Edge-aware Image Smoothing with Relative Wavelet Domain Representation
Authors: Huiqing QI
Xiaoliu Luo
Tingting Li
Fang Li
Venue: ICLR 2025 Poster (2025)
Keywords: Image smoothing
Wavelet transformation
Relative wavelet domain representation
Edge-preserving
Non-convex optimization
Primary area: optimization

Abstract: Image smoothing is a fundamental technique in image processing, designed to eliminate perturbations and textures while preserving dominant structures. It plays a pivotal role in numerous high-level computer vision tasks. More recently, both traditional and deep learning-based smoothing methods have been developed. However, existing algorithms frequently encounter issues such as gradient reversals and halo artifacts. Furthermore, the smoothing strength of deep learning-based models, once trained, cannot be adjusted for adapting different complexity levels of textures.  These limitations stem from the inability of previous approaches to achieve an optimal balance between smoothing intensity and edge preservation. Consequently, image smoothing while maintaining edge integrity remains a significant challenge. To address these challenges, we propose a novel edge-aware smoothing model that leverages a relative wavelet domain representation. Specifically, by employing wavelet transformation, we introduce a new measure, termed Relative Wavelet Domain Representation (RWDR), which effectively distinguishes between textures and structures. Additionally, we present an innovative edge-aware scale map that is incorporated into the adaptive bilateral filter, facilitating mutual guidance in the smoothing process. This paper provides complete theoretical derivations for solving the proposed non-convex optimization model. Extensive experiments substantiate that our method has a competitive superiority with previous algorithms in edge-preserving and artifact removal. Visual and numerical comparisons further validate the effectiveness and efficiency of our approach in several applications of image smoothing.

Published as a conference paper at ICLR 2025
EDGE-AWARE IMAGE SMOOTHING WITH RELATIVE WAVELET DOMAIN REPRESENTATION
Huiqing Qi
Xiaoliu Luo
School Electronic & Electrical Engineering Nanyang Technological University huiqing. q @ntuenuuuu.mmm edu.
College of Science
Chongqing University of Technology
luoxiaol wwt.iuu..dcuieecc. .dw.uut.edd...omm
Tingting Li
Fang Li
School Mathematics & Information Science Zhengzhou University of Light Industry 522055000068stu. ecnu. edu.cn on
School of Mathematical Sciences, Key Laboratory of MEA(Ministry of Education)
Shanghai Key Laboratory of PMMP
East China Normal University fl dei0mmhhh... ecnu. edu. cn
ABSTRACT
Image smoothing is fundamental technique in image processing, designed eliminate perturbations and textures while preserving dominant structures. It plays a pivotal role in numerous high-level computer vision tasks. More cently, both traditional and deep learning-based smoothing methods have been developed. However, existing algorithms frequently encounter issues such gra- dient reversals and halo artifacts. Furthernore, the smoothing strength of deep learning- based models, once trained, cannot be adjusted for adapting different complexity levels of textures. These limitations stem from the inability of previ- ous approaches to achieve an optimal balance between smoothing intensity and edge preservation. Consequently, image smoothing while maintaining edge in- tegrity remains a significant challenge. novel edge-aware smoothing model that leverages relative wavelet domain rep- resentation. Specifically, by employing wavelet transformation, we introduce a measure, termed Relative Wavelet Domain Representation (RWDR), which effectively distinguishes between textures and structures. Additionally, we present innovative edge-aware scale map that is incorporated into the adaptive bilateral filter, facilitating mutual guidance in the smoothing process. This paper provides complete thcoretical derivations for solving the proposed non-convex optimiza- tion model. Extensive experiments substantiate that our method has a competitive superiority with previous algorithms in edge-preserving and artifact removal. Vi- sual and numerical comparisons further validate the effectiveness and efficiency of our approach several applications of image smoothing.
address these challenges, we propose a
INTRODUCTION
Edge-preserving smoothing (EPS) represents a fundamental challenge computer vision Luo al. (2024b) 2025 2024a) and computational graphics. The objective of image smoothing is to eliminate minor perturbations and non-essential textures while maintaining prominent edges and structural features Qi et al. (2024a). EPS has garnered significant attention due to its wide range of applica- tions, including image decomposition Song et al. (2017); |: Liu et al. (2019), image texture removal Xu et al. (2012), high dynamic range (HDR) tone mapping Anand Swamy & Shylashree/ (2023), detail enhancement Xu et I (0022b), texture transfer Gupta et al. (2022), and clip-art compression artifacts removal Li al. (0024). Over the past few decades, numerous image smoothing algorithms have been developed, which can be broadly classified into local information-
methods, global approaches. However, existing methods often face limitations related to edge preservation and the reduction of visual artifacts such as halos He al. (2012) and gradient reversal Gastal & Oliveira (2012). as illustrated in Figure To address
information-bain methods, and deep learning-basse
"Corresponding Author

---

Published as a conference paper at ICLR 2025
these challenges, Liu et al. (0188) has explored embedding bilateral filters within least squares mod- els. More recently, the incorporation of soft clustering into bilateral filtering has been introduced Yang et al. (2021). Additionally, studies Li al. (2023); He et al. (2022) have combined relative total variation with bilateral filters.
Drawing inspiration from multi-model embedding methods, we propose novel mutual guidance EPS model. Specifically, we first derive a guidance image by solving a non-convex function, which leverages newly introduced relative wavelet domain representation. We then integrate the proposed edge-aware scale map of the guidance image into a bilateral filter. This edge-awareness technique enhances the algorithm's 's ability to preserve edges effectively. In a nutshell, the primary contri- butions of this work are as follows: (1) We introduce novel global information-bast relative measure with wavelet transformation, termed Relative Wavelet Domain Representation (RWDR), which more effectively distinguishes textures from primary structures and preserves weaker edges. (2) We integrate the proposed edge-aware scale map into an adaptive bilateral filter for mutual guid- ance in smoothing, which significantly reduces gradient reversal and halo artifacts.
(3) Extensive
experimental results demonstrate that our model outperforms competing algorithms both visual and numerical evaluations across range of smoothing applications. This indicates that our proposed model achieves an improved balance between smoothing strength and edge preservation.
(a) Input
(b) RTV
Ghosh
(d) Tight frame
(c) Ours
Figure I: Image detail enhancement. The left-bottom part of each plot presents the smoothed image, and the right-top part shows the detail enhancement image that is boosted by four detail layers. The detail layer is extracted by subtracting the smoothed image from the input image. Results of (b), (c), and (d) suffer halos and grandient reversal artifacts.
All theoretical derivations and more EPS application experiments that have been omitted for space appear in the supplementary material.
RELATED WORK
Local information-based models. Methods of this category take the weighted average of neighbor- ing pixels' values in a local window as an output. These approaches are known as filter-based meth- ods, which include bilateral filter (BF) Tomasi & Manduchil (1998), joint bilateral filter Petschnigg et al. (2004) and rolling guidance filter Zhang et al. (2014a). There are also other filter-based meth- ods such as weighted median filter Ma et al. (2013); Zhang et al. (2014b), guided filterHe et al. (2012), tree filter Bao et al. (2013), recursive filter Tu & Chien (2021) 1). and mutually guided filtering Guo et al. (2017). The fixed size a local window at the central pixel limits the performance of lo- cal information-
methods. Therefore, several scale-adaptive bilateral filters Xu et al. (002a)); Ruhela et al. (2022): Ghosh et al. (2019): [ong et al. (2019) and structure/scale- aware filtering meth- ods Kaplan& & Erer Gupta al. 2022 have been proposed improve the performance. However, they cause halo artifacts He et al. (2012) and gradient reversals Gastal & Oliveira (2012) in EPS application tasks.
2

---

Published as a conference paper at ICLR 2025
Global informmttion-assee models. Algorithm in this category solve the image smoothing problem via finding a global solution to the specific objective function. The well-known approaches are the total variation based methods, which include weighted least squares smoothing Min et al. (2014).
gradient [1 norm smoothing Pang et al. (2015), and gradient Lo norm smoothing Xu et al.
2011
Many other global information-basee approaches have recently been proposed, including the re lative total variation (RTV) smoothing/Xu et al. (2012) and its variants Liu et al. (2016): Yu et al. (2022) Li
al. (2023) Qi al. (2022). Methods built on RTV are sensitive to edges aligned with horizontal and vertical directions. Directional relative total variation (DRTV) smoothing Zhou et al. (2019) has been proposed overcome this drawback. Generally, global information -aased approaches are based on solving extensive linear system to the global optimization objective function, which is high computational and time cost. Global information-bart methods have tradeoff between time cost and smoothing performance.
Deep learning-based models. These methods are built on different deep convolutional neural net- works (DCNN) for smoothing Zhu et al. (201)). These methods are end-to-end smoothing frame- works Lu el al. (2018); Pan et al. (2018). The main drawback of the end-to-end based methods is that obtaining the paired training data is challenging. Therefore, unsupervised learning smoothing Fan et al.] (2018) has been proposed to overcome the lack of paired training data. Some other deep learning methods for smoothing include filter operator approximation methods Chen et al. (207)))))) et al. (2019) via deep neural networks. Generally, methods in this category also suffer from artifacts.
BACKGROUND
3.1 ADAPTIVE BILATERAL FILTER
The adaptive bilateral filter (ABF) proposed by Ghosh et al.) (2019) builds upon the traditional bi- lateral filter (BF) Tomasi & Manduchi (1998) for image smoothing. Unlike the BF, which utilizes a fixed spatial kernel, the ABF employs a box function where the spatial kernel's scale varies each pixel. Ghosh et al. (2019) emphasize that the local scale of the range kernel plays crucial role in preserving texture details. As a result, they introduced scale map applied to each pixel in the filter. Given input image I, the output image S in ABF is expressed as:
Sp EgER,ED) Go. (1) 11))
EgERg Go, (1 T
where Go,, denotes a Gaussian range kernel with a standard deviation o,- Stated simply, the aggre- gation pixel (I,y) is perfored over local window (p) centered at p, and
(p) (x + ki y + k2). up S ki, h/2 S ups
where wpp denotes the scale at pixel which is from the scale map.
The ABF has a fast approximation version that can achieve (1) complexity in Ghosh et al. (2019) However, it produces results with gradient reversals and halos in the image detail enhancement. The gradient reversals are caused by the sharpened edges in the smoothed output, while halos are typically caused by large 01 and 02 values, it can be observed in Figure cc).
3.2 WAVELET TRANSFORM TIGHT FRAME
The tight frame is an important concept in wavelet transformation, widely used in image processing. For more details, refer to Shen (2010); Dong et al. (2010). Firstly, we present the definition of tight frame. H. be a Hilbert space. A sequence In C. H can be considered a tight frame for H
| = Σ I. for any E H.
(3)
There are two associated operators, which the analysis operator and the synthesis operator. And they are expressed as
W : r H - (0, E (N),, WT an, (2(N) Ea,,, H.
(4)
Then, Fn is tight frame if and only if WTW I, where I: H. > H is the identical operator. denotes the natural number set. The discrete tight frame is widely used based on the theory of tight frames. Cai et al. (2014) used undecimal tight frame in the image restoration. Given filter
3

---

Published as a conference paper at ICLR 2025
E (Z), the linear convolution operator C. (2) - (2( as
(Cau](n) = (a * (n)) Σ a(n - k))(()) Ve E (Z)).
(5)
kEZ
Given a set of filters {ar)== C (Z)),, the corresponding analysis operator and synthesis operator written follows
(-.)) Coz(-))) (-)(", W = [Co1 Cag. ... ,
(6)
The rows of W forms tight frame for (Z) if and only if WTW = I. Z is the rational number set. When constructing tight frames, the unitary extension principle proposed a full constraint condition Ron & Shen (1997). One of typically used full unitary extension principle conditions Ron & Shen (197)) is expressed
Σ Σ i=1 nEZz
a, + ());(n) = on. for any k E z².
With such full unitary extension principle condition, the well-known B-spline tight frame filters are used in many image applications Chai & Shen (2007); Cai et al. (2008 2009). These filters written as follows
1 a1 = 1==211,21) 02 (1,0, -1) 03 (-1,2,
V2
(8) The listed filters satisfy the full condition Equation 7 For the image smoothing task, we only need the 2D framelet filters fa, 8 aa,) that is generated via the tensor product X of filters in Equation 8 The tight frame is sensitive to the weak edges that locate along diagonal directions. However, the tight frame also causes halos in the output, as shown Figure [d). In summary, the tight frame is specific class in wavelet transformation.
METHODOLOGY
4.1 RELATIVE WAVELET DOMAIN REPRESENTATION
Given image L, the non-convex smoothing problem with relative wavelet representation is ex- pressed as:
D(p) W()) +
arg min
ESs 1p)) +
(9)
where À denotes a weight value that controls the smoothing strength. E is a positive small constant, D(p)
which avoids division by zero. W(D)+ is the proposed relative wavelet domain representation. Dpp) and W (p) are written as:
K
D(p)
Σ Un-a pp))
(wa))ab w NE ((P))
Σ
Σ
gy-g (wx S)
(10)
k=1
k=1
(xpp Fgg )2 (Up Vgy
9p.4 X exp
(11)
202
where g belongs R(p) that is rectangle region centred at pixel p. (x., y.) denote the pixel index. wa denotes tight frame filters of wavelet transformtation in Equation 8 K denotes the number of tight frame filters. The smoothing performance of the proposed model is highly dependent on the RWDR, which effectively distinguishes between main structures and textures. The RWDR serves as regularization term, assigning larger values to texture regions and smaller values the structural areas.
We compare Dpp) + e) with other regularization term maps in Figure The left-up region of each image denotes the corresponding regularization term map, while the right-bottom region of each image presents the corresponding final texture removal result. Obviously, Figure 2 shows RoG Cai et al. (2017), imRTV Yu et al. (2022). and RTV methods tend to divide some textures into edges. The dRTV blurred these junction regions between textures and structures and also loses some main structures. In contrast, the map of our RWDR clearly identifies the structures and the textures. Meanwhile, comparing these final smoothed results in Figure our approach provides a better pregerinss smoothed image than other methods.
4

---

Published as a conference paper at ICLR 2025
Based on Equation Equation 11 obtain the intermediate smoothed image via the following optimization objective function, expressed as
(s, - 1p)² + A EnERGh) Fp.q Σ 1(vxs)yll S
min
(12)
Ek=1 I RpE Inna (wg +
The soluion Equation conjugate gradient (PCG) via iteraive mode. More details are in Appednix A. The global solution Equation is the inermediate smoothed result S", used the following mutually guided adaptive bilateral filter.
be obtained by solving linear system with the preconditioned
(a) Input
(b) RoG
(c) dRTV
(d) imRTV
(c)
(f) Ours
Figure 2: Relative wavelet domain representation comparison. Results of (b) RoG, (c) dRTV, (d)
imRTV, (e) RTV, (f) Ours. The left-up part of each image shows the relative feature map, and the right-bottom part is the corresponding smoothed image.
4.2 MUTUALLY GUIDED ADAPTIVE BILATERAL FILTER
Drawing inspiration from muGIF Guo between the proposed RWDR and adaptive bilateral filter. It means that the outputs of RWDR and adaptive bilateral filters act as cross inputs simultaneously. Therefore, the proposed mutually guided adaptive bilateral filter is built on Equation [ expressed as:
2017). we use the mechanism of mutual guidance
FT+1 EmEN, Go,, (1S$) 1)
(13)
EgEN, Go. (15) St
where t denotes the iterative number. (p) is local window that centered at p (i.j). We also use Equation obtain the scale map. expressed
Ny(p) = {(i + kı,j + k2) wp S ki, K2 S wp).
(14)
important note the subtle distinction between (p and Rp(p) in Equation The term (p) is dynamic and evolves with each iteration. The subsequent scale map is dependent on the previously smoothed image; thus, the previous iteration produces a poorly smoothed image, the subsequent iteration is likely to exacerbate the eтror, often resulting in the blurring of weak edges. However, we propose a novel edge-aware approach derive the scale map, effectively mitigat- ing the issue of blurred weak edges and enhancing edge-preserving capabilities. Additionally, the Fourier approximation of the mutually guided adaptive bilateral filter achieves O(1) computational complexity, demonstrated in Appendix B.
4.3 EDGE-AWARE SCALE MAP
The scale map plays a critical role in Equation 14 Ghosh et al. (2019) introduce a method for deriving the scale map through the erosion of the gradient map. In contrast, propose novel

---

Published as a conference paper at ICLR 2025
(a) Input
(b) Edge
(c) E
(d)
(c) Smoothed output
Figure 3: Illustration an edge-aware map. (b) Edges, (c) E map. Notice that я map (d) covers all
edges in (b). acts edge-aware map that improve edge-preserving ability, shown in (e).
edge-aware method for computing the scale map. Given an input image I, the intermediate feature map E is defined as:
E'
Σ Gaa Vell. (ENs(P)
(15)
Go. denotes Gaussian range kernel. Ng(p) 8-neighborhood that centered at pixle p. V; is the gradient operator. The E map facilitates the extraction of key structural information, illus- trated FigureBlc) Subsequently, we define the general gradient magnitude as follows:
gh / (V.))² (V,1)² i {1, 4). j= -
(16)
Oh can capture the local edge feature information. Then, coefficient based on can be obtained by
(17)
A=1 9h exp(gn/>")+
is constant. is positive value, which avoids division by zero. provides small value in the structure regions while providing large value in the texture ones. acts as weight map obtain the edge-aware feature map which is expressed as
(18) We show π map in Figure dj. As can be observed in Figure B()),, it is ground truth edges. Obivi- ously, æ provides small values texture regions while large values are structure regions. We use such a map to obtain the scale map, which can help to improve the edge-pres preserving ability in the smoothed output. The scale wp map should be designed to satisfy that small along sharp edges and large in smooth regions. Then, Wip expressed as
π E(1 m).
Wp Tmin (Tmaz Tmin) exp( -
(19)
Ae a positive value to control the transition rate from Tmin to mmaz- We constrain wp in an interval ['min= Tmaz]- Wp is rounded off to be the final edge-aware scale map. The effects of scale maps obtained by using Equation 19 and Ghosh model are shown in Figure 4
(a) Input
(b) in Ghosh
(c) S in Ghosh
Ours
(e) Ours S
Figure 4: Adaptive scale maps (Tmaz 15). (a) The input image, (b) w in Ghosh model, (c) smoothed image corresponding (b), (d) scale map w, (e) smoothed image of our model. Note that w is able to maintain more edge information than that of w in (b). The more edge information in w, the more edges and details can preserved. Refer to the enlarged areas of (c) and (e).
4.4 ALGORITII
For the sake of completeness, we briefly present our final model. which is built upon the RWDR and the mutually guided edge-aware adaptive bilateral filter. The proposed method is formulated as

---

Published as a conference paper at ICLR 2025
follows:
Egemps pn Ets (wos")) Et= Enerip In
St = arg ging' - + - 22
(20)
EgeN,in Go, -
Ft+1
(g), Go, -1;1:-
When / is assigned at I, F is initialized to the input image I. F and S update each other via mutually guided. We draft the scheme of the mutually guided edge-aware RWDR smoothing model in Algorithm]
Algorithm Mutually Guided Edge-aware RWDR
Input: Image I, Te. ",a Ae, Tminn Tmazs 0, E, X, K, ai Cr.
Initialization: p0 = I.
1: Calculate E' via Equation
15
2: Calculate m via Equation
3: Calculate ıT with m and E via Equation 18
4: Calculate wp with
Equation 19
5: for 1= to T do
6: Update Sa with pt-1 via Equation
7: Update F*+1 with S and up via Equation 13
8: end for
Output: Smoothed image S.
EXPERIMENTS
To assess the performance and effectiveness of our approach, we conduct a comparative analysis against several existing methods, including BF Tomasi & Manduchi (1998). LO Xu al. WLS Min et al. (2014), RGF Zhang et al. 22)14)). ILS et al. (2020). EP/SP Liu et al.
(201
2021
dRTV Jeon et al. ) D al. (2019), imRTV Yu al. (2022), RoG Cai et
2017
muGIF Guo et al. (2017). RTV et al. (012)) Ghosh Ghosh et al. (2019), Resnet Lim et al.
2017
and VDCNN Kim et al. (2016). We utilize five no-reference image quality objective evaluation metrics: BRISQUE Mittal et al. 2012). PIQE Venkatanath et al. (015), SSEQ Liu et al. (2014),
| al. (2015). and CEIQ Yan al. (219) to compare performance on image detail enhancement and HDR tone mapping tasks. The smaller values of BRISQUE and PIQE denote higher-quality images. While the larger values of SEQ, ILNIQUE, and CEIQ indicate higher-quality images. Mcanwhile, CEIQ is specifically designed to measure the effects of the HDR tone mapping task. We utilized the original code and recommended parameters provided by the respective authors for comparison. For all the evaluated approaches, the parameters were fine-tuned to achieve an optimal balance between smoothing performance and edge preservation. Our code will be available on my github)
Texture removal.
We evaluate the texture removal performance of our model in comparison to other
methods, illustrated in Figure 5 Notably, the enlarged regions (b), (c), (d), (D, of Figures
5 reveal instances of incomplete texture removal. Additionally, the enlarged yellow rectangles in Tigures e) and (h) indicate that the detailed edges are blurred. Furthermore, Figures 5[g). (i)
(k), and (m) demonstrate suboptimal performance in edge preservation. Although Figures 5( and (n) exhibit competitive bnge-prrseriing capabilities, Figure(5]nn introduces halo artifacts along the edges, as indicated by the enlarged magenta region. In contrast, our proposed method achieves superior balance between texture smoothing and edge preservation compared to other approaches.
Image detail enhancement. It aims to enhance high-frequency regions by incorporating detail layer into the input image. The core this technology involves extracting the high-frequency detail layer by subtracting the smoothed image from the original input. Consequently, the effectiveness of detail enhancement is closely linked to the quality of the smoothed image, particularly with respect to prominent edges. Two extreme scenarios can arise: artifacts in the echail-enhanced image, and another where sharp edges result in gradient reversals in the enhanced output. In this study, enhance the details by adding four detail layers to the
where blurred edges introduce halo

---

Published as a conference paper at ICLR 2025
(a) Input
BF
(c)
(d) WLS
(c) RGF
( ILS
EP/SP
(h) dRTV
(i) DRTV
imRTV
(k) RoG
(I) muGIF
(m) RTV
(n) Ghosh
(o) Ours
Figure 5: (a) The input image. textures are removed by (b) BF, (c) LO, (d) WLS, (e) RGF, (f) ILS, (g) EP/SP. (h) dRTV, (i) DRTV, (j) imRTV, (k) RoG, (I) muGIF, (m) RTV, (n) Ghosh, (o) Ours. Refer these highlighted boxes, our method can get better tradeoff between the smoothing strength and the edge-preserving than other algorithms.
original image. We present the smoothed images alongside their corresponding detail-enhanced versions in Figure 6 In the enlarged magenta and green regions of each image, Figures 6]b) (f) exhibit halo artifacts along the edges. Figure ((g) displays a slight halo effect within the green enlarged rectangle. Notably, our model demonstrates significant reduction in the occurrence of halo artifacts. Additionally, as indicated by the enlarged rectangle in the lower-left corner of each image, our model effectively preserves weak edges. We present corresponding numerical results in Table [ One can see that our mode obtains the best index values.
Table 1: No-reference image quality evaluation metrics of Figure 6
Methods
BRISQUE,
PIQE
SSEQ 1
ILNIQUE
LO
10.6974
35.6176
24.7566
117.81
ILS
23.5255
37.0118
17.7950
122.43
DRTV
8.5623
33.5745
25.1130
124.46
RoG
14.9437
34.6882
24.5738
124.32
muGIF
20.6328
36.6683
25.1297
121.44
dRTV
15,4301
25.1630
9.9543
124.49
Ours
7.6912
24.9118
26.7940
128.84
Compression artifact removal. The technology for removing compression artifacts is employed to mitigate blocking artifacts in compressed clip-art images. When an image compressed at a low bit rate using standard JPEG encoding, it may result in compression artifacts along sharp edges and staircase artifacts in homogeneous regions.Il In this study, we employ a 10% compression rate for the clip-art images. Figure 7 iillustrates the smoothed results obtained from various methods. Notably, the upper right part of Figure (a)) displays the original clip-art image, while the lower left part shows the compressed version. Observations indicate that Figures 7b). (c), and (d) exhibit blurred edges
8

---

Published as a conference paper at ICLR 2025
(a) Input
(b) LO
Ic) ILS
(d) DRTV
(e) RoG
(I) muGIF
(g) dRTv
(h) Ours
Figure 6: Image detail enhancement. (a) The input image. is enhanced with the detail layer 4x boosted by (b) LO, (c) ILS, (d) DRTV, (e) RoG, (f) muGIF, (g) dRTV, (h) Ours. The cyan highlighted box denotes the details the smoothed image, magenta and green boxes are the details of the enhanced image. In these highlighted rectangles,
method can reduce halo artifacts. In addition, performs better in edge-preserving than other state-of-the-ar algorithms.
in the smoothed images, highlighted in the right box of each image. Additionally, Figures ( and (1)) present staircase artifacts near the edges. Furthermore, Figure (g) displays colorful halos along the edges, as indicated by the left red-highlighted box. In contrast, our approach effectively eliminates staircase artifacts and halos while demonstrating strong edge-preserving capabilities. We also employ two widely recognized quality metrics, PSNR and SSIM, to assess the images result- ing from compression artifact removal, as presented in Table Higher values of PSNR and SSIM indicate superior quality in the smoothed images. Our proposed model demonstrates the best per- formance, achieving the highest PSNR value of 27.10 and an SSIM value of 0.991. In summary, both numerical and visual comparisons indicate that the proposed algorithm outperforms others in the removal of compression artifacts.
Table 2: PSNR(dB) and SSIM comparison.
Metric WLS RoG RGF muGIF EP/SP imRTV
BF
Ghosh
LO
ILS
DRTV Ours
PSNR 20.55 26.43 26.21 25.97
26.68
26.80
21.44
20.51 25.91
21.67
25.80
27.10
SSIM 0.937 0.978 0.974
0.977 0.980
0.979
0.924
0.894
0.976 0.909 0.975
0.991
Ablation Study. To assess the capability of RWDR in distinguishing between textures and struc- tures, conduct an ablation study on RWDR in Figure g The model deployed without RWDR has mistreated texture as structure, leading to removing texture uncleanly. In contrast, To assess the capability of the edge-aware scale map in edge preservation, we conduct an ablation study on the edge-aware aware scale map in Figure [ The model deployed without the edge-awang aware scale map has smoothed textures cleanly while making main structures and edges lost and blurred. As observed in the enlarged red- and blue-highlight areas, our model achieves the best visual effects.
CONCLUSIONS AND LIMITATIONS
This study introduces mutually guided edge-aware smoothing model based on relative wavelet do- main representation. The RWDR serves novel measure for effectively differentiating between textures and structures. The proposed edge-aware scale map is integrated into an adaptive bilateral filter, providing mutual guidance the RWDR during the smoothing process. The solution of the

---

Published as a conference paper at ICLR 2025
(a) Input
(b) imRTV
(c) WLS
(d) DRTV
(e) EP/SP
(1) muGIF
(g) Ghosh
(h) Ours
Figure 7: Compresion artifact removal results. The left-bottom part of (a) is the compressed JEPG image, and the right-top part is the original image. Results are obtained by (b) imRTV, (c) WLS, (d) DRTV, (e) EP/SP, (f) muGIF, (g) Ghosh, (h) Ours. Referring to the marked boxes, results of (b), (c), and (d) suffer from blurred edges. Results of (e), (f), and (g) have staircase artifacts.
(a) Input
(b) GT
(c) w/o
(d) w/ (Ours)
Figure Visual effects the ablation experiment on the RWDR.
(a) Input
(b) GT
(c) w/o
(d) w/ (Ours)
Figure 9; Visual comparison of the ablation experiment on the edge-aware scale map.
proposed model is supported by a complete theoretical guarantee. Extensive comparisons demon- strate that the proposed method outperforms existing algorithms in mitigating gradient reversals, staircase artifacts, and halos. Whether compared to traditional image smoothing techniques or deep learning-based approaches, our method consistently achieves a superior balance between smoothing strength and edge preservation.
Limitations. Traditional smoothing methods, including our model, are limited in their ability to address long-range texture dependencies. This implies that our model has constraints when dealing with irregular multiscale textures. A promising direction for future work would be to integrate the proposed algorithm into a neural network, such as Vision Mamba Zhu et al. (2024). for enhanced image smoothing. Exploring the potential of combining traditional models with neural networks could provide valuable insights into achieving more effective texture smoothing while maintaining superior edge preservation.
10

---

Published as a conference paper at ICLR 2025
ACKNOWLEDGEMENT
This work is supported by Natural Science Foundation of Shanghai (No. 222R14195000, Science and Technology Commission of Shanghai Municipality (No. 22DZ2229014), Fundamental Re- search Funds for the Central Universities. This paper is also supported by Chongqing University of Technology Research Startup Funding (2023ZDZ032); Science and Technology Research Pro- gram of Chongqing Municipal Education Commission (KIQN202401 138); Science and Technology Innovation Key R&D Program of Chongqing (CSTBB024TAAD-SXX002)
REFERENCES
AS Anand Swamy and Shylashree. Multiscale decomposition of hdr images using the edge- preserving filters. In ICMCMLT. loI., pp. 573-600, 2023.
Linchao Bao, Yibing Song, Qingxiong Yang, Hao Yuan, and Gang Wang. Tree filtering: Efficient structure-preserving smoothing with a minimum spanning tree. IEEE Trans. Image Process., 23 (2):555-56,, 2013.
Bolun Cai, Xiaofen Xing, and Xiangmin Xu. Edge/structure preserving smoothing via relativity-of- gaussian. In IEEE Int. Conf. Image Process., pp. 250-254, 2017.
Jian-Feng Cai, Raymond H Chan, and Zuowei Shen. A framelet-based image inpainting algorithm. Appl. Comput. Harmon. Anal., 24(2):131-149 2008.
Jian-Feng Cai, Hui Ji, Chaoqiang Liu, and Zuowei Shen. Blind motion deblurring from a single image using sparse approximation. In Proc. IEEE Conf. Comput. Vis. Pattern Recognit, pp. 104- 111, 2009.
Jian-Feng Cai, Hui Ji, Zuowei Shen, and Gui-Bo Yc. Data-driven tight frame construction and image denoising. Appl. Comput. Harmon. Anal. 37 2014.
Anwei Chai and Zuowei Shen. Deconvolution: A wavelet frame approach. Numer: Math., 106(4): 529-587, 2007.
Qifeng Chen, Jia Xu, and Vladlen Koltun. Fast image processing with fully-convolutional networks. In IEEE Int. Conf. Comput. Vis., pp. 2497-2506, 2017.
Bin Dong, Zuowei Shen, al. Mra-based wavelet frames and applications. IAS Lect. Notes Ser: Sum. Program Math. Image Process. Park City Math. Inst., 19:9-158, 2010.
Qingnan Fan, Jiaolong Yang, David Wipf, Baoquan Chen, and Xin Tong. Image smoothing via unsupervised learning. ACM Trans. Graph., 37(6):1-14, 2018.
Eduardo SL. Gastal and Manuel M Oliveira. Adaptive manifolds for real-time high-dimensional filtering. ACM Trans. Graph., 31(4):1-13, 2012.
Sanjay Ghosh, Ruturaj G Gavaskar, Debasisha Panda, and Kunal N Chaudhury. Fast scale-adaptive bilateral texture smoothing. IEEE Trans. Circuits Syst. Video Technol., 30(7):001--2266, 2019.
Xiaojie Guo, Yu Li, and Jiayi Ma. Mutually guided image filtering. In Proc. ACM Int. Conf. Multimed., pp. 1283-1290, 2017.
Bhupendra Gupta, Subir Singh Lamba, et al. Structure-aware adaptive bilateral texture filtering. Digit. Signal Process., 103:10 2022.
Kaiming He, Jian Sun, and Xiaoou Tang. Guided image filtering, IEEE Trans. Pattern Anal. Mach. Intell., 3576):1977-1099, 2012.
Lei He, Yongfang Xie, Shiwen Xie, and Zhipeng Chen. Strueture-rreserving texture smoothing via scale-aware bilateral total variation. IEEE Trans. Circuits Syst. Video Technol., 2022.
Junho Jeon, Hyunjoon Lee, Henry Kang, and Seungyong Lee. Scale-aware Lirueture-preserving texture filtering. In Comput. Graph. Forum., volume 35, pp. 77-86, 2016.
11

---

Published as a conference paper at ICLR 2025
Nur Huseyin Kaplan and Erer. Scale aware remote sensing image enhancement using rolling guidance. J. Vis. Commun. Image Represent., 80:103315.
Jiwon Kim, Jung Kwon Lee, and Kyoung Mu Lee. Accurate image super-resolution using very deep convolutional networks. Proc. IEEE Conf. Comput. Vis. Puttern Recognit., pp. 1646- 1654, 2016.
Jun Li, Yuxuan Han, Yin Gao, Qiming Li, and Sumei Wang. An enhance relative total variation with bf model for edge-preserving image smoothing. IEEE Trans. Circuits Syst. Video Technol. 2023.
Tingting Li, Fang Li, and Huiqing Qi. A weibull gradient prior for image restoration. Journal of Computational and Applied Mathematics, 439: 15594, 2024.
Yijun Li, Jia-Bin Huang, Narendra Ahuja, and Ming-Hsuan Yang. Joint image filtering with deep convolutional networks. IEEE Trans. Puttern Anal. Mach. Intell., 1999 2019.
Bee Lim, Sanghyun Son, Heewon Kim, Seungjun Nah, and Kyoung Mu Lee, Enhanced deep resid- ual networks for single image super-resolution. In Proc. IEEE Conf. Comput. Vis. Pattern Recog- nit., pp. 136-144, 2017.
Lixiong Liu, Bao Liu, Hua Huang, and Alan Conrad Bovik. No-reference image quality assessment based on spatial and spectral entropies. Signal processing: Image communication, 29(8):856-833, 2014.
Qiegen Liu, Biao Xiong, Dingcheng Yang, and Minghui Zhang. A generalized relative total variation method for image smoothing. Multimed. Tools. Appl., 75:7909-7930. 2016.
Wei Liu, Pingping Zhang, Xiaogang Chen, Chunhua Shen, Xiaolin Huang, and Jie Yang. Embedding bilateral filter in least squares for efficient edge-preserving image smoothing. IEEE Trans. Circuits Syst. Video Technol. 30(1):23-35, 2018
Wei Liu, Pingping Zhang, Xiaolin Huang, Jie Yang, Chunhua Shen, and lan Reid. Real-time image smoothing via iterative least squares. ACM Trans. Graph., 39(3):1-24 2020.
Wei Liu, Pingping Zhang, Yinjie Lei, Xiaolin Huang, Jie Yang, and Michael Ng. A generalized framework for edge-preserving and structure- pressevin image smoothing. IEEE Trans. Pattern Anal. Mach. Intell., 44 (4(10):6631-6666 2021.
Xianming Liu, Deming Zhai, Yuanchao Bai, Xiangyang Ji, and Wen Gao. Contrast enhancement via dual graph total variation-based image decomposition. IEEE Trans. Circuits Syst. Video Technol., 30(8):2463-2476 2019.
Kaiyue Lu, Shaodi You, and Nick Barnes. Deep texture and structure aware filtering network for image smoothing. In Eur. Conf. Comput. Vis., pp. 217-233, 2018
Xiaoliu Luo, Zhao Duan, Anyong Qin, Zhuotao Tian, Ting Xie, Taiping Zhang, and Yuan Yan Tang. Laser-wise mutual information meta-learning network for few-shot segmentation. IEEE Transactions Neural Networks and Learning Systems, 2024a.
Xiaoliu Luo, Zhuotao Tian, Taiping Zhang, Bei Yu, Yuan Yan Tang, and Jiaya Jia. Pfenet++: Boost- ing seo- semantic segmentation with the noise-filtered context-aware prior mask. IEEE Trans- actions on Pattern Analysis & Machine Intelligence, 46(02):1273-1288) 2024b.
Xiaoliu Luo, Ting Xie, Weisen Qin, Zhao Duan, Jin Tan, and Taiping Zhang. Combining hierarchi- sparse representation with adaptive prompt for few-shot segmentation. Expert Systems with Applications, 260: 125377, 2025.
Yi-Si Luo, Xi-Le Zhao, Tai-Xiang Jiang, Yu-Bang Zheng, and Yi Chang. Hyperspectral mixed noise removal via spatial-spectral constrained unsupervised deep image prior. IEEE J. Sel. Top. Appl. Earth Obs Remote Sens., 14:9435-9442 99449 2021.
Ziyang Ma, Kaiming He, Yichen Wei, Jian Sun, and Enhua Wu. Constant time weighted median filtering for stereo matching and beyond. In IEEE Int. Conf. Comput. Vis., pp. 49-56, 2013.
12

---

Published as a conference paper at ICLR 2025
Dongbo Min, Sunghwan Choi, Jiangbo Lu, Bumsub Ham, Kwanghoon Sohn, and Minh N Do. Fast global image smoothing based on weighted least squares. IEEE Trans. Image Process., 23(12): 5638-5653. 2014.
Anish Mittal, Anush Krishna Moorthy, and Alan Conrad Bovik. No-reference image quality assess- ment in the spatial domain. 21(12):4695-44 -4708, 2012.
Jinshan Pan, Sifei Liu, Deqing Sun, Jiawei Zhang, Yang Liu, Jimmy Ren, Zechao Li, Jinhui Tang, Huchuan Lu, Yu-Wing Tai, et al. Learning dual convolutional neural networks for low-level vision. In Proc. IEEE Conf. Comput. Vis. Pattern Recognit., pp. 3070-3079, 2018.
Xueshun Pang, Suqi Zhang, Junhua Gu, Lingling Li, Boying Liu, and Huaibin Wang. Improved 10 gradient minimization with 11 fidelity for image smoothing. PLoS One., 10(9):1-10, 2015.
Georg Petschnigg, Richard Szeliski, Mancesh Agrawala, Michael Cohen, Hugues Hoppe, and Ken- taro Toyama. Digital photography with flash and no-flash image pairs. ACM Trans. Graph., 23 (3):664-672) 2004.
Huiqing Qi. Shengli Tan, and Zhichao Li. Anisotropic weighted total variation feature fusion net- work for remote sensing image denoising. Remote Sensing, 14(24):6300, 2022.
Huiqing Qi, Fang Li, Peng Chen, Shengli Tan, Xiaoliu Luo, and Ting Xie. Edge-preserving image restoration based on weighted anisotropic diffusion model. Pattern Recognition Letters, 184: 80-88, 2024a. ISSN 0167-8655.
Huiqing Qi, Shengli Tan, and Xiaoliu Luo. Self-supervised dual generative networks for edge- preserving image smoothing. IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 7215-7219, 20244.
Amos Ron and Zuowei Shen. Affine systems inl2 (rd): the analysis of the analysis operator. J. Funct. Anal., 148(2):408-447, 1997.
Riya Ruhela, Bhupendra Gupta, and Subir Singh Lamba. An efficient approach for texture smooth- ing by adaptive joint bilateral filtering. Vis. Comput. pp. 1-15, 2022.
Zuowei Shen. Wavelet frames and image restorations. In Proc. Int. Congr. Math., pp. 2834-2863, 2010.
Chengfang Song, Chunxia Xiao, Ling Lei, and Haigang Sui. Scale-adaptive structure-preserving texture filtering. In Comput. Graph. Forum., volume 38, pp. 149-158, 2019.
Jinjoo Song, Heeryon Cho, Jungho Yoon, and Sang Min Yoon. Structure adaptive total variation minimization-basse image decomposition. IEEE Trans. Circuits Syst. Video Technol., 28(9): 2164-2176, 2017.
Carlo Tomasi and Roberto Manduchi. Bilateral filtering for gray and color images. In IEEE Int. Conf. Comput. Vis., pp. 839-846, 1998.
Wei-Chih Tu and Shao-Y Yi Chien. Two-way recursive filtering. IEEE Trans. Circuits Syst. Video Technol., 31 31(11):4255 4268, 2021.
Narasimhan Venkatanath, D Pranecth, Maruthi Chandrasekhar Bh, Sumohana Channappayya, and Swarup S Medasani. Blind image quality evaluation using perception based features. In The first national conference on communications, pp. 1-6, 2015.
Huiqin Xu, Zhongrong Zhang, Yin Gao, Haizhong Liu, Feng Xie, and Jun Li. Adaptive bilateral texture filter for image smoothing. Front. Neurorobot, 16. 2022a.
Kai Xu, Huaian Chen, Chunmei Xu, Yi Jin, and Changan Zhu. Structure-texture aware network for low-light image enhancement. IEEE Trans. Circuits Syst. Video Technol., 32(8):49834496 2022b.
Li Xu. Cewu Lu, Yi Xu. and Jiaya Jia. Image smoothing via gradient minimization. In Proc. SIGGRAPH Asia Conf., pp. 1-12, 2011.
13

---

Published as a conference paper at ICLR 2025
Li Xu, Qiong Yan, Yang Xia, and Jiaya Jia. Structure extraction from texture via relative total variation. ACM Trans. Graph,, 31(6):1-10, 2012.
Jia Yan, Jie Li, and Xin Fu. No-reference quality assessment of contrast-distorted images using contrast enhancement. arXiv e-prints, art. arXiv:1904.08879 2019.
Yang Yang, Hongjun Hui, Lanling Zeng. Yan Zhao, Yongzhao Zhan, and Tao Yan. image filtering based on soft clustering. IEEE Trans. Circuits Syst. Video Technol., 32(7):4150- 4162, 2021.
Futing Yu, Tangyou Liu, and Xiaoyuan Zhang. Main structure extraction algorithm based on relative total variation model. In Int. comput. sci. eng. technol., pp. 97-103, 2022.
Lin Zhang, Lei Zhang, and Alan C Bovik. A feature-enriched completely blind image quality evaluator. (8):2579-2591, 2015.
Qi Zhang, Xiaoyong Shen, Li Xu, and Jiaya Jia. Rolling guidance filter. In Eur: Conf. Comput. Vis., pp. 815-830, 2014a.
Qi Zhang, Li Xu, and Jiaya Jia. 100+ times faster weighted median filter (wmf). In Proc. IEEE Conf. Comput. Vis. Pattern Recognit. pp. 2830 2830-2837, 2014b.
Pucheng Zhou, Jie Zhang, Mogen Xue, and Lei Bao. Directional relative total variation for Structure-texture decomposition. IET Image Process., 13(11)11835-185, 2019.
Feida Zhu, Zhetong Liang, Xixi Jia, Lei Zhang, and Yizhou Yu. A benchmark for edge-preserving image smoothing. IEEE Trans. Image Process., 2857):355-3370, 2019.
Lianghui Zhu, Bencheng Liao, Qian Zhang, Xinlong Wang, Wenyu Liu, and Xinggang Wang. Vision mamba: Efficient visual representation learning with bidirectional state space model. In Forty-first International Conference on Machine Learning, 2024.
14

---

Published as a conference paper at ICLR 2025
This supplementary document paper provides in-depth account of mathematical analysis and presents additional numerical results. The structure of this supplement is organized as follows. Section A.I presents a mathematical analysis of the solution to the proposed non-convex problem utilizing a relative wavelet domain representation. In Section A.2 we discuss the computational complexity and convergence properties of the Fourier approximation applied to the mutually guided edge-aware adaptive bilateral filter. Section B provides detailed descriptions of the parameter set- tings, runtime analysis, and presents five additional experimental results as well as fail scenarios.
A THEORETICAL ANALYSIS OF RWDR
A.1 NUMERICAL SOLUTION TO RWDR
This subsection primarily presents the numerical solution for the proposed RWDR non-convex prob- lem. To begin, we revisit the formulation of the RWDR, which is expressed as:
arg min E(s, 1p)² + ARWDR(p).
(21)
And we have
D(p) Egerips In.a E (wr
RwDR(p) W(p) +8
(22)
Σ EvER(n) I0.a ( (wxS), +
Then, we note that when the two sum operators exchange the order, there is no differencen from the original formal. Therefore, we can get
D(v) w(P) +E
Σ Ek=1 EyeRUp) 9p (wg
Σ P
K -k=1 EneRioo Inn Sw. +
(23)
K
EneR(n) Ind (ux k=
EQER(D) (wk +E
k=
To simplify, we let
En EgER(n R()) Ir (w
R
k= Eg R(p) mp.9 (wk. (wk +
Ip S)
(24)
Σ Eip In S) (w + E.
9 PE p()(()
Σ 2kg (w) S)
where denotes constant value avoid division by zero. The 2kg: 2ky can be written as follows:
Wkq Epen()) wips+e (G.r E Go. (e
(25)
Zkg 1(,,3),l+e,"
G. is Gaussian filter with standard deviation a. Therefore, obtain the matrix form of Equation 21 expressed as:
K (V's - VV) - v2) + X W CVTCCUUZZCCV
(26)
V., Vy the vector form of output image S and input image I, respectively. C2 is the Toeplitz matrices from the discrete gradient operators with a forward difference. Un and Z4 are diagonal matrices with values of Un [
uks and Ze[i, il Zki, respectively.
The form of Equation 26 can be solved via a special iterative optimization procedure. boils down solving a linear system via iterative mode, which expressed as
(1 ++
V1,
(27)
where 1 is the identity matrix. R' (CKKCCCT (CU and 1 + XR is the symmetric positive definite Laplacian matrix. In this case, many efficient solvers are available for We adopt the preconditioned conjugate gradient (PCG) solve it.
15

---

Published as a conference paper at ICLR 2025
A.2 FOURIER APPROXIMATION OF MUTUALLY GUIDED ABF
In this section, we present the Fourier approximation of mutually guided adaptive bilateral filter with our edge-aware scale map. For completeness, we review Equation] 13 expressed as
F EnEN, Go., (1S5) 1
(28)
EgEN,(P) Go.. (11y% li
where Go, is Gaussian range kernel with a standard deviation or. To simplify, let
(() G.,, (1) = exp( 20
(29)
In this study, we adopt the Fourier expansion of Equation filter, which deployed with an edge-aware scale map. Specifically. Equation can be approximated in the following format:
29 to approximate the adaptive bilateral
((t)
Σ enexp(rnut),
(30)
where 72 = -1, v = x/T. N is the order of Fourier expansion, and on is the coefficient. t denotes the difference of pixel values in the local window s2p- We can get the range of t E (-.,..., ,,..,TT}, obtained via:
T max max - (y)
(31)
SE gEBp
Therefore, we take the approximation used in Equation 28 which results in the approximation of j+t1 (p). It is written as:
fet (p))
(32)
Q())
where
H(p) = Σ (() ('()s) (g),
(33)
pESp
and
@(p) Σ
(0S - (v))
(34)
gE
Then, we further have
(T)) Σ m= n c, exp(Tnn:S
(p))h, (xp((ros"(p)))b (p),
(35)
Q(p)) Em exp((nuuS(()))
where I (p) and qn (p) are expressed as follows:
1n = Even, St (())p(((7m))) qp)(p) = Eyea, (xp(TneS*
(36)
In this study, hn (p) and gn (p) can be calculated by adding each pixel value of 2pp However, the computation cost is expensive. To simplify, we get hn (p) and (p) via recursive. Firstly, we let m(g) be the integrated element of pixel p. Then, we have
m(g) = S* (4)exp(TneS* (q)).
(37)
And also can obtain the integral image M (p) at the center pixel p.
y
((p) = M (z. v)
Σ
m(ki, k2).
(38)
k kg=1
where (Ir, y) is the coorddinate of the center pixel p, k2 are the coordinates from the edge-aware scale map wp-
we compute Equation 38 by using recursion at pixel (a + 1, y + 1) as follows:
M(x + 1, 1) m(a 1, + 1) + M (x: 1,y) M(x, + 1) M(,,y).
(39)
By adopting this way. given an edge-aware scale map Wp: we can get (p) as follows:
hn (p) M (Ir + Wpp y - M Wp 1,
(40)
( + wp; wp - 1) + M (xr wp 1, Wp - 1).
16

---

Published as a conference paper at ICLR 2025
Similarly, 4n (p) can be calculated. In summary, tional complexity in a recursive manner. We consider the approximation error that is defined as
implement Equation 28
(1) computa-
||*+1 f*
|| max[ (p) - f (p) fo......
(41)
It shows the largest difference in the pixxel values between the original and approximating outputs. Since (p) and (p)) are calculated exactly, the error mainly comes from the approximation of the Gaussian range kernel. Therefore, we have
|| - مااا max (()) (O.....T)).
(42)
According to previous work Ghosh et al. 2019)) gives the guarantee condition of convergence for Equation 41 bounded by Equation 42 The condition is expressed as
20E
|| f+t 1loe S
(43)
where 7) (1)m and is the dynamic range and Tmaz the maximum value in wy. In con- clusion, we have the complete theoretical guarantee for the convergence of Fourier approximation in our mutually guided edge-aware ABF.
B ADDITIONAL NUMERICAL RESULTS
B. PARAMETER SETTINGS
For the scheme presented in Algorithm| 1 the input parameters are de, T, 7, Xe: Tmins Tmans S, X, K, a,, and o, that need to be initialized in advance. de. 7, des E, and K are fixed parameters. We empirically found that smoothed results are not sensitive to these parameters. The default reasonable fixed settings are de = 3, 7 = 10/255, A = 4, = 5e - 4, and K = 4, which are suitable for guarantee that the minimized value in the edge-aware
each test image. We have set rmin scale map is close to 1. However, Tmas is changed as the width of textures for different images. The recommended value of mmaz lies in the range [5, 15]. We empirically find that different strengths of weak edges can be preserved with different values of o,.. The recommened range of a, is [10, 30). For our experiments,
have used a 3. and À is in the range [0, 0.008). We utilized aj and as of Equation in our all experiments. It is worth noting that it is hard to automatically determine parameters in the image smoothing problem. Therefore, we recommend that users tune parameters in real-world applications via our proposed approach.
30/6
(a) Input
(b) MSE
(c) Iteration 1.
(d) Iteration 4.
(c) Iteration 8.
Figure 10: Iteration number exploration. (a) The input image, (b) the mean square difference of gradients between adjacent outputs, (c) the smoothed image after iteration, (d) the smoothed image after 4 iterations, (c) the smoothed image after iterations. We note that there has a minimum of iterations in (b), which denotes the best output. We also get the same conclusion by (d).
The parameter T is crucial for the quality of smoothed images. A larger value of T can cause over-smoothing, while a small value can not remove all textures. We can obtain a satisfactory result with suitable stop the smoothing process. The effects of on the final output in our model are presented in Figure TO We have conducted our method with 8 iterations. Figure 10(b) shows the mean square difference of gradient magnitudes between adjacent smoothed results. evident that the mean square difference is minimum when T = 4. Figure (0(c)-(e) present the smoothed results of our model at T = 1, 4, 8. Figure [0[c) shows it cannot ensure sufficient smoothing strength with small iteration number. However, large iteration value causes artifacts and outliers, shown in Figure e)... The best-smoothed result with T 4 is shown in Figure d) Considering the effect of over-smoothing set the max iteration number T 4 in all experiments.
17

---

Published as a conference paper at ICLR 2025
B.2 RUNNING TIME
We compare the running time of different methods under parameters recommended by correspond- ing authors. We slightly tune the parameters of those methods to achieve satisfactory performance. This paper uniform metric Luo et al. (2021) for smoothing strength to compare running time fairly. Given an input image and final adjacent smoothed outputs , the metric is defined
Re --
(44)
where denotes Frobenius norm. We conduct the speed comparison on the image with a size of 768 1024 under Re 0.00001. The average running time is shown in Table Our method is faster than muGIF, RoG, Ghosh. and EP/SP while slower than ILS, RGF, and RTV. Our model needs solve large linear system, achieving the same speed ILS, RGF, and RTV.
Table 3: Running Time (in Seconds) of Different Methods.
Methods
ILS
RGF
RTV
muGIF
Time (s)
0.59
0.75
4.65
10.31
Methods
RoG
EP/SP
Ghosh
Ours
Time (s)
11.15
14.42
7.60
6.08
(a) Input
(b) WL.
(c) RGF
(d) DRTV
(e) RoG
(1) muGIF
(g) RTV
(h) Ghosh
(i) Tight frame
(j) Ours
Figure 11: Image texture removal results of different methods. Parameters of compared methods tuned to achieve similar smoothing effect in the highlighted green boxes. Results of (c), (f), (g). (h), and (i) suffer from texture removal not thoroughly. referring to marked yellow boxes. For highlighted magenta rectangles, except for the tight frame and our method, the rest models have blurred edges.
B.3 TEXTURE REMOVAL
The objective of texture removal is to distinguish and eliminate complex textures while preserving the primary structures. Figure II presents the smoothed outcomes produced by various methods. Each plot in Figure 11 contains three enlarged regions, where is evident that models such WLS. RGF, DRTV, RoG. muGIF, RV, and Ghosh blur edges or structural details. Although the tight frame model better preserves edges, it fails to completely remove textures. contrast, the proposed model effectively preserves sharp edges while thoroughly eliminating textures. Further comparisons be- tween our model and a deep learning-based method are shown in Figure VDCNN Kim et al.) (2016), deep-learning-base texture removal method, produces the result shown in Figure c2. While VDCNN reduces textures, it does not entirely remove them, as observed in the two enlarged regions. Figure 12 b) displays the result from the RTV model, which successfully removes textures but at the cost of blurring edges, as shown in the two enlarged rectangles. contrast, our model clearly removes textures while preserving the primary structures. The key advantage of our ap- proach is its ability to achieve a superior balance between smoothing strength and edge preservation compared to other algorithms.
18

---

Published as a conference paper at ICLR 2025
(a) Input
(b) RTV
(c) VDCNN
(d) Ours
Figure 12: (a) The input image. Texture removal results of (b) RTV, (c) VICNN, (d) Ours. As
observed the highlighted boxes, our model can effectively preserve edges, which substantiates that the proposed method can get better tradeoff' between smoothing strength and edge sigee-preerriig than the compared methods.
(a) Input
(b) RGF
(c) Resnet
(d) VDCNN
Wardh Mip
(c) EP/SP
(f) Ghosh
(g) RTV
(h) Ours
Figure 13: Image smoothed results of (b) RGF, (c) Resnet. (d) VDCNN, (e)EP/SP, (f) Ghosh, (g)
RTV, (h) Ours. The right-bottom part of each image is the ID signal smoothed result corresponding to the yellow line in the green marked box. The blue line the input signal, while the red the smoothed result. We note that our method has better-smoothed output than other algorithms. Meanwhile, the proposed model climinates staircase artifacts.
B.4 NATURAL IMAGE SMOOTHING
The task of natural image smoothing serves effective benchmark for evaluating model's ability to balance smoothing intensity with edge preservation. Figure 13] displays the results of natural image smoothing along with corresponding 1D signal comparisons. The lower-left rectangle in each image is an enlarged view of the region within the green- box, while the lower-right rectangle shows the ID signal comparison along the yellow line within each enlarged region. In Figure 13 b). the RGF method produces a smoothed output that suffers from excessive blurring. Figures 13 c) to g)) exhibit staircase artifacts in homogeneous regions, which are also evident as abrupt changes along the middle of the red lines in the ID plots. In contrast, our proposed method effectively eliminates staircase artifacts, delivering the most refined smoothing result among all the methods compared.
B.5 DETAIL ENILACNCEMENT
Figure [4 illustrates additional results detail enhancement. Figure [4[a) a) shows the original input image. Notably, Figures 14/b) and d) exhibit gradient reversals in the output images, as high- lighted within the enlarged green rectangles. Figures 14 c), 14(e), e), and 14 f) show similar gradient present all the methods shown in Figures [4[b) through f). It is also worth noting that the two deep learning-bast based methods (ResNet and VDCNN) fail achieve optimal balance between edge preservation and smoothing.
reversal effects. Additionally, halo artifacts along the edges
19

---

Published as a conference paper at ICLR 2025
In contrast, the proposed method produces results that comparable to, and in some cases better than, the other approaches this scenario. We present corresponding numerical results in Table One can see that mode obtains the best index values.
(a) Input
(b) RGF
(c) Resnet
(d) muGIF
(e) VDCNN
(l) Ours
Figure 14: (a) The image. Image detail enhancement results of (b) RGF, (c) Resnet, (d) muGIF, (e) VDCNN, (f) Ours. We use four detail layers to boost the input image in this case. Note that our model can reduce halo artifacts and weaken the gradient reversals (referring to the marked boxes).
Table 4: No-reference image quality evaluation metrics of Figure 14
Methods
BRISQUE,
PIQE
SSEQ f
ILNIQUE1
RGF
28.8617
30.1807
23.7323
117.56
ResNet
22.3801
36.4881
26.4587
120.12
muGIF
16.0524
30.2385
19.8289
123.25
VDCNN
28.9499
35.4522
26.1938
123.85
Ours
9.6503
26.8804
31,5996
126.78
B.6 HDR TONE MAPPING
HDR tone mapping is key application of image smoothing, requiring the extraction of detail layer Liu et al. 2018) from the HDR input image. The smoothed result is then used to generate a low dynamic range (LDR) image through nonlinear mapping. The final lone-mapped output is produced by combining the nonlincarly transformed smoothed image with the detail layer. However, since HDR tone mapping relies the high-frequency detail layer, it is also prone to gradient reversals and halo effects. Figure 15 presents the HDR tone mapping results from different methods. In Figure 15 gradient reversals and artifacts along the edges evident, highlighted by the green and yellow boxes. Figure d) suffers from pronounced halos, especially in the two highlighted regions, where strong white halos are visible along the edges. The other methods avoid gradient reversal artifacts and halo effects in their outputs. We present corresponding numerical results of this HDR tone mapping task in Table 4 One can see that our mode obtains the best index values. Overall, our proposed method demonstrates competitive performance compared to other approaches in this task.
B.7 TEXTURE TRANSFERRING
The goal of texture transfer is to generate target images with new texture patterns, which can include regular, near-regular, and irregular textures. One of the main challenges in deep learning methods for texture removal the difficulty of acquiring paired training datasets. Texture transfer offers solution to this problem. Training datasets can be created by transferring the texture pattern from source image (as shown in Figure 16(a) to target image (as shown in Figure [6(b)). essence, the texture transfer process, described in Ghosh al. (2019), involves three steps. First, both the source and target images are smoothed. Second, the texture layer is extracted by subtracting the smoothed source image from the original source image, followed by extracting the texture pattern from a local window (illustrated in I6g a)). Finally, the texture patch is resized to the target image. The transferred output obtained by adding the smoothed target image to the resized texture
20

---

Published as a conference paper at ICLR 2025
(a) LO
(b) dRTv
(c) RGF
(d) Ghosh
(c) RoG
(f) muGIF
(g) RTV
(h) Ours
Figure 15: HDR tone mapping. Results of (a) LO, (b) dRTv, (c) RGF, (d) Ghosh, (c) RoG, (f)
muGIF, (g) RTV, (h) Ours. We note that the LO model produces staircase artifacts near edges. The
result of (d) shows heavy halo artifacts. On the contrary, model has comparable results in HDR tone mapping over the other methods.
Table 5: No-reference image quality evaluation metrics of Figure 15
Methods
BRISQUEJ
PIQE
SSEQ
ILNIQUE
CEIQ
LO
24.8086
28.6853
12.3523
129.59
2.6947
dRTv
23.7972
26.9161
16.8464
129.16
2.6867
RGF
24.3680
27.2629
15.6265
126.34
2.7728
Ghosh
23.7344
38.8357
15.3685
125.49
2.0085
RoG
28.1463
28.8937
12.1442
135.53
2.8496
muGIF
22.8831
32.4163
13,2606
132.94
2.6515
RTV 20.7134
30.0749
16.1376
133.17
2.6804
Ours
14.8449
24.8736
17.9714
135.69
2.8867
layer, as shown in Figure [6.c). More recently, Qi et (2024b) have created a paired dataset for edge-preserving smoothing tasks via leveraging texture transfer technology.
(a) Source
(b) Target
(c) Transferred
Figure 16: Texture transfer. (a) Source images, (b) Target images, (c) Texture transferred images. We copy textures of the marked boxes in (a) to fusion with target images (b), and the final results are shown (c).
21

---

Published as a conference paper at ICLR 2025
Input
(b) Fail Case
Input
(d) Fail Case
Figure 17: Visual effects of our model the fail scenario.
B.8
THE ANALYSIS OF FAILURE CASES
textures in Figure 17 a) and (c) are irregular and relatively large in scale. Therefore, smoothing these textures faces the challenge of dealing with the long-range depen- dency problem. In other words, it is difficult to remove them from the context of the current pixel. Our approach can remove some these textures, but cannot completely remove them. Our model's performance relies on the edge-aware scale map, while these long-range textures are mistreated as edges and kept. Even our model is deployed with Wavelet domain features for smoothing. still to fail in this scenario.
As can be seen from Figure 17
22


Reviews:
[{"review_id": "qRMqKNWnBp", "reviewer": "Reviewer_bdtH", "summary": "The main contribution of this work is the introduction of RWDR that effectively distinguishes textures from primary structures and preserves weaker edges. Additionally, the paper proposes an innovative edge-aware scale map method that dynamically adjusts scale based on the image structure, resulting in clearer distinctions between structure and texture. Experimental results demonstrate that the proposed approach provides superior edge-preserving smoothing compared to existing methods.", "questions": "Could the authors provide an online demo to allow users to test the method easily? While it’s not essential for acceptance, it would add value for potential users.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "1. This paper introduces relative wavelet domain representation into bilateral filtering, which is reasonable and novel.\n\n2. The method achieves superior visual results compared to previous studies. \n\n3. The paper includes comprehensive theoretical derivations, technical descriptions, and runtime analysis of the algorithm.", "weaknesses": "1. The paper provides extensive visual results, but I’m curious how different algorithms are objectively evaluated based on visual quality. The authors should consider comparing performance on downstream tasks with objective metrics. A user study could also statistically confirm the advantages of the proposed method.\n\n2. As a new method, it likely performs well in certain scenarios. However, I am more interested in its robustness and stability. In other words, can the authors provide a lower bound for the algorithm's performance? In which scenarios might it fail? Additionally, how sensitive is the algorithm to parameter changes?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "nkMxihkZup", "reviewer": "Reviewer_kXjP", "summary": "The author introduces a mutually guided edge-aware smoothing model based on relative wavelet domain representation. Their proposed RWDR serves as a novel measure for effectively differentiating between textures and structures.", "questions": "Please check the weakness section.", "limitations": "", "rating": 6, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The solution of the proposed model is supported by a complete theoretical guarantee, which is a strong point.\n2. Extensive experiments prove that the proposed method outperforms existing algorithms in mitigating gradient reversals,\n staircase artifacts, and halos and achieves a superior performance in balancing smoothing strength and edge preservation.", "weaknesses": "1. Though the authors support their claims by extensive qualitative results, but they should also provide the quantitative results to validate their points in the main paper or at least in the supplementary. For instance, the authors can include PSNR (Peak Signal-to-Noise Ratio), SSIM (Structural Similarity Index), or MSE (Mean Squared Error) on standard synthetic benchmark datasets and LPIPS, MUSIQ, NIQE, MANIQA for real-world tasks. This would allow for a more objective comparison with existing method.\n2. The method section needs to be refined, as mentioned in Fig1 (that the detail enhancement image is boosted by four detail layers), this statement is not explained in the method section, how the four detail layers are being generated , is it from the wavelet decomposition?\n3. The paper has lacks ablation study. The authors have given mathematical proofs of choosing the particular operations like RWDR and the edge-aware scale map, they should also try to prove the effectiveness of each proposed component  on the overall model.\n4. The authors should also try to report the results on some real-world applications in Super-Resolution (RealSR, DrealSR,RealLR200), denoising (SIDD, DND) that would further prove the use of the proposed model, if the time permits and should also check on synthetic SR datasets like Manga109, Urban100, and BSD68 (for denoising).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "cEkB4FHUR3", "reviewer": "Reviewer_M1ZN", "summary": "In this article, the author reviews image smoothing methods based on local information, global information, and deep learning, and discusses the limitations of current image smoothing techniques when dealing with image textures and image structural edges. To address this issue, first, the author proposes a novel edge-aware smoothing model that more effectively distinguishes between image textures and image structures through relative wavelet domain representation (RWDR). Second, the author reintroduces edge-aware scale maps into bilateral filters to improve image edges during the smoothing process. Finally, the author demonstrates the superiority of this method in texture preservation and artifact removal after image smoothing through comprehensive theoretical derivations and experimental results compared to other algorithms.", "questions": "1.In the experimental part of this paper, there is a predominance of qualitative analysis of images. However, due to the significant subjective factors inherent in qualitative experiments, supplementing with more quantitative experiments would enhance the persuasiveness of the results; 2.Image smoothing operations, as one of the fundamental image processing tasks, play a crucial role in various visual tasks. However, the paper seems to lack exploration of specific visual tasks (for example, in super-resolution tasks, the textures and structures preserved after image smoothing are vital for image reconstruction).", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1.This paper proposes a relative wavelet domain representation and an edge-aware smoothing model, achieving certain progress in image smoothing technology; 2.The paper utilizes extensive theoretical proofs to establish a mathematical model for the relative wavelet domain. The experimental validation is well-supported by theory; 3.The writing of this paper is relatively fluent and conforms to the standards of English academic writing.", "weaknesses": "1.In the experimental part of this paper, there is a predominance of qualitative analysis of images. However, due to the significant subjective factors inherent in qualitative experiments, supplementing with more quantitative experiments would enhance the persuasiveness of the results; 2.Image smoothing operations, as one of the fundamental image processing tasks, play a crucial role in various visual tasks. However, the paper seems to lack exploration of specific visual tasks (for example, in super-resolution tasks, the textures and structures preserved after image smoothing are vital for image reconstruction).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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:

Decision: Accept (Poster)