1 Introduction
Image restoration tasks, such as deblurring and denoising, are illposed problems, whose solution requires effective image priors. In the last decades, several natural image priors have been proposed, including total variation RUDIN1992259 , gradient sparsity priors Fergus:2006:RCS , models based on image patches Buades2005NLM , and Gaussian mixtures of local filters Portilla2003SMG , just to name a few of the most successful ideas. See Figure 1
for a visual comparison of some popular priors. More recently, deep learning techniques have been used to construct generic image priors.
Here, we propose an image prior that is directly based on an estimate of the natural image probability distribution. Although this seems like the most intuitive and straightforward idea to formulate a prior, only few previous techniques have taken this route
Levin2011NID. Instead, most priors are built on intuition or statistics of natural images (e.g., sparse gradients). Most previous deep learning priors are derived in the context of specific algorithms to solve the restoration problem, but it is not clear how these priors relate to the probability distribution of natural images. In contrast, our prior directly represents the natural image distribution smoothed with a Gaussian kernel, an approximation similar to using a Gaussian kernel density estimate. Note that we cannot hope to use the true image probability distribution itself as our prior, since we only have a finite set of samples from this distribution. We show a visual comparison in Figure
1, where our prior is able to capture the structure of the underlying image, but others tend to simplify the texture to straight lines and sharp edges.We formulate image restoration as a Bayes estimator, and define a utility function that includes the smoothed natural image distribution. We approximate the estimator with a bound, and show that the gradient of the bound includes the gradient of the logarithm of our prior, that is, the Gaussian smoothed density. In addition, the gradient of the logarithm of the smoothed density is proportional to the meanshift vector Comaniciu:2002:MSR , and it has recently been shown that denoising autoencoders (DAEs) learn such a meanshift vector field for a given set of data samples JMLR:v15:alain14a ; bigdeli2017image . Hence we call our prior a deep meanshift prior
, and our framework is an example of Bayesian inference using deep learning.
We demonstrate image restoration using our prior for noiseblind deblurring, superresolution, and image demosaicing, where we solve Bayes estimation using a gradient descent approach. We achieve performance that is competitive with the state of the art for these applications. In summary, the main contributions of this paper are:

A formulation of image restoration as a Bayes estimator that leverages the Gaussian smoothed density of natural images as its prior. In addition, the formulation allows us to solve noiseblind restoration problems.

An implementation of the prior, which we call deep meanshift prior, that builds on denoising autoencoders (DAEs). We rely on the observation that DAEs learn a meanshift vector field, which is proportional to the gradient of the logarithm of the prior.

Image restoration techniques based on gradientdescent risk minimization with competitive results for noiseblind image deblurring, superresolution, and demosaicing.
Input  Our prior  BM3D dabov2006image  EPLL Zoran:2011:LMN  FoE roth2005fields  SF schmidt2014shrinkage 

2 Related Work
Image Priors.
A comprehensive review of previous image priors is outside the scope of this paper. Instead, we refer to the overview by Shaham et al. RottShaham2016 , where they propose a visualization technique to compare priors. Our approach is most related to techniques that leverage CNNs to learn image priors. These techniques build on the observation by Venkatakrishnan et al. venkatakrishnan2013plug that many algorithms that solve image restoration via MAP estimation only need the proximal operator of the regularization term, which can be interpreted as a MAP denoiser meinhardt2017learning . Venkatakrishnan et al. venkatakrishnan2013plug build on the ADMM algorithm and propose to replace the proximal operator of the regularizer with a denoiser such as BM3D or NLM. Unsurprisingly, this inspired several researchers to learn the proximal operator using CNNs chang2017one ; zhang2017learning ; xiao2017discriminative ; meinhardt2017learning . Meinhardt et al. meinhardt2017learning
consider various proximal algorithms including the proximal gradient method, ADMM, and the primaldual hybrid gradient method, where in each case the proximal operator for the regularizer can be replaced by a neural network. They show that no single method will produce systematically better results than the others.
A key difference to our approach is that, in the proximal techniques, the relation between the proximal operator of the regularizer and the natural image probability distribution remains unclear. In contrast, we explicitly use the Gaussiansmoothed natural image distribution as a prior, and we show that we can learn the gradient of its logarithm using a denoising autoencoder.
Romano et al. romano2016little designed a prior model that is also implemented by a denoiser, but that does not build on a proximal formulation such as ADMM. Interestingly, the gradient of their regularization term boils down to the residual of the denoiser, that is, the difference between its input and output, which is the same as in our approach. However, their framework does not establish the connection between the prior and the natural image probability distribution, as we do. Finally, Bigdeli and Zwicker bigdeli2017image formulate an energy function, where they used a Denoising Autoencoder (DAE) network for the prior, similar as in our approach, but they do not address the case of noiseblind restoration.
Noise and KernelBlind Deconvolution.
Kernelblind deconvolution has seen the most effort recently, while we support the fully (noise and kernel) blind setting. Noiseblind deblurring is usually performed by first estimating the noise level and then restoration with the estimated noise. Jin et al. Jin:2017:NBD
proposed a Bayes risk formulation that can perform deblurring by adaptively changing the regularization without the need of the noise variance estimate. Zhang et al.
NIPS2013_4864 ; NIPS2014_5566 explored a spatiallyadaptive sparse prior and scalespace formulation to handle noise or kernelblind deconvolution. These methods, however, are tailored specifically for image deconvolution. Also, they only handle the noise or kernelblind case, but not fully blind.3 Bayesian Formulation
We assume a standard model for image degradation,
(1) 
where is the unknown image, is the blur kernel, is zeromean Gaussian noise with variance , and is the observed degraded image. We restore an estimate of the unknown image by defining and maximizing an objective consisting of a data term and an image likelihood,
(2) 
Our core contribution is to construct a prior that corresponds to the logarithm of the Gaussiansmoothed probability distribution of natural images. We will optimize the objective using gradient descent, and leverage the fact that we can learn the gradient of the prior using a denoising autoencoder (DAE). We next describe how we define our objective by formulating a Bayes estimator in Section 3.1, then explain how we leverage DAEs to obtain the gradient of our prior in Section 3.2, describe our gradient descent approach in Section 3.3, and finally our image restoration applications in Section 4.
3.1 Defining the Objective via a Bayes Estimator
A typical approach to solve the restoration problem is via a maximum a posteriori (MAP) estimate, where one considers the posterior distribution of the restored image , derives an objective consisting of a sum of data and prior terms by taking the logarithm of the posterior, and maximizes it (minimizes the negative logposterior, respectively). Instead, we will compute a Bayes estimator for the restoration problem by maximizing the posterior expectation of a utility function,
(3) 
where denotes the utility function (e.g., a Gaussian). This is a generalization of MAP (where the utility is a Dirac impulse) and the utility function typically encourages its two arguments to be similar.
Ideally, we would like to use the true data distribution as the prior . But we only have data samples, hence we cannot learn this exactly. Therefore, we introduce a smoothed data distribution
(4) 
where
has a Gaussian distribution with zeromean and variance
, which is represented by the smoothing kernel . The key idea here is that it is possible to estimate the smoothed distribution or its gradient from sample data. In particular, we will need the gradient of its logarithm, which we will learn using denoising autoencoders (DAEs). We now define our utility function as(5) 
where we use the same Gaussian function
with standard deviation
as introduced for the smoothed distribution . This penalizes the estimate if the latent parameter is far from it. In addition, the term penalizes the estimate if its smoothed density is lower than the true density of the latent parameter. Unlike the utility in Jin et al. Jin:2017:NBD , this approach will allow us to express the prior directly using the smoothed distribution .By inserting our utility function into the posterior expected utility in Equation (3) we obtain
(6) 
where the true density canceled out, as desired, and we introduced the variable substitution .
We finally formulate our objective by taking the logarithm of the expected utility in Equation (6), and introducing a lower bound that will allow us to split Equation (6) into a data term and an image likelihood. By exploiting the concavity of the function, we apply Jensen’s inequality and get our objective as
(7) 
Image Likelihood.
We denote the image likelihood as
(8) 
The key observation here is that our prior expresses the image likelihood as the logarithm of the Gaussiansmoothed true natural image distribution , which is similar to a kernel density estimate.
Data Term.
Given that the degradation noise is Gaussian, we see that Jin:2017:NBD
(9) 
which will allow us to address noiseblind problems as we will describe in detail in Section 4.
3.2 Gradient of the Prior via Denoising Autoencoders (DAE)
A key insight of our approach is that we can effectively learn the gradients of our prior in Equation (8) using denoising autoencoders (DAEs). A DAE is trained to minimize Vincent:2008:ECR
(10) 
where the expectation is over all images and Gaussian noise with variance , and indicates that the DAE was trained with noise variance . Alain et al. JMLR:v15:alain14a show that the output of the optimal DAE (by assuming unlimited capacity) is related to the true data distribution as
(11) 
where the noise has a Gaussian distribution with standard deviation . This is simply a continuous formulation of meanshift, and corresponds to the smoothing kernel in our prior, Equation (8).
To obtain the relation between the DAE and the desired gradient of our prior, we first rewrite the numerator in Equation (11) using the Gaussian derivative definition to remove , that is
(12) 
where we used the Leibniz rule to interchange the operator with the integral. Plugging this back into Equation (11), we have
(13) 
One can now see that the DAE error, that is, the difference between the output of the DAE and its input, is the gradient of the image likelihood in Equation (8). Hence, a main result of our approach is that we can write the gradient of our prior using the DAE error,
(14) 
3.3 Stochastic Gradient Descent
We consider the optimization as minimization of the negative of our objective and refer to it as gradient descent. Similar to Bigdeli and Zwicker bigdeli2017image , we observed that the trained DAE is overfitted to noisy images. Because of the large gap in dimensionality between the embedding space and the natural image manifold, the vast majority of training inputs for the DAE lie at a distance very close to from the natural image manifold. Hence, the DAE cannot effectively learn meanshift vectors for locations that are closer than to the natural image manifold. In other words, our DAE does not produce meaningful results for input images that do not exhibit noise close to the DAE training .
To address this issue, we reformulate our prior to perform stochastic gradient descent steps that include noise sampling. We rewrite our prior from Equation (
8) as(15)  
(16)  
(17) 
where , we used the fact that two Gaussian convolutions are equivalent to a single convolution with a Gaussian whose variance is the sum of the two, and we applied Jensen’s inequality again. This leads to a new, even lower bound for the prior, which we call . Note that the bound proposed by Jin et al. Jin:2017:NBD corresponds to the special case where and .
We address our DAE overfitting issue by using the new, lower bound with . Its gradient is
(18) 
In practice, computing the integral over is not possible at runtime. Instead, we approximate the integral with a single noise sample, which leads to the stochastic evaluation of the gradient of the prior as
(19) 
where . This addresses the overfitting issue, since it means we add noise each time before we evaluate the DAE. Given the stochastically sampled gradient of the prior, we apply a gradient descent approach with momentum that consists of the following steps:
(20) 
where is the update step for at iteration , is the running step, and and are the momentum and stepsize.
2 NB:  1.  2.  3. 

NA:  1.  2.  3. 
KE:  4. ]  5.  6. 
2 
4 Image Restoration using the Deep MeanShift Prior
We next describe the detailed gradient descent steps, including the derivatives of the data term, for different image restoration tasks. We provide a summary in Table 1. For brevity, we omit the role of downsampling (required for superresolution) and masking.
NonBlind Deblurring (NB).
The gradient descent steps for nonblind deblurring with a known kernel and degradation noise variance are given in Table 1, top row (NB). Here denotes the Toeplitz matrix of the blur kernel .
NoiseAdaptive Deblurring (NA).
When the degradation noise variance is unknown, we can solve Equation (9) for the optimal (since its independent of the prior), which gives
(21) 
By plugging this back into the equation, we get the following data term
(22) 
which is independent of the degradation noise variance . We show the gradient descent steps in Table 1, second row (NA), where adaptively scales the data term with respect to the prior.
Noise and KernelBlind Deblurring (NA+KE).
Gradient descent in noiseblind optimization includes an intuitive regularization for the kernel. We can use the objective in Equation (22) to jointly optimize for the unknown image and the unknown kernel. The gradient descent steps to update the image remain as in Table 1, second row (NA), and we take additional steps to update the kernel estimate, as in Table 1, third row (KE). Additionally, we project the kernel by applying and after each step.
5 Experiments and Results
Our DAE uses the neural network architecture by Zhang et al. zhang2016beyond
. We generated training samples by adding Gaussian noise to images from ImageNet
deng2009imagenet . We experimented with different noise levels and found to perform well for all our deblurring and superresolution experiments. Unless mentioned, for image restoration we always take 300 iterations with step length and momentum . The runtime of our method is linear in the number of pixels, and our implementation takes about seconds per iteration for one megapixel on an Nvidia Titan X (Pascal).2  Levin levin2007image  Berkeley arbelaez2011contour  

Method  :  2.55  5.10  7.65  10.2  2.55  5.10  7.65  10.2 
FD krishnan2009fast  30.03  28.40  27.32  26.52  24.44  23.24  22.64  22.07  
EPLL Zoran:2011:LMN  32.03  29.79  28.31  27.20  25.38  23.53  22.54  21.91  
RTF6 schmidt2016cascades *  32.36  26.34  21.43  17.33  25.70  23.45  19.83  16.94  
CSF schmidt2014shrinkage  29.85  28.13  27.28  26.70  24.73  23.61  22.88  22.44  
DAEP bigdeli2017image  32.64  30.07  28.30  27.15  25.42  23.67  22.78  22.21  
IRCNN zhang2017learning  30.86  29.85  28.83  28.05  25.60  24.24  23.42  22.91  
EPLL Zoran:2011:LMN + NE  31.86  29.77  28.28  27.16  25.36  23.53  22.55  21.90  
EPLL Zoran:2011:LMN + NA  32.16  30.25  28.96  27.85  25.57  23.90  22.91  22.27  
TVL2 + NA  31.05  29.14  28.03  27.16  24.61  23.65  22.90  22.34  
GradNet 7S Jin:2017:NBD  31.43  28.88  27.55  26.96  25.57  24.23  23.46  22.94  
Ours  29.68  29.45  28.95  28.29  25.69  24.45  23.60  22.99  
Ours + NA  32.57  30.21  29.00  28.23  26.00  24.47  23.61  22.97  
2 
5.1 Image Deblurring: NonBlind and NoiseBlind
In this section we evaluate our method for image deblurring using two datasets. Table 2 reports the average PSNR for 32 images from the Levin et al. levin2007image and 50 images from the Berkeley arbelaez2011contour segmentation dataset, where 10 images are randomly selected and blurred with 5 kernels as in Jin et al. Jin:2017:NBD . We highlight the best performing PSNR in bold and underline the second best value. The upper half of the table includes nonblind methods for deblurring. EPLL Zoran:2011:LMN + NE uses a noise estimation step followed by nonblind deblurring. Noiseblind experiments are denoted by NA for noise adaptivity. We include our results for nonblind (Ours) and noiseblind (Ours + NA). Our noise adaptive approach consistently performs well in all experiments and in average we achieve better results than the state of the art. Figure 2 provides a visual comparison of our results. Our prior is able to produce sharp textures while also preserving the natural image structure.
Ground Truth  EPLL Zoran:2011:LMN  DAEP bigdeli2017image  GradNet 7S Jin:2017:NBD  Ours  Ours + NA 

5.2 Image Deblurring: Noise and KernelBlind
We performed fully blind deconvolution with our method using Levin et al.’s levin2007image dataset. In this test, we performed 1000 gradient descent iterations. We used momentum and step size for the unknown image and momentum and step size for the unknown kernel. Figure 3 shows visual results of fully blind deblurring and performance comparison to state of the art (last column). We compare the SSD error ratio and the number of images in the dataset that achieves error ratios less than a threshold. Results for other methods are as reported by Perrone and Favaro perrone2016logarithmic . Our method can reconstruct all the blurry images in the dataset with errors ratios less than 3.5. Note that our optimization performs endtoend estimation of the final results and we do not use the common two stage blind deconvolution (kernel estimation, followed by nonblind deconvolution). Additionally our method uses a noise adaptive scheme where we do not assume knowledge of the input noise level.
Ground Truth  Blurred with 1% noise  Ours (blind)  SSD Error Ratio 

5.3 Superresolution
To demonstrate the generality of our prior, we perform an additional test with single image superresolution. We evaluate our method on the two common datasets Set5 Bevilacqua:2012:LCS and Set14 Zeyde:2010:SIS for different upsampling scales. Since these tests do not include degradation noise (), we perform our optimization with a rough weight for the prior and decrease it gradually to zero. We compare our method in Table 3. The upper half of the table represents methods that are specifically trained for superresolution. SRCNN Dong:2016:ISR and TNRD chen2017trainable have separate models trained for scales, and we used the model for to produce the results. VDSR Kim:2016:AIS and DnCNN3 zhang2016beyond have a single model trained for scales, which we also used to produce results. The lower half of the table represents general priors that are not designed specifically for superresolution. Our method performs on par with state of the art methods over all the upsampling scales.
2  Set5 Bevilacqua:2012:LCS  Set14 Zeyde:2010:SIS  

Method  scale:  
Bicubic  31.80  28.67  26.73  25.32  28.53  25.92  24.44  23.46  
SRCNN Dong:2016:ISR  34.50  30.84  28.60  26.12  30.52  27.48  25.76  24.05  
TNRD chen2017trainable  34.62  31.08  28.83  26.88  30.53  27.60  25.92  24.61  
VDSR Kim:2016:AIS  34.50  31.39  29.19  25.91  30.72  27.81  26.16  24.01  
DnCNN3 zhang2016beyond  35.20  31.58  29.30  26.30  30.99  27.93  26.25  24.26  
DAEP bigdeli2017image  35.23  31.44  29.01  27.19  31.07  27.93  26.13  24.88  
IRCNN zhang2017learning  35.07  31.26  29.01  27.13  30.79  27.68  25.96  24.73  
Ours  35.16  31.38  29.16  27.38  30.99  27.90  26.22  25.01  
2 
5.4 Demosaicing
We finally performed a demosaicing experiment on the dataset introduced by Khashabi et al. khashabi2014joint . This dataset is constructed by taking RAW images from a Panasonic camera, where the images are downsampled to construct the ground truth data. Due to the down sampling effect, in this evaluation we train a DAE with noise standard deviation. The test dataset consists of 100 noisy images captured by a Panasonic camera using a Bayer color filter array (RGGB). We initialize our method with Matlab’s demosic function. To get even better initialization, we perform our initial optimization with a large degradation noise estimate () and then perform the optimization with a lower estimate (). We summarize the quantitative results in Table 4. Our method is again on par with the state of the art. Additionally, our prior is not trained for a specific color filter array and therefore is not limited to a specific subpixel order. Figure 4 shows a qualitative comparison, where our method produces much smoother results compared to the previous state of the art.
2 Matlab malvar2004high  RTF khashabi2014joint  Gharbi et al. gharbi2016deep  Gharbi et al. gharbi2016deep f.t.  SEM klatzer2016learning  Ours 
33.9  37.8  38.4  38.6  38.8  38.7 
2 
Ground Truth  RTF khashabi2014joint  Gharbi et al. gharbi2016deep  SEM klatzer2016learning  Ours 

6 Conclusions
We proposed a Bayesian deep learning framework for image restoration with a generic image prior that directly represents the Gaussian smoothed natural image probability distribution. We showed that we can compute the gradient of our prior efficiently using a trained denoising autoencoder (DAE). Our formulation allows us to learn a single prior and use it for many image restoration tasks, such as noiseblind deblurring, superresolution, and image demosaicing. Our results indicate that we achieve performance that is competitive with the state of the art for these applications. In the future, we would like to explore generalizing from Gaussian smoothing of the underlying distribution to other types of kernels. We are also considering multiscale optimization where one would reduce the Bayes utility support gradually to get a tighter bound with respect to maximum a posteriori. Finally, our approach is not limited to image restoration and could be exploited to address other inverse problems.
Acknowledgments.
MJ and PF acknowledge support from the Swiss National Science Foundation (SNSF) on project 200021153324.
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