel interactions and global information modeling.\n\n**Overall Novelty**: FourierMamba combines spatial zigzag scanning and channel Mamba scanning to form a versatile deraining framework, outperforming MambaIR and FreqMamba. This dual-domain frequency modeling represents a significant advancement in image restoration, far beyond a mere scanning improvement.\n\n**R2**: Response to “Experiments Lack Convincing Evidence”\n\nJoint training on Rain13k demands greater network generalization, often yielding lower performance than models trained separately on individual datasets. Except for FreqMamba, results in Table 1 reflect joint training, whereas FreqMamba’s paper conflates the two approaches. For a fair comparison with FreqMamba, we provide results under identical experimental settings in Appendix Tables 11 and 16, showing improvements of 0.54 dB and 0.71 dB on Rain13k and SPA datasets, respectively, with our method.\n\n**R3**: Response to “Performance Lacking Competitiveness on Certain Datasets”\n\nAST was trained on the enhanced SPAD version[1] of the SPA dataset, making direct comparison unfair. Thus, we retrained AST on the SPA dataset using parameters from its paper, yielding the following results in Tab1. For low-light enhancement, FourierMamba targets image deraining, with experiments in this task only validating its generalizability, lacking specialized designs like MambaLLIE’s Retinex prior. Hence, it is reasonable that FourierMamba underperforms MambaLLIE in low-light enhancement. Future work will explore tailoring FourierMamba for this task.\n\n**R4**: Response to “Suboptimal Inference Speed”\nIt was noted that FourierMamba’s inference time lags behind Restormer’s. This stems from PyTorch’s extensive CUDA optimizations for attention-based operators, which Mamba currently lacks. However, our method’s FLOPS (22.5) is significantly lower than Restormer’s (174.7), suggesting that further CUDA optimization of Mamba operators will enhance inference speed. Meanwhile, compared to other Mamba-based methods, ours achieves a superior balance of inference efficiency and performance.\n\n**R5**: Response to “Issues with Supplementary Material”\n\nThe review highlighted a missing table reference in Section A.10 and an incorrect citation for FADformer. We apologize for these oversights and will correct them in the revised manuscript: Section A.10 will explicitly reference Table 12, and the FADformer citation will be updated to the correct source. We appreciate your meticulous feedback.\n\n| Method | PSNR  | SSIM   |\n|--------|-------|--------|\n| AST    | 48.49 | 0.9924 |\n| Ours   | 49.18 | 0.9931 |\n\n[1] Learning Weather-General and Weather-Specific Features for Image Restoration Under Multiple Adverse Weather Conditions [CVPR 2023]", "claims_and_evidence": "The quantitative and qualitative results demonstrate the effectiveness of the proposed method.", "methods_and_evaluation_criteria": "The authors do experiments on standard datasets and metrics. In addition, the evaluation is performed using non-reference metrics.", "theoretical_claims": "In addition to the foundational equations for the Fourier transform, no other theoretical claims are provided.", "experimental_designs_or_analyses": "This paper closely follows the FreqMamba (MM'24) method. However, there is an inconsistency between these two methods regarding using datasets. Specifically, it seems that FreqMamba trains separate models for each deraining dataset while the proposed method is trained on a mixed dataset, Rain13k. The authors also mention this issue in the paper. Will this inconsistency impact the performance?", "relation_to_broader_scientific_literature": "This paper introduces an image deraining architecture incorporating the zigzag scanning method with Mamba. The network achieves promising performance on some image restoration datasets.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "The proposed method introduces a new scanning method for image deraining to model correlations between Fourier frequencies. State-of-the-art performance is achieved on some image restoration tasks.\n\nThe reviewer's concerns are as follows:\n\n1. The proposed network mainly follows that of FreMamba. The novelty of the method only lies in using a different scanning method for spatial-dimensional frequency features. Using Mamba to scan the channel features is direct and not novel in image restoration. Overall, the novelty is a little bit limited.\n\n2. The experiments are not convincing, as the proposed method employs a different training strategy for the dataset compared to FreqMamba.\n\n3. On some additional datasets, the performance is not competitive. For example, on SPA-Data, AST achieves 49.51 dB, which is much higher than the proposed method. For low-light enhancement, the proposed method is inferior to MambaLLIE while using more parameters.\n\n4. The authors provide the speed comparisons in the supplementary material. The reviewer finds that the proposed method does not run fast in the deraining domain.\n\n[AST] Adapt or Perish: Adaptive Sparse Transformer with Attentive Feature Refinement for Image Restoration, CVPR24. \n\n[MambaLLIE] MambaLLIE: Implicit Retinex-Aware Low Light Enhancement with Global-then-Local State Space, NeurIPS24.", "other_comments_or_suggestions": "n/a", "questions_for_authors": "Please refer to the entries above.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "f2jxbgWQkp", "reviewer": "Reviewer_Rwtp", "summary": "The authors propose a novel framework, FourierMamba, which integrates Fourier priors with a state-space model to associate different frequencies in the Fourier domain for image deraining.\n\n## update after rebuttal\nThe authors' second-round response has addressed most of my concerns. I now find the motivation of the paper to be reasonable and the experiments to be sufficiently thorough. I have decided to raise my score.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**R1**: Clarification on Motivation\n\nMotivation: Mamba, a state-space model, offers global modeling and linear complexity, making it ideal for sequential data. Frequency information in the Fourier domain is inherently global, and Mamba’s sequence modeling efficiently captures inter-frequency dependencies. In contrast, Transformers incur high computational complexity (O(n²)) for long sequences, while 3×3 convolutions are limited by their receptive fields in capturing global frequency data. Thus, integrating Mamba with the Fourier domain is a targeted signal-processing design, leveraging frequency globality and computational efficiency to enhance deraining performance innovatively.\nComparative Experiments: Addressing the suggestion, we conducted experiments comparing Transformer and 3×3 convolution on the Rain200L dataset (Table 1). Results demonstrate that Mamba’s scanning approach effectively balances performance and efficiency.\n\n**R2**: Response to “Lack of Quantitative Evaluation in Real-World Scenarios”\n\nWe have included no-reference quantitative evaluation results for the Internet test set without ground truth (GT) from the SPA dataset, as presented in Table 2.\n\n**R3**: Response to “Experimental Design and Analysis”\n\nReal-world deraining results are presented in Figures 13–16, with quantitative outcomes in Tables 13–15, demonstrating adaptability and generalizability to real rain due to Fourier priors and Mamba’s efficient global modeling.\n\n**R4**: Response to “Issues with Mamba in the Fourier Domain”\n\nMamba’s local pixel forgetting in the spatial domain stems from its sequence modeling, potentially overlooking local details, while spatial misalignment relates to scanning order. In the Fourier domain, however, information is globally represented as frequencies, rendering local forgetting less prominent, as frequency components reflect overall image properties. Moreover, Fourier priors decompose the image into frequencies, aiding Mamba in modeling global information comprehensively and mitigating misalignment. Experimental results (Figures 5 and 9) show FourierMamba preserves background details and removes rain streaks effectively, confirming these issues are alleviated in the Fourier domain. This analysis will be added to the revision.\n\n**R5**: Response to “Unclear Connection Between Preliminary Equations and Figure 3 Modules”\n\nWe acknowledge the lack of clarity and will refine the Preliminary section in the revised manuscript. The equations therein introduce Fourier transform fundamentals, laying the theoretical groundwork for subsequent frequency modeling. The FFT operation in Figure 3 directly applies these principles, converting images from the spatial to the Fourier domain and separating amplitude and phase. We will explicitly link these equations to the FFT module in Figure 3, enhancing logical coherence and eliminating perceived redundancy.\n\n**R6**: Response to “Issues with Feature Maps in Figure 7”\n\nFigure 7 compares feature maps of FreqMamba and our method, revealing that our approach more effectively focuses on and localizes rain streak-related features. Consequently, our derain results (Fig5-6, Fig9-16) retains background information while minimizing residual rain streaks compared to FreqMamba.\n\n**R7**: Response to “Missing Entries in Table 1”\n\nTable 1 references the table in FreqMamba. Missing data primarily result from unavailable open-source code or discrepancies between open-source implementations and the corresponding papers, leading us to directly adopt reported results from the papers. In the future, we will strive to complete missing entries or provide explanations as feasible.\n\n**R8**: Response to “Formatting and Presentation Issues”\n\nIn the revised manuscript, we will adjust the font sizes of tables and figures for consistency.\n\n**R9**: Response to “Layout Issues in Figure 6”\n\nOur method’s advantages over others in real-world scenarios are observable through enlarged patches. We will revise the layout of Figure 6 in the updated manuscript. Additional visual results for real-world deraining can be found in Supplementary Figures 13–16.\n\n\n\n| Method      | PSNR  | SSIM   | Flops | Params |\n|-------------|-------|--------|-------|--------|\n| 3*3 Conv    | 40.24 | 0.9887 | 17.44 | 21.98  |\n| Transformer | 42.21 | 0.9896 | 18.98 | 72.49  |\n| Ours        | 42.27 | 0.9908 | 17.62 | 22.56  |\n\n| Method      | BRIQUE ↓ | NIQE ↓ | SSEQ ↓ |\n|-------------|----------|--------|--------|\n| Rainy Input | 28.517   | 5.095  | 28.280 |\n| MPRNet      | 34.733   | 5.144  | 33.765 |\n| Restormer   | 32.288   | 4.851  | 31.789 |\n| IDT         | 27.042   | 4.536  | 28.314 |\n| DRSformer   | 26.080   | 4.531  | 27.954 |\n| FADformer   | 25.959   | 4.760  | 26.667 |\n| Freqmamba   | 26.172   | 4.890  | 27.387 |\n| Ours        | 25.827   | 4.682  | 26.423 |", "claims_and_evidence": "The motivation behind combining Mamba with the Fourier space needs to be discussed in greater depth. The results shown in Figure 1 only include 1×1 convolution and previous scanning in Fourier space. However, more commonly used alternatives, such as Transformers and stacked 3×3 convolutions, should also be considered and discussed.", "methods_and_evaluation_criteria": "The authors lack quantitative evaluation in real-world scenarios, yet the ultimate goal of image processing is to apply it to real-world situations. Quantitative assessment of real rain images without ground truth (GT), using no-reference evaluation metrics, is crucial for demonstrating the method's practical applicability.", "theoretical_claims": "n/a", "experimental_designs_or_analyses": "If the authors provide more real-world examples and analyze the contributions of Fourier and Mamba in these scenarios, this work would  more convincing.", "relation_to_broader_scientific_literature": "n/a", "essential_references_not_discussed": "n/a", "other_strengths_and_weaknesses": "Strengths:\n1.The authors combine Mamba with the Fourier space, which is an interesting idea. \n2. The proposed method achieves state-of-the-art performance on multiple image deraining datasets.\n3. This work is evaluated on multiple image restoration tasks beyond image deraining, further demonstrating the model's generalization capability and effectiveness.\n\nWeaknesses:\n1.The Mamba network suffers from two issues: local pixel forgetting and spatial misalignment. I am curious about how these issues manifest in the Fourier domain. Would they still occur, or do the Fourier priors help mitigate these problems?\n2.The equations in the Preliminary section are not clearly linked to the modules in Figure 3. In other words, these equations are not directly utilized later in the paper, making them appear redundant. The module primarily applies FFT for phase and amplitude separation, which has a weak connection to the equations presented in the Preliminary section.\n3.The authors should include more examples from real-world scenarios to further demonstrate the model's generalization ability.\n4.Some references need to be updated, changing citations from arXiv papers to the officially accepted versions of the papers.\n5.The double-line and triple-line tables in the manuscript should be standardized for consistency.\n6.In Figure 7, I noticed that besides the rain streaks, the feature maps also contain window edge textures. Could this lead to an issue where the Fourier space mistakenly treats window edges as having similar frequency characteristics to rain streaks, potentially causing background degradation?\n7.I am curious why many entries in Table 1 are missing. As far as I know, these methods have publicly available source codes. Could the authors clarify why the results for these methods are not reported?\n8.The font size in tables and figures is inconsistent (e.g., Tables 2, 3, and 4). The authors should standardize the font size across all tables and figures for consistency.", "other_comments_or_suggestions": "As mentioned earlier, real-world evaluation is a crucial aspect of validating image deraining methods. However, Figure 6 is too small, making it difficult to discern the differences between different methods. The authors may consider removing some of the early baseline methods to improve clarity and focus on more recent approaches.", "questions_for_authors": "n/a", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This paper proposes FourierMamba, an image deraining framework that combines Mamba-based state-space modeling with Fourier domain representations. The key contribution lies in a zigzag scanning strategy that captures inter-frequency dependencies, enhancing both spatial and channel-wise frequency modeling. The approach is well-motivated and grounded in both signal processing and recent modeling techniques.
The reviewers found the method to be original and effective, with strong empirical results across multiple datasets. The paper is clearly written and supported by thorough experiments and ablation studies. Some concerns were raised about the novelty of individual components and training consistency, but the authors addressed these points with detailed rebuttals and clarifications.

Overall, this is a well-executed and solid contribution to frequency-domain image restoration. I recommend acceptance.
Decision: Accept (poster)