Original: S OLVING I NVERSE P ROBLEMS IN M EDICAL I MAGING WITH S CORE -B ASED G ENERATIVE M ODELS
Revised: S OLVING I NVERSE P ROBLEMS IN M EDICAL I MAGING WITH S CORE -B ASED G ENERATIVE M ODELS

Original: Anonymous authors
Revised: Anonymous authors

Original: Paper under double-blind review
Revised: Paper under double-blind review

Original: A BSTRACT
Revised: A BSTRACT

Original: Reconstructing medical images from partial measurements is an important inverse problem in Computed Tomography (CT) and Magnetic Resonance Imaging (MRI).
Revised: Reconstructing medical images from partial measurements is an important inverse problem in Computed Tomography (CT) and Magnetic Resonance Imaging (MRI).

Original: Existing solutions from machine learning typically train a model to directly map measurements to medical images, with a training dataset comprising paired images and measurements.
Revised: Existing solutions based on machine learning typically train a model to directly map measurements to medical images, leveraging a training dataset of paired images and measurements.

Original: These measurements are  synthesized from images using a ﬁxed physical model of the measurement process, which consequently limits the generalization capability of models to unknown measurement processes.
Revised: These measurements are typically synthesized from images using a ﬁxed physical model of the measurement process, which hinders the generalization capability of models to unknown measurement processes.

Original: To address this issue, we propose a fully unsupervised technique for inverse problem solving, leveraging the recently introduced score-based generative models.
Revised: To address this issue, we propose a fully unsupervised technique for inverse problem solving, leveraging the recently introduced score-based generative models.

Original: Speciﬁcally, we  train a score-based generative model on medical images to capture their prior distribution.
Revised: Speciﬁcally, we ﬁrst train a score-based generative model on medical images to capture their prior distribution.

Original: Given a measurement and a physical model of the measurement process, we propose a sampling method to reconstruct an image consistent with both the prior and the observed measurement.
Revised: Given measurements and a physical model of the measurement process at test time, we introduce a sampling method to reconstruct an image consistent with both the prior and the observed measurements.

Original: Our method does not assume a ﬁxed measurement process during training, and can thus be ﬂexibly adapted to different measurement processes at test time.
Revised: Our method does not assume a ﬁxed measurement process during training, and can thus be ﬂexibly adapted to different measurement processes at test time.

Original: Empirically, we observe comparable or better performance to supervised learning techniques in several medical imaging tasks, including sparse-view CT andundersampled MRI, while demonstrating remarkably better generalization to unknown measurement processes.
Revised: Empirically, we observe comparable or better performance to supervised learning techniques in several medical imaging tasks in CT and MRI, while demonstrating signiﬁcantly better generalization to unknown measurement processes.

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: Computed Tomography (CT) and Magnetic Resonance Imaging (MRI) are commonly used imaging tools for medical diagnosis.
Revised: Computed Tomography (CT) and Magnetic Resonance Imaging (MRI) are commonly used imaging tools for medical diagnosis.

Original: Reconstructing CT and MRI images from raw measurements (sinograms for CT and k-spaces for MRI) are well-known inverse problems.
Revised: Reconstructing CT and MRI images from raw measurements (sinograms for CT and k-spaces for MRI) are well-known inverse problems.

Original: Speciﬁcally, measurements in
Revised: Speciﬁcally, measurements in

Original: CT are given by X-ray projections of an object from various directions, and measurements in MRI are obtained by inspecting the Fourier spectrum of an object with magnetic ﬁelds.
Revised: CT are given by X-ray projections of an object from various directions, and measurements in MRI are obtained by inspecting the Fourier spectrum of an object with magnetic ﬁelds.

Original: However, since obtaining the full sinogram for CT causes excess ionizing radiation for patients, and measuring the full k-space of MRI is very time-consuming, it has become important to reduce the number of measurements in CT and MRI.
Revised: However, since obtaining the full sinogram for CT causes excessive ionizing radiation for patients, and measuring the full k-space of MRI is very time-consuming, it has become important to reduce the number of measurements in CT and MRI.

Original: In many cases, only partial measurements, such as sparse-view sinograms and downsampled k-spaces, are available.
Revised: In many cases, only partial measurements, such as sparse-view sinograms and downsampled k-spaces, are available.

Original: Due to this loss of information, the inverse problems in CT and MRI are often ill-posed, making image reconstruction especially challenging.
Revised: Due to this loss of information, the inverse problems in CT and MRI are often ill-posed, making image reconstruction especially challenging.

Original: With the rise of machine learning, many methods (Zhu et al., 2018; Mardani et al., 2017;
Revised: With the rise of machine learning, many methods (Zhu et al., 2018; Mardani et al., 2017;

Original: Shen et al., 2019; Würﬂ et al., 2018; Ghani & Karl, 2018;
Revised: Shen et al., 2019; Würﬂ et al., 2018; Ghani & Karl, 2018;

Original: Wei et al., 2020) have been proposed to perform medical image reconstruction given a small number of measurements.
Revised: Wei et al., 2020) have been proposed for medical image reconstruction using a small number of measurements.

Original: Most of these methods are supervised learning techniques.
Revised: Most of these methods are supervised learning techniques.

Original: They learn to directly map partial measurements to medical images, by training on a large dataset comprising pairs of CT/MRI images and measurements.
Revised: They learn to directly map partial measurements to medical images, by training on a large dataset comprising pairs of CT/MRI images and measurements.

Original: These measurements need to be synthesized from medical images with a ﬁxed physical model of the measurement process.
Revised: These measurements need to be synthesized from medical images with a ﬁxed physical model of the measurement process.

Original: However, when the measurement process changes, such as using a different number of CT projections or different downsampling ratio of MRI k-spaces, we need to re-collect the paired dataset with the new measurement process and re-train the model.
Revised: However, when the measurement process changes, such as using a different number of CT projections or different downsampling ratio of MRI k-spaces, we have to re-collect the paired dataset with the new measurement process and re-train the model.

Original: This prevents models from generalizing effectively to  measurement processes that are unknown at the time of training, leading to counter-intuitive instabilities such as more measurements causing worse performance (Antun et al., 2020).
Revised: This prevents models from generalizing effectively to new measurement processes, leading to counter-intuitive instabilities such as more measurements causing worse performance (Antun et al., 2020).

Original: In this work, we sidestep this difﬁculty completely by proposing unsupervised methods that do not require a paired dataset for training, and therefore are not restricted to a ﬁxed measurement process.
Revised: In this work, we sidestep this difﬁculty completely by proposing unsupervised methods that do not require a paired dataset for training, and therefore are not restricted to a ﬁxed measurement process.

Original: Our main idea is to learn the prior distribution of medical images with a generative model in order to infer the lost information due to partial measurements.
Revised: Our main idea is to learn the prior distribution of medical images with a generative model in order to infer the lost information due to partial measurements.

Original: Speciﬁcally, we propose to train a score-based generative model (Song & Ermon, 2019; 2020; Song et al., 2021) on medical images as the data prior, due to its strong performance in image generation (Ho et al., 2020; Dhariwal & Nichol, 2021).
Revised: Speciﬁcally, we propose to train a score-based generative model (Song & Ermon, 2019; 2020; Song et al., 2021) on medical images as the data prior, due to its strong performance in image generation (Ho et al., 2020; Dhariwal & Nichol, 2021).

Original: Given a measurement observation, and a physical model ofthe measurement process at test time, we can incorporate them into many existing sampling algorithmsof score-based generative models to create image samples that areboth consistent with the observed measurements andcompatible with the estimated data prior.
Revised: Given a trained score-based generative model, we provide a family of sampling algorithms to create image samples that are consistent with the observed measurements and the estimated data prior, leveraging the physical measurement process.

Original: Once our model is trained, it can be used to solve any inverse problem within the same image domain, as long as the mapping from images to measurements is linear, which holds for a large number of medical imaging applications.
Revised: Once our model is trained, it can be used to solve any inverse problem within the same image domain, as long as the mapping from images to measurements is linear, which holds for a large number of medical imaging applications.

Original: We evaluate the performance of our method on several tasks inmedical imaging, with a focus on image reconstruction for sparse-view CT andundersampled MRI.
Revised: We evaluate the performance of our method on several tasks in CT and MRI.

Original: Empirically, we observe comparable or better performance compared to supervised learning counterparts, even when evaluated with the same measurement process in their training.
Revised: Empirically, we observe comparable or better performance compared to supervised learning counterparts, even when evaluated with the same measurement process in their training.

Original: In addition, we are able to uniformly surpass all baselines when changing the number of measurements, e.g ., using a different number of projections in sparse-view CT or changing the k-space downsampling ratio in undersampled MRI.
Revised: In addition, we are able to uniformly surpass all baselines when changing the number of measurements, e.g ., using a different number of projections in sparse-view CT or changing the k-space downsampling ratio in undersampled MRI.

Original: Moreover, we show that by plugging in a different measurement process, we can use a single model to perform both sparse-view CT reconstruction and metal artifact removal for CT imaging with metallic implants.
Revised: Moreover, we show that by plugging in a different measurement process, we can use a single model to perform both sparse-view CT reconstruction and metal artifact removal for CT imaging with metallic implants.

Revised: To the best of our knowledge, this is the ﬁrst time that generative models are reported successful on clinical CT data.

Original: These empirical results indicate that our method is a competitive alternative to supervised techniques in medical image reconstruction and artifact removal, and has the potential to be a universal tool to solve many inverse problems within the same image domain.
Revised: Collectively, these empirical results indicate that our method is a competitive alternative to supervised techniques in medical image reconstruction and artifact removal, and has the potential to be a universal tool for solving many inverse problems within the same image domain.

Original:  B ACKGROUND 2. L INEAR INVERSE PROBLEMS
Revised:  B ACKGROUND 2. L INEAR INVERSE PROBLEMS

Original: The goal in an inverse problem is to recover an unknown signal from a set of observed measurements.
Revised: An inverse problem seeks to recover an unknown signal from a set of observed measurements.

Original: Speciﬁcally, suppose x P R n is an unknown signal, and y P R m “ A x  is an observation given by m linear measurements, where the measurement acquisition process is represented by a linear operator A P R m ˆ n .
Revised: Speciﬁcally, suppose x P R n is an unknown signal, and y P R m “ A x ` (cid:15) is a noisy observation given by m linear measurements, where the measurement acquisition process is represented by a linear operator A P R m ˆ n , and (cid:15) P R n represents a noise vector.

Original: Solving a linear inverse problem amounts to ﬁnding theunderlying signal x that satisﬁes A x “ y .
Revised: Solving a linear inverse problem amounts to recovering the signal x from its measurement y .

Original: It is typically desirable to keep the number of measurements m small.

Original: However, without further assumptions, the problem is ill-deﬁned when m ă n , so we additionally assume that x is sampled from aknown prior distribution p p x q .
Revised: Without further assumptions, the problem is ill-deﬁned when m ă n , so we additionally assume that x is sampled from a prior distribution p p x q .

Original: In this probabilistic formulation, the observation and signal are connected through a measurement distribution p p y | x q “ δ py ´
Revised: In this probabilistic formulation, the measurement and signal are connected through a measurement distribution p p y | x q “ q (cid:15) p

Original:  A x q , where δ p¨q is the Dirac delta function.
Revised: y ´ A x q , where q (cid:15) denotes the noise distribution of (cid:15) .

Original: We can solve the inverse problem by sampling from the posterior distribution p p x | y q, which is deﬁned given p p x q and p p y | x q .
Revised: Given p p y | x q and p p x q , we can solve the inverse problem by sampling from the posterior distribution p p x | y q .

Original: Examples of linear inverse problems in medical imaging include image reconstruction for CT and
Revised: Examples of linear inverse problems in medical imaging include image reconstruction for CT and

Original: MRI.
Revised: MRI.

Original: In both cases, the signal x is a medical image.
Revised: In both cases, the signal x is a medical image.

Original: The measurement y in CT is a sinogram formed by X-ray projections of the image from various angular directions (Buzug, 2011), while the measurement y in MRI consists of spatial frequencies in the Fourier space of the image (a.k.a.
Revised: The measurement y in CT is a sinogram formed by X-ray projections of the image from various angular directions (Buzug, 2011), while the measurement y in MRI consists of spatial frequencies in the Fourier space of the image (a.k.a.

Original: the k-space in the MRI community) (Vlaardingerbroek & Boer, 2013).
Revised: the k-space in the MRI community) (Vlaardingerbroek & Boer, 2013).

Original: 2. S CORE - BASED GENERATIVE MODELS
Revised: 2. S CORE - BASED GENERATIVE MODELS

Original: When solving inverse problems in medical imaging, we are given a likelihood function p p y | x q and aims to sample from the posterior distribution p p x | y q .
Revised: When solving inverse problems in medical imaging, we are given an observation y , the measurement distribution p p y | x q and aim to sample from the posterior distribution p p x | y q .

Original: However, the prior distribution p p x q is unknown. Given a dataset t x  1  , x  2  , ¨ ¨ ¨ , x  N  u „ p p x q, we can train probabilistic generative models to infer the prior distribution p p x q and generate samples from it.
Revised: The prior distribution p p x q is typically unknown, but we can train generative models on a dataset t x p 1 q , x p 2 q , ¨ ¨ ¨ , x p N q u „ p p x q to estimate this prior distribution.

Original: These generative models are useful for sampling from the posterior p p x| y q when combined with knowledge of the likelihood p p y | x q by virtue of Bayes’ rule.
Revised: Given an estimate of p p x q and the measurement distribution p p y | x q , the posterior distribution p p x | y q can be determined through Bayes’ rule.

Original: In particular, we propose to estimate the prior distribution of medical images using the recently introduced score-based generative models (Song & Ermon, 2019; Ho et al., 2020; Song et al., 2021), whose iterative sampling procedure makes it especially easy for controllable generation conditioned on an observation y .
Revised: We propose to estimate the prior distribution of medical images using the recently introduced scorebased generative models (Song & Ermon, 2019; Ho et al., 2020; Song et al., 2021), whose iterative sampling procedure makes it especially easy for controllable generation conditioned on an observation y .

Original: Speciﬁcally, we adopt the formulation of score-based generative models in Song et al.
Revised: Speciﬁcally, we adopt the formulation of score-based generative models in Song et al.

Original: (2021), where we leverage a Markovian diffusion process to progressively perturb data to noise, and then smoothly convert noise to samples of the data distribution by estimating and simulating its time reversal.
Revised: (2021), where we leverage a Markovian diffusion process to progressively perturb data to noise, and then smoothly convert noise to samples of the data distribution by estimating and simulating its time reversal.

Original: The reverse process requires estimating scores , i.e ., gradients of the logarithmic probability density functions, from a training dataset.

Original: We provide an illustration of this generative modeling framework in Fig.
Revised: We provide an illustration of this generative modeling framework in Fig.

Original: Perturbation process Suppose the dataset is sampled from an unknown data distribution p p x q .
Revised: Perturbation process Suppose the dataset is sampled from an unknown data distribution p p x q .

Original: For datapoints from p p x q , we perturb them with a stochastic process over a time horizon r 0 , 1 s , governed by a linear stochastic differential equation (SDE) of the following form
Revised: We perturb datapoints with a stochastic process over a time horizon r 0 , 1 s , governed by a linear stochastic differential equation (SDE) of the following form

Original: [Equation]
Revised: [Equation]

Original: where f : r 0 , 1 s Ñ R , g : r 0 , 1 s Ñ R , t w t P R n u t Pr 0 , 1 s denotes a standard Wiener process (a.k.a., Brownian motion), and t x t P R n u t Pr 0 , 1 s symbolizes the trajectory of random variables in the stochastic process.
Revised: where f : r 0 , 1 s Ñ R , g : r 0 , 1 s Ñ R , t w t P R n u t Pr 0 , 1 s denotes a standard Wiener process (a.k.a., Brownian motion), and t x t P R n u t Pr 0 , 1 s symbolizes the trajectory of random variables in the stochastic process.

Original: We further denote the marginal probability distribution of x t as p t p x q , and the transition distribution from x 0 to x t as p 0 t p x t | x 0 q .
Revised: We further denote the marginal probability distribution of x t as p t p x q , and the transition distribution from x 0 to x t as p 0 t p x t | x 0 q .

Original: By deﬁnition, we clearly have p 

Revised: By deﬁnition, we clearly have p

Original: p x q ” p p x q .
Revised: p x q ” p p x q .

Original: Moreover, the transition density p 0 t p x t | x 0 q is always a conditional linear Gaussian distribution, taking the form p 0 t p x t | x 0 q “ N p x t | α p t q x 0 , β 2 p t q I q where α : r 0 , 1 s Ñ R and β : r 0 , 1 s Ñ R can be derived analytically from f p t q and g p t q (Särkkä & Solin, 2019).

Original: The functions f p t q and g p t q are speciﬁcally chosen such that for any initial distribution p 

Revised: Moreover, the functions f p t q and g p t q are speciﬁcally chosen such that for any initial distribution p

Original: p x q , the distribution at the end of the perturbation process, p 1 p x q , is close to a pre-deﬁned noise distribution π p x q .
Revised: p x q , the distribution at the end of the perturbation process, p 1 p x q , is close to a pre-deﬁned noise distribution π p x q .

Revised: In addition, the transition density p 0 t p x t | x 0 q is always a conditional linear

Revised: Gaussian distribution, taking the form p 0 t p x t | x 0 q “ N p x t | α p t q x 0 , β 2 p t

Revised: q

Revised: I q where α : r 0 , 1 s Ñ R and β : r 0 , 1 s Ñ R can be derived analytically from f p t q and g p t q (Särkkä & Solin, 2019).

Original: Examples of such SDEs include Variance Exploding (VE), Variance Preserving (VP), and subVP SDEs proposed in Song et al.
Revised: Examples of such SDEs include Variance Exploding (VE), Variance Preserving (VP), and subVP SDEs proposed in Song et al.

Original: We will use the VE SDE throughout our experiments.
Revised: We found VE SDEs performed the best in our experiments.

Original: Reverse process By reversing the perturbation process in Eq.
Revised: Reverse process By reversing the perturbation process in Eq.

Original: (1), we can start from a noise sample x 1 „ p 1 p x q and gradually remove the noise therein to obtain a data sample x 0 „ p 0 p x q ” p p x q .
Revised: (1), we can start from a noise sample x 1 „ p 1 p x q and gradually remove the noise therein to obtain a data sample x 0 „ p 0 p x q ” p p x q .

Original: Crucially, the time reversal of Eq.
Revised: Crucially, the time reversal of Eq.

Original: (1) is given by the following reverse-time SDE (Anderson, 1982;
Revised: (1) is given by the following reverse-time SDE (Song et al., 2021)

Original: Haussmann & Pardoux, 1986)

Original: [Equation]
Revised: [Equation]

Original: where t ¯ w t u t Pr 0 , 1 s denotes a standard Wiener process in the reverse-time direction, and d t represents an inﬁnitesimal negative time step, since the above SDE must be solved backwards from t “ 1 to t “ 0 .
Revised: where t ¯ w t u t Pr 0 , 1 s denotes a standard Wiener process in the reverse-time direction, and d t represents an inﬁnitesimal negative time step, since the above SDE must be solved backwards from t “ 1 to t “ 0 .

Original: The quantity ∇ x t log p t p x t q is known as the score function of p t p x t q .
Revised: The quantity ∇ x t log p t p x t q is known as the score function of p t p x t q .

Original: By the deﬁnition of time reversal, the trajectory of the reverse stochastic process given by Eq.
Revised: By the deﬁnition of time reversal, the trajectory of the reverse stochastic process given by Eq.

Original: (2) is t x t u t Pr 0 , 1 s , same as the one from the forward SDE in Eq.
Revised: (2) is t x t u t Pr 0 , 1 s , same as the one from the forward SDE in Eq.

Original: Sampling Given an initial sample from p 1 p x q , as well as scores at each intermediate time step, ∇ x log p t p x q , we can simulate the reverse-time SDE in Eq.
Revised: Sampling Given an initial sample from p 1 p x q , as well as scores at each intermediate time step, ∇ x log p t p x q , we can simulate the reverse-time SDE in Eq.

Original: (2) to obtain samples from the data distribution p 

Revised: (2) to obtain samples from the data distribution p

Original: p x q ” p p x q .
Revised: p x q ” p p x q .

Original: In practice, the initial sample is approximately drawn from π p x q since π p x q « p 1 p x q , and the scores are estimated by training a neural network s θ p x , t q (named the score model ) on a dataset t x  1  , x  2  , ¨ ¨ ¨ , x  N  u „ p p x q with denoising score matching (Vincent, 2011; Song et al., 2021), such that s θ ˚ p x , t q « ∇ x log p t p x q , where θ ˚ denotes the optimal model parameters.
Revised: In practice, the initial sample is approximately drawn from π p x q since π p x q « p 1 p x q , and the scores are estimated by training a neural network s θ p x , t q (named the score model ) on a dataset t x p 1 q , x p 2 q , ¨ ¨ ¨ , x p N q u „ p p x q with denoising score matching (Vincent, 2011; Song et al., 2021), i.e ., solving the following objective [Equation] where U r 0 , 1 s denotes a uniform distribution over r 0 , 1 s .

Revised: The theory of denoising score matching ensures that s θ ˚ p x , t q « ∇ x log p t p x q .

Revised: After training this score model, we plug it into Eq.

Original: As an example, we provide one sampling method in Algorithm 1, which uses the Euler-Maruyama discretization to solve the reverse-time SDE in Eq.
Revised: (2) and solve the resulting reverse-time SDE [Equation] for sample generation. One sampling method is to use the Euler-Maruyama discretization for solving

Original: (2) with s θ ˚ p x , t q and π p x q plugged in.
Revised: Eq. (3), as given in Algorithm 1.

Original: Other sampling methods include annealed Langevin dynamics (ALD, Song & Ermon, 2019),  and PredictorCorrector samplers (Song et al., 2021).
Revised: Other sampling methods include annealed Langevin dynamics (ALD, Song & Ermon, 2019), probability ﬂow ODE solvers (Song et al., 2021), and Predictor-Corrector samplers (Song et al., 2021).

Revised: Require: N 1: ˆ x 1 „ π p x q , ∆ t Ð 1 N 2: for i “ N ´ 1 to 0 do 3: t Ð i ` 1 N 4: ˆ x t ´ ∆ t Ð ˆ x t ´ f p t q ˆ x t ∆ t 5: ˆ x t ´ ∆ t Ð ˆ x t ´ ∆ t ` g p t q 2 s θ ˚ p ˆ x t , t q ∆ t 6: z „ N p 0 , I q 7: ˆ x t ´ ∆ t Ð ˆ x t ´ ∆ t ` g p t q?

Revised: ∆ t z 8: return ˆ x

Revised: Require: N , y , λ 1: ˆ x 1 „ π p x q , ∆ t Ð 1 N 2: for i “ N ´ 1 to 0 do 3: t Ð i ` 1 N 4: ˆ y t „

Revised: p 0 t p y t | y q 5: ˆ x t Ð T ´ 1 r λ Λ P ´ 1 p Λ q ˆ y t

Revised: ` p 1 ´ λ q ` Λ T ¨ ˆ x t ` p I ´ Λ q T ˆ x t s 6: ˆ x t ´ ∆ t Ð ˆ x t ´ f p t q ˆ x t ∆ t 7: ˆ x t ´ ∆ t Ð ˆ x t ´ ∆ t ` g p t q 2 s θ ˚ p ˆ x t , t q ∆ t 8: z „ N p 0 , I q 9: ˆ x t ´ ∆ t Ð ˆ x t ´ ∆ t ` g p t q?

Original:   S OLVING INVERSE PROBLEMS WITH SCORE - BASED GENERATIVE MODELS
Revised: ∆ t z 10: return ˆ x  S OLVING INVERSE PROBLEMS WITH SCORE - BASED GENERATIVE MODELS

Original: With score-based generative modeling, we can train a score model s θ ˚ p x , t q to estimate the prior distribution  p p x q of medical images and generate samples from it.
Revised: With score-based generative modeling, we can train a score model s θ ˚ p x , t q to generate unconditional samples from the the prior distribution of medical images p p x q .

Original: To solve inverse problems however,[Table] [Table] we will need to sample from the posterior p p x | y q .
Revised: To solve inverse problems however, we will need to sample from the posterior p p x | y q .

Original: This can be done by conditioning the  stochastic process t x t u t Pr 0 , 1 s on an observation y , which is denoted by t x t | y u t Pr 0 , 1 s .[-For this conditional stochastic process, the initial marginal distribution is given by p 0 p x 0 | y q9 p 
 p x 0 q p p y | x 0 q , where p p y | x 0 q “ δ p y ´ A x 0 q .-]
Revised: This can be accomplished by conditioning the original stochastic process t x t u t Pr 0 , 1 s on an observation y , yielding a conditional stochastic process t x t | y u t Pr 0 , 1 s .[-For this conditional stochastic process, the initial marginal distribution is given by p 0 p x 0 | y q9 p 
 p x 0 q p p y | x 0 q , where p p y | x 0 q “ δ p y ´ A x 0 q .-]

Original: Through the transition density p 0 t p x t | x 0 q , we deﬁne the marginal distribution for t ą 0 by p t p x t | y q : “ ş p 0{+
+} p x 0 | y q  p0 t p xt | x 0 q d x 0 .
Revised: We denote the marginal distribution at t as p t p x t | y q , and our goal is to sample from p {+
+} p x 0 | y q , the same distribution as p p x | y q by deﬁnition.

Original: Much like sampling with Eq.
Revised: Much like generating unconditional samples by solving the reverse-time SDE in Eq.

Original: (2), we can reverse the conditional stochastic process t x t | y u t Pr 0 , 1 s to sample from the posterior distribution p 

Revised: (2), we can reverse the conditional stochastic process t x t | y u t Pr 0 , 1 s to sample from the posterior distribution p

Original: p x 0 | y q by solving the following conditional reverse-time SDE (Song et al., 2021):
Revised: p x 0 | y q by solving the following conditional reverse-time SDE (Song et al., 2021):

Original: [Equation]
Revised: [Equation]

Original: The conditional score function ∇ x t log p t p x t | y q is a critical part of Eq.
Revised: The conditional score function ∇ x t log p t p x t | y q is a critical part of Eq.

Original: (3), yet it is non-trivial to compute.
Revised: (4), yet it is non-trivial to compute.

Original: One solution is to estimate the score function by training a new score model s θ ˚ p x t , y , t q that explicitly depends on y (Song et al., 2021; Dhariwal & Nichol, 2021), such that s θ ˚ p x t , y , t q « ∇ x t log p t p x t | y q .
Revised: One solution is to estimate the score function by training a new score model s θ ˚ p x t , y , t q that explicitly depends on y (Song et al., 2021; Dhariwal & Nichol, 2021), such that s θ ˚ p x t , y , t q « ∇ x t log p t p x t | y q .

Original: However, this requires paired data tp x i , y i qu Ni “ 1 for training and has the same drawbacks as supervised learning techniques.
Revised: However, this requires paired data tp x i , y i qu Ni “ 1 for training and has the same drawbacks as supervised learning techniques.

Revised: We do not consider this approach in this work.

Original: An  alternative is to approximate the conditional score function [-by combining the likelihood p p y | x 0 q , transition density p 
 t p x t | x 0 q , and-] an unconditional score model s θ ˚ p x t , t q « ∇ x t log p t p x t q  .
Revised: An unsupervised alternative is to approximate the conditional score function [-by combining the likelihood p p y | x 0 q , transition density p 
 t p x t | x 0 q , and-]with an unconditionallytrained score model s θ ˚ p x t , t q « ∇ x t log p t p x t q and the measurement distribution p p y | x q .

Original: Many existing works (Song et al., 2021; Kawar et al., 2021; Kadkhodaie & Simoncelli, 2020; Jalal et al., 2021) can be considered as implementations of this idea.
Revised: Many existing works (Song et al., 2021; Kawar et al., 2021; Kadkhodaie & Simoncelli, 2020; Jalal et al., 2021) have implemented this idea in different ways.

Original: However,  Kawar et al.
Revised: However, the methods in Kawar et al.

Original: (2021) and
Revised: (2021) and

Original: Kadkhodaie & Simoncelli (2020) both require computing the singular value decomposition (SVD) of A P R m ˆ n , which can be difﬁcult for many measurement processes in medical imaging.
Revised: Kadkhodaie & Simoncelli (2020) both require computing the singular value decomposition (SVD) of A P R m ˆ n , which can be difﬁcult for many measurement processes in medical imaging.

Original: The method proposed in Jalal et al.
Revised: The method proposed in Jalal et al.

Original: (2021) is only designed for a speciﬁc sampling methodof score-based generative models called annealed Langevin dynamics (ALD, Song & Ermon, 2019), which proves to be inferior to more advanced sampling algorithms such as Predictor-Corrector methods (Song et al., 2021).
Revised: (2021) is only designed for a speciﬁc sampling method called annealed Langevin dynamics (ALD, Song & Ermon, 2019), which proves to be inferior to more advanced sampling algorithms such as Predictor-Corrector methods (Song et al., 2021).

Original: Finally, due to the inevitable errors in approximating the conditional score function, a sample ˆ x obtained from these approaches is not guaranteed to satisfy y

Original: “ A ˆ x exactly, and therefore may not constitute a valid solution to the inverse problem.

Original: In what follows, we propose a new conditional sampling approach for inverse problem solving with score-based generative models.
Revised: In what follows, we propose a new conditional sampling approach for inverse problem solving with score-based generative models.

Original: Our method is computationally efﬁcient forhigh resolution medical image reconstruction,  is applicable to a large family of iterative sampling methods for score-based generative models, and is guaranteed togive valid solutions that are consistent with A and y . 
Revised: Our method is computationally efﬁcient for medical image reconstruction, and is applicable to a large family of iterative sampling methods for score-based generative models. At a high level, we ﬁrst train an unconditional score model s θ ˚ p x , t q on medical images without assuming any measurement process. Given an observation y at test time, we form a stochastic process t y t u t Pr 0 , 1 s by adding appropriate noise to y . We then discretize the reverse-time

Revised: SDE in Eq.

Revised: (3) with existing unconditional samplers for s θ ˚ p x , t q , while incorporating the conditional information from y with a proximal optimization step to generate intermediate samples that are consistent with t y t u t Pr 0 , 1 s .

Original: 3. A CONVENIENT FORM OF THE LINEAR MEASUREMENT PROCESS
Revised: 3. A CONVENIENT FORM OF THE LINEAR MEASUREMENT PROCESS

Revised: Many different measurement processes in medical imaging share same components of computation.

Revised: For example, sparse-view CT reconstruction and metal artifact removal for CT both involve computing the same Radon transform.

Revised: Similarly, MRI measurement processes require computing the same spatial Fourier transform regardless of different downsampling ratios.

Original: Our method relies on a special formulation of A that is easy to obtainfor applications in medical image reconstruction.
Revised: To rigorously characterize this structure of measurement processes, we propose a special formulation of A that is efﬁcient to obtain in medical imaging applications.

Original: Without loss of generality, we assume that the linear operator A has full rank, i.e ., rank p
Revised: Without loss of generality, we assume that the linear operator A has full rank, i.e ., rank p

Original: A q “ min p n, m q “ m , whereby an alternative formulation of A can be given below:
Revised: A q “ min p n, m q “ m . The result below gives the alternative formulation of A :

Original: Proposition 1.
Revised: Proposition 1.

Original: If rank p A q “ m , then there exist an invertible matrix T P R n ˆ n , and a diagonal matrix Λ P t 0 , 1 u n ˆ n with tr p Λ q “ m , such that A “ P p Λ q T .
Revised: If rank p A q “ m , then there exist an invertible matrix T P R n ˆ n , and a diagonal matrix Λ P t 0 , 1 u n ˆ n with tr p Λ q “ m , such that A “ P p Λ q T .

Original: Here P p Λ q P t 0 , 1 u m ˆ n is an operator that, when multiplied with any vector a P R n , reduces its dimensionality to m by removing each i -th element of a for i “ 1 , 2 , ¨ ¨ ¨ , n if Λ ii “ 0 .
Revised: Here P p Λ q P t 0 , 1 u m ˆ n is an operator that, when multiplied with any vector a P R n , reduces its dimensionality to m by removing each i -th element of a for i “ 1 , 2 , ¨ ¨ ¨ , n if Λ ii “ 0 .

Original: We illustrate this decomposition for CT/MRI in Fig.
Revised: We illustrate this decomposition for CT/MRI in Fig.

Original: It is useful because many measurement processes  share the same T , even though they correspond to different A .
Revised: Many measurement processes in medical imaging share the same T , even if they correspond to different A .

Original: For example, T represents thesame Radon transform and Fourier transform in sparse-view CT and undersampled MRI, regardless of the number of measurements, i.e ., CT projections and k-space downsampling ratios.
Revised: For example, T corresponds to the Radon transform and Fourier transform in sparse-view CT and undersampled MRI respectively, regardless of the number of measurements, i.e ., CT projections and k-space downsampling ratios.

Original: Sparse-view CT reconstruction and metal artifact removal(MAR) for CT images also share the same T  (see Fig.
Revised: For both sparse-view CT reconstruction and metal artifact removal for CT images, the operator T is the Radon transform (see Fig.

Original: Intuitively, diag p Λ q can be viewed as a subsampling mask on the sinogram/k-space, and P p Λ q subsamples the sinogram/k-space into an observation y with a smaller size according to this subsampling mask.
Revised: Intuitively, diag p Λ q can be viewed as a subsampling mask on the sinogram/k-space, and P p Λ q subsamples the sinogram/k-space into an observation y with a smaller size according to this subsampling mask.

Original: In addition, we note that
Revised: In addition, we note that

Original: T
Revised: T

Original: ´ 1 can be efﬁciently implemented with the inverse Radon transform or the inverse Fourier transform in CT/MRI applications.
Revised: ´ 1 can be efﬁciently implemented with the inverse Radon transform or the inverse Fourier transform in CT/MRI applications.

Original: 3. I NCORPORATING THE MEASUREMENT OBSERVATION INTO AN UNCONDITIONAL SAMPLING PROCESS
Revised: 3. I NCORPORATING A GIVEN OBSERVATION INTO AN UNCONDITIONAL SAMPLING PROCESS

Original: In order to solve a linear inverse problem, we hope to sample from the posterior distribution p 

Revised: In order to solve a linear inverse problem, we hope to sample from the posterior distribution p

Original: p x 0 | y q , which can be done once the conditional stochastic process t x t | y u t Pr 0 , 1 s is known.
Revised: p x 0 | y q , which can be done once the conditional stochastic process t x t | y u t Pr 0 , 1 s is known.

Original: However, sampling t x t | y u t Pr 0 , 1 s involves solving the conditional reverse-time SDE in Eq.
Revised: However, sampling t x t | y u t Pr 0 , 1 s involves solving the conditional reverse-time SDE in Eq.

Original:  and we have already discussed various challenges of this approach.
Revised: (4), and we have already discussed various challenges of this approach.

Original: To bypass the difﬁculty of directly sampling t x t | y u t Pr 0 , 1 s , we consider another stochastic process that is related to t x t | y u t Pr 0 , 1 s but is much easier to sample from.

Original: Speciﬁcally, we introduce t y t

Original: u t Pr 0 , 1 s , which is deﬁned in reference to t x t u t Pr 0 , 1 s with y t

Original: “ A x t .

Original: Unlike t x t | y u t Pr 0 , 1 s , the conditional stochastic process t y t | y u t Pr 0 , 1 s is [-fully tractable. By deﬁnition, we have p 0 p y 
 | y q “ δ p y 
 ´ y q .-]
Revised: To bypass the difﬁculty of solving for t x t | y u t Pr 0 , 1 s , we consider a related stochastic process that is [-fully tractable. By deﬁnition, we have p 0 p y 
 | y q “ δ p y 
 ´ y q .-]much easier to sample from.

Original: Recall that p 

Revised: Recall that p

Original: t p x t | x 0 q “ N p x t | α p t q x 0 , β 2 p t
Revised: t p x t | x 0 q “ N p x t | α p t q x 0 , β 2 p t

Original: q I q where α p t q and β p t q can be derived from f p t q and g p t q (Song et al., 2021).
Revised: q I q where α p t q and β p t q can be derived from f p t q and g p t q (Song et al., 2021).

Original: Since y t“ A x t , we have p t p y t | y q “ p 0  tp y t[-| y 
-]
Revised: Given the unconditional stochastic process t x t u t Pr 0 , 1 s , we deﬁne t y t[-| y 
-]

Original:  q {+(cid:15) . Unlike t x t | y u t Pr 0 , 1 s , the conditional stochastic process t y t | y u t Pr 0 , 1 s is fully tractable. First, we have y 
+} “ 
Revised: u t Pr 0 , 1 s , where y t “ A x t ` α p t q {+(cid:15) . Unlike t x t | y u t Pr 0 , 1 s , the conditional stochastic process t y t | y u t Pr 0 , 1 s is fully tractable. First, we have y 
+} “ A x 0 ` α p 0 q (cid:15) “ A x 0 ` (cid:15) “ y .

Original:  N p y t | α p t q y , β 2 p t q AA T q  .
Revised: Since p 0 t p x t | x 0 q “ N p x t | α p t q x 0 , β 2 p t q I q , we have x t “ α p t q x 0 ` β p t q z , where z P R n „ N p 0 , I q . By deﬁnition, y t “ A x t ` α p t q (cid:15) , so we have y t “ A p α p t q x 0 ` β p t q z q ` α p t q (cid:15) “ α p t qp y ´ (cid:15) q ` β p t q A z ` α p t q (cid:15) “ α p t q y ` β p t q A z .

Original: Therefore, we can easily generate a sample ˆ y t „ p t p y t | y q by ﬁrst drawing zP R m „ N p 0 , I q and then computing ˆ y t “ α p t q y ` β p t q A z .
Revised: Therefore, we can easily generate a sample ˆ y t „ p t p y t | y q by ﬁrst drawing z „ N p 0 , I q and then computing ˆ y t “ α p t q y ` β p t q A z .

Original: How can we use t y t | y u t Pr 0 , 1 s to approximately sample from t x t | y u t Pr 0 , 1 s ? Our approach modiﬁes an existing iterative sampling algorithm designed for the unconditional stochastic process t x t u t Pr 0 , 1 s  .
Revised: The key of our approach is to modify any existing iterative sampling algorithm designed for the unconditional stochastic process t x t u t Pr 0 , 1 s so that the samples are consistent with t y t | y u t Pr 0 , 1 s .

Original: In general, an iterative sampling process of score-based generative models selects a sequence of time steps t 0 “ t 0 ă t 1 ă  ¨ ¨ ¨ ă t N “ 1 u and iterates according to
Revised: In general, an iterative sampling process of score-based generative models selects a sequence of time steps t 0 “ t 0 ă t 1 ă [Equation] ¨ [Equation]

Revised: ă t N “ 1 u and iterates according to

Original: [Equation]
Revised: [Equation]

Original: where ˆ x t N „ π p x q , z i „ N p 0 , I q , and θ ˚ denotes the parameters in an unconditional score model s θ ˚ p x , t q .
Revised: where ˆ x t N „ π p x q , z i „ N p 0 , I q , and θ ˚ denotes the parameters in an unconditional score model s θ ˚ p x , t q .

Original: Here  hi takes a noisy sample ˆ x t i and reduces the noise therein to generate ˆ x t i ´ 1 , using the unconditional score model s θ ˚ p x , t q .
Revised: Here the iteration function h takes a noisy sample ˆ x t i and reduces the noise therein to generate ˆ x t i ´ 1 , using the unconditional score model s θ ˚ p x , t q .

Original: Samples obtained by this procedure t ˆ x t i u Ni “ 0 constitute an approximation of t x t u t Pr 0 , 1 s , where the last sample ˆ x t 0 can be viewed as an approximate sample from p 0 p x q . Most existing sampling methods for score-based generative models ﬁt into this iterative sampling framework, including Algorithm 1, ALD (Song & Ermon, 2019), and Predictor-Corrector samplers (Song et al., 2021).
Revised: For example, for the Euler-Maruyama sampler detailed in Algorithm 1, this iteration function is given by

Original: [Equation]
Revised: [Equation]

Original: [Equation]
Revised: Samples obtained by this procedure t ˆ x t i u Ni “ 0 constitute an approximation of t x t u t Pr 0 , 1 s , where the last sample ˆ x t 0 can be viewed as an approximate sample from p 0 p x q . Most existing sampling

Original: To enforce the constraint implied by t y t | y u t Pr 0 , 1 s , we prepend an additional step to the iteration rule in Eq.
Revised: To enforce the constraint implied by t y t | y u t Pr 0 , 1 s , we prepend an additional step to the iteration rule in Eq.

Original: (4), leading to
Revised: (5), leading to

Original: [Equation]
Revised: [Equation]

Original: where ˆ x t N „ π p x q , ˆ y t i „ p t i p y t i | y q , and 0 ď λ ď 1 is a hyper-parameter to be discussed later.
Revised: where ˆ x t N „ π p x q , ˆ y t i „ p t i p y t i | y q , and 0 ď λ ď 1 is a hyper-parameter.

Original: The iteration function k p¨ , ˆ y t i , λ q : R n Ñ R n fuses t x t u t Pr 0 , 1 s and t y t | y u t Pr 0 , 1 s by generating ˆ x 1 t i from ˆ x t i in each sampling step, with constraints ˆ x 1 t i « ˆ x t i and A ˆ x 1 t i « ˆ y t i .
Revised: We provide an illustration of this process in Fig.

Original: Formally, we simultaneously minimize the distance between ˆ x 1 t i and ˆ x t i , and the distance between ˆ x 1 t i and the hyperplane t x P R n | Ax “ ˆ y t i u , with a hyperparameter 0 ď λ ď 1 balancing between the two:
Revised: The iteration function k p¨ , ˆ y t i , λ q : R n Ñ R n promotes data consistency by solving a proximal optimization step (Hammernik et al., 2021) that simultaneously minimizes the distance between ˆ x 1 t i and ˆ x t i , and the distance between ˆ x 1 t i and the hyperplane t x P R n | Ax “ ˆ y t i u , with a hyperparameter 0 ď λ ď 1 balancing between the two:

Original: [Equation]
Revised: [Equation]

Original: where we choose the norm (cid:107) a (cid:107) 2 T : “ (cid:107) T a (cid:107) 22 to simplify our theoretical analysis.
Revised: where we choose the norm (cid:107) a (cid:107) 2 T : “ (cid:107) T a (cid:107) 22 to simplify our theoretical analysis.

Original: The following result gives a closed-form solution to the optimization problem in Eq.
Revised: The following result gives a closed-form solution to the optimization problem in Eq.

Original: (7), which also deﬁnes our iteration function k : p ˆ x t i , ˆ y t i , λ q ÞÑ ˆ x 1 t i :
Revised: (8), which also deﬁnes our iteration function k : p ˆ x t i , ˆ y t i , λ q ÞÑ ˆ x 1 t i :

Original: Theorem 1.
Revised: Theorem 1.

Original: The solution of Eq.
Revised: The solution of Eq.

Original: (7) can be given by
Revised: (8) can be given by

Original: [Equation]
Revised: [Equation]

Original: where P ´ 1 p Λ q : R m Ñ R n denotes any right inverse of P p Λ q .
Revised: where P ´ 1 p Λ q : R m Ñ R n denotes any right inverse of P p Λ q .

Original: See Fig.
Revised: See Fig.

Original: 3 for an illustration of the function ˆ x 1 t i “ k p
Revised: 3 for an illustration of the function ˆ x 1 t i “ k p

Original: ˆ x t i , ˆ y t i , λ q .
Revised: ˆ x t i , ˆ y t i , λ q .

Original: The right inverse P ´ 1 p Λ q increases the dimensionality of a vector a P R m to n by putting its entries on every index i of an n -dimensional vector where Λ ii “ 1 .
Revised: The right inverse P ´ 1 p Λ q increases the dimensionality of a vector a P R m to n by putting its entries on every index i of an n -dimensional vector where Λ ii “ 1 .

Original: Recall that in sparse-view CT or undersampled MRI, diag p Λ q represents a subsampling mask, and P p Λ q subsamples the full sinogram/k-space to generate the observation y .
Revised: Recall that in sparse-view CT or undersampled MRI, diag p Λ q represents a subsampling mask, and P p Λ q subsamples the full sinogram/k-space to generate the observation y .

Original: In this case, P ´ 1 p Λ q pads the observation y
Revised: In this case, P ´ 1 p Λ q pads the observation y

Original: so that it has the same size as the full sinogram/k-space.
Revised: so that it has the same size as the full sinogram/k-space.

Original: When λ “ 0 , ˆ x 1 t i “ k p ˆ x t i , ˆ y t i , 0 q “ ˆ x t i completely ignores the constraint
Revised: When λ “ 0 , ˆ x 1 t i “ k p ˆ x t i , ˆ y t i , 0 q “ ˆ x t i completely ignores the constraint

Original: A ˆ x 1 t i “ ˆ y t i , in which case our sampling method in Eq.
Revised: A ˆ x 1 t i “ ˆ y t i , in which case our sampling method in Eq.

Original: (6) performs unconditional generation from the data prior.
Revised: (7) performs unconditional generation.

Original: On the other hand, when λ “ 1 , ˆ x 1 t i “ k p ˆ x t i , ˆ y t i , 1 q satisﬁes A ˆ x 1 t i “ ˆ y t i exactly but incorporates less information from ˆ x t i , and hence the data prior as well.{+measurement is noisy, we choose 
 ă λ ă 1 to allow slackness in the constraint+}
Revised: On the other hand, when λ “ 1 , ˆ x 1 t i “ k p ˆ x t i , ˆ y t i , 1 q satisﬁes A ˆ x 1 t i “ ˆ y t i exactly. When the {+measurement is noisy, we choose 
 ă λ ă 1 to allow slackness in the constraint+}

Original: Due to the ill-posedness of the inverse problem, prior knowledge from the data distribution isnecessary to obtain good reconstruction images, and we ﬁnd it important to tune λ to balance betweenthe constraint ˆ x 1 t i « ˆ x t i and A ˆ x 1 t i « ˆ y t
Revised: A ˆ x 1 t i “ ˆ y t i . The value of λ is important for balancing between ˆ x 1 t i « ˆ x t i and A ˆ x 1 t i « ˆ y t

Original: i .
Revised: i .

Revised: In practice, we use Bayesian optimization to tune this λ automatically on a held-out validation dataset.

Original: At the last iteration, we replace ˆ x t 0 with k p ˆ x t 1 , y , 1 q  to guarantee a valid solution to the inverse problem.
Revised: When the measurement process contains no noise, we replace ˆ x t 0 with k p ˆ x t 0 , y , 1 q at the last sampling step to guarantee A ˆ x t 0 “ y .

Original: In summary, our method given in Eq.
Revised: In summary, our method given in Eq.

Original: (6) introduces minimal modiﬁcations to an existing iterative sampling method of score-based generative models.
Revised: (7) introduces minimal modiﬁcations to an existing iterative sampling method of score-based generative models.

Original: For example, we can convert the sampler in
Revised: For example, we can convert the sampler in

Original: Algorithm 1 to an inverse problem solver in Algorithm 2 by adding/modifying just three lines of pseudo-code.
Revised: Algorithm 1 to an inverse problem solver in Algorithm 2 by adding/modifying just three lines of pseudo-code.

Original: Unlike the concurrent work Jalal et al.
Revised: Unlike the concurrent work Jalal et al.

Original: (2021), our method is not limited to annealed
Revised: (2021), our method is not limited to annealed

Revised: [Figure]

Original: Langevin dynamics (ALD).
Revised: Langevin dynamics (ALD).

Original: As demonstrated in our experiments, we outperform Jalal et al.
Revised: As demonstrated in our experiments, we outperform Jalal et al.

Original: (2021) even with the same ALD sampler, and can widen the performance gap further by using more[Figure] advanced approaches like the Predictor-Corrector sampler (Song et al., 2021).
Revised: (2021) even with the same ALD sampler, and can widen the performance gap further by using more advanced approaches like the Predictor-Corrector sampler (Song et al., 2021).

Original: Unlike Kadkhodaie &
Revised: Unlike Kadkhodaie &

Original: Simoncelli (2020); Kawar et al. (2021), we rely on the efﬁcient alternative representation of A given in Section 3.1, and do not require expensive SVD computation.
Revised: Simoncelli (2020); Kawar et al. (2021), we rely on the efﬁcient alternative representation of A given in Section 3.1, and do not require expensive SVD computation.

Original: Finally, with the post-processing step k p ˆ x t 1 , y , 1 q , we are guaranteed to provide a valid solution to the inverse problem.

Original:  E XPERIMENTS
Revised:  E XPERIMENTS

Original: We aim to answer the following questions in this section: (1) Can we directly compete with best-inclass supervised learning techniques for the same measurement process used in their training, even though our approach is fully unsupervised?
Revised: We aim to answer the following questions in this section: (1) Can we directly compete with best-inclass supervised learning techniques for the same measurement process used in their training, even though our approach is fully unsupervised?

Original: (2) Can our method generalize better to new measurement processes?
Revised: (2) Can our method generalize better to new measurement processes?

Original: (3) How do we fare against other unsupervised approaches based on score-based generative models?
Revised: (3) How do we fare against other unsupervised approaches?

Original: To study these questions, we experiment on several tasks in medical imaging, including sparse-view CT reconstruction, metal artifact removal (MAR) for CT, and undersampled MRI reconstruction.
Revised: To study these questions, we experiment on several tasks in medical imaging, including sparse-view CT reconstruction, metal artifact removal (MAR) for CT, and undersampled MRI reconstruction.

Original: More experimental details are provided in Appendix B.
Revised: More experimental details are provided in Appendix B.

Original: Datasets We consider two datasets for CT experiments.
Revised: Datasets We consider two datasets for CT experiments.

Original: The ﬁrst is the Lung Image Database
Revised: The ﬁrst is the Lung Image Database

Original: Consortium (LIDC) image collection dataset (Armato III et al., 2011; Clark et al., 2013) where we slice the original 3D CT volumes to obtain 130304 2D images of resolution 320 ˆ 320 for training.
Revised: Consortium (LIDC) image collection dataset (Armato III et al., 2011; Clark et al., 2013) where we slice the original 3D CT volumes to obtain 130304 2D images of resolution 320 ˆ 320 for training.

Original: The second is the Low Dose CT (LDCT) Image and Projection dataset (Moen et al., 2021) that contains CT scans of multiple anatomic sites, including head, chest, and abdomen, from which we generate 47006 2D image slices of resolution 512 ˆ 512 for training.
Revised: The second is the Low Dose CT (LDCT) Image and Projection dataset (Moen et al., 2021) that contains CT scans of multiple anatomic sites, including head, chest, and abdomen, from which we generate 47006 2D image slices of resolution 512 ˆ 512 for training.

Original: We simulate CT measurements (sinograms) with a parallel-beam geometry using projection angles equally distributed acrossdegrees.
Revised: We simulate CT measurements (sinograms) with a parallel-beam geometry using projection angles equally distributed acrossdegrees.

Original: For MAR experiments, we follow Yu et al.
Revised: For MAR experiments, we follow Yu et al.

Original: (2020) to synthesize metal artifacts.
Revised: (2020) to synthesize metal artifacts.

Original: For undersampled MRI experiments, we use the Brain Tumor Segmentation (BraTS) 2021 dataset (Menze et al., 2014;
Revised: For undersampled MRI experiments, we use the Brain Tumor Segmentation (BraTS) 2021 dataset (Menze et al., 2014;

Original: Bakas et al., 2017), where we slice 3D MRI volumes to get 297270 images of resolution 240 ˆ 240 as the training dataset.
Revised: Bakas et al., 2017), where we slice 3D MRI volumes to get 297270 images of resolution 240 ˆ 240 as the training dataset.

Original: We simulate MRI measurements with Fast Fourier Transform using a single-coil setup, and follow Zbontar et al.
Revised: We simulate MRI measurements with Fast Fourier Transform using a single-coil setup, and follow Zbontar et al.

Original: ; Knoll et al.
Revised: ; Knoll et al.

Original: (2020) to undersample the k-space with an equispaced Cartesian mask.
Revised: (2020) to undersample the k-space with an equispaced Cartesian mask.

Original: The performance is measured on 1000 test images with peak signal-to-noise ratio (PSNR) and structural similarity (SSIM).
Revised: The performance is measured on 1000 test images with peak signal-to-noise ratio (PSNR) and structural similarity (SSIM).

Original: Standard techniques in medical imaging We include two standard learning-free techniques as baselines for sparse-view CT reconstruction.
Revised: Standard techniques in medical imaging We include two standard learning-free techniques as baselines for sparse-view CT reconstruction.

Original: The ﬁrst is ﬁltered back projection on sparse-view sinograms, which is denoted by “FBP”.
Revised: The ﬁrst is ﬁltered back projection on sparse-view sinograms, which is denoted by “FBP”.

Original: The second is an iterative reconstruction method with total variation regularization called FISTA-TV (Beck & Teboulle, 2009).
Revised: The second is an iterative reconstruction method with total variation regularization called FISTA-TV (Beck & Teboulle, 2009).

Original: For MAR experiments, we include another learning-free baseline called linear interpolation (LI, Kalender et al., 1987).
Revised: For MAR experiments, we include another learning-free baseline called linear interpolation (LI, Kalender et al., 1987).

Original: Supervised learning baselines For sparse-view CT on both LIDC and LDCT, we include cGAN (Ghani & Karl, 2018), Neumann (Gilton et al., 2019), and SIN-4c-PRN (Wei et al., 2020) as

Original: [Table] [Table]
Revised: [Table] [Table]

Original:  supervised learning baselines.
Revised: Supervised learning baselines For sparse-view CT on both LIDC and LDCT, we include cGAN (Ghani & Karl, 2018), Neumann (Gilton et al., 2019), and SIN-4c-PRN (Wei et al., 2020) as supervised learning baselines.

Original: We follow the settings in Wei et al.
Revised: We follow the settings in Wei et al.

Original: and train all methods with 23 projection angles.
Revised: and train all methods with 23 projection angles.

Original: For MAR, we use cGANMAR (Wang et al., 2018) and SNMAR (Yu et al., 2020) as the baselines.
Revised: For MAR, we use cGANMAR (Wang et al., 2018) and SNMAR (Yu et al., 2020) as the baselines.

Original: For undersampled MRI on BraTS, we compare against Cascade DenseNet (Zheng et al., 2019) and DuDoRNet (Zhou & Zhou, 2020), which are both trained with a 8 ˆ acceleration factor by measuring only 1 { 8 of the full k-space.
Revised: For undersampled MRI on BraTS, we compare against Cascade DenseNet (Zheng et al., 2019) and DuDoRNet (Zhou & Zhou, 2020), which are both trained with a 8 ˆ acceleration factor by measuring only 1 { 8 of the full k-space.

Original: Unsupervised learning baselines  We compare with several existing methods that apply score-based generative models to inverse problem solving.
Revised: Unsupervised learning baselines For unsupervised techniques, so far only score-based generative models have witnessed success on clinic data. We compare with several existing methods that apply score-based generative models to inverse problem solving.

Original: Speciﬁcally, we consider the “Langevin” approach proposed in Jalal et al.
Revised: Speciﬁcally, we consider the “Langevin” approach proposed in Jalal et al.

Original: [Table] and the “Score SDE” method in Song et al.
Revised: (2021), and the “Score SDE” method in Song et al.

Original: [Table] where the former is limited to annealed Langevin dynamics (ALD) sampling [Table] and the latter was proposed as a theoretical possibility without experiments.
Revised: (2021), where the former is limited to annealed Langevin dynamics (ALD) sampling, and the latter was proposed as a theoretical possibility without experiments.

Original: We only focus on undersampled MRI for these baselines, since it is the only medical imaging problem ever tackled with score-based generative models before our work.
Revised: We only focus on undersampled MRI for these baselines, since it is the only medical imaging problem ever tackled with score-based generative models before our work.

Original: All methods share the same score models and only differ in terms of inference.
Revised: All methods share the same score models and only differ in terms of inference.

Original: We make sure all sampling algorithms have comparable number of iteration steps ( N in Eqs.
Revised: We make sure all sampling algorithms have comparable number of iteration steps ( N in Eqs.

Original: (4) and (6)).
Revised: (5) and (7)).

Original: [Table]
Revised: [Table]

Original: Competing with supervised learning approaches Thanks to the outstanding sample quality of score-based generative models, we can achieve comparable or better performance than best-in-class supervised learning methods even for the same measurement process used in their training.
Revised: Competing with supervised learning approaches Thanks to the outstanding sample quality of score-based generative models, we can achieve comparable or better performance than best-in-class supervised learning methods even for the same measurement process used in their training.

Original: As shown in Table 2, we outperform the top supervised learning technique
Revised: As shown in Table 2, we outperform the top supervised learning technique

Original: SIN-4c-PRN on sparse-view CT reconstruction by a signiﬁcant margin, on both the LIDC and LDCT datasets.
Revised: SIN-4c-PRN on sparse-view CT reconstruction by a signiﬁcant margin, on both the LIDC and LDCT datasets.

Original: Our results with 20 measurements are even better than supervised learning counterparts with 23 measurements.
Revised: Our results with 20 measurements are even better than supervised learning counterparts with 23 measurements.

Original: In Fig.
Revised: In Fig.

Original: 4, we provide a visual comparison of the reconstruction quality for various methods, where it is clear to see that our method can recover more details faithfully.
Revised: 4, we provide a visual comparison of the reconstruction quality for various methods, where it is clear to see that our method can recover more details faithfully.

Original: From results in Table 3, we also outperform the top supervised learning method SNMAR on metal artifact removal.
Revised: From results in Table 3, we also outperform the top supervised learning method SNMAR on metal artifact removal.

Original: As shown in Fig.
Revised: As shown in Fig.

Original: 7, our method generates images with less artifacts and preserves the structure better.
Revised: 7, our method generates images with less artifacts and preserves the structure better.

Original: For undersampled MRI reconstruction results given in Tables 1 and 3, our method is ranked the 2nd for the case of 8 ˆ acceleration, with comparable performance to the top supervisedlearning method DuDoRNet.
Revised: For undersampled MRI reconstruction results given in Tables 1 and 3, our method is ranked the 2nd for the case of 8 ˆ acceleration, with comparable performance to the top supervised method DuDoRNet.

Original: Generalizing to different number of measurements Since our approach is fully unsupervised, we can naturally apply the same score model to different measurement processes.
Revised: Generalizing to different number of measurements Since our approach is fully unsupervised, we can naturally apply the same score model to different measurement processes.

Original: We ﬁrst consider changing the number of measurements at the test time, e.g ., using different number of projection angles (resp.
Revised: We ﬁrst consider changing the number of measurements at the test time, e.g ., using different number of projection angles (resp.

Original: different acceleration factors) for sparse-view CT (resp.
Revised: different acceleration factors) for sparse-view CT (resp.

Original: undersampled MRI) reconstruction.
Revised: undersampled MRI) reconstruction.

Original: As shown in Table 1 and Fig.
Revised: As shown in Table 1 and Fig.

Original: 5 (Left) , we achieve the best performance on undersampled MRI for both 24 ˆ and 4 ˆ acceleration factors, whereas DuDoRNet fails to generalize when the acceleration factor changes.
Revised: 5 (Left) , we achieve the best performance on undersampled MRI for both 24 ˆ and 4 ˆ acceleration factors, whereas DuDoRNet fails to generalize when the acceleration factor changes.

Original: The other supervised learning approach Cascade DenseNet demonstrates limited adaptability by building a model architecture inspired by the physical measurement process of MRI, but fails to yield top-level performance.
Revised: The other supervised learning approach Cascade DenseNet demonstrates limited adaptability by building a model architecture inspired by the physical measurement process of MRI, but fails to yield top-level performance.

Original: For sparse-view CT reconstruction, all supervised learning methods struggle to generalize to different projection angles, as shown in Fig.
Revised: For sparse-view CT reconstruction, all supervised learning methods struggle to generalize to different projection angles, as shown in Fig.

Original: 5 (Center) .
Revised: 5 (Center) .

Original: Generalizing to different measurement processes in CT We can perform both sparse-view CT reconstruction and metal artifact removal (MAR) with a single score model trained on CT images.
Revised: Generalizing to different measurement processes in CT We can perform both sparse-view CT reconstruction and metal artifact removal (MAR) with a single score model trained on CT images.

Original: These two tasks are inverse problems in CT imaging with different measurement processes A , but they share the same T in the decomposition of Proposition 1.
Revised: These two tasks are inverse problems in CT imaging with different measurement processes A , but they share the same T in the decomposition of Proposition 1.

Original: We provide a visualization of the measurement process corresponding to MAR in Fig.
Revised: We provide a visualization of the measurement process corresponding to MAR in Fig.

Original: As shown in Table 3, we can outperform supervised learning techniques speciﬁcally designed and trained for MAR, while using the same score model used in sparse-view CT reconstruction on LIDC.
Revised: As shown in Table 3, we can outperform supervised learning techniques speciﬁcally designed and trained for MAR, while using the same score model used in sparse-view CT reconstruction on LIDC.

Original: Comparing against existing score-based methods We compare our method against Langevin (Jalal et al., 2021) and Score SDE (Song et al., 2021) for undersampled MRI reconstruction on BraTS.
Revised: Comparing against existing score-based methods We compare our method against Langevin (Jalal et al., 2021) and Score SDE (Song et al., 2021) for undersampled MRI reconstruction on BraTS.

Original: Two variants of our approach are considered, which respectively use annealed Langevin dynamics (ALD) and the Predictor-Corrector (PC) sampler for score-based generative models as the backend.
Revised: Two variants of our approach are considered, which respectively use annealed Langevin dynamics (ALD) and the Predictor-Corrector (PC) sampler for score-based generative models as the backend.

Original: We denote the former by “ALD + Ours”, and the latter by “PC + Ours” (our default method for all other experiments).
Revised: We denote the former by “ALD + Ours”, and the latter by “PC + Ours” (our default method for all other experiments).

Original: Recall that Langevin uses ALD as the sampler, same as “ALD + Ours”.
Revised: Recall that Langevin uses ALD as the sampler, same as “ALD + Ours”.

Original: All results are provided in Fig.
Revised: All results are provided in Fig.

Original: 5 (Right) .
Revised: 5 (Right) .

Original: We observe that “ALD + Ours” uniformly outperform Langevin and Score
Revised: We observe that “ALD + Ours” uniformly outperform Langevin and Score

Original: SDE across all numbers of measurements in the experiment.
Revised: SDE across all numbers of measurements in the experiment.

Original: Moreover, “PC + Ours” can further improve “ALD + Ours”, demonstrating the power of switching to more advanced sampling methods of score-based generative models in our proposed approach.
Revised: Moreover, “PC + Ours” can further improve “ALD + Ours”, demonstrating the power of switching to more advanced sampling methods of score-based generative models in our proposed approach.

Original:  C ONCLUSION
Revised:  C ONCLUSION

Original: We propose a new method to solve linear inverse problems with score-based generative models.
Revised: We propose a new method to solve linear inverse problems with score-based generative models.

Original: Our method is fully unsupervised, requires no paired data for training, can ﬂexibly adapt to different measurement processes at test time, and only requires minimal modiﬁcations to a large number of existing sampling methods of score-based generative models.
Revised: Our method is fully unsupervised, requires no paired data for training, can ﬂexibly adapt to different measurement processes at test time, and only requires minimal modiﬁcations to a large number of existing sampling methods of score-based generative models.

Original: Empirical results demonstrate that our method can match or outperform existing supervised learning counterparts on image reconstruction for sparse-view CT and undersampled MRI, and has better generalization to new measurement processes, such as using a different number of projections or downsampling ratios in CT/MRI, and tackling both sparse-view CT reconstruction and metal artifact removal with a single model.
Revised: Empirical results demonstrate that our method can match or outperform existing supervised learning counterparts on image reconstruction for sparse-view CT and undersampled MRI, and has better generalization to new measurement processes, such as using a different number of projections or downsampling ratios in CT/MRI, and tackling both sparse-view CT reconstruction and metal artifact removal with a single model.

Original: A P ROOFS
Revised: A P ROOFS

Original: Proposition 1.
Revised: Proposition 1.

Original: If rank p A q “ m , then there exist an invertible matrix T P R n ˆ n , and a diagonal matrix Λ P t 0 , 1 u n ˆ n with tr p Λ q “ m , such that A “ P p Λ q T .
Revised: If rank p A q “ m , then there exist an invertible matrix T P R n ˆ n , and a diagonal matrix Λ P t 0 , 1 u n ˆ n with tr p Λ q “ m , such that A “ P p Λ q T .

Original: Here P p Λ q P t 0 , 1 u m ˆ n is an operator that, when multiplied with any vector a P R n , reduces its dimensionality to m by removing each i -th element of a for i “ 1 , 2 , ¨ ¨ ¨ , n if Λ ii “ 0 .
Revised: Here P p Λ q P t 0 , 1 u m ˆ n is an operator that, when multiplied with any vector a P R n , reduces its dimensionality to m by removing each i -th element of a for i “ 1 , 2 , ¨ ¨ ¨ , n if Λ ii “ 0 .

Original: Proof.
Revised: Proof.

Original: Let A “ p a T 1 , a T 2 , ¨ ¨ ¨ , a T m q P R m ˆ n .
Revised: Let A “ p a T 1 , a T 2 , ¨ ¨ ¨ , a T m q P R m ˆ n .

Original: Since A has full rank, the row vectors t a 1 , a 2 , ¨ ¨ ¨ , a m u are linearly independent.
Revised: Since A has full rank, the row vectors t a 1 , a 2 , ¨ ¨ ¨ , a m u are linearly independent.

Original: We can therefore extend them to a total of n linearly independent vectors, i.e ., t a 1 , a 2 , ¨ ¨ ¨ , a m , b 1 , ¨ ¨ ¨ , b n ´ m u .
Revised: We can therefore extend them to a total of n linearly independent vectors, i.e ., t a 1 , a 2 , ¨ ¨ ¨ , a m , b 1 , ¨ ¨ ¨ , b n ´ m u .

Original: Due to the linear independence, we know T “ p a T 1 , a T 2 , ¨ ¨ ¨ , a T m , b T 1 , ¨ ¨ ¨ , b T n ´ m q P R n ˆ n has full rank and is invertible.
Revised: Due to the linear independence, we know T “ p a T 1 , a T 2 , ¨ ¨ ¨ , a T m , b T 1 , ¨ ¨ ¨ , b T n ´ m q P R n ˆ n has full rank and is invertible.

Original: Next, we deﬁne
Revised: Next, we deﬁne

Original: [Equation]
Revised: [Equation]

Original: where diag converts a vector to a diagonal matrix.
Revised: where diag converts a vector to a diagonal matrix.

Original: Clearly tr p Λ q “ m and A “ P p Λ q T , which completes the proof.
Revised: Clearly tr p Λ q “ m and A “ P p Λ q T , which completes the proof.

Original: Lemma 1.
Revised: Lemma 1.

Original: Let P ´ 1 p Λ q : R m Ñ R n be any right inverse of P p Λ q : R n Ñ R m .
Revised: Let P ´ 1 p Λ q : R m Ñ R n be any right inverse of P p Λ q : R n Ñ R m .

Original: For any u P R n and ˆ y t P R m , we have
Revised: For any u P R n and ˆ y t P R m , we have

Original: [Equation]
Revised: [Equation]

Original: Proof.
Revised: Proof.

Original: By the deﬁnition of P p Λ q , we have P p Λ q “ P p Λ q Λ , and
Revised: By the deﬁnition of P p Λ q , we have P p Λ q “ P p Λ q Λ , and

Original: [Equation]
Revised: [Equation]

Original: a P R n , b P R n :
Revised: a P R n , b P R n :

Original: P
Revised: P

Original: [Equation]
Revised: [Equation]

Original: To prove the “if” direction
Revised: To prove the “if” direction

Original: [Equation]
Revised: [Equation]

Original: we note that
Revised: we note that

Original: [Equation]
Revised: [Equation]

Original: To prove the “only if” direction
Revised: To prove the “only if” direction

Original: [Equation]
Revised: [Equation]

Original: we have
Revised: we have

Original: [Equation]
Revised: [Equation]

Original: where (i) is due to the property in Eq.
Revised: where (i) is due to the property in Eq

Revised: [Equation]

Original: This completes the proof for both directions.
Revised: This completes the proof for both directions.

Original: Theorem 1.
Revised: Theorem 1.

Original: The solution of Eq.
Revised: The solution of Eq.

Original: (7) can be given by
Revised: (8) can be given by

Original: [Equation]
Revised: [Equation]

Original: where P ´ 1 p Λ q : R m Ñ R n denotes any right inverse of P p Λ q .
Revised: where P ´ 1 p Λ q : R m Ñ R n denotes any right inverse of P p Λ q .

Original: Proof.
Revised: Proof.

Original: The optimization objective function in Eq.
Revised: The optimization objective function in Eq.

Original: (7) can be written as
Revised: (8) can be written as

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Figure]
Revised: [Figure]

Original: Since Au “ ˆ y t , we have P p Λ q T u “ ˆ y t and equivalently Λ T u “ Λ P ´ 1 p Λ q ˆ y t due to Lemma 1.
Revised: Since Au “ ˆ y t , we have P p Λ q T u “ ˆ y t and equivalently Λ T u “ Λ P ´ 1 p Λ q ˆ y t due to Lemma 1.

Original: This constraint does not restrict the value of p I ´ Λ q T u .
Revised: This constraint does not restrict the value of p I ´ Λ q T u .

Original: Therefore, when Au “ ˆ y t , we have
Revised: Therefore, when Au “ ˆ y t , we have

Original: [Equation]
Revised: [Equation]

Original: min
Revised: min

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: This simpliﬁes the optimization problem in Eq.
Revised: This simpliﬁes the optimization problem in Eq.

Original: (7) to min
Revised: (8) to min

Original: [Equation]
Revised: [Equation]

Original: which is minimizing a quadratic function of z .
Revised: which is minimizing a quadratic function of z .

Original: The optimal solution z ˚ is thus in closed form:
Revised: The optimal solution z ˚ is thus in closed form:

Original: [Equation]
Revised: [Equation]

Original: According to the deﬁnition, ˆ x 1 t “ z ˚ , whereby the proof is completed.
Revised: According to the deﬁnition, ˆ x 1 t “ z ˚ , whereby the proof is completed.

Original: B A DDITIONAL EXPERIMENTAL DETAILS B. A DDITIONAL RESULTS
Revised: B A DDITIONAL EXPERIMENTAL DETAILS B. A DDITIONAL RESULTS

Original: In Fig.
Revised: In Fig.

Original: 6, we provide SSIM results versus the number of measurements for multiple methods and tasks.
Revised: 6, we provide SSIM results versus the number of measurements for multiple methods and tasks.

Original: In general, the SSIM curves have very similar trends to the PSNR curves in Fig.
Revised: In general, the SSIM curves have very similar trends to the PSNR curves in Fig.

Original: We additionally provide a visualization of metal artifact removal results in Fig.
Revised: We additionally provide a visualization of metal artifact removal results in Fig.

Original: B. T HE TASK OF METAL ARTIFACT REMOVAL
Revised: B. T HE TASK OF METAL ARTIFACT REMOVAL

Original: Metallic implants in an object can cause strong metal artifacts in CT imaging.
Revised: Metallic implants in an object can cause strong metal artifacts in CT imaging.

Original: As shown in Fig.
Revised: As shown in Fig.

Original: 8, the source of artifacts come from extremely bright regions in the sinogram, called metal traces.
Revised: 8, the source of artifacts come from extremely bright regions in the sinogram, called metal traces.

Original: To reduce or ideally remove metal artifacts from a CT image, we remove metal traces from the sinogram and leverage the data prior to complete the sinogram.
Revised: To reduce or ideally remove metal artifacts from a CT image, we remove metal traces from the sinogram and leverage the data prior to complete the sinogram.

Original: As a result, metal artifact removal can be viewed as an inverse problem, where the measurement process gives the full sinogram except for the metal trace region, and our goal is to reconstruct the full CT image using this partially known sinogram, which will be artifact-free assuming perfect inpainting of the sinogram.
Revised: As a result, metal artifact removal can be viewed as an inverse problem, where the measurement process gives the full sinogram except for the metal trace region, and our goal is to reconstruct the full CT image using this partially known sinogram, which will be artifact-free assuming perfect inpainting of the sinogram.

Original: B. D ETAILS OF DATASETS
Revised: B. D ETAILS OF DATASETS

Original: CT datasets We conduct experiments of 2D CT image reconstruction on two datasets.
Revised: CT datasets We conduct experiments of 2D CT image reconstruction on two datasets.

Original: First, the
Revised: First, the

Original: Lung Image Database Consortium image collection (LIDC) (Armato III et al., 2011; Clark et al., 2013) consists of diagnostic and lung cancer screening thoracic computed tomography (CT) scans for lung cancer detection and diagnosis, which contains 1018 cases.
Revised: Lung Image Database Consortium image collection (LIDC) (Armato III et al., 2011; Clark et al., 2013) consists of diagnostic and lung cancer screening thoracic computed tomography (CT) scans for lung cancer detection and diagnosis, which contains 1018 cases.

Original: Second, the Low Dose CT Image and Projection dataset (LDCT) (Clark et al., 2013; Moen et al., 2021) involves CT images of multiple anatomic sites, including 99 head CT scans, 100 chest CT scans, and 100 abdomen CT scans.
Revised: Second, the Low Dose CT Image and Projection dataset (LDCT) (Clark et al., 2013; Moen et al., 2021) involves CT images of multiple anatomic sites, including 99 head CT scans, 100 chest CT scans, and 100 abdomen CT scans.

Original: Note that for the LDCT dataset, we only use the full-dose CT images in our experiments.
Revised: Note that for the LDCT dataset, we only use the full-dose CT images in our experiments.

Original: In CT image processing, we convert the Hounsﬁeld units from dicom ﬁles to the attenuation coefﬁcients and set the background pixels to zero.
Revised: In CT image processing, we convert the Hounsﬁeld units from dicom ﬁles to the attenuation coefﬁcients and set the background pixels to zero.

Original: Then, 2D CT images are sliced from 3D CT volumes.
Revised: Then, 2D CT images are sliced from 3D CT volumes.

Original: The sinograms are simulated from 2D CT images based on parallel-beam geometry with different number of projection angles that are equally distributed across 180 degrees.
Revised: The sinograms are simulated from 2D CT images based on parallel-beam geometry with different number of projection angles that are equally distributed across 180 degrees.

Original: MRI dataset The Brain Tumor Segmentation (BraTS) 2021 dataset (Menze et al., 2014; Bakas et al., 2017) collected for the image segmentation challenge contains 2000 cases (8000 MRI scans), where each case has four different MR contrasts: native (T1), post-contrast T1-weighted (T1Gd),
Revised: MRI dataset The Brain Tumor Segmentation (BraTS) 2021 dataset (Menze et al., 2014; Bakas et al., 2017) collected for the image segmentation challenge contains 2000 cases (8000 MRI scans), where each case has four different MR contrasts: native (T1), post-contrast T1-weighted (T1Gd),

Original: T2-weighted (T2), and T2 Fluid Attenuated Inversion Recovery (T2-FLAIR).
Revised: T2-weighted (T2), and T2 Fluid Attenuated Inversion Recovery (T2-FLAIR).

Original: For each 3D MR volume, we extract 2D slices from 3D volumes and simulate k-space data by Fast Fourier Transform.
Revised: For each 3D MR volume, we extract 2D slices from 3D volumes and simulate k-space data by Fast Fourier Transform.

Original: To reconstruct MR images, we follow Knoll et al.
Revised: To reconstruct MR images, we follow Knoll et al.

Original: (2020); Zbontar et al.
Revised: (2020); Zbontar et al.

Original: (2018) to undersample k-space data with an equispaced Cartesian mask, where the center k-space is fully sampled while the left k-space is under-sampled by equispaced columns.
Revised: (2018) to undersample k-space data with an equispaced Cartesian mask, where the center k-space is fully sampled while the left k-space is under-sampled by equispaced columns.

Original: B. D ETAILS OF SCORE - BASED GENERATIVE MODELS
Revised: B. D ETAILS OF SCORE - BASED GENERATIVE MODELS

Original: We use the NCSN++ model architecture in Song et al.
Revised: We use the NCSN++ model architecture in Song et al.

Original: (2021), and perturb the data with the Variance
Revised: (2021), and perturb the data with the Variance

Original: Exploding (VE) SDE.
Revised: Exploding (VE) SDE.

Original: Instead of generating samples according to the numerical SDE solver in
Revised: Instead of generating samples according to the numerical SDE solver in

Original: Algorithm 1, we use the Predictor-Corrector (PC) sampler as described in Song et al.
Revised: Algorithm 1, we use the Predictor-Corrector (PC) sampler as described in Song et al.

Original: (2021) since it generally has better performance for VE SDEs.
Revised: (2021) since it generally has better performance for VE SDEs.

Original: In PC samplers, the predictor refers to a numerical solver for the reverse-time SDE while the corrector can be any Markov chain Monte Carlo (MCMC) method that only depends on the scores.
Revised: In PC samplers, the predictor refers to a numerical solver for the reverse-time SDE while the corrector can be any Markov chain Monte Carlo (MCMC) method that only depends on the scores.

Original: One such MCMC method considered in this work is
Revised: One such MCMC method considered in this work is

Original: Langevin dynamics, whereby we transform any initial sample x p 0 q to an approximate sample from p t p x q via the following procedure:
Revised: Langevin dynamics, whereby we transform any initial sample x p 0 q to an approximate sample from p t p x q via the following procedure:

Original: [Equation]
Revised: [Equation]

Original: Here N P N ą 0 , (cid:15) ą 0 , and z p i q „ N p 0 , I q .
Revised: Here N P N ą 0 , (cid:15) ą 0 , and z p i q „ N p 0 , I q .

Original: The theory of Langevin dynamics guarantees that in the limit of N Ñ 8 and (cid:15) Ñ 0 , x p N q is a sample from p t p x q under some regularity conditions.
Revised: The theory of Langevin dynamics guarantees that in the limit of N Ñ 8 and (cid:15) Ñ 0 , x p N q is a sample from p t p x q under some regularity conditions.

Original: Note that
Revised: Note that

Original: Langevin dynamics only requires the knowledge of ∇ x log p t p x q , which can be approximated using the time-dependent score model s θ ˚ p x , t q .
Revised: Langevin dynamics only requires the knowledge of ∇ x log p t p x q , which can be approximated using the time-dependent score model s θ ˚ p x , t q .

Original: In PC samplers, each predictor step immediately follows multiple consecutive corrector steps, all using the same s θ ˚ p x , t q evaluated at the same t .
Revised: In PC samplers, each predictor step immediately follows multiple consecutive corrector steps, all using the same s θ ˚ p x , t q evaluated at the same t .

Original: This jointly ensures that our intermediate sample at t is approximately distributed according to p t p x q .
Revised: This jointly ensures that our intermediate sample at t is approximately distributed according to p t p x q .

Original: As shown in Song et al.
Revised: As shown in Song et al.

Original: (2021), PC sampling often outperforms numerical solvers for the reverse-time SDE, especially when the forward SDE in Eq.
Revised: (2021), PC sampling often outperforms numerical solvers for the reverse-time SDE, especially when the forward SDE in Eq.

Original: (1) is a VE SDE.
Revised: (1) is a VE SDE.

Original: In order to use PC samplers for inverse problem solving, our modiﬁcation is similar to the change made in Algorithm 2 for Algorithm 1.
Revised: In order to use PC samplers for inverse problem solving, our modiﬁcation is similar to the change made in Algorithm 2 for Algorithm 1.

Original: Speciﬁcally, we run line 4 & 5 in Algorithm 2 before every corrector or predictor step.
Revised: Speciﬁcally, we run line 4 & 5 in Algorithm 2 before every corrector or predictor step.

Original: When comparing our approach to previous methods with score-based generative models, we use the same score model to isolate the confounding factors in model training and architecture design.
Revised: When comparing our approach to previous methods with score-based generative models, we use the same score model to isolate the confounding factors in model training and architecture design.

Original: Moreover, we make sure the total cost of sampling is comparable across different methods.
Revised: Moreover, we make sure the total cost of sampling is comparable across different methods.

Original: For the
Revised: For the

Original: ALD sampler used in Jalal et al.
Revised: ALD sampler used in Jalal et al.

Original: (2021), we use 700 noise scales with 3 steps of Langevin dynamics per noise scale, resulting in a total of 700 ˆ 3 “ 2100 steps that require score function evaluation.
Revised: (2021), we use 700 noise scales with 3 steps of Langevin dynamics per noise scale, resulting in a total of 700 ˆ 3 “ 2100 steps that require score function evaluation.

Original: For the PC sampler, we use 1000 noise scales and 1 step of Langevin dynamics per noise scale, totalling 1000 ` 1000 “ 2000 steps of score model evaluation.
Revised: For the PC sampler, we use 1000 noise scales and 1 step of Langevin dynamics per noise scale, totalling 1000 ` 1000 “ 2000 steps of score model evaluation.

Original: For all methods, we tune the step size (cid:15) in Langevin dynamics, as well as λ in Eq.
Revised: For PC samplers, the step size (cid:15) in Langevin dynamics is determined by a signal-to-noise ratio η . For all methods, we tune η and λ in Eq.

Original: (7) with 100 steps of Bayesian optimization on a validation dataset, and report the results on the test dataset with the optimal parameters.
Revised: (8) with 100 steps of Bayesian optimization on a validation dataset, and report the results on the test dataset with the optimal parameters.

Revised: We use the ax-platform toolkit for Bayesian optimization.

Revised: The optimal parameters in our experiments are given by

Revised: • Sparse-view CT on LIDC 320 ˆ 320 : η “ 0 .

Revised: 246 , λ “ 0 . 841 .

Revised: • Metal artifact removal on LIDC 320 ˆ 320 : η “ 0 . 209 , λ “ 0 . 227 .

Original: {+• Sparse-view CT on LDCT 
 ˆ 512 : η “ 0 . 4 , λ “ 0 . 72 . • Accelerated MRI on BraTS 240 ˆ 240 : η “ 0 . 577 , λ “ 0 . 982 .+} B. T RAINING DETAILS OF BASELINE MODELS B.5. B ASELINE MODELS FOR SPARSE - VIEW CT RECONSTRUCTION
Revised: {+• Sparse-view CT on LDCT 
 ˆ 512 : η “ 0 . 4 , λ “ 0 . 72 . • Accelerated MRI on BraTS 240 ˆ 240 : η “ 0 . 577 , λ “ 0 . 982 .+} B. T RAINING DETAILS OF BASELINE MODELS B.5. B ASELINE MODELS FOR SPARSE - VIEW CT RECONSTRUCTION

Original: FBP Filtered back projection (FBP) is a standard way for CT image reconstruction, which simply put the projections (sinogram) back to the image space based on the corresponding projection angles and geometry to get an approximated estimation of the unknown image.
Revised: FBP Filtered back projection (FBP) is a standard way for CT image reconstruction, which simply put the projections (sinogram) back to the image space based on the corresponding projection angles and geometry to get an approximated estimation of the unknown image.

Original: Usually, a high-pass ﬁlter, ramp ﬁlter is used to eliminate the blurring during this process.
Revised: Usually, a high-pass ﬁlter, ramp ﬁlter is used to eliminate the blurring during this process.

Original: In our experiments, we conduct FBP on sparse-view sinograms using the torch radon toolbox (Ronchetti, 2020).
Revised: In our experiments, we conduct FBP on sparse-view sinograms using the torch radon toolbox (Ronchetti, 2020).

Original: FISTA-TV FISTA-TV is a fast iterative shrinkage-thresholding algorithm (FISTA) for solving linear inverse problems in image processing (Beck & Teboulle, 2009).
Revised: FISTA-TV FISTA-TV is a fast iterative shrinkage-thresholding algorithm (FISTA) for solving linear inverse problems in image processing (Beck & Teboulle, 2009).

Original: It adopts a total variation (TV) term as the regularization in the optimization procedure.
Revised: It adopts a total variation (TV) term as the regularization in the optimization procedure.

Original: Each optimization iteration involves a matrixvector multiplication followed by a shrinkage-threshold step.
Revised: Each optimization iteration involves a matrixvector multiplication followed by a shrinkage-threshold step.

Original: In experiments, FISTA is implemented using the tomobar toolbox (Kazantsev & Wadeson, 2020) with the regularization using the CCPi regularisation toolkit (Kazantsev et al., 2019).
Revised: In experiments, FISTA is implemented using the tomobar toolbox (Kazantsev & Wadeson, 2020) with the regularization using the CCPi regularisation toolkit (Kazantsev et al., 2019).

Original: We run 300 iterations for reconstructing each CT image with regularization parameter 0.001.
Revised: We run 300 iterations for reconstructing each CT image with regularization parameter 0.001.

Original: Considering the nature of iterative reconstruction in FISTA, it is quite natural to generalize this method to different number of projections for reconstructing CT images.
Revised: Considering the nature of iterative reconstruction in FISTA, it is quite natural to generalize this method to different number of projections for reconstructing CT images.

Original: In experiments of generalizing to different number of measurements, FISTA method takes as input the sinogram with different numbers of projections and the corresponding angles for these input projections for the iterative procedure.
Revised: In experiments of generalizing to different number of measurements, FISTA method takes as input the sinogram with different numbers of projections and the corresponding angles for these input projections for the iterative procedure.

Original: cGAN
Revised: cGAN

Original: Conventional iterative CT reconstruction algorithms like FISTA are typically slow due to their iterative nature.
Revised: Conventional iterative CT reconstruction algorithms like FISTA are typically slow due to their iterative nature.

Original: Ghani & Karl (2018) proposed to cast sparse-view CT reconstruction as a sinogram inpainting problem.
Revised: Ghani & Karl (2018) proposed to cast sparse-view CT reconstruction as a sinogram inpainting problem.

Original: Speciﬁcally, it used a conditional generative adversarial network (cGAN) to ﬁrst complete the sinogram data prior to reconstructing CT images, thereby avoiding the costly iterative tomographic processing.
Revised: Speciﬁcally, it used a conditional generative adversarial network (cGAN) to ﬁrst complete the sinogram data prior to reconstructing CT images, thereby avoiding the costly iterative tomographic processing.

Original: However, the imperfect sinogram inpainting may further cause image artifacts.
Revised: However, the imperfect sinogram inpainting may further cause image artifacts.

Original: Speciﬁcally, cGAN model takes zero-padded sparse-view sinogram withprojections as input and generates the completed full-angle sinogram with 180 projections.
Revised: Speciﬁcally, cGAN model takes zero-padded sparse-view sinogram withprojections as input and generates the completed full-angle sinogram with 180 projections.

Original: The cGAN model was implemented using PyTorch (Paszke et al., 2019) and trained using a batchsize of 64 and learning rate of 0.0001 with 50 epochs in total.
Revised: The cGAN model was implemented using PyTorch (Paszke et al., 2019) and trained using a batchsize of 64 and learning rate of 0.0001 with 50 epochs in total.

Original: In experiments of generalizing to different number of measurements, we deployed the trained cGAN model by zero-padding sparse-view sinogram with different numbers of projections to full-view sinogram as the input.
Revised: In experiments of generalizing to different number of measurements, we deployed the trained cGAN model by zero-padding sparse-view sinogram with different numbers of projections to full-view sinogram as the input.

Original: After obtaining the output inpainted sinogram, we replace the corresponding projections in the output based on the ground truth projections in the input.
Revised: After obtaining the output inpainted sinogram, we replace the corresponding projections in the output based on the ground truth projections in the input.

Original: Finally, the images were reconstructed from the overlayed sinogram.
Revised: Finally, the images were reconstructed from the overlayed sinogram.

Original: Note that we trained the model using 23 projections and tested it on other projection settings to evaluate the generalization.
Revised: Note that we trained the model using 23 projections and tested it on other projection settings to evaluate the generalization.

Original: SIN -4c-PRN To further reduce the artifacts in both sinogram and image space, SIN-4c-PRN (Wei et al., 2020) proposed a two-step sparse-view CT reconstruction model.
Revised: SIN-4c-PRN To further reduce the artifacts in both sinogram and image space, SIN-4c-PRN (Wei et al., 2020) proposed a two-step sparse-view CT reconstruction model.

Original: It involves a sinogram inpainting network (SIN) to generate super-resolved sinograms with different number of projections, and then a post-processing reﬁning network (PRN) to further remove image artifacts.
Revised: It involves a sinogram inpainting network (SIN) to generate super-resolved sinograms with different number of projections, and then a post-processing reﬁning network (PRN) to further remove image artifacts.

Original: Both networks are connected through a ﬁltered back-projection operation (FBP).
Revised: Both networks are connected through a ﬁltered back-projection operation (FBP).

Original: Speciﬁcally, SIN model takes 23view sinogram as input to ﬁstly upsample to full-view sinogram and then generate sinograms through network for 23, 45, 90, 180 projections respectively.
Revised: Speciﬁcally, SIN model takes 23view sinogram as input to ﬁstly upsample to full-view sinogram and then generate sinograms through network for 23, 45, 90, 180 projections respectively.

Original: FBP transforms these generated sinograms to image space, which was then concatenated and feed into PRN model for reﬁnement.
Revised: FBP transforms these generated sinograms to image space, which was then concatenated and feed into PRN model for reﬁnement.

Original: The framework was implemented using PyTorch (Paszke et al., 2019) while FBP operation was implemented using .
Revised: The framework was implemented using PyTorch (Paszke et al., 2019) while FBP operation was implemented using .

Original: SIN model was trained using a batchsize of 20 and learning rate of 0.0001, while PRN model was trained using a batchsize of 15 and learning rate of 0.0001.
Revised: SIN model was trained using a batchsize of 20 and learning rate of 0.0001, while PRN model was trained using a batchsize of 15 and learning rate of 0.0001.

Original: Considering that LIDC dataset is much larger than LDCT dataset, the SIN-4c-PRN model was trained for 30 epochs on LIDC dataset and 50 epochs on LDCT dataset.
Revised: Considering that LIDC dataset is much larger than LDCT dataset, the SIN-4c-PRN model was trained for 30 epochs on LIDC dataset and 50 epochs on LDCT dataset.

Original: To deploy the trained SIN model to different numbers of measurements, the sinograms with various number of projections are taken as the input for SIN model to generate multi-view sinograms, which were also overlayed with corresponding ground truth projections in inputs.
Revised: To deploy the trained SIN model to different numbers of measurements, the sinograms with various number of projections are taken as the input for SIN model to generate multi-view sinograms, which were also overlayed with corresponding ground truth projections in inputs.

Original: The generated multi-view sinograms are then used for PRN model inference.
Revised: The generated multi-view sinograms are then used for PRN model inference.

Original: Since SIN-4c-PRN model involves the dual-domain learning in both sinogram and image spaces to remove artifacts, and generates multi-scale sinograms during sinogram inpainting, it shows a better generalization to different numbers of measurements compared with cGAN model as shown in
Revised: Since SIN-4c-PRN model involves the dual-domain learning in both sinogram and image spaces to remove artifacts, and generates multi-scale sinograms during sinogram inpainting, it shows a better generalization to different numbers of measurements compared with cGAN model as shown in

Original: Figure 5 and Figure 6.
Revised: Figure 5 and Figure 6.

Original: Neumann Meanwhile, in another parallel direction, researchers proposed to learn the regularizer used in optimization from training data, outperforming traditional regularizers.
Revised: Neumann Meanwhile, in another parallel direction, researchers proposed to learn the regularizer used in optimization from training data, outperforming traditional regularizers.

Original: Speciﬁcally, Gilton et al.
Revised: Speciﬁcally, Gilton et al.

Original: (2019) presented an end-to-end, data-driven method for learning a nonlinear regularizer for solving inverse problems inspired by the Neumann series, called Neumann network.
Revised: (2019) presented an end-to-end, data-driven method for learning a nonlinear regularizer for solving inverse problems inspired by the Neumann series, called Neumann network.

Original: Neumann network was implemented using PyTorch (Paszke et al., 2019).
Revised: Neumann network was implemented using PyTorch (Paszke et al., 2019).

Original: Due to GPU memory constraints, the model training used the batchsize of 5 on LIDC dataset and the batchsize of 2 on LDCT dataset.
Revised: Due to GPU memory constraints, the model training used the batchsize of 5 on LIDC dataset and the batchsize of 2 on LDCT dataset.

Original: The initial learning rate was 0.00001 with an exponential learning rate decay.
Revised: The initial learning rate was 0.00001 with an exponential learning rate decay.

Original: The network was trained with 15 training epochs on both datasets.
Revised: The network was trained with 15 training epochs on both datasets.

Original: B.5. B ASELINE MODELS FOR UNDERSAMPLED MRI RECONSTRUCTION
Revised: B.5. B ASELINE MODELS FOR UNDERSAMPLED MRI RECONSTRUCTION

Original: DuDoRNet Zhou & Zhou (2020) proposed a dual domain recurrent network (DuDoRNet) to simultaneously recover k-space data and images for MRI reconstruction, in order to address aliasing artifacts in both frequency and image domains.
Revised: DuDoRNet Zhou & Zhou (2020) proposed a dual domain recurrent network (DuDoRNet) to simultaneously recover k-space data and images for MRI reconstruction, in order to address aliasing artifacts in both frequency and image domains.

Original: The original model in Zhou & Zhou (2020) also embedded a deep T1 prior to make use of fully-sampled short protocol (T1) as complementary information.
Revised: The original model in Zhou & Zhou (2020) also embedded a deep T1 prior to make use of fully-sampled short protocol (T1) as complementary information.

Original: For a fair comparison with other supervised learning approaches, in our experiments, we do not include this additional information but train the DuDoRNet model without T1 prior.
Revised: For a fair comparison with other supervised learning approaches, in our experiments, we do not include this additional information but train the DuDoRNet model without T1 prior.

Original: The
Revised: The

Original: DuDoRNet was trained using a batchsize of 6 and a learning rate of 0.0005 with 5 training epochs.
Revised: DuDoRNet was trained using a batchsize of 6 and a learning rate of 0.0005 with 5 training epochs.

Original: In experiments of generalizing to different number of measurements, we trained the model with an acceleration factor of 8 and deployed the trained model to other acceleration factors during testing.
Revised: In experiments of generalizing to different number of measurements, we trained the model with an acceleration factor of 8 and deployed the trained model to other acceleration factors during testing.

Original: Speciﬁcally, for inference, we use different Cartesian masking function corresponding to different acceleration factors or down-sampling ratios to sub-sample the k-space data for the network input with the corresponding initial reconstructed image with zero-padding k-space.
Revised: Speciﬁcally, for inference, we use different Cartesian masking function corresponding to different acceleration factors or down-sampling ratios to sub-sample the k-space data for the network input with the corresponding initial reconstructed image with zero-padding k-space.

Original: Cascade DenseNet To reconstruct de-aliased MR images from under-sampled k-space data, Zheng et al.
Revised: Cascade DenseNet To reconstruct de-aliased MR images from under-sampled k-space data, Zheng et al.

Original: (2019) proposed a cascaded dilated dense network (CDDN) for MRI reconstruction, based on stacked dense blocks with residual connections while using the zero-ﬁlled MR image as inputs.
Revised: (2019) proposed a cascaded dilated dense network (CDDN) for MRI reconstruction, based on stacked dense blocks with residual connections while using the zero-ﬁlled MR image as inputs.

Original: Speciﬁcally, they used a two-step data consistency layer for k-space correction, and replaced corresponding phase-coding lines of the generated image with the original sampled k-space data after each block.
Revised: Speciﬁcally, they used a two-step data consistency layer for k-space correction, and replaced corresponding phase-coding lines of the generated image with the original sampled k-space data after each block.

Original: In experiments, we trained the model using a batchsize of 8 and a learning rate of 0.0001, with 5 epochs on BraTS dataset.
Revised: In experiments, we trained the model using a batchsize of 8 and a learning rate of 0.0001, with 5 epochs on BraTS dataset.

Original: In experiments of generalizing to different number of measurements, we trained the model with an acceleration factor of 8 and deployed the trained model to other acceleration factors during testing.
Revised: In experiments of generalizing to different number of measurements, we trained the model with an acceleration factor of 8 and deployed the trained model to other acceleration factors during testing.

Original: Similarly, different masking functions corresponding to different acceleration factors were used to sub-sample k-space data to get network inputs.
Revised: Similarly, different masking functions corresponding to different acceleration factors were used to sub-sample k-space data to get network inputs.

Original: From results, we observe that
Revised: From results, we observe that

Original: Cascaded DenseNet generalizes better to more measurements than DuDoRNet as shown in Figureand Figure 6.
Revised: Cascaded DenseNet generalizes better to more measurements than DuDoRNet as shown in Figureand Figure 6.

Original: B.5. B ASELINE MODELS FOR METAL ARTIFACT REMOVAL
Revised: B.5. B ASELINE MODELS FOR METAL ARTIFACT REMOVAL

Original: LI One straightforward way for reducing metal artifacts is to complete or inpaint the metal-affected missing regions in sinogram directly through linear interpolation (Kalender et al., 1987).
Revised: LI One straightforward way for reducing metal artifacts is to complete or inpaint the metal-affected missing regions in sinogram directly through linear interpolation (Kalender et al., 1987).

Original: This method does not need any network training.
Revised: This method does not need any network training.

Original: However, the imperfect completion of sinogram may introduce secondary artifacts to the reconstructed image.
Revised: However, the imperfect completion of sinogram may introduce secondary artifacts to the reconstructed image.

Original: In our experiments setting, to ﬁt for the practical applications in real world, we assume the ground truth metal trace and mask information are unknown, which can only be estimated by a rough thresholding in artifacts-affected images.
Revised: In our experiments setting, to ﬁt for the practical applications in real world, we assume the ground truth metal trace and mask information are unknown, which can only be estimated by a rough thresholding in artifacts-affected images.

Original: We use the estimated metal mask and metal trace for linear interpolation baseline.
Revised: We use the estimated metal mask and metal trace for linear interpolation baseline.

Original: cGANMAR
Revised: cGANMAR

Original: Wang et al.
Revised: Wang et al.

Original: (2018) proposed a conditional generative adversarial network (cGAN)- based approach for metal artifacts reduction (MAR) in CT.
Revised: (2018) proposed a conditional generative adversarial network (cGAN)- based approach for metal artifacts reduction (MAR) in CT.

Original: Speciﬁcally, cGANMAR network learns the mapping directly from the artifacts-affected CTs to artifacts-free CTs through reﬁnement in image space.
Revised: Speciﬁcally, cGANMAR network learns the mapping directly from the artifacts-affected CTs to artifacts-free CTs through reﬁnement in image space.

Original: The cGANMAR model was implemented using PyTorch (Paszke et al., 2019) and was trained with the batchsize of 64 and the learning rate of 0.0001.
Revised: The cGANMAR model was implemented using PyTorch (Paszke et al., 2019) and was trained with the batchsize of 64 and the learning rate of 0.0001.

Original: The network was trained with 400 epochs.
Revised: The network was trained with 400 epochs.

Original: SNMAR
Revised: SNMAR

Original: Yu et al.
Revised: Yu et al.

Original: (2020) proposed a sinogram completion neural network (SinoNet) to recover the metal-affected projections.
Revised: (2020) proposed a sinogram completion neural network (SinoNet) to recover the metal-affected projections.

Original: Especially, it leveraged the learning in both sinogram domain and image domain by using a prior network to generate a good prior image to guide sinogram learning.
Revised: Especially, it leveraged the learning in both sinogram domain and image domain by using a prior network to generate a good prior image to guide sinogram learning.

Original: Note that in original setting, SNMAR required linear interpolated sinogram and CT as inputs and used ground truth metal trace and mask information to generated them.
Revised: Note that in original setting, SNMAR required linear interpolated sinogram and CT as inputs and used ground truth metal trace and mask information to generated them.

Original: But in our method, we assume the ground truth metal trace and mask information are unknown according to practical scenario and estimate it by a rough thresholding, which will introduce estimation errors.
Revised: But in our method, we assume the ground truth metal trace and mask information are unknown according to practical scenario and estimate it by a rough thresholding, which will introduce estimation errors.

Original: In SNMAR experiments, we still follow the original setting to guarantee the best performance of this baseline method for a strong comparison.
Revised: In SNMAR experiments, we still follow the original setting to guarantee the best performance of this baseline method for a strong comparison.

Original: We trained the SNMAR using the batchsize of 64 and the learning rate of 0.0001, with a total of 100 training epochs.
Revised: We trained the SNMAR using the batchsize of 64 and the learning rate of 0.0001, with a total of 100 training epochs.