IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 19, NO. 10, OCTOBER 20102669
Efficient Fourier-Wavelet Super-Resolution
M. Dirk Robinson, Cynthia A. Toth, Joseph Y. Lo, and Sina Farsiu
Abstract—Super-resolution (SR) is the process of combining
multiple aliased low-quality images to produce a high-resolu-
tion high-quality image. Aside from registration and fusion of
low-resolution images, a key process in SR is the restoration and
denoising of the fused images. We present a novel extension of the
combined Fourier-wavelet deconvolution and denoising algorithm
ForWarD to the multiframe SR application. Our method first uses
a fast Fourier-base multiframe image restoration to produce a
sharp, yetnoisy estimateof thehigh-resolution image.Our method
then applies a space-variant nonlinear wavelet thresholding that
addresses the nonstationarity inherent in resolution-enhanced
fused images. We describe a computationally efficient method
for implementing this space-variant processing that leverages the
efficiency of the fast Fourier transform (FFT) to minimize com-
plexity. Finally, we demonstrate the effectiveness of this algorithm
for regular imagery as well as in digital mammography.1
Index Terms—Digital X-ray imaging, multiframe deblurring,
super-resolution (SR), wavelets, denoising.
high-resolution high-quality image at a resolution greater than
the sampling rate of the detector. SR has received much atten-
tion in recent years in the image processing community. We
refer the interested reader to – for a broad review of re-
cent algorithmic development in this area.
Aside from registration and fusion of low-resolution images,
a key process in SR is the restoration and denoising of fused
images. In this paper, we propose an efficient restoration and
denoising method that is a novel multiframe extension of the
Fourier wavelet regularized deconvolution (ForWarD) algo-
rithm , which considers the nonstationarity of the multiframe
reconstruction process. The algorithm’s efficiency stems from
separating the multiframe deconvolution or restoration step
from the wavelet-based denoising step allowing us to achieve
UPER-RESOLUTION (SR) is the process of combining
multiple aliased low-resolution (LR) images to produce a
Manuscript received January 19, 2009; revised August 31, 2009; accepted
February 03, 2010. Date of publication May 10, 2010; date of current version
September 17, 2010. This work was supported in part by Siemens Healthcare.
The associate editor coordinating the review of this manuscript and approving
it for publication was Dr. Arun Ross.
M. D. Robinson is with Ricoh Innovations, Menlo Park, CA 94025 USA
C. A. Toth and S. Farsiu are with the Departments of Ophthalmology and
Biomedical Engineering, Duke University, Durham, NC 27710 USA (e-mail:
J. Y. Lo is with the Departments of Radiology and Biomedical Engineering,
Duke University, Durham, NC 27710 USA (e-mail: email@example.com).
Color versions of one or more of the figures in this paper are available online
Digital Object Identifier 10.1109/TIP.2010.2050107
1Preliminary results of this work were presented at ICIP, October 2008 .
nonlinear denoising in a noniterative fashion. Furthermore, we
describe how to efficiently implement the algorithm to address
thecomputational complexity associated with thenonstationary
noise processes inherent to multiframe reconstruction.
In this paper, we also explore an application of this algo-
rithm to digital mammography. While relatively new, digital
mammography is rapidly replacing film-based mammography
for the screening and diagnosis of early carcinomas in women.
Solid-state detectors have demonstrated improved performance
in terms of specificity and sensitivity over film-based imaging
for certain groups of women such as those with dense breast
Unlike film-based mammography, digital mammography
provides the opportunity to directly apply sophisticated digital
processing techniques without the need for a secondary film
scanning process. An ideal digital mammography system ex-
poses the patient to the minimum amount of radiation required
to accomplish the screening task. Digital mammography sys-
tems face the same design tradeoff between image resolution,
signal-to-noise ratio (SNR), and illumination or radiation
exposure level as those found in any digital imaging system.
Shrinking the pixel dimension at the detector increases sam-
pling resolution at the expense of dynamic range and SNR.
While improved SNR and dynamic range may be obtained by
combining multiple images, increasing total radiation beyond
the standard dosage is undesirable for the safety of the patient.
Alternatively, using large detector pixels improves both the dy-
namic range and the SNR of the system at the obvious expense
of resolution. Digital mammography imaging systems typically
choose the highest resolution which supports a minimum
To overcome the said quality tradeoffs, we propose digitally
combining multiple low-dosage images, each containing spa-
tial shifts. This motion may be the result of patient movement,
intentional dithering of the detector, vibration in the imaging
system, or small movement of the imaging gantry. In practice,
the motion contained in the captured images is a combination of
low-resolution (LR) images.
ApplyingSR algorithmsto digital mammographyhas two in-
herent challenges. The captured low-resolution images are typ-
ically of size 10 megapixels and larger. Thus, algorithmic ef-
ficiency is very important due to the sheer size of the recon-
structed images, which could range from 40 to 160 megapixels
depending on the resolution enhancement factor. Also, to min-
imize total radiation exposure, we must use lower than normal
dosages of X-ray exposure for capturing each frame. Therefore,
the captured data has extremely low peak SNR (PSNR). For ex-
ample, Fig. 1 compares a high dosage X-ray image (computed
1057-7149/$26.00 © 2010 IEEE
2670 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 19, NO. 10, OCTOBER 2010
Fig. 1. Mammogram X-ray images from a phantom breast containing a pentagram-shaped set of micro-calcification cluster. (a) High dosage at 226mAs, faintly
showing the nodules (PSNR??? dB). (b) Extremely low-dosage at 11.3 mAS used in the proposed multiframe reconstruction scheme (PSNR?? dB), where the
nodules are almost undetectable. The total dosage of using 15 of these frames (??????? ? ??? mAs) is still less than the high dosage. (c) Restoration combining
the 15 low-dosage frames, clearly demonstrating the pentagram-shaped set of micro-calcification cluster.
that the effectiveness of the two-stage restoration and denoising
algorithm allows us to provide high-resolution, high contrast,
and low noise images at very low radiation dosages. Further-
more, our results suggest improved detection rates of texture re-
sembling small calcification in breast tissue. These results sug-
gest new tradeoffs in designing digital mammogram systems.
In Section II, we describe the forward imaging model and
the problem of SR. In Section III, we describe the multiframe
variant of the ForWarD algorithm we employ to restore and
denoise the reconstructed images. Section IV presents experi-
mental results using this new approach and Section V outlines
some future directions of this work.
dB) with the very low exposure images (computed
dB) used in our multiframe scheme. We demonstrate
II. BACKGROUND: SR AND EFFICIENT DEBLURRING
In this section, we establish the background required for
the development of our efficient multiframe SR algorithm in
Section III. We introduce our imaging model and since our
method is a combination of the multiframe SR and ForWarD
deblurring algorithms, we also review these two concepts.
Moreover, we review a novel multiframe motion estimation
algorithm  that is used to produce the results in Section IV.
We believe that due to some algorithmic similarities, this brief
review of the motion estimation technique facilitates the study
of the method described in Section III.
A. Imaging Model
The captured LR X-ray images are often very large and
may contain complicated relative motions due to patient mo-
tion. However, following several other popular SR methods
–, we consider the translational (or pure rotational)
motion models. To better justify and extend the application
of this model, in , we introduced a novel joint motion
estimation and SR approach in a tile-based fashion. That is,
instead of considering a global translational motion model, we
assume that each LR image is made of a set of small sized
2In this work, the PSNR was computed numerically as ????
?????. In experiments on real images, ? is the grayscale difference
between the minimum and maximum signal regions and ? is the noise standard
deviation estimated from flat regions. In simulated experiments, ? is the RMSE
error between the estimated and ground truth image.
tiles (blocks). These tiles move independently in a translational
(or pure rotational) model. When using tile-based processing,
the relative shifts between data sets is better approximated by
the translational (or rotational) motion models. Furthermore,
applying the algorithm to small tiles reduces the memory
requirements of the multiframe reconstruction algorithm. The
motion estimation process involves identifying image tiles
from different LR images corresponding to a particular region
of interest. In , we introduced a joint estimation technique,
in which matching blocks of different LR frames are optimally
To simplify the notations, without the loss of generality, all
formulas used in this paper correspond to the reconstruction of
a single HR tile (a full image is reconstructed by stitching a set
of such HR tiles).
image tiles by the vector
. These noisy LR input image tiles
high-resolution image tile denoted by . The forward model re-
tile is given by
in which the vector
without loss of generality) samples of the captured image
vector. The captured image is undersampled with re-
spect to an unknown high-resolution image
, by a factor of
represents samples of the unknown
high-resolution image tile
represents the blurring associated with the imaging
system. In X-ray imaging, this blurring is due to the geometry
of the radiating illumination as well as the scattering of light
in the object material. In each set of tiles, we approximate
this effect by a spatially-invariant point spread function (PSF)
. Although, the PSF can be different in different tile
sets. The warping operator
represents the subpixel spatial shifts between similar tiles in
the captured images. The spatial shifting is described by the
represents (assumed square
, are ordered as a
in each dimension.
similarly ordered. The
th frame. In our model, we
ROBINSON et al.: EFFICIENT FOURIER-WAVELET SUPER-RESOLUTION2671
assume that these spatial shifts are continuous values in the
. This corresponds to the range of subpixel
motions in the captured images. Without loss of generality, we
assume that the tile
defines the coordinate system of the
high-resolution image and, hence, we only have to estimate
the unknown motion parameters for the remaining
The downsampling operator
the undersampling of the detector. Finally,
represents the noise inherent in the analog-to-digital conver-
sion. For our purposes, we assume this noise to be uncorrelated
zero-mean noise with standard deviation .
B. Classic Maximum A Posteriori SR Reconstruction
The general problem of SR is to combine
images and estimate the high-resolution image . An important
category of solutions to the SR problem is the maximum a pos-
teriori (MAP) and closely related methodologies, which is dis-
cussed in this section. The MAP methods are based on the con-
struction of a cost function
, which is the summation of two
distinct terms. One is the data penalty term
the closeness of data to the estimates. The other is the regular-
, which represents the prior information about
the unknown high-resolution (HR) image
Early MAP-based SR methods assumed that the motion vec-
tors were accurately estimated in a separate process and the
noise model was Gaussian , , which justifies the ap-
plication of quadratic data penalty terms. As for the regular-
ization term, these techniques most frequently employed the
quadratic Tikhonov style regularization despite its tendency to
reduce edge contrast. The resulting cost function is in the form
, which measures
weighting scalar. When
known HR image, then this cost function produces the ideal
Wiener filter estimate of the unknown image. This MAP func-
tional has the advantage of being quadratic, which means that
the estimate image is a linear function of the input measure-
ments and is, thus, easy to compute.
Through the years, application of more advanced prior func-
such as Adaptive Kernel regression  which gener-
alizes popular priors such as Tikhonov and Bilateral Total-Vari-
ation (B-TV) , have produced higher quality estimates. For
example, the B-TV cost function is defined as
is often a spatial high-pass operator and
is the exact covariance of the un-
is a constant . The parameter
responding Bilateral filter kernel. The Bilateral filter and its pa-
rameters are extensively discussed in , .
is a set of integer pixel shifts and
defines the size of the cor-
These advanced regularization functions are not quadratic
(nonlinear estimators) and, hence, require more compu-
tationally-complex iterative minimization strategies. Such
nonquadratic functionals can, however, preserve many impor-
tant features of images such as edges. Also, MAP-based robust
SR techniques (e.g., , , and ) are able to reduce the
effect of outliers such as motion estimation error.
Practical tests show that using a separate motion estimation
process, specially in low-SNR cases, is suboptimal. Therefore,
the critical issue of joint SR and motion estimation problem has
been the topic of several papers (e.g., –). A simplified
MAP formulation of this problem has the form
tion vector distribution may also be added to the previously
mentioned cost function . The previously mentioned ap-
proaches are commonly solved in an iterative fashion and are
techniques are generally computationally more complex than
LR images is small or when the motion of most LR frames are
mating the unknown motion vectors and the aliasing free image
ages containing aliasing artifacts.
. Note that, additional priors on mo-
C. Problem of Joint Motion and Image Estimation
In this subsection, we briefly review an alternative approach
variable-projection principal , which we described in detail
inourrecent publication.Whilemotion-estimationis notthe
simplifies the material described in later sections of this paper.
 and rewrite the imaging model as
tion process will then be formulated as
is the unknown HR blurry image. The optimiza-
which is typically assumed to be stationary. A typical solution
to the previously mentioned problem is the cyclic coordinate-
descent method , in which in each iteration one unknown
variable is updated based on the estimate of the other unknown
variable in the previous iteration.
is the covariance matrix of the unknown signal,
D. Efficient Joint Estimation Using Variable Projections
Separable Nonlinear Least Squares problem , in our Vari-
2672 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 19, NO. 10, OCTOBER 2010
linear parameters (motion-vectors) are known. Consequently,
the estimate of the set of linear parameters
is computed as
Note that, unlike the cyclic coordinate-descent method, we re-
quire no iterations between the sets of parameters since we do
not explicitly calculate (7). Indeed, a direct approach to max-
imize (10) involves inverting a large matrix of size
which is computationally challenging for even small
image tiles. In , we described a series of numerical tricks
to speed up the process. One trick is solving the problem in
the Fourier domain and taking advantage of the spectral folding
phenomenon in aliased images.
To simplify the derivation, we reformulate the image cap-
ture model (19) in the Fourier domain. We use the over-script
“ ” to denote the Fourier domain representation. For example,
the th LR image is given by
tial frequencies are indexed by
discrete Fourier transform of the observed image
Similarly, the high-resolution spatial frequency coordinates are
Because the shift
operator is spatially-invariant, its Fourier
representation is a diagonal matrix defined as
, where the LR spa-
term is the base of the
The downsampling operator
its Fourier representation is not diagonal. The downsampling
operator is, however, periodic and is conveniently represented
is not spatially-invariant and so
corresponds to samples of the spectrum of the captured image
The form of matrix
justifies a decoupling of the observed
spatial frequency components in the Fourier domain. In other
words, we can consider each LR spatial frequency component
represents the Kronecker matrix product,
vector of all ones, and
dimension. In the Fourier domain, the vector
represents the identity
and ) as an independent observation model
matrices are constructed as
and the vectors are constructed according to
A single spatial frequency in the captured image is a function of
the original signal content and summation of the
spectral components. This demonstrates that the reconstruction
can be appliedto each collectionof aliased spectral components
independently. Thus, a singlet set of
can be estimated as
density (PSD) function.
Finally, the motion estimation function of (10) simplifies to
are samples of the signal’s power spectral
Estimating the motion vectors using (18) and the high-resolu-
tion image using (15) is significantly faster than using the di-
rect matrix form of (10) and (7). The simplified form requires
small matrices of size
verting one very large
similar acceleration techniques was described in .
So far, ignoring the PSF effects, we have studied a com-
putationally efficient multiframe joint motion estimation and
as opposed to in-
matrix of (7). A set of
ROBINSON et al.: EFFICIENT FOURIER-WAVELET SUPER-RESOLUTION2673
SR technique in the presence of aliasing. In the next subsec-
tion, ignoring the aliasing effects, we study an efficient single
frame restoration (deblurring and denoising) method. We com-
bine these two techniques in Section III.
E. Single Frame Deblurring Using ForWarD Algorithm
In this section, we consider the problem of restoring the
contrast lost due to the blurring (PSF) inherent to the imaging
system. To achieve this, we briefly review the fast two-step
ForWarD deblurring algorithm . The ForWarD algorithm
combines a Fourier-based regularized deconvolution algorithm
with a wavelet-based denoising post processing step. As de-
scribed in the following, in the first step, a Fourier domain
implementation of the Wiener filter reduces blur artifacts
while suboptimally magnifying the noise. In the second step, a
wavelet-based denoising process reduces the noise artifacts.
The basic ForWarD algorithm addresses the problem of
restoring an unknown image signal, which has been blurred by
a spatially-invariant operator such as a point spread function
(PSF) and corrupted by a stationary noise distribution as in the
The noise is assumed to have a stationary distribution. The first
step of the ForWarD algorithm involves inverting the spatially-
invariant blurring operator
using a regularized sharpening
filter such as the Wiener filter
process is formulated as
, . Such estimation
ment noise (typically
. For example,in the case of the normal and low-dosage im-
ages shown in Fig. 1, the noise standard deviation corresponds
to about 20% and 65% of the maximum signal intensity, respec-
tively. Indeed, under certain conditions the Wiener filter can be
regarded as a Tikhonov regularization functional , which
was described in Section II-B. While in the common Wiener
, in ForWarD algorithm, weighting
factor is chosen such that
choice of the weighting factor often enhances sharpness at the
expense of substantial noise amplification.
The Wiener filter can be implemented efficiently in the
Fourier domain using fast Fourier transforms (FFT). Imple-
mentation of the Wiener filter in the Fourier domain is founded
on the assumption that the blurring operator represented by
is spatially-invariant and that both the noise and signal are
stationary random processes. In this case, the blurring oper-
ator can be represented by its transfer function
computed as the Fourier transform of the system’s PSF. Also,
the signal statistics are represented by a power spectral density
. The application of the Wiener
is the covariance matrix of the stationary measure-
) and is a weighting factor
. Such perceivably suboptimal
filter is performed in the Fourier domain producing an estimate
of the original signal spectrum
This estimate is converted back into the spatial domain using
FFT operators to obtain the estimate
The second step in the ForWarD process applies a nonlinear
denoising filter to the wavelet transform of the sharpened image
for the purpose of eliminating the residual noise amplified
by the restoration filter
step is based on adaptive thresholding in the wavelet domain.
The redundant (no downsampling) wavelet transform  is
applied to the sharpened image by convolving the image with
a set of scaling and wavelet functions, represented by the ma-
and , producing a set of scale coefficient images
and wavelet coefficient images
ficients, or wavelet shrinkage, eliminates the noisy artifacts in-
troduced during the deblurring step according to
. This nonlinear denoising
. Reducing the wavelet coef-
represents the noise variance at the th wavelet space for the
After shrinking the wavelet coefficients in this manner, the
inverse wavelet transform produces the final denoised estimate
of the original image . We refer the reader to the original work
of  for a more complete explanation of this process including
visualization of the images at the various steps in the algorithm.
Conceptually similar to the Wiener filter, the wavelet
shrinkage reduces the observed wavelet coefficients as a
function of the local SNR of the wavelet coefficients. The
performance of the wavelet shrinkage depends on the ability
to predict the SNR for the wavelet coefficients. Because of
the wavelet’s spatial locality property, the wavelet shrinkage
provides efficient local, signal-dependent denoising.
The value of the noise variance for each wavelet function is
given by the covariance of the residual noise
domain. The covariance matrix of the noise in the wavelet do-
main is given by
. Because the Wiener filter is spa-
tially-invariant, the residual noise covariance
otherwords, theresidualnoise isstationarydue tothespatial in-
variance of the Wiener filter. Since the redundant wavelet filter
 is also spatially-invariant,theresidual noise power ina par-
ticular wavelet space is uniform over the entire image . The
residual noise power in the wavelet domain is computed effi-
ciently in the Fourier domain by way of
identifies the pixel location,
is the wavelet signal power.
in the wavelet
is circulant. In
2674 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 19, NO. 10, OCTOBER 2010
This must be computed once for each wavelet filter used in the
The accuracy of the wavelet thresholding depends on accu-
In practice, this signal information must be estimated from
the captured image data. In , the authors propose a simple
process for estimating the signal power term
The authors estimate the wavelet signal power by first very
coarsely denoising the sharpened image
simple hard thresholding wavelet denoising approach to obtain
the coarsely denoised image using a different set of scaling and
soft thresholding. We denote the wavelet coefficient images for
this different wavelet transform by
applied to the wavelet and scaling coefficients according to
. They propose a
than those used for the
. A hard thresholding
provides the coarsely denoised image. The
transform applied to the hard thresholded wavelet coefficients
reader to  for more information).
Next, the wavelet transform using the original wavelet func-
to produce the coefficients
term is the input
The spatially-varying standard deviation of the signal’s wavelet
coefficients is estimated to be the value of the coefficients of the
coarsely denoised image, or
This estimate of the signal power is used in (23). The ForWarD
approach to deconvolution has been applied to several deconvo-
lution problems with success and has been used as a benchmark
for evaluating the success of other single frame deconvolution
algorithms . In the next section, we propose a multiframe
extension of this algorithm, which considers the aliasing and is-
sues inherent to the SR problem.
III. MULTIFRAME IMAGE RESTORATION AND
The goal of the multiframe SR problem, aside from reducing
ring inherenttothe imagingsystem. Toachievethis, wederivea
novel multiframe variant of the fast two-step ForWarD method
. The authors of the original ForWarD algorithm have re-
cently proposed a new version of this algorithm that addresses
the multiframe deblurring problem  for nonaliased images
with stationary noise model. In this section, we introduce an ex-
tension of this multiframe algorithm, which considers the non-
stationarity inherent to the SR problem. Fig. 2 presents a flow
chart representation of this multiframe ForWarD (MForWard)
Fig. 2. Flow chart representation of the MForWard algorithm.
The efficiency of the original ForWarD algorithm , as well
as the more recent multiframe version , is founded on the
stationarity of the noise as well as the spatial invariance of the
blurring operators. The stationarity assumption breaks down in
the case of multiframe SR.
and aliased images onto a higher resolution sampling grid. The
multiframe Wiener filter producing a sharp estimate of
variant of (21) given by
the motion vectors
ForWarD algorithm, we typically use values of
tend to sharpen the images at the expense of increased noise
amplification and ringing artifacts in the resolution-enhanced
and are defined in (8) and (9). Here, we assume that
are estimated from (10). Similar to the
ROBINSON et al.: EFFICIENT FOURIER-WAVELET SUPER-RESOLUTION2675
Fig. 3. Grid on the left shows the 4?4 grid of HR image sample locations
and the number of LR measurements (assuming integer sampling offsets) per
sample location (no blur case as in the shift-and-add image reconstrution de-
scribed in ). In the sharpened image ? ?, the locations with fewer measure-
ments will have higher residual noise variance. For example, the grid on the
right shows the residual noise variance for an image reconstructed using an un-
infinite noise variance.
filter of (28), the covariance matrix of the residual noise error
Effective denoising and artifact removal using the wavelet
shrinkage method requires accurate estimates of the residual
noise power in the different wavelet filter domains. The noise
covariance matrix in the wavelet coefficient domain of the
is given by
stationary model, the residual noise power
the diagonal of this covariance matrix. Estimating this single
value is computed efficiently in (24) using FFT operations.
Unlike the standard ForWarD algorithm, the multiframe es-
timate of the deblurred image
field which is not stationary. In the case of multiframe recon-
struction, ignoring the border pixels, the residual noise field
is periodic with a period of
must compute the residual noise powers
which correspond to the
locations. We use
HR sampling locations (note Fig. 3).
The spatially-varying residual noise power depends on the
collection of motion vectors . That is, even though the noise
fields of the captured LR images
amounts of data. Because of this, the residual noise powers
in the wavelet domain are not uniform and cannot
be computed using (24).
To see this effect more clearly, we present a simple example.
Suppose that we capture twelve LR images with a downsam-
pling factor of
in both the vertical and horizontal di-
mensions. Furthermore, suppose that the captured images are
offset by integer numbers of pixels in the high-resolution grid
with the number of offsets per HR grid location shown in Fig. 3.
To simplify the analysis, we assume that the signal covariance
matrix is given by
free of blur
. In this case, after applying the multiframe
Wiener filter, the covariance matrix of the residual noise field
is a diagonal matrix. The terms along the diagonal corre-
spond to the residual noise variance in the reconstructed image.
. In the case of the
is constant along
has a residual random error
. For each wavelet filter, we
to index the
are stationary, different
and that the imaging system is
The HR pixels in the sharpened image
ments will have much higher noise variance as indicated on the
right side of Fig. 3. Indeed, in the more general case, the motion
vectors do not fall perfectly onto grid locations.
Once we obtain the
spatially-varying noise powers, we
employ the same pattern of the traditional ForWarD algorithm
using the coarse denoising by the hard thresholding of (25) to
estimate the wavelet coefficient energies followed by the soft
thresholding of (23). The only difference being that when we
apply the hard or soft thresholding of (23) and (25), we do not
use the same noise variance for every pixel. Instead, we use the
spatially-varying noise powers corresponding to the HR grid lo-
erties of the sharpened image
denoising. The next subsection explains a computationally effi-
cient approach to this issue.
lacking LR measure-
when performing the wavelet
B. Efficient Fourier MForWarD
One key advantage of the original ForWarD algorithm is its
to compute (22) and (24) eliminates the need to explicitly con-
struct the extremely large matrices to deconvolve the image and
to estimate the residual noise variances. The MForWarD algo-
trix in (30) directly is computationally prohibitive due to the
size of the images. We now describe an efficient implementa-
tion of the MForWarD algorithm which leverages FFT opera-
tions, analogous to the method used in the case of the stationary
We operate in the the Fourier domain as we did in
Section II-C, where we originally defined many of the matrices.
The only additional matrix is that of the blur operator
is spatially-invariant and, hence, is diagonal in the Fourier
In the case of multiframe Wiener filtering, we again consider
each spatial frequency component indexed by
independent observation model given by
A single spatial frequency in the captured image is a function of
the original signal content and summation of the
spectral components. This demonstrates that the reconstruction
can be appliedto each collectionof aliased spectral components
2676 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 19, NO. 10, OCTOBER 2010
Estimates of the
high-resolution image from the multiple measurements of the
observed spatial frequencies
spatial frequency components in the
are obtained via
resolution image in this fashion is significantly faster than using
the direct matrix form of (28). Implementing the multiframe
Wiener filter in the Fourier domain requires inverting
matrices of size
as opposed to inverting one very large
matrix of (28). In a practical sense, any im-
ages larger than 200 by 200 pixels would require inordinate (for
today’s machines) amounts of processing to invert the B matrix
if usingthedirectmatrix formulation.This approachenablesef-
ficient implementation of a multiframe resolution enhancement
in the Fourier domain. After estimating the sharpened image ,
the image is converted back into the spatial domain using an in-
verse FFT to obtain the multiframe filtered image in the spatial
We can also apply a similar technique to accelerate the cal-
culation of the
residual noise powers in the wavelet spaces
, which are required for the wavelet denoising step.
Because the residual noise field is not stationary, its statistics
are not completely characterized by a power spectral density
covariance matrix of the residual noise field spec-
tral components associated with the LR spatial frequency set
is defined in (8). As in Section II-C, estimating the high-
To estimate the residual noise power for a given wavelet filter
at a particular grid location, we compute
the diagonal matrix
and “Tr” represents the trace operator and
represents the samples of the wavelet filter spectral response.
Equation (35) provides an efficient method for computing the
residual noise powers in the wavelet domain for use in wavelet
denoising. This step must be performed twice; once for the soft
thresholding wavelet functions
olding wavelet set
and once for the hard thresh-
Fig. 4. Comparison of linear SR techniques for increasing the resolution of 12
simulated LR frames by a factor of ? ? ?: (a) original image; (b) captured
image; (c) linear, strong prior; and (d) linear, weak prior.
Fig. 5. Solid curve shows the PSNR performance versus regularization using
standard linear regularization. The optimal weighting is around ? ? ??? for the
regularization for the MForWarD algorithm using different thresholding values
?. The MForWarD algorithm shows superior peak PSRN performance over the
IV. EXPERIMENTAL RESULTS
In this section, we perform two sets of experiments demon-
strating the capability of the MForWarD algorithm. The first
section describes experiments using simulated general image
data. These results demonstrate the broad applicability of the
MForWarD algorithm. The second section describes some ex-
perimental results using real data captured by a digital mammo-
gram system on a phantom breast.
ROBINSON et al.: EFFICIENT FOURIER-WAVELET SUPER-RESOLUTION2677
A. Simulation-Based Experiments
For the first set of experiments, we construct a simple sim-
ulation example which highlights the advantages of the MFor-
WarD algorithm for general imaging. We construct a simulated
data sequence by downsampling the 400 by 400 pixel portion
of a high-resolution shown in Fig. 4(a) using the subsample lo-
cation shown in Fig. 3. We show only a cropped portion of the
image to highlight the detail in the experiment. We simulate the
optical blur using a simpleheavy-tailed point spread function of
is the normalizing constant. We use a
also added noise to the LR images to create an effective SNR of
about 26 dB. An example of the simulated LR image is shown
in Fig. 4(b).
We reconstructed the HR image using three different ap-
proaches representing the different classes of SR algorithms.
In all these simulated cases, the motion was assumed to be
perfectly known. The first method is the completely linear
approach embodied by the multiframe Wiener filter of (28),
or the Fourier implementation of (33). This is perhaps the
least computationally complex approach to multiframe SR.
Using (33) produces an estimate of the HR image in a single,
noniterative step and requires inverting several very small
matrices. The drawback to this linear approach, however, is
that the regularization of the multiframe linear filter trades off
sharpness in the final image for noise reduction. For example,
Fig. 4(c) shows an example of the linear reconstruction when
using a power spectral density (PSD) function of the form
term controls the rate of decay of the PSF, and
value of 1.2 pixels. We
The reconstructed image shows poor contrast to maintain
minimal noise amplification. Alternatively, if we apply the
linear reconstruction with a weaker prior
reconstructed image shows improved contrast restoration at the
expense of noise amplification as seen in Fig. 4(d). This is the
classic tradeoff inherent to linear restoration algorithms.
The black curve of Fig. 5 shows the PSNR tradeoff as a func-
tion of regularization weighting strength. For a small
PSNR is quite poor due to weak regularization of the poorly
conditioned system. When
increases beyond 4.5, the PSNR
slowly degrades as the estimate becomes overly smoothed. If
the signal’s PSD was perfectly defined by (38), then the PSNR-
maximizing weighting parameter would be near
dashed curves show PSNR versus regularization weighting for
the MForWarD algorithm using different thresholding param-
. While the performance varies considerably for large
valuesof ,thepeak PSNRnear
parameter .Whenwe applyourMForWarDalgorithm,weem-
ploy 2-tap Daubechie filters for the soft thresholding wavelet
functions and 6-tap Daubechie filters for the coarse denoising
by way of hard wavelet coefficient thresholding.
. The regularization weighting of (33) is.
Fig. 6. Comparison of image quality for different regularization techniques
using PSNR-optimal settings: (a) linear regularization (???? ? ????? dB);
(b) B-TV (???? ? ????? dB); and (c) MForWarD (???? ? ???? dB).
Fig. 6 compares the PSNR-optimal parameter settings for
three different algorithms. Fig. 6(a) shows the PSRN-optimal
linear regularization setting, which has a PSNR of 23.90 dB at
. The image shows a reasonable balance between con-
trast and noise gain, but still maintains a considerable amount
of noise in the flat sky region. Fig. 6(b) shows the PSNR-op-
timal B-TV algorithm of (3) having 23.96 dB. In this experi-
a reasonably good job of preserving contrast while eliminating
the noise in the flat regions. Fig. 6(c) shows the MForWarD
algorithm at 24.3 dB using a thresholding value of
The image preserves much of the contrast while eliminating the
is the minimal computational overhead required for the wavelet
denoising. Running on an Intel Core-2-Duo 2.2 GHz processor,
the Fourier restoration requires 4.3 s of computation time. Per-
for a total of about 6.6 s. In contrast, the B-TV algorithm, repre-
sentative of the large class of iterative techniques, requires only
0.5 s per iteration, but requires at least 60 steepest descent iter-
ations (30 s) before approaching a limiting image quality.
. The algorithm does
B. Real X-Ray Data Experiments
In this section, we apply our multiframe reconstruction
and restoration algorithm to real images captured on an ex-
perimental X-ray imaging system. Our experimental imaging
system is based on a Mammomat NovationTOMO digital
mammography prototype system (Siemens Medical Solutions,
2678 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 19, NO. 10, OCTOBER 2010
Fig. 7. Scatter plot shows the set of estimated motions ?? ? ? on the HR image
grid. Some high-resolution pixels have as many as four measurements whereas
others have none underscoring the need for a spatially-varying denoising
Erlangen, Germany),3stationed at the Duke University Medical
Center. The system uses a stationary selenium-based detector
m pixels. Pixels with this size correspond to a Nyquist
sampling rate of 5.6 line pairs per millimeter (lp/mm). We use
a CIRS model 11A breast phantom (CIRS Inc., Norfolk, VA)
to test our SR algorithms. We introduce shifts in the image
by two methods. First, we allow the x-ray tube to rotate by
1 degree. Second, we manually move the breast phantom
to introduce motion into the system. This manual motion is
completely uncontrolled. Our dataset consists of 15 frames at
the low dosage level of 11.3 mAs at 28 kVp tube voltage. As
a point of reference, we also acquire a single frame at a more
typical dosage of 226 mAs at 28 kVp tube voltage (Fig. 1). The
breast phantom includes several testing features including a
pair of resolution bar charts. We focus on the results of the test
resolution chart to explore the contrast performance of the mul-
tiframe imaging system. We apply our algorithm to 100
pixel tiles in the captured image to estimate 400
high-resolution images (enhancement
We modeled our system PSF as a heavy-tailed exponential
energy distribution with
PSNR, we calculated the standard deviation in a textureless
region of the phantom. We also measured the difference in
grayscale values between for the registration bars in the res-
olution chart to get an approximate PSNR value of 3 dB.
We fit the
terms of (38) to the periodogram obtained by
averaging the spectral energy distribution over the collection of
LR images. We employed 2-tap Daubechie filters for the the
soft thresholding wavelet functions and 6-tap Daubechie filters
for the coarse denoising by way of hard wavelet coefficient
Fig. 7 shows a scatter plot of the set of estimated motions
on the HR image grid. The grid reflects the number of
. To get a measure of the
3Caution: Investigational Device. Limited by U.S. Federal law to investiga-
tional use. The information about this product is preliminary. The product is
under development and is not commercially available in the U.S.; and its future
availability cannot be ensured.
Fig. 8. (a) High dosage LR image (226 mAs); (b) low-dosage LR image
(11.3 mAs); (c) motion compensated average of LR frames (no resolution
enhancement); and (d) Multiframe reconstruction image ? ? of (7).
image samples per pixel in the reconstructed image. The ex-
ample shows some pixel estimates combining as many as four
measurements, whereas other pixels have no measurements un-
derscoring the spatial variability of the residual noise variance.
Fig. 8 gives a visual example of the SNR for an image ob-
SNR after combining multiple low-exposure images. We focus
on the portion of the resolution chart beyond the Nyquist rate
for the imaging system (5.6 lp/mm). The numbers indicate the
ically high radiation dosage of 226 mAs. The bar targets clearly
show aliasing artifacts. The second image Fig. 8(b) shows the
same portion of the resolution chart captured at a much lower
SNR of the captured images at such low illuminating radiation.
The third image Fig. 8(c) shows the result of averaging the mo-
tion compensated LR frames without enhancing the resolution
or sampling rate of the system. Interestingly, the multiframe av-
erage appears to have an approximately equivalent SNR to the
single frame captured at a normal radiation dosage, although it
contains some blur as a result of averagingframes with subpixel
sampling offsets. The fourth image Fig. 8(d) shows the recon-
after registering the collection of images using
the multiframe algorithm described in Section II. The recon-
structed image shows a restored resolution above the Nyquist
rate inherent to the detector. The image is, however, still noisy
and has low contrast. The effective SNR seems comparable to
that of the single image captured under high dosage Fig. 8(a).
ROBINSON et al.: EFFICIENT FOURIER-WAVELET SUPER-RESOLUTION2679
Fig. 9. Different restoration techniques applied on the low-dosage sequence
illustrated in Fig. 8(b): (a) multiframe sharpened image ? ?; (b) basic ForWarD ? ?
applied to multiframe sharpened image; and (c) MForWarD ? ?. The MForWarD
algorithm provides superior noise removal while preserving contrast.
Fig. 9 shows the resulting images after applying the sharp-
shows the result
after applying the multiframe Wiener sharp-
ening filter of(28). Theimageshows improvedcontrast within-
creasedsharpness, butalso amplified noise.Theimage Fig.9(b)
to the Wiener sharpened image Fig. 9(a). The hard threshold
value was chosen to be
. The resulting image preserves
the contrast around the bar chart signal locations while elimi-
The image still contains some residual noise. The final image
Fig. 9(c) shows the result after applying the MForWarD algo-
rithm including the nonstationary noise power computations.
The hard threshold for the MForWarD algorithm was also
. Including the spatially-varying noise powers improves the
noise removal over the basic ForWarD algorithm Fig. 9(b).
To get an another perspective on the effects of the multiframe
restoration and denoising, we plot slices through the resolution
test chart region as indicated in Fig. 10 (top). Fig. 10 shows
slices through several images. The top curve shows the slice
pixel motion. The slice shows some aliasing as well as lost con-
trast for the bars about the Nyquist sampling rate of 5.6 lp/mm.
The second graph shows a slice through the multiframe recon-
structed image . The reconstruction eliminates the aliasing ar-
of the detector. The signal strength above 8 lp/mm, however, is
Fig. 10. Top: input low-dosage image [a zoomed in version of which was il-
lustrated in Fig. 8(b)]. The arrow marks the sampling slice through the resolu-
tion chart. Bottom: five curves show slices through the horizontal resolution bar
charts. The Nyquist rate of the system corresponds to 5.6 lp/mm. The top slice
shows a slice through an interpolated average of the captured images showing
aliasing artifacts and lost contrast. The second slice through ? ? shows enhanced
resolution beyond the Nyquist rate, but poor contrast. The third slice through ? ?
shows restored contrast but with noise amplification. The first three left ellip-
soids in the fourth slice through the basic Forward reconstruction mark the am-
plified noise regions. The rightmost ellipsoid marks the lost resolution region.
The bottom slice through the MForWarD estimate ? ? shows contrast preserva-
tion with significantly less noise.
very weak due to the blurring inherent to the imaging system.
sharping restores contrast out to the 12 lp/mm, more than twice
the Nyquist rate, but at the expense of noise amplification. The
has some noise amplified regions. The bottom slice shows
the bar chart signal regions.
The final goal of digital mammography is the detection and
diagnosis of cancerous lesions in the breast tissue. The breast
phantom contains small grains of calcium for predicting the
diagnostic capability of the imaging system for calcifications
in the breast. The calcium grains range from 400
m down to
2680 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 19, NO. 10, OCTOBER 2010
Fig. 11. Table of images shows the low-dosage image (first column), mul-
tiframe average (second column), standard dosage image (third column) and
synthetic calcifications are clearly visible in all of the MForWarD images.
196 m. Fig. 11 shows the performance of the MForWarD al-
gorithm for enhancing the contrast of these small deposits. The
grains of calcium carbonite. In fact, the grains are visible even
when the grains are nearly indistinguishable from noise in the
single images captured at a standard dosage.
In this paper, we have proposed a novel method for restoring
and denoising super-resolved low dosage X-ray images in a fast
multiframe variant of the ForWarD algorithm of . The pro-
posed Fourier multiframe restoration and wavelet denoising al-
proving the extremely poor SNR of low-dosage images. The
experimental results confirm that multiframe imaging can pro-
vide an alternative in the SNR versus resolution tradeoff for
digital mammography. We note that our restoration algorithm
can be easily modified to further enhance the quality of other
Shift-and-Add based SR techniques , .
The design of future X-ray imaging systems would benefit
from a systematic analysis of the resolution and SNR required
for mammographic screening and diagnosis. Currently, there
are no publicly available databases of multiframe aliased digital
mammography images, only digitized analog mammography
images. Upon approval of an institutional review board (IRB),
we intend to apply this technique to imagery collected from pa-
tients in the clinical setting. At this point, we hope to demon-
strate the ability of this technique to improve image quality on
real digital mammography images.
In the future, one might explore the fundamental tradeoffs
between radiation exposure, number of frames, and reconstruc-
tion performance. Furthermore, extensions to the ForWarD al-
gorithm which include more sophisticated redundant wavelet
techniques such as curvelets  or ridgelets  might show
even better performance. Recent research has shown that use
of more sophisticated wavelets have been found to improve the
quality in other medical imaging applications . Future re-
tiframe imaging on the higher-level segmentation or detection
tasks associated with digital mammography.
Our MATLAB software implementation of the noted algo-
rithms is in part based on the ForWarD software, developed by
Dr. R. D. Neelamani of the Digital Signal Processing group at
Rice University (available at http://www.dsp.rice.edu/software/
ward.shtml). The authors would like to thank Prof. P. Milanfar
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trical engineering from the University of California,
Santa Cruz, in 2004, where he developed super-reso-
He is currently the Manager of the Digital Optics
Research group at Ricoh Innovations, Inc., Menlo
Park, CA, researching practical applications of
computational imaging technology. His work ranges
from optical imaging systems design to image
processing algorithm research.
Cynthia A. Toth received the M.D. degree from the
Medical College of Pennsylvania.
She completed ophthalmology residency training
at Geisinger Medical Center and a Fellowship in Vit-
reoretinal Diseases and Surgery at University of Cal-
ifornia, Davis. She was Chief of the Retina Service
at Wilford Hall USAF Medical Center. Joining Duke
University Medical Center Faculty, Durham, NC, as
a vitreoretinal surgeon in 1993, she is now Professor
of Ophthalmology and of Biomedical Engineering.
She heads the Duke Advanced Research in SDOCT
Imaging (DARSI) Laboratory. She is the author of over 100 peer-reviewed pub-
lications and multiple book chapters, and an inventor on 11 patents.
Dr. Toth is a Diplomate of the American Board of Ophthalmology, Fellow of
the American Academy of Ophthalmology, and a member of the Retina Society
and Club Jules Gonin.
Joseph Y. Lo received the B.S.E. and Ph.D. degrees
in biomedical engineering from Duke University,
and biomedical engineering, faculty of the medical
physics graduate program of Duke University/Duke
ical Center. His research interests span many topics
in the advanced imaging and management of cancer,
including breast tomosynthesis imaging, computer-
aided diagnosis and risk prediction, radiation therapy treatment planning, bioin-
formatics, and digital image processing.
Sina Farsiu received the B.Sc. degree in electrical
engineering from Sharif University of Technology,
Tehran, Iran, in 1999, the M.Sc. degree in biomed-
ical engineering from the University of Tehran,
Tehran, in 2001, and the Ph.D. degree in electrical
engineering from the University of California, Santa
Cruz (UCSC), in 2005.
He was a Postdoctoral Scholar at UCSC from
2006 to 2007 and a postdoctoral research associate in
the Department of Ophthalmology, Duke University,
Durham, NC, from 2007 to 2009. He is currently
an Assistant Professor in the Departments of Ophthalmology and Biomedical
Engineering, and Fitzpatrick Institute for Photonics at Duke University, where
he is the director of the Vision and Image Processing (VIP) laboratory. His
technical interests include robust image enhancement and reconstruction,
automatic segmentation, computer aided surgery, ocular imaging and analysis,
optical coherence tomography, and digital X-ray imaging.