Learning Bilevel Layer Priors for Single Image Rain ...

IEEE SIGNAL PROCESSING LETTERS, VOL. 26, NO. 2, FEBRUARY 2019 307

Learning Bilevel Layer Priors for Single Image RainStreaks Removal

Pan Mu , Jian Chen, Risheng Liu , Member, IEEE, Xin Fan , Member, IEEE, and Zhongxuan Luo

Abstract—Rain streaks removal is an important issue of the out-door vision system and recently has been investigated extensively.In the past decades, maximum a posterior and network-basedarchitecture have been attracting considerable attention for thisproblem. However, it is challenging to establish effective regular-ization priors and the cost function with complex prior is hard tooptimize. On the other hand, it is still hard to incorporate data-dependent information into conventional numerical iterations. Topartially address the above limits and inspired by the leader–follower gaming perspective, we introduce an unrolling strategyto incorporate data-dependent network architectures into the es-tablished iterations, i.e., a learning bilevel layer priors method tojointly investigate the learnable feasibility and optimality of rainstreaks removal problem. Both visual and quantitative comparisonresults demonstrate that our method outperforms the state of theart.

Index Terms—Learning-based optimization, rain streaksremoval, deep unrolling, bilevel layer priors.

I. INTRODUCTION

IN REAL-WORLD scenarios, especially for outdoor scenes,the impact of rain has always been an annoying and inevitable

interference, which would significantly changes or lowers thecontent of images [1]. These things often happen when onerecords an event happening at a square using a smart phone, asurveillance camera monitors a street, or an autonomous vehicledrives on a road that one has to process unclear images, in

Manuscript received October 22, 2018; revised December 7, 2018; acceptedDecember 16, 2018. Date of publication December 24, 2018; date of currentversion January 9, 2019. This work was supported in part by the National NaturalScience Foundation of China (NSFC) under Grant 61672125, Grant 61562062,Grant 61733002, Grant 61632019, Grant 61806057, and Grant 61572096, andin part by the Fundamental Research Funds for the Central Universities. Theassociate editor coordinating the review of this manuscript and approving it forpublication was Dr. Daniel P. K. Lun. (Corresponding author: Risheng Liu.)

P. Mu is with the School of Mathematical Sciences, Dalian University of Tech-nology, Dalian 116024, China, and also with the Key Laboratory for UbiquitousNetwork and Service Software of Liaoning Province 116024, China (e-mail:,[email protected]).

J. Chen, R. Liu, and X. Fan are with the Key Laboratory for UbiquitousNetwork and Service Software of Liaoning Province, Dalian 116024, China,and also with DUT-RU International School of Information Science & En-gineering, Dalian University of Technology, Dalian 116024, China (e-mail:,[email protected]; [email protected]; [email protected]).

Z. Luo is with the DUT-RU International School of Information Science& Engineering, Dalian University of Technology, Dalian 116024 China, withthe Key Laboratory for Ubiquitous Network and Service Software of LiaoningProvince, Dalian 116024, China, with the School of Mathematical Sciences,Dalian University of Technology, Dalian 116024, China, and also with theInstitute of Artificial Intelligence, Guilin University of Electronic Technology,Guilin 541004, China (e-mail:,[email protected]).

Digital Object Identifier 10.1109/LSP.2018.2889277

rainy days. Hence, it is essential to find effective methods forremoving rain streaks. A rainy image O is often consideredas linear combination of a rain-free background B and a rainstreaks layer R that can be described as O = B + R whereO ∈ RM ×N , B ∈ RM ×N and R ∈ RM ×N . The goal of rainstreaks removal is to estimate B, R from a given image O,restoring the scene without rain streaks. Several methods havebeen proposed to remove rain streaks in images [2]–[11]. Thesecan be roughly divided into two types: multiple image/video-based and single image methods.

For video-based methodologies, it is more easier to removerain streaks by inter-frame information [10]–[21]. In the pastyears, more physical intrinsic properties of rain streaks havebeen explored and formulated in algorithm designing. But thesemethods are significantly aided by the temporal content of video.

In this work, we focus on removing rain from a single im-age which is significantly more challenging since less informa-tion is available when removing rain streaks. For single imagerain removal methods, traditional model and prior based meth-ods (see [3], [6], [22]–[25]) play important role. For example,Kang et al. [6] decomposed an image into the low- and high-frequency parts using bilateral filter and applied an morpho-logical component analysis (MCA for short) based dictionarylearning and sparse coding to separate rain streaks from thebackground. Gu et al. [22] gave a joint convolution prior modelas a sparsity-based method (JCAS) to tackle with rain streaksremoval problem, Du et al. [26] introduced gradient domain toremove rain streaks and Liu et al. [27] proposed an deep layerpriors model for rain streaks removal. But, the separation in de-tail layer is challenging, always tending to either over-smooththe background or remain noticeable rain steaks. Recently, net-work based methods tend to be popular, such as deep derainnet (DN) [28], deep detail network (DDN) [4], contextualizeddilated network [24], dense network with multi-stream (DID-MDN) [29] and so on. However, it is hard to handle images fulldetails of the background. As shown in Fig. 1, the result of [22]and [4] remain a lot of rain streaks under the real world rainyimages.

Inspired by the idea of bilevel optimization [30], [31] andthe well-known leader-follower theory, we introduce a flexi-ble energy function model to characterize feasibility constraintthat incorporate implicit constraints into optimization models tojointly investigate the feasibility and optimality for challengingrain streaks removing problem. Different from most existingoptimization models which often build one single objective andtry to figure out an optimal solution with designed priors such

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308 IEEE SIGNAL PROCESSING LETTERS, VOL. 26, NO. 2, FEBRUARY 2019

Fig. 1. An example real-world rainy image and the comparison result.

Fig. 2. An illustrative diagram of the proposed algorithm framework.

that their objectives reach the best, we add new domain knowl-edge that related to the rain streaks removal area as a feasibilityconstraint in this work. Then a learning-based bilevel layer pri-ors model is formulated for rain streaks removal problem (seeFig. 2 for illustration). The bilevel model (leader-follower form)provides a collaborative mechanism to jointly investigate thefeasibility and optimality which provide us a learnable frame-work for the research problem. Further, feasibility constraint infollower subproblem is a flexible part which can be designedbased on characterize of the research problem, such as, sparsity,low-rank and so on. Deep strategy to learn the feasibility con-straint is applied to further improve the practical performance.

The contributions of this work are summarized as:1) For traditional methods, it is still hard to design effective

and simple regularize priors. To overcome this difficult,we introduce a bilevel based joint learning model withsimple priors and implicit feasibility constraints, that is,minimizing an energy function for rain streaks removaltask. Further, this enable us to avoid establishing explicitconstraints which are indeed challenging.

2) For a single network method, it is unexplainable whencompared with traditional methods and may not capableof learning all types of variations. This work remedy theregret by introducing a traditional leader-follower modelunrolling a learning based feasibility constraint. This ap-proach keeps background details when removing muchmore rain streaks.

3) Extensive experiments are conducted on synthetic and realdatasets. Qualitative and quantitative comparisons withexisting methods are presented. The experiments show

that our method achieves state-of-the-art results on all thetested datasets.

II. OUR METHOD

In this section, we first give the general maximize joint prob-ability of the two desired layers model using maximum a poste-riori (MAP) which can be used in many removing rain streakstasks [9], [27], [32]. Then, we propose a flexible learning bilevellayer priors model for rain streaks removal problem, illustratehow to solve this model and provide a specific algorithm scheme.

A. Basic Problem Model

Maximum the probability of the background layer and the rainlayer using the MAP, i.e., p (B,R|O) ∝ p (O|B,R) · p(B) ·p(R) with the assumption that the two layers B and R areindependent, can be equivalently to the following form by thenegative log function manipulation

minB ,R∈Ω

12‖O − B − R‖2

2 + Ψ(B) + Φ(R), (1)

where Ω := {B,R|0 ≤ Bi ,Ri ≤ Oi ,∀i ∈ [1,M × N ]}. Thefirst term ‖O − B − R‖2

2 aims to maintain the fidelity betweenthe input image and the recovered image. The other two reg-ularization terms Ψ(B) and Φ(R) are proposed to model ourimage priors on B and R for rain streaks removal. However, itis difficult to establish a priori model for rain and backgroundimages accurately. Then we propose the following bilevel learn-able model with simple priors information to estimate rain andbackground efficiently.

B. Learning Bilevel Layer Priors Model

Drawing on the idea of bilevel optimization [14] and thewell-known Stackelberg’s leader-follower competition theory ineconomics [34], we introduce a flexible energy function modelto characterize feasibility constraint that incorporates implicitconstraints into optimization models to jointly investigate thefeasibility and optimality for the challenging rain streaks re-moval problem. That is, we would like to incorporate residualtype convolution neural networks (CNNs) into each charac-terized follower subproblem to further improve the practicalperformance. The proposed model is

Leader (L) : minB ,R

12‖O−B−R‖2

2 +Ψ(B)+Φ(R),

Follower (F) : s.t., (B,R)∈arg min (F (B, ·), G(R, ·)) ,(2)

where Ψ(B) and Φ(R) represent two general priors; F (B, ·)and G(R, ·) are two feasibility constraints form which can becharacteristics about B of the background information and Rof the rain streaks feature. In the follower part, we first adoptcoding model to approximate the latent images B ≈ Dβ andR ≈ Dγ, where D is a given dictionary1 and β, γ respectivelydenote the sparse codes of B,R on D. Then, we summarizefunctions of the proposed model in Table I.

1We follow standard settings in image processing to define D as the inversewavelet transform in our problem.

MU et al.: LEARNING BILEVEL LAYER PRIORS FOR SINGLE IMAGE RAIN STREAKS REMOVAL 309

Fig. 3. Visual comparison of different rain streaks removal methods on two synthetic datasets, i.e., the rain streaks rendering technique in [33] (the top row) and[9] (the bottom row). The obtained rain streaks are placed at the top right-hand corner of each method.

TABLE ITHE DEFINITION OF FUNCTIONS IN OUR PROPOSED MODEL (2)

For the leader subproblem in Eq. (2), we apply the half-quadratic splitting technique [35] to obtain the following schemeby introducing auxiliary variables U and V:

(Bk+1

l ,Rk+1l

)=

(∇T U + O − RI + ρ2∇T ∇I

,ρ1V + O − B

(1 + ρ1)I

), (3)

where U = [U1 ,U2 ]T , ∇ = [∇x ,∇y ]T and ∇x , ∇y are gradi-ents in two directions. I is a unit matrix. Discarding the variablesunrelated to U and V respectively yields:⎧⎨

⎩

Uk+1 = arg minU

λ1‖U‖1 +ρ2

2‖U −∇B‖2

2 , (4)

Vk+1 = arg minV

λ1‖V‖1 +ρ1

2‖V − R‖2

2 . (5)

Obviously, the above two parts are classic LASSO problemswhich can be efficiently obtained by the shrinkage opera-tor [36] that the definition of which on scalars is Tt>0(x) :=(x)sgn max (|x| − t, 0). The extension of the shrinkage opera-tor to vectors and matrices is simply applied element-wise.

For the follower subproblem, we would like to incorpo-rate specially designed network architectures into each char-acterized constraint iteration to further improve the practicalperformance. Specifically, the network-based building blocksfor background layer and rain streaks layer are denoted asD(·;θD) and R(·;θR), where θD and θR are learnable pa-rameters. For the background layer, the learning network D isa denoiser as stated in [37] which extracts the natural imagewell. For the network architecture R, we just adopt a CNNswhich consist of 7 dilated convolution layers with 64 kernelsof the size 3 × 3. The ReLU and batch normalization layersare also incorporated into the network. In training process,we take rainy image and synthetic rain layer as the degradedclean image pair. We crop training data into small patches of

Fig. 4. Illustration of the effect of learning constraint part. The top and bottomrow are the results of background layer (with PSNR and SSIM scores) and thecorresponding rain streaks layer respectively. The left column is the result ofthe leader subproblem, i.e., model (1) and the middle column is the result underfollower subproblem. The right column shows the result of bilevel model (2).

size 80 × 80 firstly, then to enhance the ability of our networkwe augment the small patches by rotating and flipping. Thenwe calculate a temporary variable B̃k = D(Bk ;θD) and R̃k =R(Bk ;θR). By further considering data-dependent proximalapproximation form Fk

μ1(B,β) = F (B,β) + μ1

2 ‖B − B̃k‖22

and Gkμ2

(R,γ) = F (R,γ) + μ22 ‖R − R̃k‖2

2 , we obtain that

Bk+1f ∈ProjΩ

(B̃k−s1∇f(B̃k ,βk )+μ1(B̃k −Bk )

),

Rk+1f ∈ProjΩ

(R̃k−s2∇g(R̃k ,γk )+μ1(R̃k −Rk )

),

(6)

where the auxiliary variables βk+1 , γk+1 can be updated by theshrinkage operator. And the variables B and R have:{

Bk+1 = αk1B

k+1l + (1 − αk

1 )Bk+1f , (7)

Rk+1 = αk2R

k+1l + (1 − αk

2 )Rk+1f , (8)

where {αki }, i = 1, 2 are sequences of real number in

(0, 1) and satisfying that limk→∞ αki → 0,

∑∞k=1 αk

i = ∞ andlimk→∞ αk+1

i /αki = 1. Then, the proposed leaning based opti-

mization approach can be summarized in Algorithm 1.

III. EXPERIMENTAL RESULTS

In this section, we firstly evaluate the contributions of con-straint part in our model, i.e., the follower subproblem inmodel (2). Then, we evaluate our method using both syn-thetic and real images, and compare the proposed approach

310 IEEE SIGNAL PROCESSING LETTERS, VOL. 26, NO. 2, FEBRUARY 2019

Fig. 5. Comparisons with state-of-the-art methods using various real-world rainy images.

Algorithm 1: Optimizing (1) via Learning Bilevel LayerPriors.

1: Input: Input image O, maximum iterations number K,and parameter {αk

i |αki ∈ (0, 1), i = 1, 2}.

2: for k = 1 : K do3: Update U, V using Eq. (4) and Eq. (5) respectively.4: Update Bl , Rl by Eq. (3).5: Update Bf , Rf using Eq. (6).6: Update B and R by Eq. (7) and Eq. (8) respectively.7: end for8: Return: B, R.

against state-of-the-art single-image rain streaks removingmethods, including Gaussian mixture model-based layer prior(GMMLP) [9], removing rain from single images via Derain-Net (DN) [28], a further work of DN, i.e., deep detail network(DDN) [4] and a joint convolution and synthesis sparse repre-sentation model (JCAS) [22] and UGSM [23]. For measuringthe performance quantitatively, we employ PSNR, SSIM andelapsed time as the metrics. All these experiments are conductedon a PC with Intel Core i7 CPU at 3.4 GHz, 32 GB RAM and aNVIDIA GTX 1070 GPU.

Model Analysis: As can be seen in Fig. 4, using only theobjective function (1), i.e., the leader subproblem, has a pooreffect on the background and the rain streaks layer containsmany noises. Meanwhile, the background layer under singlefollower part performs smooth and the rain streaks is not sharp.While, adding the follower part to the objective optimizationhelps to recover background efficiently. This is because thefollower subproblem provide an efficient feasibility constraintfor the leader subproblem to perform better.

Synthetic Images: We conduct our experiments on twodatabases: one is provided by using photorealistic renderingof rain streaks [33] with 7 synthesized rain images and the otherone is obtained by Dr. Yu Li with the rain streaks renderingtechnique in [9] which includes 12 synthesized rain images andthe comparison quantitative results are shown in Table II. Whatneeds to be explained is that we produce the results of existentcomparison methods using authors’ code and tuning its parame-ters to achieve the best. The quantitative results shown in Table IIdemonstrate that our method achieves the best (highest) PSNRand SSIM scores.

TABLE IIAVERAGED QUANTITATIVE COMPARISON (I.E., PSNR, SSIM AND TIME(S))

BETWEEN OUR METHOD AND STATE-OF-THE-ART RAIN REMOVAL METHODS

ON TWO TEST DATASET, I.E., DATASET1 (7 SYNTHETIC IMAGES) [33] AND

DATASET2 (12 SYNTHETIC IMAGES) [9]

Further, Fig. 3 shows visual comparisons for two synthesizedrainy images. As can be observed, the GMMLP [9] still containsrain streaks and tends to over-smooth the background details andJCAS [22] sometimes fails to capture the rain streaks which is forthe weakness of model prior. Other deep learning methods [4],[28] preserve most of the details in the background but remainsome streaks residual because the distributions of rain streaks intest are different from training. In contrast, the proposed methodremoves the rain streaks while keeping more image details inthe background layer.

Real Images: The performance of proposed method is alsoevaluated on several real-world images and the results are shownin Fig. 5. Again, DN, DDN and JCAS tend to retain excessiverain streaks, while GMMLP tends to keep rain streaks for imagesand over-smooth background details. Qualitatively, as can beseen in Fig. 5, our method achieves the best vision results interms of effectively removing the rain streaks while preservingthe scene details.

IV. CONCLUSION

In this letter, we proposed an unrolling strategy to incorpo-rate data-dependent network architectures into the establishediterations, i.e., the learning bilevel layer priors model for rainstreaks removal problem. Experiments both on the syntheticdatasets and the challenging real images showed that the pro-posed method has better visual performance and quantitativescores against other state-of-the-art methods.

MU et al.: LEARNING BILEVEL LAYER PRIORS FOR SINGLE IMAGE RAIN STREAKS REMOVAL 311

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