Solving Inverse Problems in Medical Imaging with Score-Based Generative Models
Authors: Yang Song
Liyue Shen
Lei Xing
Stefano Ermon
Venue: ICLR 2022 Poster (2022)
Keywords: score-based generative modeling
inverse problems
sparse-view CT
undersampled MRI
metal artifact removal
diffusion

Abstract: Reconstructing medical images from partial measurements is an important inverse problem in Computed Tomography (CT) and Magnetic Resonance Imaging (MRI). 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. 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. To address this issue, we propose a fully unsupervised technique for inverse problem solving, leveraging the recently introduced score-based generative models. Specifically, we first train a score-based generative model on medical images to capture their prior distribution. 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. Our method does not assume a fixed measurement process during training, and can thus be flexibly adapted to different measurement processes at test time. 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.

Published as a conference paper at ICLR 2022
SOLVING INVERSE PROBLEMS IN MEDICAL IMAGING WITH SCORE-BASED GENERATIVE MODELS
Yang Song", Liyue Shen", Lei Xing & Stefano Ermon Stanford University
[yangsong@cs) 1 lyuee,, l 0, ermon@cs) -staforrdeedu
ABSTRACT
Reconstructing medical images from partial measurements is an important inverse problem Computed Tomography (CT) and Magnetic Resonance Imaging (MRI). Existing solutions based on machine learning typically train a model to directly map measurements to medical images, leveraging training dataset of paired images and measurements. These measurements are typically synthesized from images using a fixed physical model the measurement process, which hinders the generalization capability of models to unknown measurement processes. To address this issue, we propose fully unsupervised technique for inverse problem solving, leveraging the recently introduced score-based generative models. Specifically, score-based generative model on medical images to capture their prior distribution. Given measurements and 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. Our method does not assume fixed measurement process during training, and can thus be flexibly adapted to different measurement processes at test time. Empirically, we observe comparable better performance to supervised learning techniques in several medical imaging tasks in CT and MRI, while demonstrating significantly better generalization to unknown measurement processes.
first train
INTRODUCTION
Computed Tomography (CT) and Magnetic Resonance Imaging (MRI) are commonly used imaging tools for medical diagnosis. Reconstructing CT and MRI images from raw measurements (sinograms for CT and k-spaces for MRI) are well-known inverse problems. Specifically, measurements in 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. However, since obtaining the full sinogram for CT causes excessive ionizing radiation for patients, and measuring the full k-space of MRI very time-consuming, has become important to reduce the number of measurements in CT and MRI. In many cases, only partial measurements, such as sparse-view sinograms and downsampled k-spaces, are available. Due to this loss of information, the inverse problems in CT and MRI are often ill ill-posed, making image reconstruction especially challenging.
With the rise of machine learning, many methods (Zhu et al., 2018; Mardani et al., 2017; Shen et al., 2019; Würfl et al., 2018; Ghani & Karl, 2018; Wei et al., 2020) have been proposed for medical image reconstruction using a small number of measurements. Most of these methods are supervised learning techniques. They learn to directly map partial measurements to medical images, by training on a large dataset comprising pairs CT/MRI images and measurements. These measurements need to synthesized from medical images with a fixed physical model of the measurement process. 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. This prevents models from generalizing effectively to new measurement processes, leading to -onnter-intuitive instabilities such more measurements causing worse performance (Antun et al., 2020).
$ Joint first authors.

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In this work, 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. 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. 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). Given trained score-based generative model, we provide 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. Once our model trained, it can be used 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.
We evaluate the performance of our method on several tasks in CT and MRI. Empirically, we observe comparable or better performance compared to supervised learning counterparts, even when evaluated with the same measurement process in their training. 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-vies -iew CT or changing the k-space downsampling ratio in undersampled MRI. Moreover, 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. To the best of our knowledge, this is the first time that generative models are reported successful on clinical CT data. Collectively, these empirical results indicate that our method is competitive alternative to supervised techniques in medical image reconstruction and artifact removal, and has the potential to be universal tool for solving many inverse problems within the same image domain.
2 BACKGROUND
2.1 LINEAR INVERSE PROBLEMS
An inverse problem seeks recover an unknown signal from a set of observed measurements. Specifically, suppose x IR' is an unknown signal, and y E RT Ax + is noisy observation given by m linear measurements, where the measurement acquisition process is represented by a linear operator A E Rm and c E R"" represents a noise vector. Solving linear inverse problem amounts to recovering the signal x from its measurement y. Without further assumptions, the problem is ill-defined when m n, we additionally assume that x is sampled from prior distribution p(x ()). In this probabilistic formulation, the measurement and signal are connected through a measurement distribution p(y x) (y Ax), where qe denotes the noise distribution of e. Given p(y x) and p(x). we can solve the inverse problem by sampling from the posterior distribution p(x y).
Examples of linear inverse problems in medical imaging include image reconstruction for CT and MRI. In both cases, the signal x is a medical image. 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 in MRI consists of spatial frequencies in the Fourier space of the image ka. the k-space in the MRI community) (Vlaardingerbroek & Boer, 2013).
2.2 SCORE-BASSE GENERATIVE MODELS
When solving inverse problems in medical imaging, we are given an observation y. the measurement distribution p(y x) and aim to sample from the posterior distribution p(x y). The prior distribution p(x) is typically unknown, but we can train generative models on a dataset x()) ..... x( ~
p(x) to estimate this prior distribution. Given an estimate of p(x) and the measurement distribution ply x), the posterior distribution p(x y) can be determined through Bayes' rule.
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 observation y. Specifically, we adopt the formulation of score- score-based generative models in Song et al. (2021), where we leverage Markovian diffusion process to progressively perturb data noise, and then smoothly convert noise to samples of the data distribution by estimating and simulating its time reversal. We provide illustration of this generative modeling framework in Fig. 1.
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Data
Forward SDE
Noise
Figure I: We can smoothly perturb images to noise by following the trajec- tory of an SDE. By esti- mating the score function Vx po(p)) with neural networks (called score mod- els), is possible approx- imate the reverse SDE and then solve it to generate im- age samples from noise.
X
dx, - [())x dt g(()dw,
X1
Score function
X0
dx, = [f)) (t)
Vx, log (x,))
]dr + g(t) dw,
X1 Prior
Samples
Reverse-time SDE
Perturbation process Suppose the dataset is sampled from unknown data distribution p(x). We perturb datapoints with a stochastic process over a time horizon [0, 1J. governed by a linear stochastic differential equation (SDE) of the following form
dx,- f(t)x, dt + g(t) dw.. where / [0,1] g. g: [0,
te [o, 1].
R. {w, R ]e denotes standard Wiener process (a.k.a., Brownian motion), and (x, RD) (E(o.) symbolizes the trajectory of random variables in the stochastic process. We further denote the marginal probability distribution of x; Pe(x), and the transition distribution from X to X2 as Pue(x; Xy). By definition, we clearly have Po(x) = p(x). Moreover, the functions f(t) and g(t) are specifically chosen such that for any initial distribution Po(x), the distribution at the end of the perturbation process, Pi (x), is close to a pre-defined noise distribution =(x). In addition, the transition density por (x,, xo) is always conditional linear Gaussian distribution, taking the form Poe Xo) = NNxx | a ((()xo (() where : [0. 1] - IR and B : [o, 1]
can be derived analytically from f(t) and g(t) (Särkkä Solin, 2019). Examples of such SDEs include Variance Exploding (VE), Variance Preserving (VP), and subVP SDEs proposed in Song et al. (2021). We found VE SDEs performed the best in our experiments.
Reverse process By reversing the perturbation process in Eq. (l), we can start from noise sample X1 (x) and gradually remove the noise therein to obtain a data sample Xy Po(x) p(x). Crucially, the time reversal of Eq. (1) is given by the following reverse-time SDE (Song et al., 2021)
dx, [](()x Vxx lx.)) di + g(() dw.,
LE [0,
(2)
where (We)ee(o.1) denotes standard Wiener process in the reverse-time direction, and dt represents an infinitesimal negative time step, since the above SDE must be solved backwards from = 1 to
t O. The quantity Xx, log Pa (x(x)) ) is known the score function of px(xx). By the definition of time reversal, the trajectory of the reverse stochastic process given by Eq. (2) is (xe)=e(o.1) same as the one from the forward SDE in Eq. (1).
Sampling Given an initial sample from (x)), as well as scores at each intermediate time step, Vx logp:(x), we can simulate the reverse-time SDE in Eq. (2) to obtain samples from the data distribution (x) = p(x). In practice, the initial sample is approximately drawn from 7(x) since (xx) ze Pi (xx, and the scores are estimated by training a neural network sø( 5x(x,t) t) (named the score model) on dataset (x(1), x(2) ..... x(A)) ~ p(x) with denoising score matching (Vincent, 2011; Song et al., 2021), ), i.e., solving the following objective
1 0 N
0* arg min
Σ E, EE-u((.
Bx
~x x(1)) po
V
x(0 (o log Pue (x
x
where 2(0, 1] denotes a uniform distribution over [0, 1]. The theory of denoising score matching ensures that $ø* (x, f) a Vx log (x). After training this score model, we plug it into Eq. (2) and solve the resulting reverse-time time SDE
dx; = [f(l)x: - ( sge (xe, 0)] di g(1) dw:, E [0, 11,
(3)
for sample generation. One sampling method is to use the Buler-Maruyee discretization for solving Eq. (3), as given in Algorithm 1. Other sampling methods include annealed Langevin dynamics (ALD, Song & Ermon, 2019), probability flow ODE solvers (Song et al., 2021), and Peddccor-Cooreccte samplers (Song al., 2021).
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Algorithm 1 Unconditional sampling
Algorithm Inverse problem solving
Require: N
Require: N.y.A X1
I: 81 Txx),
π T(x),
2: for N 1 to do
2: for N- - 1 to do
3: t - ++1 N
3:
f + i+1
N
4:
Y: Purlye ly)
5:
T DAP
MAPP1((A)) (1- -1-
(I
4:
-t c 8, f(1)x,A1
6:
& c 8, -
5:
X1-A1 81-A1 A1 g(()"((')))))))"")))"(" (Rx, 1)
7:
X1-A1 c- 81- +g(t)²
((t))=Ro 80x (8e, t)
6:
20,1)
8:
z~,(0,,1)
7:
X1-A1 X1- + g(t)vAizz
9:
X1-A1 Ai gg(t)Aiiz
8:
return No
10:
return X0
3 SOLVING INVERSE PROBLEMS WITH SCORE-BASED GENERATIVE MODELS
With score-based generative modeling, we can train a score model sø* (x, t) to generate unconditional samples from the the prior distribution of medical images p(x). To solve inverse problems however, we will need to sample from the posterior p(x y). This can be accomplished by conditioning the original stochastic process (xfo on observation y. yielding conditional stochastic
process (x, [o. We denote the marginal distribution at Px y), and our goal is to sample from (x y), the same distribution as p(x y) by definition. Much like generating unconditional samples by solving the reverse-time SDE in Eq. (2), we can reverse the conditional stochastic process {x, [o..j] sample from the posterior distribution Po y) by solving the following conditional timerse-time SDE (Song et al., 2021):
dx,- [s(t)x: v(()3Vx, log |y] de dwr, E [0,
(4)
The conditional score function Vx, lo y) is critical part of Eq. (4), yet it non-trivial to compute. One solution is to estimate the score function by training new score model Sø* yx,,,, that explicitly depends on y (Song et al., 2021: Dhariwal & Nichol, 2021). such that Sge Yx y, x Vx, log Pe x, y)- However, this requires paired data y() y for training and has the same drawbacks as supervised learning techniques. We do not consider this approach in this work.
An unsupervised alternative is to approximate the conditional score function with an unconditionally- trained score model 8g* (x,, t) z Xx, log Pt (x)) and the measurement distribution p(y x). Many existing works (Song et al., 2021; Kawar et al., 2021; Kadkhodaie & Simoncelli, 2020; Jalal et al., 2021) 1) have implemented this idea in different ways. However, the methods in Kawar et al. (2021) and Kadkhodaie & Simoncelli (2020) both require computing the singular value decomposition (SVD) of A E T which can be difficult for many measurement processes in medical imaging. The method proposed in Jalal et al. (2021) only designed for specific sampling method called annealed Langevin dynamics (ALD, Song & Ermon, 2019), which proves to be inferior to more advanced sampling algorithms such Predictor-Corrector methods (Song et al., 2021).
In what follows, we propose a new conditional sampling approach for inverse problem solving with score-based generative models. 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 Sø* (x, f) on medical images without assuming any measurement process. Given an observation at test time, we form stochastic process (y: (e by adding appropriate noise to y. We then discretize the reverse-time SDE Eq. (3) with existing unconditional samplers for sø* (x, t). while incorporating the conditional information from y with a proximal optimization step to generate intermediate samples that are consistent with {yr)sejo.1]-
3.1 CONVENIENT FORM OF THE LINEAR MEASUREMENT PROCESS
Many different measurement processes in medical imaging share same components of computation. For example, sparse-view CT reconstruction and metal artifact removal for CT both involve computing the same Radon transform. Similarly, MRI measurement processes require computing the same spatial Fourier transform regardless of different downsampling ratios. To rigorously characterize this structure of measurement processes, propose special formulation of A that is efficient obtain
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A = P(ATT
A = P(A)T
T
P(A)
T
P(A)
y
y
X
Sinogram
diag(A) A)
X
kspace
diag(A)
Figure 2: Linear measurement processes for sparse-view CT (left) and undersampled MRI (right).
in medical imaging applications. Without loss of generality, we assume that the linear operator A has full rank, i.e., rank(A) () = min(n, m) m. The result below gives the alternative formulation of A: Proposition 1. If rank(A) = m, then there exist an invertible matrix T E R=x" and a diagonal matrix A ε {0, 1 j with tr(A) m, such that A = P(A)T. Here P(A) {0, 1] mmmx is an operator that, when multiplied with any vector E R=, reduces its dimensionality to by removing each i-th element of a for i 1, n if Au = 0.
We illustrate this dccomposition for CT/MRI in Fig. 2. Many measurement processes in medical imaging share the same T, even if they correspond to different A. 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. For both sparse-view CT reconstruction and metal artifact removal for CT images, the operator T is the Radon transform (see Fig. 8). Intuitively, diag( A) can be viewed as a subsampling mask on the sinogran/k-space, and PAA) subsamples the sinogram/k-space into observation y with smaller size according to this subsampling mask. In addition, we note that T can be efficiently implemented with the inverse Radon transform or the inverse Fourier transform in CT/MRI applications.
3.2 INCORPORATING A GIVEN OBSERVATION INTO AN UNCONDITIONAL SAMPLING PROCESS
In what follows, we show that the decomposition in Proposition 1 provides an efficient way to generate approximate samples from the conditional stochastic process (x, t) with an unconditional score model 8g* (x, f). The basic idea to "hijack" the unconditional sampling process of score- based generative models to incorporate an observed measurement
As we have already discussed, it is difficult directly solve (x; yhee[..1] for sample generation. To bypass this difficulty, we first consider a related stochastic process that is much easier to sample from. Recall that Poe (x, xx) = Ax; a<()xo, (²( where at)) and (()) can be derived from f(t) and g(t) (Song et al., 2021). Given the unconditional stochastic process {xr [xjo,,,, (o, where y: Ax, + a(t)c. Unlike (x, y) the conditional stochastic process fyr I 1h is fully tractable. First, we have yo = Axo + a(0) = Axo + c y. Since Poe(x, | Xn) = N(x; (9)))) we have
define
a(txxo + 3(t)z, where E IR" ~ NN(0, T). By definition, yt Ax, a(t)e, so we have y, A(a(t)x + ((t)z) + a(f) at) c) + A()) a(t)e a(t)y + Az. Therefore, can easily generate a sample ý. Pr y) by first drawing ~ NNO,, I) and then computing y, = a(t)y + (z)
The key of our approach is to modify any existing iterative sampling algorithm designed for the unconditional stochastic process (xe)te(o.1) so that the samples are consistent with {yz y In general, an iterative sampling process of based generative models selects sequence of time steps {0- = to < h < .... < iN 1} and iterates according to
سt,-, h 1e(t, Zi, 80* (8t,, 44), i =
.1
(5)
where Xex (x)), 2, No I). and 0* denotes the parameters in unconditional score model Sga (x, t). Here the iteration function takes a noisy sample X,, and reduces the noise therein to generate K;,,,, using the unconditional score model Søx (x, t). For example, for the Euler-Maruyama sampler detailed in Algorithm 1, this iteration function is given by
h(Xi, Zi, S0s (Kt, t+)) = x;, - f + g(t,) s0= (&t, +)/N + g(t))////N
Samples obtained by this procedure HHo constitute an approximation of (xt]xe(o.1). where the last sample Ri can be viewed as an approximate sample from px(x). Most existing sampling
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naise
(1 (1-A)4
T
I A
results
X
(A
YtN
tsr
S1L
y
AA
Unconditional generation Inverse problem solving
Figure 3: (Left) An overview of our method for solving inverse problems with score-based generative models. (Right) An illustration about how combine &r and y to form X
methods for score-based generative models are instances of this iterative sampling paradigm, including Algorithm ALD (Song& & Ermon, 2019), probability flow ODEs (Song et al., 2021) and Predictor- Corrector samplers (Song et al., 2021).
To enforce the constraint implied by (ye yhe(o.1), we prepend an additional step the iteration rule in Eq. (5), leading to
_ h( Zi, 8g (xt,;)))
N, 1
where Key (x), Sr. Pr, (yr: y). and 0 & À < 1 is hyper-parameter. We provide an illustration of this process in Fig. 3. The iteration function k(-, Vr,, A) IR" - R" promotes data consistency by solving a proximal optimization step (Nesterov, 2003; Boyd al., 2004; Hammernik et al., 2021) that simultaneously minimizes the distance between X,, and &t, and the distance between xt, and the hyperplane fır T Ax Sr. with a hyperparameter & A & balancing between the two:
X, = arg min (1 A) ||= & =EB
2 min MEll
|| - u1 s.t. Au = Et.
Recall that A P(A]T according Proposition In the equation above we choose the norm ||a|| = |Tall 2 to simplify our theoretical analysis. The decomposition in Proposition allows us to derive closed-fore form solution the optimization problem in Eq. (8), as given below:
Theorem 1. The solution of Eq. (8) can be given by
L -TPP
(1 AATTX, + + J)
where P P-((A): R. I. denotes any right inverse P(A).
See Fig. 3 for illustration of the function X =k(Ss,, -e, The right inverse P-'(A) increases the dimensionality of a vector a E R" to n by putting its entries on every index of an n-diiennional vector where A 1. Recall that in sparse-view CT undersampled MRI, diag( (A) represents subsampling mask, and PP subsamples the full sinogram/k-space to generate the observation y. In this case, (A) pads the observation y that it has the same size as the full sinogram/k-space.
When À- = 0, x k( Fr.. St. case our sampling method in Eq. (7) performs unconditional generation. On the other hand, when = 1, & = ( 1) satisfies A& = yt, exactly. When the measurement is noisy, we choose 0 < A < to allow slackness the constraint A&; = Yt.- The value of is important for balancing between x s Xt, and Ax', Yt.- In practice, we use Bayesian optimization to tune this À automatically a validation dataset. When the measurement process contains noise, replace Xi, with k ((E,,, y, 1) at the last sampling step to guarantee Axi, = y.
X,, completely ignores the constraint AX = Yr. in which
In summary, our method given in Eq. (7) introduces minimal modifications to existing iterative sampling method of score- score-based generative models. For example, we can convert the sampler in Algorithm 1 to inverse problem solver in Algorithm 2 by adding/modifying just three lines of

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PSNR: 15.32, SSIM: 6.796 PSNR: 17.76, SSIM: 0.454 PSNR: 17 17.60, SSIM: 0.471 PSMR: 27.86. SSIM: o. 0.908 PSNR: 35.57, S5 0.929
PSNR: 20.30, SSIM: 0.778 PSAR: 22.04, SSIM: 0.552
PSNR: 22.78, SSIM: 0.603
PSMR: 31.76, S5IM: 2.882 PSNR 35,23, SS 0.912
(a) FISTA-TV
(b) cGAN
(c) Neumann
(d) SIN-4c-PRN
(c) Ours
(f) Ground truth
Figure 4: Examples of sparse-view CT reconstruction results on LIDC 320 x 320 (Top row) and
LDCT 512 x 512 (Bollom row), all with 23 projections. You may zoom in to view more details.
pseudo-code. Unlike the concurrent work Jalal al. (2021), our method Langevin dynamics (ALD). As demonstrated in our experiments, we outperform Jalal et al. (2021) even with the same ALD sampler, and can widen the performance gap further by using more advanced approaches like the Predictor Correcto sampler (Song et al., 2021). Unlike Kadkhodaie & Simoncelli (2020); Kawar al. (2021), we rely on the efficient alternative representation of A given in Section 1, and do not require expensive SVD computation.
limited to annealed
EXPERIMENTS
We aim to answer the following questions in this section: (1) Can we directly compete with best-in- class supervised learning techniques for the same measurement process used in their training, even though our approach is fully unsupervised? (2) Can our method generalize better to new measurement processes? (3) How do we fare against other unsupervised approaches? 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. More experimental details provided in Appendix B.
Datasets consider two datasets for CT experiments. The first is the Lung Image Database Consortium (LIDC) image collection dataset (Armato III et al., 2011 1; Clark et al., 2013) where slice the original 3D CT volumes to obtain 130304 2D images of resolution 320 x 320 for training. The second is the Low Dose CT (LDCT) Image and Projection dataset (Moen 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 512 512 for training. We simulate CT measurements (sinograms) with a parallel-beam geometry using projection angles equally distributed across 180 degrees. For MAR experiments, we follow Yu al. (2020) synthesize metal artifacts. For undersampled MRI experiments, we use the Brain Tumor Segmentation (BraTS) 2021 dataset (Menze al., 2014; Bakas et al., 2017), where we slice 3D MRI volumes get 297270 images of resolution 240 x 240 as the training dataset. We simulate MRI measurements with Fast Fourier Transform using a single-coil setup, and follow Zbontar al. (2018); Knoll et al. (2020) to undersample the k-space with equispaced Cartesian mask. The performance is measured on 1000 test images with peak signal-to-noise ratio (PSNR) and structural similarity (SSIM).
Standard techniques in medical imaging We include two standard learning-free techniques as baselines for sparse-view CT reconstruction. The first is filtered back projection on sparse-view sinograms, which is denoted by "FBP". The second is iterative reconstruction method with total variation regularization called FISTA-TV (Beck & Teboulle, 2009). For MAR experiments, we include another learning-free baseline called linear interpolation (LII, Kalender al., 1987).

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Table 1: Results for undersampled MRI reconstruction on BraTS. First two methods are supervised learning techniques trained with 8x acceleration. The others are unsupervised techniques.
24x Acceleration
8x Acceleration
4x Acceleration
Method
PSNR1
SSIM
PSNR1
SSIM
PSNR1
SSIM
Cascade DenseNet
23.39+2.17
0.765=20042
28.35=2.30
0.845+0.038
30.97
0.902=00028
DuDoRNet
18.4613.05
0.662
37.88
0.985
30.5314.13
0.891=0.071
Score SDE
27.83+2.73
0,849 :0.998
35,04
0.943+
37.55+2.08
0.960=00013
Langevin
28.8013.11
0.873
36.44
0.952
38.7612.32
0.966-00012
Ours 29.42:3.03
0.880
37.63
0.958
39.91
0.965
Table 2: Results for sparse-view CT reconstruction on LIDC and LDCT. FISTA-TV is a standard
iterative reconstruction method that does not need training. cGAN, Neumann, and SIN-4c-PRN are supervised learning techniques trained with 23 projection angles.
LIDC 320
320
LDCT 512 x 512
Method
Projections
PSNR
SSIM
PSNR1
SSIM
FBP
23
10.1
0.230=0.072
10.11
0.302+00078
FISTA-TV
23
20.08
0.799=00061
21.88:44.42
0.850=0.067
cGAN
23
19.83
0.479
19.90
0.545
Neumann
23
17.18
0.454
18.83
0.525
SIN-4c-PRN
23
30.48
0.895
34.82
0,877
10
29. 29.52:2263
0.803
28.96:44.41
0,849
Ours
20
+..60
0.895 00008
36.80
0.936
23
35.24:2.71
0.905 0.905
37.41 14.62
0.941 00.05
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. We follow the settings in Wei et al. (2020) and train all methods with 23 projection angles. For MAR, we use eGANMAR (Wang et al., 2018) and SNMAR (Yu et al., 2020) as the baselines. For undersampled MRI on BraTS, we compare against Cascade DenseNet (Zheng al., 2019) and DuDoRNet (Zhou & Zhou, 2020), which are both trained with a 8x acceleration factor by measuring only 1/8 of the full k-space.
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. Specifically, we consider the "Langevin" approach proposed in Jalal et al. (2021), and the "Score SDE" method in Song et al. (2021), where the former limited to annealed Langevin dynamics (ALD) sampling, and the latter was based on crude approximation to the conditional score function Vx, log P. yy) in Eq. (4), and was proposed as a theoretical possibility in Appendix 1.4 of Song et al. (2021) without experiments. We only focus undersampled MRI for these baselines, since it is the only medical imaging problem ever tackled with score- based generative models before our work. All methods share the same score models and only differ in terms of inference. We make sure all sampling algorithms have comparable number of iteration steps (N in Eqs. (5) and (7)).
Competing with supervised learning approaches Thanks to the outstanding sample quality of score-based generative mod- els, we can achieve comparable or better performance than best-in-class supervised learning methods even for the same measurement process used in their training. As shown in Ta- ble 2, we outperform the top supervised learning technique SIN-4c-PRN on sparse-view CT reconstruction by a significant margin, both the LIDC and LDCT datasets. Our results with
Table MAR results on LIDC.
Method LI
PSNR1
SSIM
26.3012.2
0,910
cGANMAR
27.2741.6
0.927+u00
SNMAR
27.28+1.5
0.937=0048
Ours
32.16:2.22
0.939:u022
20 measurements are even better than supervised learning counterparts with 23 measurements. In Fig. 4, we provide visual comparison of the reconstruction quality for various methods, where is clear to see that our method can recover more details faithfully. From results in Table 3, we also outperform the top supervised leaming method SNMAR metal artifact removal. As shown
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90
DuDoRNNtt
FISTA-TV
50
SDE
DanseNot
45
Langarii
Langevin
SIN-oo-RR
ALD Ours
SDE
PSNR
Ours.
PSNR
-10
Ours
Ours
20
10
10 15
20
100
10
20
Acceleration factor
CT projections
Acceleration fastor
Figure 5: Performance vs. numbers of measurements. Shaded areas represent standard deviation. (Left) MRI on BraTS. (Center) CT on LIDC. (Right) Comparing score-based generative models for undersampled MRI reconstruction on BraTS.
in Fig. 7, our method generates images with less artifacts and preserves the structure better. For undersampled MRI reconstruction results given in Tables 1 and 3, our method is ranked the 2nd for the case of 8x acceleration, with comparable performance to the top supervised method DuDoRNet.
Generalizing to different number of measurements Since our approach is fully unsupervised, we naturally apply the same score model to different measurement processes. We first consider changing the number of measurements at the test time, e.g., using different number of projection angles (resp. different acceleration factors) for sparse-view CT (resp. undersampled MRI) reconstruction. As shown in Table and Fig. 5 (Left), we achieve the best performance on undersampled MRI for both 24x and xx acceleration factors, whereas DuDoRNet fails generalize when the acceleration factor changes. The other supervised learning approach Cascade DenseNet demonstrates limited adaptability by building model architecture inspired by the physical measurement process of MRI, but fails to yield top-level performance. For sparse- view CT reconstruction, all supervised learning methods struggle to generalize to different projection angles, as shown Fig. 5 (Center).
Generalizing to different measurement processes in CT We can perform both sparse- -ieww CT reconstruction and metal artifact removal (MAR) with a single score model trained on CT images. 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. We provide a visualization of the measurement process corresponding to MAR in Fig. 8. As shown in Table 3, supervised learning techniques specifically designed and trained for MAR, while using the same score model used in sparse-vies view CT reconstruction LIDC.
outperform
Comparing against existing score-based methods We compare our method against Langevin (Jalal al., 2021) and Score SDE (Song et al., 2021) for undersampled MRI reconstruction BraTS. Two variants of our approach are considered, which respectively use annealed Langevin dynamics (ALD) and the Predictor-Correctt denote the former by "ALD Ours", and the latter by "PC + Ours" (our default method for all other experiments). Recall that Langevin uses ALD as the sampler, same as "ALD Ours". All results are provided in Fig. 5 (Right). We observe that "ALD + Ours" uniformly outperform Langevin and Score SDE across all numbers of measurements in the experiment. Moreover, "PC Ours" can further improve "ALD + Ours", demonstrating the power of switching to more advanced sampling methods of score-based generative models our proposed approach.
(PC) sampler for score-based generative models the backend. We
5 CONCLUSION
We propose a new method to solve linear inverse problems with score-based generative models. 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. Empirical results demonstrate that our method can match or outperform existing supervised learning counterparts on image reconstruction for sparse- 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 single model.

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AUTHOR CONTRIBUTIONS
Yang Song designed the project, wrote the paper, and ran all experiments for score-based generative models. Liyue Shen preprocessed data, ran all baseline experiments, and helped write the paper. Lei Xing and Stefano Ermon supervised the project, provided valuable feedback, and helped edit the paper.
ACKNOWLEDGMENTS
YS is supported by the Apple PhD Fellowship in AI/L. LS is supported by the Stanford Bio-X Graduate Student Fellowship. This research was supported by NSF (#1651566 #1522054, #1733686), ONR (N000141912144) and Google TPU Research Cloud. This research was also supported by NIH/NCI (IR01 CA256890 and IR01 CA227713).
AFOSR (FA95501910024), ARO IN911NF211---122
Sloan Fellowship,
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A PROOFS
Proposition 1. If (A) m, then there exist invertible matrix T E RR and a diagonal matrix A E {0, 1 with tr(A) m, such that A = P(A)T. Here P(A) E {0. 1 is an operator that, when multiplied with any vector R", reduces its dimenstonality to by removing each i-th element of a for 1,2, n in As = O.
Proof. Let A (a az a R Since A has full rank, the vectors {a1, 02.... am } are linearly independent. We can therefore extend them to total of n linearly independent vectors, i.., {ar
am. bi.... bn-m). Due the linear independence, know T = (a), 02. ... a bj. ... b)-m) E Ruxn has full rank and is invertible. Next, we define
a2 a2;
A diag 1, 1,0.0.... ,0 0),
where diag converts a vector to diagonal matrix. Clearly tr(A) m and A P(A)T, which completes the proof.
Lemma 1. Let (-1(A): IRm R be any right inverse of P(A) R Rm. For any I and St R. we huve
P(A)Tu= = ATu= AP (
Proof. By the definition of P(A), we have P(A) P(A)A, and
V R" b R" P(A)a = P(A)6 = Aa= Ab.
(10)
To prove the "if" direction, we note that
ATu = AP AP (A)):
P(A)ATu (AA)
P(A)TU- P(A) P(A)Tu= = Sr.
To prove the "onlyy if" direction, have
P(A)Tu ÿr
P(A)Tu = P(A)P-
ATu AA)
where (i) is due the property in Eq. (10). This completes the proof both directions.
Theorem 1. The solution of Eq. (8) can be given by
* T [AAP (A)8.. + (1 A)AT + - A)T8;,
(9)
where P- (A) m IR.h denotes any right inverse P(A).
Proof. The optimization objective function in Eq. (8) can be written as
(1 >) - x; 1 A| 2112-
=(1 X) >|TT T + À |Tz Tull2
=(1- - A) IT= Tx + > >|AT(( u) + (I A)T(z u2)13
=(1 1T= -T&el1 + À AAT 11)3==u)11 [I A)T(2 24)1
-(1- TT= -T&1l1 + AT= AP-1(A)); + -][[I-A)((z 24
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1.0
1.0
1.00
0.0
0.8
0.55
FISTA-TV
OGAN
BIN-4o-PRN
SS1N
0.00
Cascodt DenoeNet
0.4
Ours
Scree STF
0.6
Langerth
SDE
0.6
PC + Outs
0.0
15
Acceloration
projections
Acceleration
Figure 6: SSIM vs. numbers of measurements. Shaded areas represent standard deviation. (Left) MRI BraTS. (Center) CT LIDC. (Right) Comparing score-based generative models for undersampled MRI reconstruction on BraTS.
PSNR: 28.02, SSIM 0.921 PSNR: 29.48 SSI: 0,950
PSMR: 27.44. SSIM: 0.947 PSNR: 34.53, SSM 0.971
(a) Metal artifact
(b) LI
(c) cGANMAR
(d) SNMAR
(e) Ours
(I) Ground truth
Figure 7: Examples of metal artifact removal LIDC. You may zoom in to view details.
Since Au = yr, we have P(A)Tu = ÿr and equivalently ATu AP-)(A)): due to Lemma 1. This constraint does not restrict the value of (I A)Tu. Therefore, when Au ýt, we have
| - x + min(1 - A)A u1/
=(1 - A) TT T& 2 + min X
| AT AP (A)9 A 2
=(1 - A) T T + AT - AP (A))
=(1 - A) AT - AT&; 1 X AT AP 'A)\ +1 (1-
A)T&:
This simplifies the optimization problem in Eq. (8) to
min(1 1 X) ATT AT + A AT= AP (A)ý: (1- X) II(I II A)T2 (I AyT
which is minimizing quadratic function of z. The optimal solution = thus in closed form:
=T - + (1 A)ATX; + AAP AAP-1 (A)r.).
According to the definition, X = z², whereby the proof is completed.
B ADDITIONAL EXPERIMENTAL DETAILS
B. ADDITIONAL RESULTS
In Fig. 6, we provide SSIM results versus the number of measurements for multiple methods and tasks. In general, the SSIM curves have very similar trends to the PSNR curves in Fig. 5. We additionally provide a visualization of metal artifact removal results Fig. 7.
B.2 THE TASK OF METAL ARTIFACT REMOVAL
Metallic implants in object can strong metal artifacts in CT imaging. As shown in Fig. the source of artifacts come from extremely bright regions in the sinogram, called metal traces. To reduce ideally remove metal artifacts from CT image, remove metal traces from the sinogram and
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A= P(A)T
X
X
T
P(A)
Sinogram
diag(A) (A)
y
Figure 8: The linear measurement process of metal artifact removal.
leverage the data prior to complete the sinogram. 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.
B.3 DETAILS OF DATASETS
CT datasets We conduct experiments of 2D CT image reconstruction on two datasets. First, the 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. 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. Note that for the LDCT dataset, we only use the full-dose CT images in our experiments. In CT image processing, convert the Hounsfield units from dicom files the attenuation coefficients and set the background pixels to zero. Then, 2D CT images are sliced from 3D CT volumes. The sinograms are simulated from 2D CT images based parallel-beam geometry with different number of projection angles that are equally distributed across 180 degrees.
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 (TI), post-contrast -eightht (T1Gd), T2-weighted (T2), and T2 Fluid Attenuated Inversion Recovery (T2-FLAIR). For each 3D MR volume, we extract 2D slices from 3D volumes and simulate k-space data by Fast Fourier Transform. To reconstruct MR images, we follow Knoll et al. (2020); Zbontar et al. (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- under-sampled by equispaced columns.
B.4 DETAILS OF SCORE-BASED GENERATIVE MODELS
We use the NCSN++ model architecture in Song et al. (2021), and perturb the data with the Variance Exploding (VE) SDE. Our training procedure follows that of Song et al. (2021). Instead of generating samples according to the numerical SDE solver in Algorithm 1, we use the Predictor-Correcde sampler as described in Song et al. (2021) since it generally has better performance for VE SDEs. In PC samplers, the predictor refers to 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. One such MCMC method considered in this work is Langevin dynamics, whereby we transform any initial sample (() to approximate sample from pt (x) via the following procedure:
(PC)
(i+1) (+) c cV log (x + 2 (1) i 0, 1, 1
(11)
Here N E Noo. e 0. and (i) I). The theory of Langevin dynamics guarantees that in the limit of N - 00 and - 0, x (N) is sample from pr (x) under some regularity conditions. Note that Langevin dynamics only requires the knowledge of Vx log (xx). which can be approximated using the dime-depeneent score model sø* (x, In PC samplers, each predictor step immediately follows multiple consecutive corrector steps, all using the same Søs (x, t) evaluated at the same t. This jointly ensures that intermediate sample / is approximately distributed according (x().. As shown
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in Song al. (2021), PC sampling often outperforms numerical solvers for the reverse-time SDE, especially when the forward SDE in Eq. (1) is a VE SDE. In order to use PC samplers for inverse problem solving, our modification is similar to the change made in Algorithm 2 for Algorithm 1.
Specifically, we run line 4 & 5 in Algorithm 2 hefore every corrector or predictor step.
When comparing
approach to previous methods with score-based generative models, we the same score model to isolate the confounding factors in model training and architecture design. Moreover, we make sure the total cost of sampling is comparable across different methods. For the ALD sampler used in Jalal et al. (2021), we use 700 noise scales with 3 steps of Langevin dynamics per noise scale, resulting a total of 700 the PC sampler, we use 1000 noise scales and step of Langevin dynamics per noise scale, totalling 1000 + 1000 2000 steps of score model evaluation.
2100 steps that require score function evaluation. For
For PC samplers, the step size c in Langevin dynamics is determined by signal-to-nol -o-nosee ratio 1). For all methods, we tune 1) and in Eq. (8) with 100 steps Bayesian optimization on a validation dataset, and report the results the test dataset with the optimal parameters. We use the ax ax-plat at form toolkit for Bayesian optimization. The optimal parameters our experiments are given by
Sparse-view CT on LIDC 320 x 320: 7) = 0.246, 0.841.
Metal artifact removal on LIDC 320 x 320: 7) = 0.209, À = 0.227.
view CT on LDCT 512 512: 17 0.4, À = 0.72.
Accelerated MRI BraTS 240 x 240: ?) 0.577, À 0.982.
B.5 TRAINING DETAILS OF BASELINE MODELS
B.5.1
BASELINE MODELS FOR -PARSE-VIEW CT RECONSTRUCTION
Filtered back projection (FBP) is 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 approximated estimation of the unknown image. Usually, a high-pass filter, ramp filter is used to eliminate the blurring during this process. In our experiments, we conduct FBP on sparse-view view sinograms using the torch radon toolbox (Ronchetti, 2020).
FBP
FISTA-TV FISTA-TV is fast iterative sarinkage-thresholle algorithm (FISTA) for solving linear inverse problems in image processing (Beck& & Teboulle, 2009). It adopts total variation (TV) term as the regularization in the optimization procedure. Each optimization iteration involves matrix- vector multiplication followed by shrinkage-threshole step. In experiments, FISTA is implemented using the tomobar toolbox (Kazantsev & Wadeson, 2020) with the regularization using the CCPi regularisation toolkit (Kazantsev et al., 2019). We run 300 iterations for reconstructing each CT image with regularization parameter 0.001. Considering the nature of iterative reconstruction in FISTA, is quite natural to generalize this method to different number of projections for reconstructing CT images. 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.
cGAN Conventional iterative CT reconstruction algorithms like FISTA are typically slow due to their iterative nature. Ghani & Karl (2018) proposed to cast sparse-view CT reconstruction as a sinogram inpainting problem. 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. However, the imperfect sinogram inpainting may further cause image artifacts. Specifically, cGAN model takes zero- pado-ppdddd sparse-view sinogram with 23 projections input and generates the completed full-angle sinogram with 180 projections. 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. In experiments of generalizing to different number of measurements, we deployed the trained cGAN model by zero prdddiadding sparse-view sinogram with different numbers of projections full-view sinogram the input. After obtaining the output inpainted sinogram, we replace the corresponding projections in the output based on the ground truth projections in the input. Finally, the images were reconstructed from the overlayed sinogram. Note
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Published as a conference paper at ICLR 2022
that we trained the model using 23 projections and tested it on other projection settings to evaluate the generalization.
-IN-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. involves a sinogram inpainting network (SIN) to generate super-resolved sinograms with different number of projections, and then a post- post-procsssing refining network (PRN) to further remove image artifacts. Both networks connected through filtered back paojeprooettion operation (FBP). Specifically, SIN model takes 23- view sinogram as input to fistly upsample to full-view sinogram and then generate sinograms through network for 23, 45, 90, 180 projections respectively. FBP transforms these generated sinograms to image space, which was then concatenated and feed into PRN model for refinement. The framework was implemented using PyTorch (Paszke al., 2019) while FBP operation was implemented using SIN model was trained using a batchsize of 20 and learning rate of 0,0001, while PRN model was trained using batchsize of 15 and learning rate of 0.0001. 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. 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. The generated multi-view sinograms are then used for PRN model inference. 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 better generalization to different numbers of measurements compared with cGAN model as shown in Figure and Figure 6.
Neumann Meanwhile, in another parallel direction, researchers proposed to learn the regularizer used in optimization from training data, outperforming traditional regularizers. Specifically, Gilton et al. (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. Neumann network was implemented using Py PoTcch (Paszke et al., 2019). Due to GPU memory constraints, the model training used the batchsize of 5 on LIDC dataset and the batchsize of 2 on LDCT dataset. The initial learning rate was 0.00001 with an exponential learning rate decay. The network was trained with 15 training epochs on both datasets.
B.5.2 BASELINE MODELS FOR UNDERSAMPLED MRI RECONSTRUCTION
DuDoRNet Zhou & Zhou (2020) proposed 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. 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. For fair comparison with other supervised learning approaches, in our experiments, we do not include this additional information but train the DuDoRNet model without TI prior. The DuDoRNet was trained using batchsize of 6 and a learning rate of 0.0005 with training epochs. In experiments of generalizing to different number of measurements, we trained the model with an acceleration factor of and deployed the trained model other acceleration factors during testing. 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.
Cascade DenseNet To reconstruct de-aliased MR images from under-sampled k-space data, Zheng et al. (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. Specifically, they used a two-step data consistency layer for k-space correction, and replaced corre- sponding phase-coding lines of the generated image with the original sampled k-space data after each block. In experiments, we trained the model using batchsize of 8 and a learning rate of 0.0001, with 5 epochs on BraTS dataset. 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. Similarly, different masking functions corresponding to different acceleration factors were used sub-sample k-space data get network inputs. From results, we observe that
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Published as a conference paper at ICLR 2022
Cascaded DenseNet generalizes better more measurements than DuDoRNet as shown in Figure and Figure 6.
B.5.3 BASELINE MODELS FOR METAL ARTIFACT REMOVAL
LI One straightforward way for reducing metal artifacts to complete or inpaint the metal-affected missing regions in sinogram directly through linear interpolation (Kalender et al., 1987). This method does not need any network training. However, the imperfect completion of sinogram may introduce secondary artifacts to the reconstructed image. 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 only be estimated by rough thresholding in artifacts- images. We use the estimated metal mask and metal trace for linear interpolation baseline.
cGANMAR Wang al. (2018) proposed a conditional generative adversarial network (cGAN)- based approach for metal artifacts reduction (MAR) in CT. Specifically, eGANMAR network learns the mapping directly from the artifacts-affected CTs to artifacts-free free CT' through refinement image space. The cGANMAR model was implemented using PyTorch (Paszke et al., 2019) and was trained with the batchsize of and the learning rate 0.0001. The network was trained with 400 epochs.
SNMAR Yu et al. (2020) proposed sinogram completion neural network (SinoNet) recover the metal-affected projections. Especially, it leveraged the learning in both sinogram domain and image domain by using prior network generate a good prior image guide sinogram learning. 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. But in our method, we assume the ground truth metal trace and mask information are unknown according to practical scenario and estimate by a rough thresholding, which will introduce estimation errors. In SNMAR experiments, we still follow the original setting to guarantee the best performance of this baseline method for strong comparison. We trained the SNMAR using the batchsize of 64 and the learning rate of 0.0001, with total of 100 training epochs.
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Reviews:
[{"review_id": "lt5RBaGaA56", "reviewer": "Reviewer_fqS4", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "The paper establish a new approach approach to solve linear inverse problems with score based generative models. In particular, given an unconditional score based generative model it is diffucult to sample directly from the conditional distribution $p(x_t|y)$ as it requires solving the conditional backward SDE. Instead, the authors propose to utlize the linear relationship between $x$ and $y$, and introduce a coupled stohcastic process $y_t$. Sampling can be done by adding an additional step that enforcing the linear relation between $x_t$ and $y_t$ at each sampling step.\n\nThe paper is well written and the idea is concise and novel. In current form it can only be applied to linear inverse problems, but I think there are certainly important applications of linear inverse problems so the contribution of this paper is substantial. I am not very familiar with the experimental evaluation of medical imaging, but I do see the paper making significant improments over previous methods based on paired training or generative models.\n\nI have several concerns:\n\n1. I think the biggest issue of the paper is lacking an motivation for the method. The method part of the paper can be re-organize a little bit to make the presentation clear. For example, the proposition 1 is somewhat absurd, as at the begining I do see why we need such a decomposition. The purpose of the extra iteration function is clear after reading it carefully, but I find it difficult to read at first look. I would suggest to first give a high level overview of the approach before section 2.1, that briefly introduces all the components of this approach and how to make them work, and them go through the details.\n\n2. Since score-based model also has the probability flow ODE formulation, and it can be sampled by solving the ODE where the randonmess only come from initialization. Can the probability flow ODE formulation be used in this task? In addition, maybe you can also discuss previous approch that uses normalizing flow as prior for the solving inverse problems such as [1]. \n\n3. Let me know if I missed anything, but what is the sampling formuation for the item \"Score SDE\" in table 1? Do you train an explicit conditional score based model? In general I think more detailed introduction to each item in the table is needed, espeically for the item \"Score SDE\" which is highly related. \n\n4. Also, do you include a study on if the result is sensitive to $\\lambda$, the coefficient for the linear constraint? \n\n[1] Invertible generative models for inverse problems: mitigating representation error and dataset bias. https://arxiv.org/abs/1905.11672", "summary_of_the_paper": "The paper introduces a method to use score-based generative model as a powerful prior when solving linear inverse problems in medical imaging. Compared to previous work that applied score models to inverse problems, this paper proposes a new conditional sampling approach.", "summary_of_the_review": "The paper is well written, with a clean idea and strong empirical results. There are some issues with the presenttaion. I will give a weak accept.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "guFK9bm1kD2", "reviewer": "Reviewer_YBf4", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "Strengths: \n1) The paper targets an interesting, important, but challenging problem in medical image reconstruction, which could be attractive to ICLR participants. \n\n2) Although I do not fully understand the proposed method (see the weaknesses below), it seems to be technically sound and efficient. \n\n3) The experimental results demonstrate the effectiveness of the proposed method, which shows comparable or even better results compared to supervised methods. \n\nWeaknesses:\n1) The writing of the method section could be improved to make it easier to understand. What is the connection between Section 3.1 and 3.2? How to obtain T? Predfined transformations? How to compute P(\\Lambda)? T and P(\\Lambda) are essential to calculate A, while they are not clearly explained in the paper. \n\n2) Maybe I missed the point, but what is the main difference between GAN-based methods and the score-based generative models? In the SDE model, is there a time-step size that should be predefined? If so, will it affect the generation result? How about the proposed method? How to set the step size? 1/N? how to choose N? \n\n3) The paper claims the efficiency of the method, then what is the computational cost of the proposed method? ", "summary_of_the_paper": "This paper provides an unsupervised approach to solve the inverse problem for reconstructing medical CT and MRI scans using score-based generative models. The proposed method was evaluated on the LIDC and BraTS datasets. Compared to existing supervised and unsupervised approaches, the proposed method demonstrates comparable or better performances in terms of PSNR and SSIM. ", "summary_of_the_review": "Although the paper has some limitations, it presents good technical contributions. Therefore, I recommend the weak acceptance for this paper. ", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "0gEpq-KHgsk", "reviewer": "Reviewer_5fGt", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "Strengths:\n\nThe overall idea - combine diffusion-based generative models with inverse problems - is well presented and the paper is easy to read. Compared to existing methods in this research direction, there is some novelty, mainly in the adapted sampling approaches.\n\nThe experimental results of the paper suggest that the proposed combination of unsupervised denoising score matching of a variance exploding SDE along with the novel sampling approach yields superior results and even outperforms supervised methods on some problems such as metal artifact removal.\n\nWeaknesses:\n\nThe definition of inverse problems in section 2.1 is quite restrictive. The Dirac delta measurements distribution prevents any incorporation of real-world noise typically observed in inverse problems in medical imaging. For instance, in MRI frequently a Gaussian measurement noise is assumed, while the electron and photon noise in CT measurement can be described by a mixture of Gaussian and Poisson noise. Hence, real-world problems do not fit this restrictive definition of inverse problems.\n\nAll proofs (Proposition 1,  Lemma 1, and Theorem 1) are trivial and do not yield new theoretical insights.\n\nAs already mentioned above, the core novelty of the proposed paper is the additional sampling step performed in Eq. (5) and defined in Theorem 1. From an optimization point of view, the update rule in Theorem 1 implements a proximal mapping, which is very frequently used in MRI reconstruction [1]. This proximal mapping minimizes the quadratic energy\n$$\n\\min_z \\frac{1}{2}\\Vert z-x\\Vert_2^2 + \\frac{\\tau}{2}\\Vert\\text{diag}(\\Lambda) T z - y\\Vert_2^2\n$$\nfor $\\tau>0$ and reads as\n$$\nz = T^\\top \\left(\\frac{Tx + \\tau \\text{diag}(\\Lambda) y}{I + \\tau \\text{diag}(\\Lambda)}\\right),\n$$\nusing the assumption $T^\\top T=I$, which is fulfilled for a proper choice of $T$ in MRI and CT reconstruction problems. The equivalence of the proximal mapping and the mapping defined in Theorem 1 can be shown by using $I = (I-\\Lambda)+\\Lambda$ and the fact that $\\frac{1}{1+\\tau}+\\frac{\\tau}{1+\\tau}$ is a convex combination. Clearly, the proximal map minimizes an energy that employs a quadratic penalization of the data fidelity, which implicitly leads to a Gaussian noise assumption of the measurement process. Consequently, the limited assumptions discussed in the first weakness are actually not relevant.\n\nThe general approach is motivated by an explicit discretization scheme of the backward SDE. However, proximal maps are related to implicit discretizations schemes. So, is the resulting algorithm a mixed discretization (semi-explicit)? What is the corresponding forward SDE? More details w.r.t. this regard would clearly improve the quality of the paper since the SDE perspective is an integral part of the entire work. \n\nSince the choice of $\\lambda$ often influences the reconstruction quality strongly, more details on determining this hyperparameter would be helpful.\n\nIn section 3.2 at the end of the second paragraph, the dimensions of $A\\in\\mathbb{R}^{m\\times n}$ and $z\\in\\mathbb{R}^{m}$ do not match in the proposed sampling. Please, correct this.\n\nThe paper misses many implementation details. For instance, the learning problem (model, loss, optimization) is not explicitly defined.\n\n[1] Hammernik, K., Schlemper, J., Qin, C., Duan, J., Summers, R. M., & Rueckert, D. (2021). Systematic evaluation of iterative deep neural networks for fast parallel MRI reconstruction with sensitivity‐weighted coil combination. Magnetic Resonance in Medicine.", "summary_of_the_paper": "The manuscript applies denoising score matching to linear inverse problems to solve compressed sensing problems in medical imaging, such as angular-undersampled CT and accelerated MRI reconstruction. Throughout the paper, the observed measurements $y$ are considered noise-free, which is reflected by a Dirac measurement distribution. To train a score function, a variance exploding SDE is considered as in previous works, e.g. Song et al. (2021). \nDuring inference, the authors incorporate a weighted projection onto measurement samples unlike previous approaches, which consider the gradient of the data distribution. This projection can be implemented in various sampling approaches such as annealed Langevin dynamics or predictor-corrector schemes. \nThe numerical results indicate that unsupervised score matching methods are well suited for inverse problems in imaging and yield superior generalization performance.", "summary_of_the_review": "Overall I think the manuscript is marginally below the acceptance threshold due to unnecessary limitations of the approach and missing theoretical justification of the proposed discretization scheme of the reverse SDE despite the outstanding numerical results. ", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}]

Author rebuttal:

Meta-review: This is an interesting paper on improving score-based conditional sampling and its use in solving inverse problems. The current method of sampling from NCSNv2 is somewhat inefficient and the authors propose a different SDE that seems to work better for conditional generation. 

The paper is applied to Computational imaging and MRI and shows very good results and reasonable comparisons with the recent state of the art. One limitation is that the measurement process is artificial and ignores specifics of MRI (real measurements and multi-coils would strengthen the paper). In any case since this is a fundamental methods paper with a solid technical innovation on score-based sampling, I recommend acceptance.
Decision: Accept (Poster)