1 Introduction
Many problems in image processing, including superresolution (Ledig et al., 2017; Haris et al., 2018), deblurring (Kupyn et al., 2019; Suin et al., 2020), inpainting (Yeh et al., 2017), colorization (Larsson et al., 2016; Zhang et al., 2016), and compressive sensing (Baraniuk, 2007), are instances of linear inverse problems, where the goal is to recover an image from potentially noisy measurements given through a known linear degradation model. For a specific degradation model, image restoration can be addressed through endtoend supervised
training of neural networks, using pairs of original and degraded images
(Dong et al., 2015; Zhang et al., 2016; Saharia et al., 2021a). However, realworld applications such as medical imaging often require flexibility to cope with multiple, possibly infinite, degradation models (Song et al., 2021b). Here, unsupervised approaches, where the degradation model is only known and used during inference, may be more desirable since they can adapt to the given problem without retraining (Venkatakrishnan et al., 2013). By learning sound assumptions over the underlying structure of images (e.g., priors, proximal operators or denoisers), unsupervised approaches can achieve effective restoration without training on specific degradation models (Venkatakrishnan et al., 2013; Romano et al., 2017).Under this unsupervised setting, priors based on deep neural networks have demonstrated impressive empirical results in various image restoration tasks (Romano et al., 2017; Ulyanov et al., 2018; Santurkar et al., 2019; Pan et al., 2020; Gu et al., 2020). To recover the signal, most existing methods obtain a priorrelated term over the signal from a neural network (e.g., the distribution of natural images), and a likelihood term from the degradation model. They combine the two terms to form a posterior over the signal, and the inverse problem can be posed as solving an optimization problem (e.g., maximum a posteriori (Calvetti & Somersalo, 2008; Romano et al., 2017)) or solving a sampling problem (e.g., posterior sampling (Bardsley, 2012; Bardsley et al., 2014; Kawar et al., 2021b)
). Then, these problems are often solved with iterative methods, such as gradient descent or Langevin dynamics, which may be demanding in computation and sensitive to hyperparameter tuning. An extreme example is found in
Laumont et al. (2021) where a “fast” version of the algorithm uses neural function evaluations (NFEs).Inspired by this unsupervised line of work, we introduce an efficient approach named Denoising Diffusion Restoration Models (DDRM), that can achieve competitive results in as low as NFEs. DDRM is a denoising diffusion generative model (Ho et al., 2020; Song et al., 2021a) that gradually and stochastically denoises a sample to the desired output, conditioned on the measurements and the inverse problem. This way we introduce a variational inference objective for learning the posterior distribution of the inverse problem at hand. We then show its equivalence to the objective of an unconditional denoising diffusion generative model (Ho et al., 2020), which enables us to deploy such models in DDRM for various linear inverse problems (see Figure 2). To our best knowledge, DDRM is the first general inverse problem solver that can efficiently produce a range of highquality, diverse, yet valid solutions for general content images.
We demonstrate the empirical effectiveness of DDRM by comparing with various competitive unsupervised methods, such as Deep Generative Prior (DGP, Pan et al. (2020)), SNIPS (Kawar et al., 2021b), and Regularization by Denoising (RED, Romano et al. (2017)). On ImageNet examples, DDRM mostly outperforms the neural network baselines under noiseless superresolution and deblurring measured in PSNR and KID (Bińkowski et al., 2018), and is at least more efficient in terms of NFEs when it is secondbest. Our advantage becomes even larger when measurement noise is involved, as noisy artifacts produced by iterative methods do not appear in our case. Over various realworld images, we further show DDRM results on superresolution, deblurring, inpainting and colorization (see Figure 1). A DDRM trained on ImageNet also works on images that are out of its training set distribution (see Figure 8).
2 Background
Linear Inverse Problems.
A general linear inverse problem is posed as
(1) 
where we aim to recover the signal from measurements , where is a known linear degradation matrix, and is an i.i.d.
additive noise with known variance.
The underlying structure of can be represented via a generative model, denoted as . Given and , a posterior over the signal can be posed as: , where the “likelihood” term is defined via Equation 1. Recovering can be done by sampling from this posterior (Bardsley, 2012), which may require many iterations to produce a good sample. Alternatively, one can also approximate this posterior by learning a model via amortized inference (i.e.
, supervised learning); the model learns to predict
given , generated from and a specific . While this can be more efficient than samplingbased methods, it may generalize poorly to inverse problems that have not been trained on.Denoising Diffusion Probabilistic Models.
Structures learned by generative models have been applied to various inverse problems and often outperform dataindependent structural constraints such as sparsity (Bora et al., 2017). These generative models learn a model distribution that approximates a data distribution from samples. In particular, diffusion models have demonstrated impressive unconditional generative modeling performance on images (Dhariwal & Nichol, 2021)
. Diffusion models are generative models with a Markov chain structure
(where), which has the following joint distribution:
After drawing , only is kept as the sample of the generative model. To train a diffusion model, a fixed, factorized variational inference distribution is introduced:
which leads to an evidence lower bound (ELBO) on the maximum likelihood objective (Song et al., 2021a).
A special property of some diffusion models is that both and
are chosen as conditional Gaussian distributions for all
, and that is also a Gaussian with known mean and covariance, i.e., can be treated asdirectly corrupted with Gaussian noise. Thus, the ELBO objective can be reduced into the following denoising autoencoder objective (please refer to
Song et al. (2021a) for derivations):(2) 
where is a parametrized neural network that aims to recover a noiseless observation from a noisy , and are a set of positive coefficients that depend on .
3 Denoising Diffusion Restoration Models
Inverse problem solvers based on posterior sampling often face a dilemma: unsupervised approaches apply to general problems but are inefficient, whereas supervised ones are efficient but can only address specific problems.
To solve this dilemma, we introduce Denoising Diffusion Restoration Models (DDRM), an unsupervised solver for general linear inverse problems, capable of handling such tasks with or without noise in the measurements. DDRM is efficient and exhibits competitive performance compared to popular unsupervised solvers (Romano et al., 2017; Pan et al., 2020; Kawar et al., 2021b).
The key idea behind DDRM is to find an unsupervised solution that also suits supervised learning objectives. First, we describe the variational objective for DDRM over a specific inverse problem (Section 3.1). Next, we introduce specific forms of DDRM that are suitable for linear inverse problems and allow pretrained unconditional and classconditional diffusion models to be used directly (Sections 3.2, 3.3). Finally, we discuss practical algorithms that are compute and memory efficient (Section 3.4).
3.1 Variational Objective for DDRM
For any linear inverse problem, we define DDRM as a Markov chain conditioned on , where
and is the final diffusion output. In order to perform inference, we consider the following factorized variational distribution conditioned on :
leading to an ELBO objective for diffsuion models conditioned on (details in Appendix A).
In the remainder of the section, we construct suitable variational problems given and and connect them to unconditional diffusion generative models. To simplify notations, we will construct the variational distribution such that for noise levels .^{1}^{1}1This is called “Variance Exploding” in (Song et al., 2021c). In Appendix B, we will show that this is equivalent to the distribution introduced in DDPM (Ho et al., 2020) and DDIM (Song et al., 2021a),^{2}^{2}2This is called “Variance Preserving” in (Song et al., 2021c).
up to fixed linear transformations over
.3.2 A Diffusion Process for Image Restoration
Similar to SNIPS (Kawar et al., 2021b)
, we consider the singular value decomposition (SVD) of
, and perform the diffusion in its spectral space. The idea behind this is to tie the noise present in the measurements with the diffusion noise in , ensuring that the diffusion result is faithful to the measurements. By using the SVD, we identify the data from that is missing in , and synthesize it using a diffusion process. In conjunction, the noisy data in undergoes a denoising process. For example, in inpainting with noise (e.g., , ), the spectral space is simply the pixel space, so the model should generate the missing pixels and denoise the observed ones in . For a general linear , its SVD is given as(3) 
where , are orthogonal matrices, and
is a rectangular diagonal matrix containing the singular values of
, ordered descendingly. As this is the case in most useful degradation models, we assume , but our method would work for as well. We denote the singular values as , and define for .We use the shorthand notations for values in the spectral space: is the
th index of the vector
, and is the th index of the vector (where denotes the Moore–Penrose pseudoinverse). Becauseis an orthogonal matrix, we can recover
from exactly by left multiplying . For each index in , we define the variational distribution as:(4)  
(5)  
where is a hyperparameter controlling the variance of the transitions, and and may depend on . We further assume that for all positive .^{3}^{3}3This assumption is fair, as we can set a sufficiently large .
In the following statement, we show that this construction has the “Gaussian marginals” property similar to the inference distribution used in unconditional diffusion models (Ho et al., 2020).
Proposition 3.1.
The conditional distributions defined in Equations 4 and 5 satisfy the following:
(6) 
defined by marginalizing over (for all ) and , where is defined as in Equation 1 with .
We place the proof in Appendix C. Intuitively, our construction considers different cases for each index of the spectral space. (i) If the corresponding singular value is zero, then does not directly provide any information to that index, and the update is similar to regular unconditional generation. (ii) If the singular value is nonzero, then the updates consider the information provided by , which further depends on whether the measurements’ noise level in the spectral space () is larger than the noise level in the diffusion model () or not; the measurements in the spectral space are then scaled differently for these two cases in order to ensure 3.1 holds.
Now that we have defined as a series of Gaussian conditionals, we define our model distribution as a series of Gaussian conditionals as well. Similar to DDPM, we aim to obtain predictions of at every step ; and to simplify notations, we use the symbol to represent this prediction made by a model^{4}^{4}4Equivalently, Ho et al. (2020) predict the noise values to subtract in order to recover . that takes in the sample and the conditioned time step . We also define as the th index of .
We define DDRM with trainable parameters as follows:
(7)  
(8)  
Compared to in Equations 4 and 5, our definition of merely replaces (which we do not know at sampling) with (which depends on our predicted ) when , and replaces with when . It is possible to learn the variances (Nichol & Dhariwal, 2021) or consider alternative constructions where 3.1 holds; we leave these options as future work.
3.3 “Learning” Image Restoration Models
Once we have defined and by choosing , and , we can learn model parameters by maximizing the resulting ELBO objective (in Appendix, Equation 10). However, this approach is not desirable since we have to learn a different model for each inverse problem (given and ), which is not flexible enough for arbitrary inverse problems. Fortunately, this does not have to be the case. In the following statement, we show that an optimal solution to DDPM / DDIM can also be an optimal solution to a DDRM problem, under reasonable assumptions used in prior work (Ho et al., 2020; Song et al., 2021a).
Theorem 3.2.
Assume that the models and are independent whenever , then when and , the ELBO objective of DDRM (details in Equation 10) can be rewritten in the form of the DDPM / DDIM objective in Equation 2.
We place the proof in Appendix C.
Even for different choices of and , the proof shows that the DDRM objective is a weighted sumofsquares error in the spectral space, and thus pretrained DDPM models are good approximations to the optimal solution. Therefore, we can apply the same diffusion model (unconditioned on the inverse problem) using the updates in Equation 7 and Equation 8 and only modify and its SVD (, , ) for various linear inverse problems.
3.4 Accelerated Algorithms for DDRM
Typical diffusion models are trained with many timesteps (e.g., 1000) to achieve optimal unconditional image synthesis quality, but sampling speed is slow as many NFEs are required. Previous works (Song et al., 2021a; Dhariwal & Nichol, 2021)
have accelerated this process by “skipping” steps with appropriate update rules. This is also true for DDRM, since we can obtain the denoising autoencoder objective in
Equation 2 for any choice of increasing . For a pretrained diffusion model with timesteps, we can choose to be a subset of the steps used in training.3.5 Memory Efficient SVD
Our method, similar to (Kawar et al., 2021b), utilizes the SVD of the degradation operator . This constitutes a memory consumption bottleneck in both algorithms, as storing the matrix has a space complexity of for signals of size . By leveraging special properties of the matrices used, we can reduce this complexity to for denoising, inpainting, super resolution, deblurring, and colorization. The detailed analyses are shown in Appendix D.
4 Related Work
Various deep learning solutions have been suggested for solving inverse problems under different settings (see a detailed survey by
Ongie et al. (2020)). We focus on the unsupervised setting, where we have access to a dataset of clean images at training time, but the degradation model is known only at inference time. This setup is inherently general to all linear inverse problems, a property desired in many realworld applications such as medical imaging (Song et al., 2021b; Jalal et al., 2021a).Almost all unsupervised inverse problem solvers utilize a trained neural network in an iterative scheme. PnP, RED, and their successors (Venkatakrishnan et al., 2013; Romano et al., 2017; Mataev et al., 2019; Sun et al., 2019) apply a denoiser as part of an iterative optimization algorithm such as steepest descent, fixedpoint, or alternating direction method of multipliers (ADMM). OneNet (Rick Chang et al., 2017) trained a network to directly learn the proximal operator of ADMM. A similar use of denoisers in different iterative algorithms is proposed in (Metzler et al., 2017; Guo et al., 2019; Laumont et al., 2021). Santurkar et al. (2019)
leverages robust classifiers learned with additional class labels.
Another approach is to search the latent space of a generative model for a generated image that, when degraded, is as close as possible to the given measurements. Multiple such methods were suggested, mainly focusing on generative adversarial networks (GANs)
(Bora et al., 2017; Daras et al., 2021; Menon et al., 2020). While they exhibit impressive results on images of a specific class, most notably face images, these methods are not shown to be largely successful when considering a more diverse dataset such as ImageNet (Deng et al., 2009). Deep Generative Prior (DGP) mitigates this issue by optimizing the latent input as well as the weights of the GAN’s generator (Pan et al., 2020).More recently, denoising diffusion models were used to solve inverse problems in both supervised (i.e., degradation model is known during training) (Saharia et al., 2021b, a; Dhariwal & Nichol, 2021; Chung et al., 2021; Whang et al., 2021) and unsupervised settings (Kadkhodaie & Simoncelli, 2021; Kawar et al., 2021a, b; Jalal et al., 2021b; Song et al., 2021b, c; Choi et al., 2021). Unlike previous approaches, diffusionbased methods can successfully recover images from measurements with significant noise. However, these methods are very slow, often requiring hundreds or thousands of iterations, and are yet to be proven on diverse datasets. Our method, motivated by variational inference, obtains problemspecific, nonequilibrium update rules that lead to highquality solutions in much fewer iterations.
ILVR (Choi et al., 2021) suggests a diffusionbased method that handles noiseless superresolution, and can run in steps. In Appendix G, we prove that when applied on the same underlying generative diffusion model, ILVR is a special case of DDRM. Therefore, ILVR can be further accelerated to run in steps, but unlike DDRM, it provides no clear way of handling noise in the measurements.
5 Experiments
Method  PSNR  KID  NFEs 

Bicubic  
DGP  
RED  
SNIPS  
DDRM  
DDRMCC 
Method  PSNR  KID  NFEs 

Blurry  
DGP  
RED  
SNIPS  
DDRM  
DDRMCC 
5.1 Experimental Setup
We demonstrate our algorithm’s capabilities using the diffusion models from (Ho et al., 2020), which are trained on CelebAHQ (Karras et al., 2018), LSUN bedrooms, and LSUN cats (Yu et al., 2015) (all pixels). We test these models on images from FFHQ (Karras et al., 2019), and pictures from the internet of the considered LSUN category, respectively. In addition, we use the models from (Dhariwal & Nichol, 2021), trained on the training set of ImageNet and , and tested on the corresponding validation set. Some of the ImageNet models require class information. For these models, we use the ground truth labels as input, and denote our algorithm as DDRM class conditional (DDRMCC). In all experiments, we use , , and a uniformlyspaced timestep schedule based on the 1000step pretrained models (more details in Appendix E). The number of NFEs (timesteps) is reported in each experiment.
In each of the inverse problems we show, pixel values are in the range , and the degraded measurements are obtained as follows: (i) for superresolution, we use a block averaging filter to downscale the images by a factor of , , or in each axis; (ii) for deblurring, the images are blurred by a uniform kernel; (iii) for colorization, the grayscale image is an average of the red, green, and blue channels of the original image; (iv) and for inpainting, we mask parts of the original image with text overlay or randomly drop of the pixels. Additive white Gaussian noise can optionally be added to the measurements in all inverse problems.
Method  PSNR  KID  NFEs 

Bicubic  
DGP  
RED  
SNIPS  
DDRM  
DDRMCC 
Method  PSNR  KID  NFEs 

Blurry  
DGP  
RED  
SNIPS  
DDRM  
DDRMCC 
Original  Lowres  DDRM () 
SNIPS  RED  DGP 
Original  Occluded  DDRM () 
row2¡ 6  

Original  Blurred  DDRM () 
Deblurring results on bedroom images. Blurred images contain noise of standard deviation
.Original  Noisy  DDRM ()  Denoised 
‘teapot’ 
row0¡ 4  
‘brown bear’ 
row0¡ 4  
Original  Grayscale  Samples from DDRMCC () 
row0¡ 6  

Original  Blurred  DDRM (20) 
5.2 Quantitative Experiments
In order to quantify DDRM’s performance, we focus on the ImageNet dataset (
) for its diversity. For each experiment, we report the average peak signaltonoise ratio (PSNR) to measure faithfulness to the original image, and the kernel Inception distance (KID)
(Bińkowski et al., 2018), multiplied by , to measure the resulting image quality.We compare DDRM (with and steps) with other unsupervised methods that work in reasonable time (requiring NFEs or less) and can operate on ImageNet. Namely, we compare with RED (Romano et al., 2017), DGP (Pan et al., 2020), and SNIPS (Kawar et al., 2021b). The exact setup of each method is detailed in Appendix F
. We used the same hyperparameters for noisy and noiseless versions of the same problem for DGP, RED, and SNIPS, as tuning them for each version would compromise their unsupervised nature. In addition, we show upscaling by bicubic interpolation as a baseline for superresolution, and the blurry image itself as a baseline for deblurring. OneNet
(Rick Chang et al., 2017) is not included in the comparisons as it is limited to images of size , and generalization to higher dimensions requires an improved network architecture.We evaluate all methods on the problems of superresolution and deblurring, on one validation set image from each of the ImageNet classes, following (Pan et al., 2020). Tables 1 and 2 show that DDRM outperforms all baseline methods, in all metrics, and on both problems with only steps. The only exception to this is that SNIPS achieves better KID than DDRM in noiseless deblurring, but it requires more NFEs to do so. DGP and DDRMCC use groundtruth class labels for the test images to aid in the restoration process, and thus have an unfair advantage.
DDRM’s appeal compared to previous methods becomes more substantial when significant noise is added to the measurements. Under this setting, DGP, RED, and SNIPS all fail to produce viable results, as evident in Tables 3 and 4 and Figure 3. Since DDRM is fast, we also evaluate it on the entire ImageNet validation set in Appendix F.
5.3 Qualitative Experiments
DDRM produces high quality reconstructions across all the tested datasets and problems, as can be seen in Figures 1, 4, 5, 6, and in Appendix H. The denoiser used in Figure 6 is the denoising diffusion function used by DDRM, where minimizes . As it is a posterior sampling algorithm, DDRM can produce multiple outputs for the same input, as demonstrated in Figure 7. Moreover, the unconditional ImageNet diffusion models can be used to solve inverse problems on outofdistribution images with general content. In Figure 8, we show DDRM successfully restoring images from USCSIPI (Weber, 1997) that do not necessarily belong to any ImageNet class (additional results in Figure 14, Appendix H).
6 Conclusions
We have introduced DDRM, a general linear inverse problem solver based on unconditional/classconditional diffusion generative models. Motivated by variational inference, DDRM only requires a few number of NFEs (e.g., 20) compared to other baselines (e.g., 1000 for SNIPS) and achieves scalability in multiple useful scenarios, including denoising, superresolution, deblurring, inpainting, and colorization. We demonstrate the empirical successes of DDRM on various problems and datasets, including general natural images outside the distribution of the observed training set. To our best knowledge, DDRM is the first unsupervised method that effectively and efficiently samples from the posterior distribution of inverse problems with significant noise, and can work on natural images with general content.
In terms of future work, apart from further optimizing the timestep and variance schedules, it would be interesting to investigate the following: (i) applying DDRM to nonlinear inverse problems, (ii) addressing scenarios where the degredation operator is unknown, and (iii) selfsupervised training techniques inspired by DDRM as well as ones used in supervised techniques (Saharia et al., 2021a) that further improve performance of unsupervised models for image restoration.
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Appendix A Details of the DDRM ELBO objective
DDRM is a Markov chain conditioned on , which would lead to the following ELBO objective (Song et al., 2021a):
(9)  
(10) 
where is the data distribution, follows Equation 1, the expectation on the right hand side is given by sampling , , , and for .
Appendix B Equivalence between “Variance Preserving” and “Variance Exploding” Diffusion Models
In our main paper, we describe our methods based on the “Variance Exploding” hyperparameters , where and
(11) 
In DDIM (Song et al., 2021a), the hyperparameters are “Variance Preserving” ones , where and
(12) 
We use the colored notation to emphasize that this is different from (an exception is ). Using the reparametrization trick, we have that:
(13)  
(14) 
where . We can divide by in both sides of Equation 13:
(15) 
Let , and let ; then from Equation 15 we have that
(16) 
which is equivalent to the “Variance Preserving” case. Therefore, we can use “Variance Preserving” models, such as DDPM, directly in our DDRM updates, even though the latter uses the “Variance Exploding” parametrization:

From , obtain predictions and .

From and , apply DDRM updates to get .

From , get .
Note that although the inference algorithms are shown to be equivalent, the choice between ”Variance Preserving” and ”Variance Exploding” may affect the training of diffusion networks.
Appendix C Proofs
See 3.1
Proof.
The proof uses a basic property of Gaussian marginals (see (Bishop, 2006) for the complete version).

If , , then .

If and , then .
First, we note that is defined from Equation 1, and thus for all :
(17) 
Case II For any and such that and , we have Equation 17 and that:
(19) 
and thus we can safely remove the dependence on via marginalization. is a Gaussian with the mean being and variance being