Efficient Blind Deblurring under High Noise Levels
EEFFICIENT BLIND DEBLURRING UNDER HIGH NOISE LEVELS
Jérémy Anger † , Mauricio Delbracio § , and Gabriele Facciolo †† CMLA, ENS Cachan, CNRS, Université Paris-Saclay, 94235 Cachan, France § IIE, Universidad de la República, Uruguay
ABSTRACT
The goal of blind image deblurring is to recover a sharp image froma motion blurred one without knowing the camera motion. Currentstate-of-the-art methods have a remarkably good performance on im-ages with no noise or very low noise levels. However, the noiselessassumption is not realistic considering that low light conditions arethe main reason for the presence of motion blur due to requiringlonger exposure times. In fact, motion blur and high to moderatenoise often appear together. Most works approach this problem byfirst estimating the blur kernel k and then deconvolving the noisyblurred image. In this work, we first show that current state-of-the-art kernel estimation methods based on the (cid:96) gradient prior can beadapted to handle high noise levels while keeping their efficiency.Then, we show that a fast non-blind deconvolution method can besignificantly improved by first denoising the blurry image. The pro-posed approach yields results that are equivalent to those obtainedwith much more computationally demanding methods. Index Terms — Image deblurring, blur kernel estimation, de-convolution, high noise
1. INTRODUCTION
Blind image deblurring is an ill-posed image restoration problemthat aims to restore a sharp image given a blurry one. Motion bluroccurs when there is relative motion between the camera and thescene during the exposure time. This phenomenon is most visible inlow light conditions, when the integration time has to be longer tocompensate for the lack of photons. The formation of a blurry imageis frequently modeled as the convolution between the sharp image u and a latent blur kernel k leading to v = u ∗ k + n, (1)where ∗ denotes the convolution, and n models acquisition noise(usually white Gaussian noise). The goal of blind image deblurringis to recover the image u without knowing k . Most methods proposea two step process: first estimating the blur kernel k and then ap-plying a non-blind deconvolution algorithm [1, 2, 3, 4]. The abovestationary kernel model can be generally extended to a non-uniformmodel [5, 3]. However, this comes at the price of a non-negligiblecomputational cost with, in general, only a minor quality improve-ment [6, 7]. Work partly financed by Office of Naval research grant N00014-17-1-2552, Agencia Nacional de Investigación e Innovación (ANII, Uruguay)grant FCE_1_2017_135458, Programme ECOS Sud – UdelaR - ParisDescartes U17E04, DGA Astrid project « filmer la Terre » n ◦ ANR-17-ASTR-0013-01, MENRT; DGA PhD scholarship jointly supported withFMJH.
Input ( σ = 10% ). Tao [8] (19.10dB) Zhong [9] (20.18dB)
Zhou [10] (20.90dB)
Pan [4] (20.97dB)
Proposed (21.66dB)
Fig. 1 : Blind deblurring under high noise. The proposed method isable to estimate the kernel and restore an high quality image.Current state-of-the-art methods, either variational [4, 11, 12]or learning based [8, 13], work very well on images with no noiseor very low noise levels. However, the noiseless assumption is notrealistic considering the low light conditions that lead to the motionblur in the first place.
Kernel estimation.
Only a handful of blind deblurring algorithmsfrom the literature consider the realistic case of moderate or highnoise. Tai et al. [14] show that denoising the image before estimat-ing the kernel leads to an oversmoothing of details in the blurry im-age and thus errors in the estimated kernel. Instead, they proposeto iteratively denoise the image and estimate the kernel. The ad-hocdenoising step uses the motion information from the kernel. Xu etal. [15] propose a two step kernel estimation. The first step only es-timates a coarse kernel. The second step uses an iterative supportrefinement of the kernel that enforces sparsity without an explicitprior. Zhong et al. [9] also observe that denoising before kernel esti-mation results in poor performance. To circumvent this, they designdirectional filters which reduce the noise level while preserving blurinformation in the orthogonal direction. The blur kernel is then re-constructed from projections using the inverse Radon transform. Panet al. [4] propose a kernel estimation method based on the (cid:96) imagegradient prior which allows high quality estimations in low noiselevel settings [6]. However, the authors indicate that the methodunder-performs in medium and high noise conditions [16]. In thispaper, we propose an adaptation of the (cid:96) -based kernel estimationmethod which is both efficient and robust to noise. Non-blind deconvolution.
Once the blurring kernel is estimated,most methods apply a non-blind deconvolution algorithm to restorethe sharp image u . The fastest deconvolution methods usually relyon image priors that do not perform well under high noise conditions a r X i v : . [ c s . C V ] M a y e.g., Total Variation). In the past decade, better image priors havebeen introduced to offer higher quality non-blind deconvolution. Forexample, EPLL [17] learns a mixture of Gaussian models to encoderepresentative patches from natural images, and proposes an iter-ative algorithm to restore the image in presence of Gaussian blur.Generic frameworks such as Plug-and-Play priors [18] and more re-cently Regularization by Denoising [19], allow to use any image de-noiser as a prior for restoration problems. Similarly, Zhong et al. [9]propose to use NL-means at each step of an iterative non-blind de-convolution, and Tai and Lin [14] incorporate a motion-aware de-noiser for blind deblurring. While these methods significantly out-perform basic priors such as TV, they are usually prohibitively slowdue to the complex optimizations involved. Other methods proposeto first inverse the blur with little regularization and then denoisethe result [20, 21]. While computationally efficient, these methodsrequire to solve the difficult problem of removing correlated noise. Contributions.
We study the robustness to noise of the kernel esti-mation method introduced by Pan et al. [4] and improve it by makingit robust to noise (up to σ = 10%) while maintaining a good per-formance in terms of quality and speed. These adaptations are notspecific to this particular method and can be included in most meth-ods that alternate between sharp image prediction and kernel estima-tion. We then propose a non-blind deconvolution method capable ofhandling moderate to high noise. The method uses denoising as apreprocessing step. While being conceptually simple, the proposedmethod is competitive with the state-of-the-art that iterate denoisinginside the algorithms, which is much more computationally demand-ing.
2. PROPOSED METHOD
The proposed method first estimates the kernel by iterating betweentwo steps: (i) sharp image prediction and (ii) kernel estimation.Then, once the kernel has been estimated, the final image is restoredusing a non-blind deconvolution algorithm.
The goal of this step is to recover the main structures of the latentsharp image using the previously estimated blur kernel and imposingadditional prior information about sharp images. One very effectiveprior is the (cid:96) gradient prior, introduced for image deblurring by Panet al. [22] in the following optimization problem arg min u (cid:107) u ∗ k − v (cid:107) + λ (cid:107)∇ u (cid:107) . (2)The energy (2) is minimized using a half quadratic splitting formu-lation, which leads to iteratively solving two sub-problems g ( t +1) = arg min g β ( t ) u (cid:107)∇ u ( t ) − g (cid:107) + λ (cid:107) g (cid:107) , (3) u ( t +1) = arg min u (cid:107) u ∗ k − v (cid:107) + β ( t ) u (cid:107)∇ u − g ( t +1) (cid:107) . (4)The closed form solution for the sub-problem (3) is the hard thresh-olding operator on the gradients of u , whereas the sub-problem (4)corresponds to the deconvolution of v with an attachment term onthe vector field g and β ( t ) u = κ t β (0) u . Unless specified and accord-ing to [4], κ is set to and β (0) u to λ . The weight λ controls theamount of details – and noise – that should be contained in u . Aftera complete sharp prediction step, the parameter λ is decreased untilit reaches the threshold λ min [4]. σ = % σ = % (a) (b) (c) (d) (e) Fig. 2 : Kernels estimated from a blurry noisy image: (a) ground-truth, (b) our result including every prior from Equation (6) with γ = 10 σ and α = 0 . , (c) setting α = 0 , (d) setting α = γ = 0 ,and (e) with a data-term formulated in a filtered domain. Notice hownoise increases as priors are removed.We observed that when the blurry image v is contaminated withnoise and λ is small, the solution u contains spikes fitting the noise.In order to have a clean estimation of u , albeit coarser, it is requiredto increase the regularization weight λ until noise is no longer in-cluded in the solution. Since the (cid:96) minimization acts as a hardthresholding, it is clear that using a larger threshold will result ina more conservative noise artifact removal. However, as the regu-larization increases, restored details that would have otherwise beenincluded are removed from the solution.To summarize, the sharp image prediction using (cid:96) step can bemade robust to noise by adapting the regularization limit λ min so thatnoise artifacts are filtered. This tuning should be performed per noiselevel. This step uses the current sharp image prediction and the blurry im-age to estimate a blur kernel. Since the support of the blur kernelis significantly smaller than the image, this problem is usually wellposed if both images are noiseless. In such conditions, simple priorsfor the kernel can be employed, leading to efficient computations.For example, a well known minimization problem for the kernel es-timation step is arg min k (cid:107) u ∗ k − v (cid:107) + γ (cid:107) k (cid:107) . (5)Variants of this energy have been proposed. For example, Cho etal. [2] showed that by formulating the data term in a filtered do-main (e.g. using image gradients) the conditioning of the problemwas improved. This speeds up convergence when using a conjugategradient algorithm but increases the weight of the frequencies mostaffected by noise. As the blurry image gets noisier, noise in the es-timation also increases, with little control. A trick often found inkernel estimation implementations [4, 12, 23], consists in filteringthe kernel values after its estimation using both a hard thresholdingand a connected component filtering, removing low amplitude noisebut also biasing the estimation.Instead, we propose to use more suited priors and kernel con-straints by minimizing arg min k,k ≥ , supp ( k ) ⊂ Ω (cid:107) u ∗ k − v (cid:107) + α (cid:107) k (cid:107) + γ (cid:107)∇ k (cid:107) , (6)where Ω is a rectangular domain covering the support of k , and γ and α are regularization parameters. The regularizers (cid:107) k (cid:107) and (cid:107)∇ k (cid:107) were motivated in Xiong [24] for their effectiveness for kernel esti-ation and the spatial constraints were studied in Almeida et al. [25].To highlight to importance of each constraint and prior, we evaluatetheir contribution by successively removing them and running thefull blind kernel estimation method for two noise levels ( σ = 2% and σ = 10% ). Results are shown in Figure 2. The kernel (2b)was estimated using Equation (6) with γ = 10 σ and α = 0 . . Wethen successively set α = 0 (2c), γ = 0 (2d), and finally use gradi-ents in the data-term [2] (2e). Notice how each prior helps removingnoise and the difference to the ground-truth is reduced (2a). Usinga filtered domain to estimate the kernel introduces errors that can beotherwise easily avoided.We propose an efficient solver for (6) based on half quadraticsplitting [26]. Our kernel estimation step iterates as follows h ( t +1) = arg min h (cid:107) u ∗ h − v (cid:107) + β k (cid:107) k ( t ) − h (cid:107) + γ (cid:107)∇ h (cid:107) (7) k ( t +1) = arg min k,k ≥ ,, supp ( k ) ⊂ Ω β k (cid:107) k − h ( t +1) (cid:107) + α (cid:107) k (cid:107) . (8)Assuming circular boundary conditions for the convolution, thesubproblem (7) can be solved efficiently using two discrete Fouriertransforms h ( t +1) = F − (cid:32) F ( u ) F ( v ) + β ( t ) k F ( k ( t ) ) |F ( u ) | + β ( t ) k + γ ( |F ( ∇ x ) | + |F ( ∇ y ) | ) (cid:33) . (9)The subproblem (8) enforces non-negativity and a given spatial sup-port for h , and its solution corresponds to a soft thresholding k ( t +1) ( x ) = (cid:40) max (cid:16) h ( t +1) ( x ) − αβ k , (cid:17) , if x ∈ Ω0 , otherwise. (10)Similarly to continuation methods, β ( t ) k starts with a low value β (0) k = 1 and is multiplied by at each iteration. The method stopswhen it reaches β ( t ) k = 10 which implies that only iterationsare required, with FFTs per iteration. In comparison, conjugategradient methods usually require iterations with FFTs per iter-ation in ideal conditions [2], but are unstable in presence of noise.Finally, even though unrealistic circular boundary conditions are as-sumed in Equation (9), we observed that the regularization terms inconjunction with an edge-tapering procedure [27] are sufficient toavoid boundary artifacts.
Coarse-to-fine scheme kernel estimation.
Alternating betweenkernel estimation and sharp image prediction allows to successfullyretrieve small kernels. A coarse-to-fine scheme is generally em-ployed to efficiently recover large kernels [2]. Our implementationis based on [28] which upscales the predicted sharp image by a fac-tor two using bicubic interpolation. However, instead of iterationsper scale as performed in [28], our method requires only iterationsby warm-starting the second one using the the previous estimationof u . This allows to reduce the number of inner iterations requiredfor the sharp prediction step by setting κ = 5 and β (0) u = 0 . in (3)and (4). These modifications constitute a significant speed-up withno loss of performance, as we show in the experimental section. Non-blind deconvolution algorithms in noiseless settings reach ingeneral high quality results. The main difficulties come from errorsin the estimated kernel or when a frequency component gets cancel Input Groundtruth Pan [4] Zhong [9] Proposed
Fig. 3 : Sample of three estimated kernels (from the dataset of Levinet al. [36]) with Gaussian noise.by the blurring kernel. Priors such as total variation [29] (TV) areefficient at reducing ringing artifacts arising from these errors andfast solvers exist [30]. However, in presence of noise, the weightassociated with the regularization has to be increased, and in the caseof total variation artifacts such as staircasing start to appear, hencethe need for more natural image priors.Given recent progress in the denoising field [20, 31, 32, 33], weargue that preprocessing the image with a denoising before non-blinddeconvolution is now a viable, and very efficient, solution against thenoise. While a direct inversion of blur on a denoised image can stillproduce ringing artifacts, using a TV prior with a low regularizationis sufficient to remove ringing while keeping a staircasing free im-age, giving it a more natural aspect than a high TV regularizationwithout denoising as preprocessing. A similar approach was studiedin Badri et al. [34].We have found that the quality gain obtained from this procedurewas quite independent of the denoiser and selected the implementa-tion from [35] of the FFDNet [33] CNN denoiser.
3. EXPERIMENTS
In what follows we present several deblurring results on syntheticand real images. We compare our results against Zhong et al. [9]which is robust to noise, Pan et al. [4] which uses the (cid:96) gradientprior and more recent blind methods [10, 8]. We first assess the per-formance of our kernel estimation method under challenging noiselevels, then show qualitative results from our non-blind deconvo-lution procedure before evaluating blind results. Finally, we com-pare blind deblurring results on a real-world image. We first assessthe performance of our kernel estimation method under challengingnoise levels, then show qualitative results from our non-blind de-convolution procedure before evaluating blind results. Finally, wecompare on a real-world image. Noise-robust kernel estimation.
In order to assess the performanceof our kernel estimation, we extend the dataset of Levin et al. [36]by adding three levels of Gaussian noise to the blurry images: , and . As a measure of quality of the estimated kernels, wecompute the root mean square error (RMSE) minimized by translat-ing the kernel by integer shifts. Table 1 shows the results for Pan etal. [4], Zhong et al. [9] and our kernel estimation on this dataset. Asexpected, in the noiseless case all kernels are well estimated. How-ever as the noise increases, the results of Pan et al. degrade quicklywhile Zhong’s and ours show robustness.ethod σ = 0% 5% 10% Pan et al. [4] 0.132 0.163 0.171Zhong et al. [9] 0.137 0.143 0.158Proposed : Comparison of kernel estimation methods on the dataset ofLevin with added noise. Kernels are registered with integer transla-tions to the ground-truth before computing the RMSE.(a) Input ( σ = 5% ) (b) Zhong et al. [9] ( . dB) (c) Without denoising ( . dB) (d) With denoising ( . dB) Fig. 4 : Non-blind deconvolution with ground-truth kernel. Regular-ization weights for the final deconvolution were optimized for PSNRover a set of 5 images.In addition to this quantitative study, we show a sample of es-timated kernels by the three methods in Figure 3 for the noise level σ = 5% . Visual inspection of the kernels are in accordance with thequantitative measure: Pan et al. show no robustness to noise, Zhonget al. kernels exhibit a correct recovering of the kernel’s shape whileour method is able to estimate sharper kernels. Non-blind deconvolution under high noise.
We proposed a non-blind deblurring method based on denoising the image before de-convolution. We compare three non-blind deconvolution methods:Zhong et al. [9], Krishnan et al. [30] (with (cid:107)∇ u (cid:107) as regulariza-tion), and our method composed of denoising using FFDNet andthe deconvolution of [30]. Figure 4 compares non-blind deconvolu-tion results using the ground-truth kernel with a noise level of .Regularization weights for all three methods are tuned for best aver-age PSNR over five images from [37] (including the image in Fig-ure (4a)). We observe that our method is able to recover more detailsthan Zhong et al. [9] while having a smoother aspect than Krishnanet al. [30] thanks to the denoising preprocessing. Blind deblurring comparison.
The previous experiments indicatedgood performance for the kernel estimation and non-blind deconvo-lution. We now validate the complete blind deblurring method andcompare against competitive methods on three levels of noise. Ta-ble 2 shows PSNR computed over 5 images from [37]. Runningtimes are also reported in Table 2 for single thread CPU executionon an Intel Xeon E5-2650. For this experiment, we set λ min = 0 . σ and γ = 200 σ , and kept α = 0 . for all noise levels. A visual com-parison of the results for σ = 10% is shown on Figure 1. In such PSNR is computed after registering the images with the ground-truth andcropping to avoid boundary effects.
Method σ = 1% 5% 10% RuntimePan et al. [4] 26.60 24.29 23.81 165sZhou et al. [10] 27.35 25.31 24.01 72sTao et al. [8] 24.99 22.76 20.28 123sZhong et al. [9] 24.39 23.84 23.38 154sProposed : Comparison of PSNR of the blind results. The reportedvalues corresponds to the average PSNR after registration over 5 im-ages of size × . Regularization parameters are tuned for bestPSNR for each noise level.(a) Input (b) Zhong et al. [9] (c) Proposed Fig. 5 : Blind deblurring of a real images from [9] (contrast enhancedfor visualization).challenging situations, most methods fail to estimate the kernel andthe deconvolution introduces ringing or regularization artifacts thatare much less present in our result. More visual results and sourcecode are available online at the project webpage . Real world images.
Figure 5 shows the results on a real-world im-age from Zhong et al. [9]. We estimated the noise standard deviationto be approximately . and applied our blind deblurring method.Even though the deblurring results are close, the method of Zhonget al. took 250s for kernel estimation and 370s for non-blind decon-volution (MATLAB implementation) while our method took 10s toestimate the kernel, 6s to denoise and 10s to deconvolve the imageof size × (C++ implementation).
4. CONCLUSION
We showed that even though kernel estimation is often understood asbeing very unstable in the presence of noise, it is possible to have ro-bust estimations. First, we showed that the (cid:96) gradient prior could beactually very robust to noise if the regularization weight is set suffi-ciently high, leading to a noiseless sharp image prediction. Then, thekernel estimation step should also take the noise into account, andwe proposed a splitting strategy to exploit spatial and non-negativityconstraints as well as two regularizations terms on the kernel. Fi-nally, for the final non-blind deconvolution, a simple and efficientway to handle high noise is simply to denoise the blurry image beforeusing deconvolution. Qualitative and quantitative results highlightedthe strength of our method when compared to other noise handlingmethods.As future work, we would like to improve the non-blind decon-volution part by using a network trained on blurry image as well asuse other restoration methods to remove JPEG compression artifactsfor example. https://goo.gl/p5Rndy . REFERENCES [1] R. Fergus, B. Singh, A. Hertzmann, S. T. Roweis, and W. T.Freeman, “Removing camera shake from a single photograph,” ACM Trans. Graph. , vol. 25, no. 3, pp. 787, 2006.[2] S. Cho and S. Lee, “Fast motion deblurring,”
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