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IEEE TMI-2015-0061
1
Abstract—Image super-resolution (SR) aims to recover
high-resolution images from their low-resolution counterparts for
improving image analysis and visualization. Interpolation
methods, widely used for this purpose, often result in images with
blurred edges and blocking effects. More advanced methods such
as total variation (TV) retain edge sharpness during image
recovery. However, these methods only utilize information from
local neighborhoods, neglecting useful information from remote
voxels. In this paper, we propose a novel image SR method that
integrates both local and global information for effective image
recovery. This is achieved by, in addition to TV, low-rank
regularization that enables utilization of information throughout
the image. The optimization problem can be solved effectively via
alternating direction method of multipliers (ADMM).
Experiments on MR images of both adult and pediatric subjects
demonstrate that the proposed method enhances the details in the
recovered high-resolution images, and outperforms methods such
as the nearest-neighbor interpolation, cubic interpolation,
iterative back projection (IBP), non-local means (NLM), and
TV-based up-sampling.
Index Terms— Image enhancement, spatial resolution, image
sampling, matrix completion, sparse learning
I. INTRODUCTION
IGH-resolution (HR) medical images provide rich
structural details that are critical for accurate image
post-processing and pathological assessment of bodily organs
[1-10]. However, image resolution is limited by factors such as
imaging hardware, signal to noise ratio (SNR), and time
constraints. Image SNR is proportional to voxel size and the
Manuscript received January 22, 2015; accepted May 17, 2015. This work
was supported in part by National Institutes of Health grants MH100217,
EB006733, EB008374, EB009634, AG041721, AG042599, MH088520, and
the National Research Foundation under Grant 2012-005741 from the Korean
government.
§
F. Shi and J. Cheng contributed equally to this work.
*
corresponding author.
F. Shi, L. Wang, and P.-T. Yap are with the Department of Radiology and
Biomedical Research Imaging Center, University of North Carolina, Chapel
Hill, NC 27599, USA (e-mails: fengshi, li_wang,
pewthian_yap@med.unc.edu).
J. Cheng is with Department of Radiology and Biomedical Research
Imaging Center, University of North Carolina, Chapel Hill, NC 27599, USA,
and also with Section on Tissue Biophysics and Biomimetics, PPITS, National
Institute of Child Health and Human Development, NIH, Bethesda, MD, USA
(e-mail: jian.cheng.1983@gmail.com).
D. Shen is with the Department of Radiology and Biomedical Research
Imaging Center (BRIC), University of North Carolina at Chapel Hill, NC
27599 USA, and also with the Department of Brain and Cognitive Engineering,
Korea University, Seoul 136-713, Korea (dinggang_shen@med.unc.edu)
square root of the number of averages in the voxel. Reducing
the voxel size from 2×2×2 mm
3
to 1×1×1 mm
3
will require 64
averages for similar SNR [11]. This requires significantly
longer scanning time, which may not be practical clinically.
A possible alternative approach to this problem is image
post-processing. For this, interpolation methods (nearest
neighbor, linear, and spine) are generally employed due to their
simplicity. However, as pointed out in [1], interpolation
methods generally blur the sharp edges, introduce blocking
artifacts in lines, and are unable to recover fine details. In view
of this, we take a super-resolution (SR) approach for resolution
enhancement of LR images [3]. Interpolation methods are not
considered as SR methods since they do not consider the image
degradation process (e.g., blurring, and down-sampling).
Multi-frame SR algorithms reconstruct a HR image from
multiple LR images [12, 13]. These LR images are typically
acquired repeatedly with slightly shifted field of view (FOV).
In this paper, we propose a single-frame SR algorithm that
requires the acquisition of only one LR image. A number of
methods have been proposed for single-image SR [3]. For
example, iterative back-projection (IBP) was proposed to
estimate the HR image by back projecting the difference
between the LR image simulated based on the estimated HR
image via imaging blur and the input LR image [14]. This
process is repeated to minimize the energy of the difference.
Non-local means (NLM) is a method proposed to take
advantage of image self-similarity [15]. Specifically, the input
LR image is first denoised and the similar patches are used to
reconstruct to a HR image. A correction step is then applied to
ensure that the down-sampled HR image is close to the
denoised LR image. The reconstruction and correction steps are
iterated in a multi-scale manner. In another work, NLM was
employed to enhance the resolution of a single LR T2 image
with the guidance from an HR T1 image [16].
Matrix completion algorithms have recently been shown to
be effective in estimating missing values in a matrix from a
small sample of known entries [17-19]. For instance, it has been
applied to the famous Netflix problem where one needs to infer
user preference for unrated movies based on only a small
number of rated movies [20]. Matrix completion methods
assume that the recovered matrix has low rank and then uses
this property as a constraint or regularization to minimize the
difference between the given incomplete matrix and the
estimated matrix. Candes et al. proved that, low-rank matrices
can be perfectly recovered from a small number of given entries
LRTV: MR Image Super-Resolution with
Low-Rank and Total Variation Regularizations
Feng Shi
§
, Jian Cheng
§
, Li Wang, Pew-Thian Yap, Senior Member, IEEE, and Dinggang Shen
*
, Senior
Member, IEEE
H
IEEE TMI-2015-0061
2
under some conditions [18]. In image processing, one
advantage of matrix completion is that the remote information
from the whole image can be utilized in image recovery.
Besides, matrix completion has been widely applied to
image/video in-painting and decoding problems. However,
low-rank regularization is limited when the matrix to be
recovered contain rows or columns that are entirely missing. As
shown in Fig. 1, although voxels occluded by the words “Low
Rank” can be recovered very well, the voxels masked by white
horizontal and vertical lines cannot be recovered. In this case,
each missing row or column will be simply filled by an
arbitrary combination of other known rows or columns to meet
the low rank requirement. This precludes the application of
low-rank regularization to SR problem where missing values
for many rows and columns need to be recovered in the process
of recovering a HR image from a LR image.
The limitation of low-rank regularization in SR problem
could be remedied by imposing the additional local spatial
consistency. Although local information may not be useful in
applications such as the Netflix problem, where different rows
(e.g., users) can be considered independently, it is valuable in
recovering images. One possible choice is total variation (TV)
[21], defined as the integral of the absolute gradients of the
image. Recently, TV has been applied to avoid the ringing and
smearing caused by interpolation [22].
In this paper, we propose a novel low-rank total variation
method, referred to as LRTV, for recovering a HR image from a
LR image. Our method 1) explicitly models the blurring effects
when an image is down-sampled, 2) combines both low-rank
and TV regularizations for more effective image recovery, and
3) works for 3D or 4D images by a tensor formulation [19]. A
preliminary version of this work was presented at a conference
[23]. The work is significantly extended in this article with
more methodological details, validations, and discussions.
Experiments on MR images of both adults and pediatric
subjects are conducted, and the proposed method is compared
to various interpolation methods, as well as the NLM and
TV-based up-sampling methods.
II. METHOD
We first describe how image degradation processes such as
blurring and down-sampling effects are modeled. We then
describe the solution for the inverse problem of recovering the
HR image from the LR image, using low-rank and TV
regularization.
A. Super-Resolution Image Reconstruction Framework
As illustrated in Fig. 2, the acquired image is affected by
factors such as motion blur, field inhomogeneity, acquisition
time, and noise.
The observation model could be mathematically formulated
as:
(1)
where T denotes the observed LR image, D is a down-sampling
operator, S is a blurring operator, X is the HR image that we
want to recover, and n represents the observation noise. In the
case of single-image SR, the HR image can be estimated using
this observation model by minimizing the following cost
function:
(2)
This is a data fidelity term for penalizing the difference
between the degraded HR image X and the observed LR image
T. Since this is an ill-posed inverse problem, regularization
terms are added to stabilize the solution. The cost function is
thus rewritten as:
(3)
where
is the regularization term often defined based on
prior knowledge. The parameter λ is used to balance the
contributions of the fidelity term and the regularization term.
B. Low-Rank Total-Variation (LRTV) Method
The proposed LRTV method is formulated as follows:
(4)
where the regularization can be separated into low-rank and
total variation terms.
and
are the respective tuning
parameters for those two terms.
Fig. 1. Recovering the missing values using low-rank matrix completion [14].
The red arrows mark the horizontal and vertical lines that the algorithm fails to
recover.
Fig. 2. Observation model in MR image acquisition process.
IEEE TMI-2015-0061
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1) Low-Rank Regularization. The rank of a matrix is a
measure of nondegenerateness of the matrix, calculated by the
maximum number of linearly independent rows or columns in
the matrix. The low-rank property implies that some rows or
columns in the matrix can be linearly represented by other rows
or columns, indicating redundant information in the matrix.
Low-rank prior can be used in matrix completion when only a
subset of elements is known [18]. Since the rank of a matrix
is a nonconvex function of , a common approach is to
approximate it using the trace norm
, which leads to a
convex optimization problem. Recently, Liu et al. extended the
low-rank regularization to higher dimensional images and
further referred to as tensor completion [19]. Basically, a
N-dimensional image
can be seen as a high-order tensor.
Since it is an NP-hard problem to compute the rank of a
high-order tensor
1
, the rank is then approximated as the
combination of trace norms of all matrices unfolded along each
dimension:
(5)
where is the number of image dimensionality. In this study,
we use 3D MR images and thus .
are parameters
satisfying
and
.
is the unfolded along
the -th dimension:
. For example, a 3D
image with size of can be unfolded into three 2D
matrices, with sizes of , , and
, respectively.
is the trace norm of the matrix
.
2) Total-Variation Regularization. Total variation was
proposed as a regularization approach to remove noise and
handle proper edges in images [21]. It is defined as the integral
of the absolute gradients of an image:
.
Minimizing TV will enforce local spatial consistency in image
recovery, remove noise, and preserve edges.
C. LRTV Optimization
We use the alternating direction method of multipliers
(ADMM) algorithm to solve the cost function in Eq. (4).
ADMM is proven to be efficient for solving optimization
problems with multiple non-smooth terms in the cost function
[24]. First, we introduce redundant variables
to
simulate in each dimension , by requiring that the unfolded
along the -th dimension
should be equal to the unfolded
along this dimension
. The new cost function is as
follows:
, subject to
, (6)
Based on ADMM in [24], the augmented Lagrangian of the
above cost function is written in below, where
are
Lagrangian parameters:
1
http://en.wikipedia.org/wiki/Tensor_rank_decomposition
(7)
We further set
, and combine the last two linear
and quadratic terms for simplicity:
(8)
According to ADMM [24], we break Eq. (8) into three
sub-problems below that could be solved by iteratively
updating the variables. In the below, denotes the current
iteration step.
Subproblem 1: Update
by minimizing:
(9)
This subproblem can be solved by gradient descent, where
the gradient of TV term is obtained from the associated
Euler-Lagrange equation [22].
Subproblem 2: Update
by minimizing:
(10)
which can be solved using a close-form solution according to
[17]:
(11)
where
is the inverse operator of
, i.e.,
.
is the Singular Value Thresholding
operator [17] using
as the shrinkage parameter.
Subproblem 3: Update
by:
(12)
Algorithm 1. Low-Rank Total Variation (LRTV) for MR Image
Super-Resolution
Input: Low-resolution image T;
Output: Reconstructed high-resolution image X;
Initialize:
a
,
,
,
Repeat
1. Update X based on Eq. (9);
2. Update M based on Eq. (11);
3. Update Y based on Eq. (12);
4. Until difference in the cost function (Eq. (8)) is less than ;
End
a
The
operator is implemented by nearest-neighbor
interpolation.
IEEE TMI-2015-0061
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III. EXPERIMENTAL RESULTS
A. Low-Rank Approximation for Brain Images
We first evaluated whether brain images can be sufficiently
characterized using their low-rank approximation. We selected
a representative 2D axial slice from the T1 MR phantom in
Brainweb
2
, which has a size of 181×181 with in-plane
resolution of 1 mm (Fig. 3). We then performed singular value
decomposition (SVD) on this image and obtain 174 non-zero
singular values. As shown in Fig. 3, singular values decrease
dramatically, with most values being close to zero. Next, we
remove the small singular values and use the top 30, 60, 90, and
120 singular values to reconstruct the image. Note that the
number of used singular values equals to the rank of the
recovered image, implying that the recovered image is a
low-rank approximation of the original image. The
reconstructed images are shown in Fig. 3 for visual inspection
and the difference maps between original and reconstructed
images are also provided. Signal-to-noise ratio (SNR) in
decibels (dB) is used to evaluate the quality of reconstruction:
, where is the original HR
image and is the recovered HR image.
The results show that, by using the top 60 singular values, the
reconstructed image has high SNR (34.0db), although small
edge information in the brain boundary is lost. When using the
top 90 or 120 singular values (out of 174), the resulting image
does not show visual differences with respect to the original
image. For the 3D Brainweb image with size 181×217×181, it
has three ranks for its three unfolded matrices that are less than
its longest image size 217. These ranks are relatively low in
comparison to the total number of elements, suggesting brain
images could be represented using their low-rank
approximations with a relatively high SNR.
2
http://www.bic.mni.mcgill.ca/brainweb/
B. Experimental Settings
We applied our method to a set of down-sampled and blurred
3D brain images and evaluated whether our method can
successfully recover the original high-resolution images. To do
that, we use HR images as ground truth, and simulate LR
images as shown in Fig. 4. Blurring was implemented using a
Gaussian kernel with a standard deviation of 1 voxel. The
blurred image was then down-sampled by averaging every 8
voxels (to simulate the partial volume effect), resulting in half
of the original resolution. The quality of reconstruction of all
methods from the input LR images was evaluated by comparing
with their corresponding original HR images.
A number of comparison methods were also employed,
including nearest-neighbor interpolation (NN), spline
interpolation (Spline), IBP based up-sampling [14], NLM
based up-sampling [15], and TV based up-sampling [22]. The
estimated HR images from all methods were compared with the
original HR image for accuracy of image recovery by using
SNR. Note that, for NLM, we used the implementation made
available by the authors
3
. TV-based up-sampling is realized by
setting
and in the proposed method and
solving only the subproblem 1. Other methods were
implemented by in-house tools.
Parameters were optimized based on a small dataset,
consisting of 5 adult and 5 pediatric images as described in the
Real Data Evaluation section. In particular,
are weights
to combine unfolded matrices along each spatial dimension in
rank computation. All dimensions are assumed to be equally
important, i.e.,
. The difference between
iterations was measured by
, and the
program was stopped when this difference is less than
. Since TV is a major component of the proposed
method, we first optimized TV by setting
from a group of
candidate values (Fig. 5). We thus chose
. Then, we
optimized the proposed method by employing the same
while setting
from a wide range of candidate values (Fig.
5). We chose the rank regularization as
. As for
3
https://sites.google.com/site/pierrickcoupe/
Fig. 3. Low-rank approximations of brain images. Top row shows the original
image, singular-value plot, and zoomed singular-value plot of indices from 141
to 181. Bottom row shows the four reconstructed images and their differences
with the original image by using top 30, 60, 90, and 120 singular values,
respectively.
Fig. 5. Parameter optimization based on ten images. SNR was used to evaluate
the reconstruction performance.
Fig. 4. Simulation of low-resolution image from high-resolution image.
IEEE TMI-2015-0061
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the input arguments, the blurring kernel (such as Gaussian
distribution with 1 voxel standard deviation) was used in IBP,
TV, and the proposed method. On the other hand, the default
parameters in NLM implementation were used [15]. Both NN
and Spline do not have any free parameters that require tuning.
We evaluate the performance of the proposed and
comparison methods by comparing the recovered HR images
with the original HR image. Besides SNR, we also employ
another image quality measurement named Structural
Similarity Index (SSIM), which is considered to be correlated
with the quality perception of the human visual system (HVS)
[25]. SSIM has been used in many image SR studies [15,
26-28]. SSIM is defined as:
,
where
and
are the mean values respectively in the
original HR image and recovered image ,
and
are the
variances,
is the covariance of two images,
and
with
and
, and is the
dynamic range of voxel values [25]. SSIM ranges from 0 to 1,
and 1 means perfect recovery.
C. Phantom Data Evaluation
We employed the T1 MR phantom from Brainweb to
evaluate the recovery performance of the proposed and
comparison methods in cases of no noise and with noise. The
phantom has image size of 181×217×181 and spatial resolution
of 1×1×1 mm
3
. We downloaded images both without noise and
with noise at different levels of 1%, 3%, 5%, 7%, and 9% of the
maximum intensity. The noise in the phantom images has
Rayleigh statistics in the background and Rician statistics in the
signal regions. For our experiment, we generated images with
2×2×2 mm
3
resolution using the above-mentioned LR image
simulation pipeline (Fig. 4), and upsample it again to 1 mm
isotropic resolution using the proposed and comparison
methods, respectively.
Fig. 6 demonstrates the results when using noisy data at level
of 3% as input image. A typical slice is shown for each of
coronal, sagittal, and axial views, and the frontal region in
sagittal view is zoomed up for better visual inspection.
Compared to the original HR image, results of NN, Spline, IBP,
and NLM appear blurry. TV provides better image
reconstruction while the proposed method shows more fine
details. Fig. 7 shows the SNR and SSIM measurements for all
methods while changing the noise level from no noise to 9%
noise level. The proposed method outperforms all other
comparison methods in all noise levels for both SNR and SSIM.
Another observation is that, the SNR improvement of the
proposed method over other methods generally reduces when
noise level increases, while the SSIM improvement maintains a
similar level when the noise level increases.
Fig. 6. Illustration of upsampling results for simulated data. LR data with 2×2×2 mm
3
resolution was upsampled to 1×1×1 mm
3
. A typical slice for each of coronal,
sagittal, and axial views is shown, and a zoom-up of frontal region in sagittal view is also provided.
Fig. 7. Results of upsampling on simulate data with different noise levels. (A)
shows the SNR result, and (B) shows the SSIM result.
IEEE TMI-2015-0061
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D. Real Data Evaluation
We further evaluated the proposed method on two publicly
available datasets. First, we randomly selected 45 adult subjects
from ADNI
4
, with 15 from Alzheimer's disease (AD), 15 from
mild cognitive impairment (MCI), and 15 from normal controls
(NC). Their ages were 75±8 years old at MRI scan. T1 MR
images were acquired with 166 sagittal slices at the resolution
of 0.95×0.95×1.2 mm
3
. Second, we also randomly selected 45
pediatric subjects from NDAR (http://ndar.nih.gov/), with age
of 11±3 years old at MRI scan. T1 MR images were acquired
with 124 sagittal slices at the resolution of 0.94×0.94×1.3 mm
3
.
Fig. 8 shows the representative image SR results of an adult
scan (upper panel) and a pediatric scan (lower panel). From left
to right, the first row of each panel shows the input image,
original HR image, the results of NN, Spline, IBP, NLM, TV,
and the proposed LRTV method. The close-up views of
selected regions are also shown for better visualization. It can
4
http://www.loni.ucla.edu/ADNI
Fig. 8. Results for upsampling an adult image (upper panel) and a pediatric image (lower panel) with different methods. In each panel, the first row shows the input
image and the results by various methods, while the second row shows the close-up views of selected regions in the first row.
Fig. 9. Boxplot of SNR and SSIM results for recovering adult data and pediatric
data using different methods. The proposed LRTV method significantly
outperforms all other comparison methods (p<0.01 using two-sample t-tests).
Fig. 10. Boxplot of SNR and SSIM results for recovering adult data in groups of
AD, MCI, and NC.
IEEE TMI-2015-0061
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be observed that the results of NN and Spline interpolation
methods show severe blurring artifacts. The contrast is
enhanced in the results of IBP, NLM, and TV up-sampling
methods, while the proposed LRTV method best preserves
edges and achieves the highest SNR values.
Quantitative results on the images of 45 adults and 45
pediatric subjects are shown in Fig. 9. Note that the original
images were used as ground truth and the input was LR image
simulated following the pipeline of Fig. 4. The proposed
method significantly outperforms all comparison methods
(p<0.01 using two-sample t-tests). Results on adult subjects
demonstrate less variance and higher accuracy than those on
pediatric subjects, which may be because the image quality is
higher in the matured brain and also clearer gyri/sulci patterns
appear in the adult images. No significant difference was found
between the adult subjects of AD, MCI, and NC, as shown in
Fig. 10.
In addition, we applied our method directly to images with
typical imaging resolution. Fig. 11 shows the results of a typical
AD subject for visual inspection, along with the close-up views
of selected representative regions. As it can be observed, the
proposed method recovers fine details and also preserves image
edges.
E. Computational Time
All programs were run in Linux environment on a standard
PC using a single thread of an Inter® Xeon® CPU (E5630 1.6
GHz). Interpolation methods are computationally efficient as it
takes around 2 seconds for NN and 10 seconds for Spline for
one 3D image. For the up-sampling methods, IBP takes 1 min,
NLM takes 16 mins, TV takes 5 mins, and the proposed method
takes about 30 mins.
IV. DISCUSSION
We have presented a novel super-resolution method for
recovering high-resolution image from a single low-resolution
image. For the first time, we show that combining low-rank and
total-variation regularizations is a viable solution to the SR
problem. This combination brings together global and local
information for effective recovery of the high-dimensional
image. Experimental results indicate that the proposed method
is able to remedy the partial volume effect and recover the fine
brain structure details from both adult and pediatric images.
Quantitative comparisons show that the proposed method
outperforms other popular methods such as the interpolation
methods based on NN and Spline, and the up-sampling
methods based on IBP, NLM, and TV.
Fig. 11. Upsampling of an image from a typical AD subject. The resolution was thus increased from 0.95×0.95×1.2 mm
3
to 0.475×0.475×0.6 mm
3
. The first row
shows the reconstructed results and other rows show the close-up views of the regions selected from the first row.
IEEE TMI-2015-0061
8
The proposed method is computationally expensive
compared with other methods. The major reason is that the
program spends most of the time solving TV, which is
implemented using gradient decent [22] in MATLAB. In future
work, we will implement a faster TV solver, such as [29], and
also accelerate the program using C++. On the other hand, we’d
like to clarify that our main purpose is to show that the
TV-based MR image reconstruction can be improved by
combining low-rank regularization. When a better TV solver is
available, the performance of the proposed method could also
be improved.
The proposed method is mainly developed for single-image
SR. In the future, we will extend the proposed method to use
multiple LR images [12, 13], training datasets [30], and also for
more applications such as 4D image recovery in functional
MRI or diffusion MRI. The code for the proposed method will
be released at our website
5
.
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