Page 1

SUPER Learning: A Supervised-Unsupervised Framework for Low-Dose CT

Image Reconstruction

Zhipeng Li1

Siqi Ye1

Yong Long1∗

Saiprasad Ravishankar2

1University of Michigan - Shanghai Jiao Tong University Joint Institute,

Shanghai Jiao Tong University†, Shanghai, China

2Departments of Computational Mathematics, Science and Engineering,

and Biomedical Engineering, Michigan State University, East Lansing, MI, USA

{zhipengli, yesiqi, yong.long}@sjtu.edu.cn, ravisha3@msu.edu

Abstract

Recent years have witnessed growing interest in machine

learning-based models and techniques for low-dose X-ray

CT (LDCT) imaging tasks. The methods can typically be

categorized into supervised learning methods and unsuper-

vised or model-based learning methods. Supervised learn-

ing methods have recently shown success in image restora-

tion tasks. However, they often rely on large training sets.

Model-based learning methods such as dictionary or trans-

form learning do not require large or paired training sets

and often have good generalization properties, since they

learn general properties of CT image sets. Recent works

have shown the promising reconstruction performance of

methods such as PWLS-ULTRA that rely on clustering the

underlying (reconstructed) image patches into a learned

union of transforms. In this paper, we propose a new

Supervised-UnsuPERvised (SUPER) reconstruction frame-

work for LDCT image reconstruction that combines the

benefits of supervised learning methods and (unsupervised)

transform learning-based methods such as PWLS-ULTRA

that involve highly image-adaptive clustering. The SUPER

model consists of several layers, each of which includes a

deep network learned in a supervised manner and an un-

supervised iterative method that involves image-adaptive

components. The SUPER reconstruction algorithms are

learned in a greedy manner from training data. The pro-

posed SUPER learning methods dramatically outperform

both the constituent supervised learning-based networks

and iterative algorithms for LDCT, and use much fewer it-

erations in the iterative reconstruction modules.

1. Introduction

X-ray computed tomography (CT) is a popular imaging

modality in many clinical and industrial applications. There

∗Yong Long is the corresponding author.

†This work was supported by NSFC (61501292).

has been particular interest in CT imaging with low X-ray

dose levels that would reduce the potential risks to patients

from radiation. However, image reconstruction at low X-

ray dose levels is challenging. Conventional X-ray CT im-

age reconstruction methods include analytical methods, and

model-based iterative reconstruction (MBIR) methods. A

classical analytical method is the filtered back-projection

(FBP) method [1]. That can be degraded excessively by

noise and streak artifacts in low-dose situations [2, 3].

MBIR methods incorporate the system physics, statis-

tical model of measurements, and typically certain sim-

ple prior information of the unknown object [4]. A typ-

ical method of this kind is the penalized weighted-least

squares (PWLS) method, for which the cost function in-

cludes a weighted quadratic data-fidelity term that models

the measurement statistics, and a penalty term called a regu-

larizer that models the prior information [5–7]. For PWLS,

various optimization approaches and regularization designs

have been exploited with efficiency and convergence guar-

antees.

Adopting appropriate prior knowledge of images for

MBIR approaches is also important to improve CT recon-

struction. More recently, with the availability of data sets of

CT images, methods based on big-data priors have gained

interest, such as dictionary learning-based techniques [8].

The dictionary can be either pre-learned from training data,

or adaptively learned with the reconstruction. In particu-

lar, the synthesis dictionary learning approaches represent a

signal or image patch as a sparse linear combination of the

atoms or columns of a learned dictionary, and have obtained

promising results in many applications [9–11]. However,

the dictionary learning based MBIR approaches are often

computationally expensive due to expensive sparse coding

(where typically NP-hard problems are optimized for esti-

mating sparse coefficients). Different from synthesis dictio-

nary learning, sparsifying transform (a generalized analysis

dictionary model) learning techniques efficiently adapt an

operator to approximately sparsify signals in transform do-

mains, and the corresponding transform sparse coding prob-

lem can be solved exactly and cheaply by thresholding [12].

Sparsifying transform learning techniques including using

doubly-sparse transforms and unions of transforms have

been applied to image reconstruction and obtained promis-

ing results [13–15].

Very recently, there has been growing interest in deep

learning approaches for medical imaging problems [16–21].

In the LDCT image reconstruction field, typical deep learn-

ing methods learn the reconstruction mapping from large

datasets of pairs of (low-dose and regular-dose) scans.

These methods include image-domain learning, sensor-

domain learning, and hybrid-domain learning. For example,

a particular image-domain learning approach is the FBP-

ConvNet scheme [19] that solves the normal-convolutional

inverse problems by applying a (learned) CNN after the di-

rect inversion that encapsulates the system physics. The

image-domain learning approaches can have many varia-

tions. For example, instead of directly working in the image

domain, one can transform the images to a specific domain

and learn in such domain the relationship between training

pairs. Kang et al. [20] designed a neural network that learns

a mapping between contourlet transform coefficients of the

low-dose input and its high-dose counterpart. This work

was later extended to learn a wavelet domain residual net-

work (WavResNet) [21].

In the sensor-domain deep learning category, Würfl et

al. [22] proposed an end-to-end neural network for low-dose

CT that maps the sinogram to the reconstructed image by

mapping the filtered back-projection algorithm to a basic

neural network. This allows one to take into account the

artifacts in the sensor-domain, e.g., the scatter and beam-

hardening artifacts, and compensate them in the learning

process. Another framework named the Automated Trans-

form by Manifold Approximation (AUTOMAP) [23] learns

a direct mapping from the measurement domain to image

domain. However, due to the high memory requirements

for storing fully connected layers, it is challenging for AU-

TOMAP to handle large scale reconstruction tasks such

as CT image reconstruction. Hybrid-domain learning ap-

proaches exploit data-fidelity terms in the neural network

architecture. The Learned ISTA (LISTA) [24] was one of

the earliest work of this kind. LISTA unfolds the itera-

tive soft-thresholding (ISTA) algorithm [25], and learns the

weight matrices and the sparsifying soft-thresholding op-

erator. Later, Yang et al. [26] proposed an ADMM-Net

which unfolds the alternating direction method of multipli-

ers algorithm for image reconstruction. Each step of the

algorithm is mapped to a neural network module. This

idea was then extended to a learnable primal-dual approach

based CNN [27]. These methods fall in the class of physics-

driven deep learning methods [28–30]. Hybrid-domain ap-

proaches also include a type that applies a plug-and-play

model. He et al. [31] applied the plug-and-play model to

the ADMM algorithm and unfolded it into a deep recon-

struction network, so that each network module is learnable

and replaceable.

Most deep learning algorithms are learned in a super-

vised manner (using task-specific cost functions) and re-

quire large training sets. However, in CT imaging, it is often

difficult to acquire large datasets of training image pairs.

Even though in the AAPM X-ray CT Low-Dose Grand

Challenge, both regular-dose and the matched quarter-

dose images were provided, only the regular-dose images

were reconstructed from real scans, while the matched

quarter-dose images were synthesized by adding noise to

the regular-dose sinogram data. Therefore, training with

smaller number of paired data (and yet generalizing) or

without reference data is highly conducive for CT image re-

construction. Moreover, different machine learning (as well

as conventional) approaches such as dictionary or transform

learning and deep learning use different types of big-data

priors and are advantageous in different ways. For example,

transform learning approaches learn general image proper-

ties and features in an unsupervised or model-based man-

ner, and can easily and effectively adapt to specific image

instances.

In this work, we propose a new image reconstruction

framework for LDCT dubbed Supervised-UnsuPERvised

(SUPER) learning. The algorithm architecture involves in-

terconnected supervised (deep network) and unsupervised

(iterative reconstruction) modules over many layers. The

architecture enables effectively leveraging different kinds

of big data learned priors for CT reconstruction. For ex-

ample, we used FBPConvNet [19] as the supervised (deep)

learned module and PWLS-ULTRA [14] as the unsuper-

vised module with a pre-learned union of transforms, which

provided both high quality image reconstruction and image-

adaptive clustering. The proposed SUPER learning used

relatively small training sets and dramatically outperformed

both deep learning and transform/dictionary learning by ef-

fectively combining task-specific and image or instance-

specific adaptivity. The proposed framework is generaliz-

able to include various constituent modules including non-

learning based algorithms, as shown in our experiments.

2. Proposed Model and Algorithms

Here, we present the proposed reconstruction model, its

motivations and interpretations, example architectures, and

training method.

2.1. Overview of the SUPER Model

We propose a novel efficient physics-driven learning

framework for CT reconstruction that effectively combines

the benefits of supervised (deep) learning and unsupervised

Supervised

Module 1

Iterative

Module 1

Supervised

Module 2

Iterative

Module 2

Supervised

Module M

Iterative

Module M

( )

SUPER Layer 1

SUPER Layer 2

SUPER Layer M

Figure 1: Overall structure of the proposed reconstruction framework.

iterative reconstruction methods. The proposed reconstruc-

tion architecture takes an initial image as input and pro-

cesses it through multiple “super” layers (Fig. 1). Each

super layer consists of a network learned in a supervised

manner (supervised module) and an iterative reconstruc-

tion method (iterative module) in sequence. The supervised

module is different in each super layer, i.e., the weights

in the supervised module are not shared among the su-

per layers. Importantly, this module is learned in a super-

vised manner (e.g., to minimize reconstruction error) to re-

move artifacts and noise. The iterative module on the other

hand iteratively optimizes a regularized image reconstruc-

tion problem using image-adaptive priors or regularizers

(e.g., the patches of the underlying image can be clustered

and sparsified in a learned union of transforms or dictionar-

ies [14]). The iterative algorithm is run for a fixed number

of iterations in each super layer.

While the supervised module removes image noise and

artifacts using a single learned network, the iterative mod-

ule could adapt various image-specific features in an MBIR

setup to further improve image quality and remove artifacts.

Importantly, the iterative module is not learned in a su-

pervised manner. The SUPER model in Fig. 1 is flexible

and could use various architectures for the supervised mod-

ule (e.g., FBPConvNet [19], WavResNet [21], etc.) and

a variety of iterative data-adaptive methods (e.g., PWLS-

ULTRA [14]). The model could be potentially used in a

variety of imaging as well as other applications.

2.2. Interpretations and Generalization

The SUPER model enables combining different kinds

of machine learned models and priors in a common recon-

struction framework. While the supervised module could

be a deep convolutional network learned from a big dataset

to optimize task-specific performance metrics, the iterative

module could exploit models learned from images using cri-

teria such as sparsity, manifold properties, etc. For exam-

ple, an operator could be pre-learned from CT images or

patches to approximately sparsify them (a.k.a. transform

learning [12]) and used to construct the regularizer for the

iterative module. Such image-based learned operators are

not typically task-sensitive and often generalize readily to

different settings and can help delineate or reconstruct vari-

ous image features.

The proposed SUPER model can also help combine

global adaptivity and image-specific adaptivity to obtain

the best of both worlds. While each supervised module

is learned from a dataset and fixed during reconstruction,

the iterative module could optimize novel and specific fea-

tures for each image being reconstructed (during training

and testing) and thus capture the diversity of images and

enable highly adaptive reconstructions. For example, dur-

ing training and testing, the iterative module could cluster

image patches differently for each image [14], or even learn

novel models such as dictionaries for each image [32].

Another interpretation of the SUPER model arises from

the perspective of iterative reconstruction. Many recent

state-of-the-art MBIR schemes involve complex noncon-

vex optimization and priors, wherein the initialization of

the algorithm is typically quite important, and better initial-

izations can lead to better reconstructions. In the SUPER

model, the iterative module is “initialized” with a different

image (i.e., output of the corresponding supervised module)

in each super layer. If the output of the supervised module

improves in quality over layers, the iterative module will

see increasingly better initializations and could thus provide

better quality outputs over layers. Moreover, the parameters

of the iterative module could also be varied over layers to

provide optimal bias-noise trade-offs. Thus, the SUPER

model could be viewed as minimizing nonconvex costs in

sequence with better initializations and parameters.

The proposed SUPER model for LDCT reconstruction

can be generalized to incorporate a variety of iterative

and MBIR techniques in the iterative modules. For ex-

ample, conventional techniques such as PWLS-EP (edge-

preserving hyperbola regularizer) [33] could be used in the

iterative module. PWLS-EP is a non-adaptive method that

penalizes the differences between neighboring pixels in the

reconstruction. We show later that combining PWLS-EP

with supervised learning in the SUPER model boosts the

performance of both methods. Note that we do not run the

PWLS-EP modules to near convergence (as it involves a

strictly convex problem and a unique minimum) but only

for multiple iterations.

2.3. Examples of SUPER Architectures

We now discuss some example SUPER models and their

properties. To illustrate the proposed approach, in this

work, we focused on the recent FBPConvNet [19] for the

supervised module. For the iterative module, we chose the

conventional PWLS-EP approach that uses a hand-crafted

prior (edge-preserving hyperbola regularizer) as well as the

learning and clustering-based PWLS-ULTRA [14]. Our ex-

periments later show that combining such supervised and it-

erative methods improves image quality over the constituent

methods. In the following, we further discuss the chosen ar-

chitectures.

2.3.1 Supervised Module

We work with FBPConvNet, which is a CNN-based image-

domain denoising architecture, originally designed for

sparse-view CT. The backbone of FBPConvNet is a U-net

[34] like CNN that takes noisy images reconstructed by the

FBP method (from low-dose scans) as input. The neural

network is trained so that its outputs closely match the ref-

erence high-quality (true) images, e.g., in an ℓ2 norm or root

mean squared error (RMSE) sense.

The traditional U-net uses a multilevel decomposition,

and employs a dyadic scale decomposition based on max

pooling. Thus, the effective filter size in the middle layers

is larger than that in the early and late layers. This scheme is

important because the filters corresponding to the Hessian

of the data fidelity term in (1) may have noncompact sup-

port. Similar to U-net, FBPConvNet employs multichannel

filters, which is the standard approach in CNNs, to increase

the capacity of the network. Compared with the traditional

U-net, FBPConvNet adopts a residual learning strategy to

learn the difference between the input and output.

2.3.2 Iterative Module

The iterative module optimizes an MBIR problem that es-

timates the linear attenuation coefficients x ∈ RNp from

the measurements y ∈ RNd . The typical PWLS approach

involves a cost function of the following form:

x = arg min

x≥0

y − Ax2

W + βR(x),

(1)

where A ∈ RNd×Np is the CT system matrix, W is the

weighting matrix related to the measurements (capturing

measurement statistics), R(x) is the regularizer, and β is

a positive scalar controlling the balance between the data-

fidelity term and the regularizer. In this paper, we used the

unsupervised learning-based PWLS-ULTRA as well as the

conventional PWLS-EP for (1).

For PWLS-EP, the regularizer R(x) can be written as

R(x) = ∑

Np

j=1 ∑k∈Nj κjκkϕ(xj − xk), where xj is the

jth pixel of x, Nj is the neighborhood of the jth pixel,

and κj and κk are analytically determined weights that en-

courage resolution uniformity [33]. The potential function

ϕ(t) is often chosen as ϕ(t) = δ2(|t/δ| − log(1 + |t/δ|)),

for δ > 0. PWLS-EP enforces approximate sparsity of gra-

dients of the image, a non-adaptive prior.

PWLS-ULTRA pre-learns a union of sparsifying trans-

forms from a dataset of image patches, and uses the learned

model during reconstruction [14]. The formulation for

learning the union of sparsifying transforms is as follows:

min

{Ωk,Zi,Ck}

K

∑

k=1

∑

i∈Ck

{ΩkXi − Zi

2

2 + ηZi 0}

+

K

∑

k=1

λkQ(Ωk), s.t. Ck ∈ g.

(2)

Here, Ωk, Ck, and Zi represent the learned transform for

the kth class, the set of indices of patches belonging to

the kth class, and the sparse coefficient of the ith training

signal or patch Xi (N training signals assumed), respec-

tively. Each signal is grouped with a corresponding best

matched sparsifying transform in (2). The set g is the set

of all possible partitions of [1 : N] into K disjoint sub-

sets. To avoid trivial solutions for Ωk, the penalty terms

Q(Ωk)

Ωk

2

F − log |detΩk| for 1 ≤ k ≤ K are

used that also control the condition number of the trans-

forms. Parameters η and λk = λ0 ∑i∈Ck

Xi

2

2 are pos-

itive weighting factors with λ0 > 0 [14]. Problem (2) is

solved efficiently using alternating optimization [14].

With the pre-learned transforms {Ωk}, the regularizer

R(x) for image reconstruction is as follows:

R(x)

min

{zj ,Ck}

K

∑

k=1

∑

j∈Ck

τj { ΩkPjx − zj

2

2 + γ2 zj 0},

where {τj} are patch-based weights to encourage uniform

spatial resolution or uniform noise (see [14]), Pj ∈ Rl×Np

is a patch extraction operator that extracts the jth patch from

x. zj is the sparse coefficient for the jth patch, and γ > 0

is a parameter controlling sparsity.

The PWLS-ULTRA problem is efficiently solved by al-

ternating between updating x (image update step), and solv-

ing for {zj,Ck} (sparse coding and clustering step). In the

image update step, where {zj,Ck} are fixed, the subprob-

lem is quadratic with non-negativity constraints, and can be

solved using fast iterative algorithms such as the relaxed

linearized augmented Lagrangian method with ordered-

subsets (relaxed OS-LALM) [35, 36]. The sparse coding

and clustering step with fixed x is solved exactly [32], with

the optimal class assignment kj for each patch given as

arg min

1≤k≤K

ΩkPjx − Hγ(ΩkPjx)2

2 + γ2 Hγ(ΩkPjx)0.

The corresponding optimal zj = Hγ(Ωkj

Pjx), where

Hγ(·) is the hard-thresholding operator that sets elements

smaller than γ to zero. The hard-thresholding can be

viewed as the non-smooth nonlinearity in PWLS-ULTRA.

The clustering could vary from image to image and iteration

to iteration in the PWLS-ULTRA algorithm.

2.4. Training and Implementation

We propose to train the SUPER model layer-by-layer

(sequentially) from a dataset of pairs of low-dose and

regular-dose CT measurements. For example, the FBP

method can be used to obtain reconstructions from the mea-

surements. The initial low-dose reconstructed images are

then used as inputs to the first supervised module, which

is trained to minimize the reconstruction error (RMSE) at

its output, with respect to the regular-dose reconstructions.

The initial images are then passed through the trained net-

work, following which the iterative algorithm in the first it-

erative module is run for each training image (in parallel) to

produce iterative reconstructions. The iterative reconstruc-

tions serve as inputs to the subsequent supervised model,

which is trained to minimize reconstruction error. The sub-

sequent supervised modules are thus learned sequentially.

Once trained, the SUPER model is readily implemented

for test data by passing initial reconstructions sequentially

through the supervised learned networks and iterative algo-

rithms.

3. Experiments

Here, we first describe our experimental setup, training

procedures, and evaluation metrics. Then we present re-

sults for the learned SUPER-ULTRA model, and compare

these with those obtained by each individual module of SU-

PER, i.e., FBPConvNet and PWLS-ULTRA. We also tested

the generalized SUPER model that replaces the unsuper-

vised learning-based PWLS-ULTRA with the non-adaptive

PWLS-EP. This scheme is dubbed FBPConvNet+EP.

3.1. Experimental Setup

We used regular-dose CT images of two patients from

the Mayo Clinics dataset established for “the 2016 NIH-

AAPM-Mayo Clinic Low Dose CT Grand Challenge” [37],

to evaluate the performance of the proposed SUPER learn-

ing. The two patient datasets (L067 and L096) contain 224

and 330 real in-vivo slice images, respectively. We sim-

ulated low-dose CT measurements yl from the provided

regular-dose images with GE 2D LightSpeed fan-beam CT

geometry corresponding to a monoenergetic source. We

projected the regular-dose images x∗ to sinograms and

added Poisson and additive Gaussian noise to them as fol-

lows:

yli = Poisson{I0e−[Ax∗]i } + N{0,σ2}.

(3)

We chose I0 = 1 × 105 photons per ray and σ = 5 in

our experiments. We approximated elements of the di-

agonal weighting matrix W of the data-fidelity in (1) by

y2

li

yli +σ2 [38]. The images are of size 512 × 512 at a resolu-

tion of 0.9766 mm×0.9766 mm, when reconstructed using

the FBP method. These reconstructed low-dose FBP im-

ages were paired with their corresponding regular-dose CT

images for training the SUPER model.

3.2. Training Procedures

In our experiments, the total number of training image

pairs was 100, of which 50 were chosen from patient L067

and 50 from patient L096. The image set was used to

train the networks of FBPConvNet, FBPConvNet+EP, and

SUPER-ULTRA. Fig. 2 shows some example (regular-dose

or reference) images from the training and testing datasets.

Different body parts are included in the datasets. In our ex-

periments, we used the default network architecture in the

FBPConvNet public implementation. For PWLS-ULTRA,

we pre-learned a union of five sparsifying transforms (cor-

responding to five classes) from twelve (regular-dose) slices

that include three slices each from four patients (L109,

L143, L192, L506).

Figure 2: Example CT images in the training (top row) and

testing (bottom row) datasets. The display window is [800

1200] HU.

We used a GTX Titan GPU graphic processor for train-

ing and testing. The union of transforms was learned ef-

ficiently using similar parameters as in [14]. For SUPER-

ULTRA and FBPConvNet+EP, the training time was about

46 hours and 13 hours, respectively, for 15 super layers.

Since the iterative reconstruction modules are relatively

computationally expensive, we chose 15 super layers to bal-

ance reconstruction quality and computational time. The

training hyper-parameters for the CNN part of FBPConvNet

were set as follows for the various models: the learning rate

decreased logarithmically from 0.001 to 0.0001; the batch-

size was 1; and the momentum was 0.99. We initialized the

filters in the various networks during training with i.i.d. ran-

dom Gaussian entries with zero mean and variance 0.005,

and initialized the bias with zero vectors.

For training FBPConvNet+EP, we ran 10 epochs of FBP-

ConvNet training (using ADAM) in each super layer to en-

sure that the learned networks capture layer-wise features.

For the constituent PWLS-EP modules, we ran 4 iterations

of the relaxed OS-LALM algorithm with 4 subsets, and set

δ = 20 HU and the regularization parameter β = 215. For

training SUPER-ULTRA, we ran 10 epochs of FBPCon-

PSNR:13.3

PSNR:20.3

PSNR:23.4

PSNR:27.1

PSNR: 29.0

PSNR: 31.7

Figure 3: Reconstructed testing image (Test #4, from patient L096) obtained by FBP, FBPConvNet, PWLS-EP, PWLS-

ULTRA, FBPConvNet + EP, SUPER-ULTRA, and regular-dose FBP. The display window is [800 1200] HU.

PSNR:13.6

PSNR: 30.3

PSNR: 23.0

PSNR: 31.6

PSNR: 31.6

PSNR: 29.6

Figure 4: Comparison of Test #2 for FBP, FBPCon-

vNet, PWLS-EP, PWLS-ULTRA, FBPConvNet + EP, and

SUPER-ULTRA (clockwise top left). The display window

is [800 1200] HU.

vNet training in each super layer along with 4 outer iter-

ations of PWLS-ULTRA with 5 inner iterations of the im-

age update step that used the relaxed OS-LALM algorithm

with 4 subsets. We set β = 5 × 103 and γ = 20 for the

PWLS-ULTRA module.

We compared our learned models with the iterative

schemes PWLS-EP and PWLS-ULTRA. Since FBPCon-

vNet+EP and SUPER-ULTRA already include the learned

FBPConvNet modules, we used smaller regularization pa-

rameters for them compared to the usual or stand-alone

PWLS-EP and PWLS-ULTRA, which worked well in our

experiments. For the stand-alone PWLS-EP iterative recon-

struction algorithm that solves a convex problem, β and δ

were set as 216 and 20, respectively, and we ran 100 iter-

ations of the OS-LALM algorithm to ensure convergence.

For the stand-alone PWLS-ULTRA, wherein the optimiza-

tion problem is nonconvex, we set β and γ as 104 and

25, respectively, and ran 1000 iterations of the alternat-

ing algorithm to ensure convergence. The above param-

eters provided optimal image quality in our experiments.

Both PWLS-EP and PWLS-ULTRA were initialized with

the simple FBP reconstructions.

3.3. Evaluation Metrics

To quantitatively evaluate the performances of the

various reconstruction models, we chose three clas-

sic metrics, namely RMSE, peak signal to noise ra-

tio (PSNR), and structural similarity index measure

(SSIM) [39]. RMSE in Hounsfield units (HU) is defined as

RMSE=√∑

Np

i=1 (xi − x∗

i )

2

/Np, where x∗ is the reference

regular-dose CT image, x is the reconstruction, and Np is

the number of pixels. We computed PSNR in decibels (dB).

3.4. Results and Comparisons

3.4.1 Visual Results

We applied the various methods to six testing slices (three

slices from L067 and three slices from L096) indexed as

Test #1 to Test #6. Figs. 3 and 4 show the reconstructions

of Test #4 (from patient L096) and Test #2 (from patient

L067), respectively, using different methods. Clearly, the

traditional FBP reconstruction shows severe artifacts, while

0

10

20

30

40

50

60

Number Iteration

20

40

60

80

100

120

140

160

180

RMSE (HU)

PWLS-EP

FBPConvNet+EP

0

10

20

30

40

50

60

Number Iteration

0

20

40

60

80

100

120

140

160

180

RMSE (HU)

PWLS-ULTRA

SUPER-ULTRA

0

10

20

30

40

50

60

Number Iteration

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

SSIM

PWLS-EP

FBPConvNet+EP

0

10

20

30

40

50

60

Number Iteration

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

SSIM

PWLS-ULTRA

SUPER-ULTRA

Figure 5: RMSE (first row) and SSIM (second row) val-

ues for Test #4 over the iterations of PWLS-EP, FBPCon-

vNet+EP, PWLS-ULTRA, and SUPER-ULTRA.

the results obtained by the other methods have much better

image quality. The stand-alone optimal PWLS-EP recon-

struction still contains noise such as in the central soft-tissue

area (Fig. 3). The (unsupervised) learning-based PWLS-

ULTRA removes noise, but the edges of soft-tissues are

more blurry in the results. The supervised learning-based

FBPConvNet achieves better trade-off between resolution

and noise compared to PWLS-EP and PWLS-ULTRA, but

many small structures were missed or distorted. With the

same training data as FBPConvNet, the learned FBPCon-

vNet+EP and SUPER-ULTRA models provide much better

reconstructions. However, FBPConvNet+EP suffers from

some streak artifacts in the central area (Fig. 3) as well as

noise generally, and SUPER-ULTRA mitigates these arti-

facts. In both Figs. 3 and 4, SUPER-ULTRA that com-

bines supervised learned networks and transform learning-

based iterative reconstructions achieved the best overall vi-

sual quality. 1

3.4.2 Quantitative Results

Fig. 5 shows the RMSE and SSIM evolution for the stand-

alone PWLS-EP and PWLS-ULTRA along with those for

FBPConvNet+EP and SUPER-ULTRA. The latter two in-

volve 15 super layers with 4 (outer) iterations in the iterative

module per layer, and thus, the RMSE and SSIM evolution

is plotted over these individual iterations. For PWLS-EP

and PWLS-ULTRA, we show the evolution of the metrics

over 60 iterations. FBPConvNet+EP and SUPER-ULTRA

clearly achieve much lower RMSE values and higher SSIM

values over layers than PWLS-EP and PWLS-ULTRA. The

plots also show faster convergence for the SUPER models.

Table. 1 shows the RMSE, PSNR, and SSIM values

1Additional comparisons between reconstructions for the other testing

slices are included in the supplement.

Super Layer 1

Super Layer 3

Super Layer 7

Super Layer 13

Figure 6: Outputs of 1st, 3rd, 7th, and 13th SUPER-ULTRA

layers with display window [800 1200] HU. The reconstruc-

tion is visually converging.

with various methods for the testing slices. The proposed

SUPER-ULTRA typically achieves significant improve-

ments in RMSE, SSIM, and PSNR over the other meth-

ods for all testing slices. Importantly, FBPConvNet+EP

and SUPER-ULTRA perform much better than PWLS-EP

and PWLS-ULTRA, respectively, and also provide more

promising results than FBPConvNet, demonstrating that the

combination of the supervised module and iterative mod-

ules in the SUPER model works well and outperforms the

individual modules.

3.4.3 Visual Quality over SUPER layers

Fig. 6 shows the output of SUPER-ULTRA after different

numbers of SUPER layers. The initial FBP image and the

final output (after 15 layers) were shown in Fig. 4. The

first several layers of SUPER-ULTRA mainly remove se-

vere noise and artifacts, while the later layers recover some

structural details.

3.4.4 Behavior of the ULTRA model in the SUPER ar-

chitecture

Next, to better illustrate the image-adaptive learned cluster-

ing in the SUPER-ULTRA model, Fig. 7 shows pixel-level

clustering results from the last super layer for test slice #1.

Since the ULTRA modules cluster image patches into spe-

cific classes, we cluster each pixel here using a majority

vote among the patches overlapping the pixel. Class 1 con-

tains most of the more uniform soft-tissues; Classes 2, 3,

and 5 contain many oriented edges (e.g., at 45-degree and

135-degree orientation); and class 4 contains most of the

Table 1: PSNR, RMSE, and SSIM of reconstructed test images for different methods.

FBP

FBPConvNet

PWLS-

EP

PWLS-

ULTRA

FBPConvNet+EP

SUPER-ULTRA

L067

Test #1

PSNR

11.0

29.8

23.5

29.3

30.5

30.9

RMSE 245.5

26.2

54.1

28.3

24.5

23.3

SSIM

0.29

0.74

0.73

0.74

0.77

0.77

Test #2

PSNR

13.6

30.3

23.0

29.6

31.6

31.6

RMSE 170.3

22.4

52.2

24.5

19.5

19.3

SSIM

0.40

0.79

0.78

0.79

0.81

0.81

Test #3

PSNR

9.5

19.9

24.4

28.6

32.2

32.1

RMSE 299.5

81.7

47.9

29.6

19.6

19.8

SSIM

0.24

0.56

0.69

0.69

0.72

0.72

L096

Test #4

PSNR

13.3

20.3

23.4

27.1

29.0

31.7

RMSE 172.6

70.7

48.7

31.9

25.5

18.9

SSIM

0.37

0.67

0.77

0.79

0.80

0.81

Test #5

PSNR

9.3

29.7

23.3

25.7

30.6

30.8

RMSE 304.4

25.8

53.8

40.6

23.3

22.7

SSIM

0.20

0.71

0.70

0.70

0.74

0.75

Test #6

PSNR

9.7

27.6

23.6

28.1

30.7

32.3

RMSE 274.2

32.3

48.6

29.0

21.5

17.8

SSIM

0.23

0.66

0.73

0.74

0.75

0.76

Class 1

Class 2

Class 3

Class 4

Class 5

Figure 7: Pixel-level clustering results in Test #2. The top row shows the transforms, with the transform rows shown as 8×8

patches. The bottom row shows the clustering results of SUPER-ULTRA with display window [800 1200] HU.

vertical edges and some horizontal edges as well as most of

the bones. The latter classes help provide sharper SUPER-

ULTRA reconstructions. The pre-learned transforms corre-

sponding to each class are also shown in Fig. 7, and contain

various directional and edge-like features.

4. Conclusions

This paper presented a new framework that combined su-

pervised learned networks and unsupervised iterative algo-

rithms for low-dose CT reconstruction. The proposed SU-

PER framework effectively combines various kinds of pri-

ors and learning methods. In particular, we studied SUPER-

ULTRA that combines (supervised) deep learning (FBP-

ConvNet) and the recent iterative (unsupervised) PWLS-

ULTRA, as well as FBPConvNet+EP (or SUPER-EP). Both

methods showed better performance and faster convergence

compared to their individual modules. FBPConvNet+EP

substantially improved the performance of PWLS-EP, while

SUPER-ULTRA typically performed the best by effectively

leveraging deep learning and transform learning. While SU-

PER model learning can exploit a variety of architectures

and algorithms for the supervised and iterative modules, a

more detailed study of various such architectures is left for

future work. We also plan to explore layer-dependent pa-

rameter selection for the iterative modules to further im-

prove performance in future work.

References

[1] L. C. Feldkamp, L. A .and Davis and J. W. Kress. Practical

cone-beam algorithm. J. Opt. Soc. Amer. A, Opt. Image Sci.,

1(6):612–619, 1984.

[2] K. Imai, M. Ikeda, Y. Enchi, and T. Niimi. Statistical

characteristics of streak artifacts on CT images: Relation-

ship between streak artifacts and ma s values. Med. Phys.,

36(2):492–499, 2009.

[3] H. Zhang, J. Ma, J. Wang, Y. Liu, H. Lu, and Z. Liang. Sta-

tistical image reconstruction for low-dose CT using nonlocal

means-based regularization. Computerized Medical Imaging

and Graphics, 38(6):423–435, 2014.

[4] J. Nuyts, B. De Man, J. A. Fessler, W. Zbijewski, and F. J.

Beekman. Modelling the physics in the iterative reconstruc-

tion for transmission computed tomography. Phys. Med.

Biol., 58(12):R63, 2013.

[5] Jeffrey A Fessler. Penalized weighted least-squares im-

age reconstruction for positron emission tomography. IEEE

transactions on medical imaging, 13(2):290–300, 1994.

[6] J-B. Thibault, K. Sauer, C. Bouman, and J. Hsieh. A three-

dimensional statistical approach to improved image quality

for multi-slice helical CT. Med. Phys., 34(11):4526–44,

November 2007.

[7] M. Beister, D. Kolditz, and W. A Kalender. Iterative recon-

struction methods in X-ray CT. Physica Medica: European

Journal of Medical Physics, 28(2):94–108, 2012.

[8] Q. Xu, H. Yu, X. Mou, L. Zhang, J. Hsieh, and G. Wang.

Low-dose X-ray CT reconstruction via dictionary learning.

IEEE Trans. Med. Imag., 31(9):1682–97, September 2012.

[9] S. Ravishankar and Y. Bresler. MR image reconstruction

from highly undersampled k-space data by dictionary learn-

ing. IEEE Trans. Med. Imag., 30(5):1028–1041, 2010.

[10] J. Mairal, M. Elad, and G. Sapiro. Sparse representation for

color image restoration. IEEE Trans. Im. Proc., 17(1):53–69,

2007.

[11] M. Elad and M. Aharon. Image denoising via sparse and

redundant representations over learned dictionaries. IEEE

Trans. Im. Proc., 15(12):3736–3745, 2006.

[12] S. Ravishankar and Y. Bresler. Learning sparsifying trans-

forms. IEEE Trans. Signal Process., 61(5):1072–1086, 2013.

[13] S. Ravishankar and Y. Bresler. Learning doubly sparse trans-

forms for images. IEEE Trans. Im. Proc., 22(12):4598–4612,

2013.

[14] X. Zheng, S. Ravishankar, Y. Long, and J. A. Fessler. PWLS-

ULTRA: An efficient clustering and learning-based approach

for low-dose 3D CT image reconstruction. IEEE Trans. Med.

Imag., 37(6):1498–1510, 2018.

[15] S. Ye, Y. Ravishankar, S.and Long, and J. A. Fessler. SPUL-

TRA: Low-dose CT image reconstruction with joint statis-

tical and learned image models. IEEE Trans. Med. Imag.,

2019. DOI: 10.1109/TMI.2019.2934933.

[16] G. Yang, S. Yu, H. Dong, G. Slabaugh, P. L. Dragotti, X. Ye,

F. Liu, S. Arridge, J. Keegan, Y. Guo, and D. Firmin. DA-

GAN: deep de-aliasing generative adversarial networks for

fast compressed sensing MRI reconstruction. IEEE Trans.

Med. Imag., 37(6):1310–1321, 2017.

[17] S. Yu, H. Dong, G. Yang, G. Slabaugh, P. L. Dragotti, X. Ye,

F. Liu, S. Arridge, J. Keegan, D. Firmin, and Y. Guo. Deep

de-aliasing for fast compressive sensing MRI. arXiv preprint

arXiv:1705.07137, 2017.

[18] J. Schlemper, G. Yang, P. Ferreira, A. Scott, L. McGill,

Z. Khalique, M. Gorodezky, M. Roehl, J. Keegan, D. Pen-

nell, D. Firmin, and D. Rueckert. Stochastic deep compres-

sive sensing for the reconstruction of diffusion tensor cardiac

MRI. In Med. Image Comput. Comput.-Assist. Interv., pages

295–303. Springer, 2018.

[19] K. H. Jin, M. T. McCann, E. Froustey, and M. Unser. Deep

convolutional neural network for inverse problems in imag-

ing. IEEE Trans. Im. Proc., 26(9):4509–22, 2017.

[20] E. Kang, J. Min, and J. C. Ye. A deep convolutional neural

network using directional wavelets for low-dose X-ray CT

reconstruction. Med. Phys., 44(10):e360–e375, 2017.

[21] E. Kang, W. Chang, J. Yoo, and J. C. Ye. Deep convolutional

framelet denoising for low-dose CT via wavelet residual net-

work. IEEE Trans. Med. Imag., 37(6):1358–1369, 2018.

[22] T. Würfl, F. C. Ghesu, V. Christlein, and A. Maier. Deep

learning computed tomography. In Med. Image Comput.

Comput. Assist. Interv., pages 432–440. Springer, 2016.

[23] B. Zhu, J. Z. Liu, S. F. Cauley, B. R. Rosen, and M. S. Rosen.

Image reconstruction by domain-transform manifold learn-

ing. Nature, 555(7697):487, 2018.

[24] K. Gregor and Y. LeCun. Learning fast approximations of

sparse coding. In Proceedings of the 27th International Con-

ference on International Conference on Machine Learning,

pages 399–406. Omnipress, 2010.

[25] A. Beck and M. Teboulle. A fast iterative shrinkage-

thresholding algorithm for linear inverse problems. SIAM

J. Imaging Sci., 2(1):183–202, 2009.

[26] Y. Yang, J. Sun, H. Li, and Z. Xu. Deep ADMM-Net for

compressive sensing MRI. In Proceedings of the 30th Inter-

national Conference on Neural Information Processing Sys-

tems, pages 10–18. Curran Associates Inc., 2016.

[27] J. Adler and O. ¨Oktem. Learned primal-dual reconstruction.

IEEE Trans. Med. Imag., 37(6):1322–1332, 2018.

[28] S. Ravishankar, I. Y. Chun, and J. A. Fessler. Physics-driven

deep training of dictionary-based algorithms for MR image

reconstruction. In 2017 51st Asilomar Conference on Sig-

nals, Systems, and Computers, pages 1859–1863, 2017.

[29] S. Ravishankar, A. Lahiri, C. Blocker, and J. A. Fessler.

Deep dictionary-transform learning for image reconstruc-

tion. In 2018 IEEE 15th International Symposium on

Biomedical Imaging (ISBI 2018), pages 1208–1212, 2018.

[30] Y. Chun and J. A. Fessler. Deep BCD-Net using identi-

cal encoding-decoding CNN structures for iterative image

recovery. In 2018 IEEE 13th Image, Video, and Multidi-

mensional Signal Processing Workshop (IVMSP), pages 1–5,

2018.

[31] J. He, Y. Yang, Y. Wang, D. Zeng, Z. Bian, H. Zhang, J. Sun,

Z. Xu, and J. Ma. Optimizing a parameterized plug-and-play

admm for iterative low-dose CT reconstruction. IEEE Trans.

Med. Imag., 38(2):371–382, 2018.

[32] S. Ravishankar and Y. Bresler. Data-driven learning of a

union of sparsifying transforms model for blind compressed

sensing. IEEE Trans. Comput. Imag., 2(3):294–309, 2016.

[33] J. H. Cho and J. A. Fessler. Regularization designs for uni-

form spatial resolution and noise properties in statistical im-

age reconstruction for 3D X-ray CT. IEEE Trans. Med.

Imag., 34(2):678–89, February 2015.

[34] O. Ronneberger, P. Fischer, and T. Brox. U-net: convolu-

tional networks for biomedical image segmentation. In Med-

ical Image Computing and Computer-Assisted Intervention,

pages 234–41, 2015.

[35] H. Nien and J. A. Fessler. Relaxed linearized algorithms for

faster X-ray CT image reconstruction. IEEE Trans. Med.

Imag., 35(4):1090–8, April 2016.

[36] D. Kim and J. A. Fessler. Generalizing the optimized gradi-

ent method for smooth convex minimization. SIAM J. Op-

tim., 28(2):1920–1950, 2018.

[37] C. McCollough. TU-FG-207A-04: Overview of the low dose

CT grand challenge. Med. Phys., 43(2):3759–60, 2016.

[38] L. Fu, T. C. Lee, S. M. Kim, A. M. Alessio, P. E. Kinahan,

Z. Q. Chang, K. Sauer, M. K. Kalra, and B. De Man. Com-

parison between pre-log and post-log statistical models in

ultra-low-dose CT reconstruction. IEEE Trans. Med. Imag.,

36(3):707–720, 2017.

[39] Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli.

Image quality assessment: from error visibility to structural

similarity. IEEE Trans. Im. Proc., 13(4):600–12, April 2004.