Concepts in MR (SENSE) - Department of Bioengineering

A Graphical Generalized
Implementation of SENSE
Reconstruction Using
Matlab
HAMMAD OMER, ROBERT DICKINSON
Department of Bioengineering, Imperial College London, United Kingdom
ABSTRACT: Parallel acquisition of Magnetic Resonance Imaging (MRI) has the potential to significantly reduce the scan time. SENSE is one of the many techniques for the
reconstruction of parallel MRI images. A generalized algorithm for SENSE reconstruction
and theoretical background is presented. This algorithm can be used for SENSE reconstruction for any acceleration factor between 2 and 8, for any Phase Encode direction
(Horizontal or Vertical), with or without Regularization. The user can select a particular
type of Regularization. A GUI based implementation of the algorithm is also given. Signal-to-noise ratio, artefact power, and g-factor map are used to quantify the quality of
reconstruction. The effects of different acceleration factors on these parameters are
also discussed. The GUI based implementation of SENSE reconstruction provides an
easy selection of various parameters needed for reconstruction of parallel MRI
images and helps in an efficient reconstruction and analysis of the quality of
reconstruction.
Ó 2010 Wiley Periodicals, Inc.
Concepts Magn Reson Part A 36A: 178–186, 2010.
KEY WORDS:
parallel MRI; SENSE reconstruction; regularization; k-space; Matlab
I. INTRODUCTION
MRI has been its long image acquisition time and as
MRI equipment is expensive; it is costly to spend too
much time on scanning one patient. Efforts to reduce
the time of MRI scan will maximize the utility of the
hospitals’ resources.
Parallel MRI is one method to reduce MRI scan
time in which multiple coils (or coil arrays) are used
to acquire MRI data in parallel. Many reconstruction
algorithms have been suggested in the recent past that
can be broadly categorized into ‘‘k-space’’ algorithms
and ‘‘image-domain’’ algorithms. The work presented
here is based on Sensitivity Encoding (SENSE)
reconstruction suggested by Preussmann et al. (1).
SENSE is an image domain technique for reconstruction of Parallel MRI which is based on the fact
that receiver sensitivity generally has an encoding
Magnetic Resonance Imaging (MRI) has been of
great value in the medical diagnostics for the past
several years and has provided a tremendous potential to identify different pathological conditions in
the human body. One major limitation of current
Received 18 September 2009; revised 16 April
2010; accepted 27 April 2010
Correspondence to: Hammad Omer; E-mail: hammad.omer05@
imperial.ac.uk
Concepts in Magnetic Resonance Part A, Vol. 36A(3) 178–186 (2010)
Published online in Wiley InterScience (www.interscience.wiley.
com). DOI 10.1002/cmr.a.20160
Ó 2010 Wiley Periodicals, Inc.
178
179
IMPLEMENTATION OF SENSE RECONSTRUCTION
effect complementary to Fourier encoding by linear
field gradients (1). Thus, the areas of the object
(being imaged) which are closer to a particular coil,
contribute more signal to the total signal collected by
the coil as compared to the parts of the object further
away from the coil. As the coils are systematically
located at different parts of the object (to be imaged),
their location captures spatial information in the
image of the object to be reconstructed.
During image acquisition by parallel imaging, the
gap between adjacent k-space lines is increased. The
k-space lines are skipped to reduce acquisition time
and this under sampling of k-space causes aliasing in
the acquired images. As each pixel location in the
aliased images has signals from more than one location of the actual image, an important step of reconstruction is to separate the signal contribution from
each pixel location of the aliased image and to allocate it to the right place in the reconstructed image.
The sensitivity map defines the weights on the basis
of which signal at each pixel location in the aliased
image must be reallocated to the right pixel location
in the reconstructed image. Provided that the coils’
sensitivity profiles are not the same at those different
locations, the weight given to each of the signal components will be different for each coil (2) thus ensuring good reconstruction.
The quality of SENSE reconstruction depends on
how accurately the sensitivity map represents the
weights which will be used to separate the signals in
the aliased image and allocate them to the respective
locations in the unwrapped image. The key to signal
separation is the fact that in each single-coil image,
signal superposition occurs with different weights
according to their local coil sensitivities (1). Owing
to its significance, proper estimation of sensitivity
maps has drawn the attention of many researchers in
the recent past, and many regularization techniques
(3–8) have been proposed recently with the main aim
of having as precise a sensitivity map as possible.
Furthermore, non-Cartesian sampling methods (spiral
or radial) have been recently proposed for even faster
image acquisition and better navigation for flow information. Further details on this can be found in (9).
from ‘‘R’’ locations, equally spaced along the subsampled direction, overlap in the image. Field of view
(FOV) reduction can be stated mathematically by saying that the R-fold FOV reduction results in an NA fold
aliased image representation as given in (1), where NA
represents the total number of signals present at location
‘‘y’’ owing to aliasing(including the actual signal of this
location). Thus, for each location ‘‘y,’’ we can write the
image signal Ij(y) as a superposition of the original signal and displaced replicates (11):
Ij ðyÞ ¼
NX
A 1
Cj ðy þ nL=RÞMðy þ nL=RÞ
n¼0
where j ¼ 0; 1; . . . . . . N c 1
½1
Here Nc is the number of elements in the coil
array, NA is the number of overlapped signals at one
location in the aliased image, and C stands for the
encoding (or sensitivity) matrix. In the above equation, Ij(y) are known because they are the acquired
aliased images (one for each coil array element). The
aliased magnetization values M(y þ nL/R) are to be
found. If Nc NA, the system of equations can be
solved to obtain M(y þ nL/R). The above equation
can be generalized for simplicity into a matrix notation. With Nc coils, I, C, and M matrices can be
defined with dimensions Nc 1, Nc NA and NA 1, respectively and then (1) can be written as:
l ¼ CM
[2]
In SENSE, the reconstruction problem is formulated as solving a set of linear equations defined in
(2) where:
2
I0 ðyÞ
3
2
MðyÞ
3
6 I ðyÞ 7
6
7
Mðy þ L=RÞ
6 1
7
6
7
I¼6
7M ¼ 6
7
4
5
4
5
Mðy þ ðNA 1ÞL=RÞ
INc1 ðyÞ
2
C0 ðyÞ
C0 ðy þ ðNA 1ÞL=RÞ
6
6
C¼6
4
CNc1 ðyÞ
3
7
7
7
5
CNc1 ðy þ ðNA 1ÞL=RÞ
II. THEORY
The main idea in SENSE is to apply knowledge of the
sensitivities of the coil elements to calculate the aliased
signal component at each point (10) and then allocate
these signals to their actual locations in the unfolded
image. If the gap between adjacent k-space lines is
increased by an acceleration factor ‘‘R,’’ the signals
Here, ‘‘I’’ represents the aliased signals (from
aliased images) obtained by the MRI scanner (the
aliased image is obtained by skipping some phase
encode lines, thus reducing the scan time), ‘‘C’’ is the
encoding matrix (also named as Sensitivity Matrix)
which contains spatial information about each coil
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OMER AND DICKINSON
and this information is used to relocate appropriate
signals to each pixel location in the reconstructed
image. ‘‘M’’ is the image to be recovered given by:
M ¼ C1 I
[3]
The inverse of matrix C in (3) can be implemented by using Moore-Penrose pseudo-inverse
given by:
h
M ¼ ðCt
i
CÞ1 Ct I
[4]
where ‘‘M’’ is the unfolded image.
correlation with itself. So, when a particular point is
weakly detected by several coil elements or well
detected by only a single element, the composite
image will see higher noise due to this autocorrelation effect (24). The quantification of this noise
amplification factor is done with ‘‘g-factor.’’ The ‘‘gfactor’’ describes how well the coil array encodes the
magnetization distribution of the object. A smaller gfactor generally indicates that the magnetization at a
given location in the object is detected by several coil
elements. Provided the noise correlation between
those elements is weak, greater sensitivity can be
recovered than if only a single element of the array
can detect the magnetization (24). The relationship
between the SNR with and without SENSE is given
by (25)
III. QUALITY OF RECONSTRUCTION
One issue with parallel acquisition is the loss of Signal-to-Noise Ratio (SNR) due to skipping some
phase encode lines. The coil design (12–15) and the
trajectory used for the k-space acquisition (16, 17)
have a significant effect on the SNR with some trajectories or coil designs giving higher SNR as compared to others. Independent information from each
channel in the RF coil array is very important
because correlations in the spatial information from
the neighboring array elements can degrade the
image quality. In fact, when many receiver coils are
used in conjunction with high acceleration factors,
the image reconstruction may become very ill-conditioned. The standard method of reconstruction given
a poorly conditioned matrix can amplify the noise in
reconstructed SENSE images. The noise amplification for a poorly conditioned matrix can be reduced
by a process called ‘‘regularization’’ (3–8). Many
techniques have been proposed for regularization
(18–23), and these regularization techniques use different ways to decrease the ill-conditioning of the
sensitivity maps.
If we examine the image from a single element of
the array, the signal values are generally confined to
a region near the coil element, but the noise values
are distributed throughout the image. If the noise
from the different elements is weakly correlated(or
incoherent), the noise in the reconstructed image will
grow as the square root of the number of elements
while the signal will grow as the number of elements,
provided the signal phases are aligned and all the
coils are equally sensitive (24).
The acceleration factor gives the number of times
the data from each coil is reused to calculate the final
unwrapped image. When this data is reused, the
noise is necessarily amplified because it has natural
SNRSENSE ¼
SNRNormal
pffiffiffi
g R
[5]
Here, the factor ‘‘R’’ implies the expected loss in
SNR that results by reducing the scan time by acceleration factor ‘‘R’’ and ‘‘g’’ is the geometry factor
which represents noise magnification that occurs
when aliasing is reconstructed. The g-factor is determined by (1): rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
h
i [6]
gi ¼
ðCt c1 CÞ1 Ct c1 C ii
ii
Here ‘‘C’’ is the Nc Nc noise correlation matrix for
the coils in which a diagonal element represents noise
variance from a single coil and an off-diagonal element
represents a noise cross-correlation between two coils
and ‘‘C’’ is the encoding matrix. This equation applies to
all pixels in the image with the same number of aliased
replicates, i.e., NA. The subscript ‘‘i’’ refers to aliased
replicate number ‘‘i’’ for that pixel and has the range
0,1,.....NA 1. Thus, the geometry factor for all pixels
that are related by aliasing at a particular location in the
aliased image can be computed by the above equation.
The noise amplification described by the g-factor is also
related to a property of the matrix ‘‘CtC1C’’ that is
inverted in (6), called its conditioning (3–8).
The g-factor is a measure of correlation between
the neighboring coils and indicates the noise magnification capability of a coil array and it depends on the
number of aliased replicates NA as well as on the coil
sensitivity difference between aliased pixels. The
sensitivity difference depends on the coil conductor
placement, the scan plane orientation, the phase
encode direction within the scan plane, and the pixel
location within the scan plane. Therefore, g-factor is
quite useful when considering how to design a coil
which is to be used for SENSE.
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IMPLEMENTATION OF SENSE RECONSTRUCTION
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Figure 1 GUI interface for parallel MRI reconstruction. [Color figure can be viewed in the
online issue, which is available at www.interscience.wiley.com.]
IV. IMPLEMENTATION
The SENSE reconstruction (4) has been implemented
using a GUI interface that allows the user to define all
the key variables for the reconstruction, as shown in
Fig. 1. The user defined variables for this application
are: (1) aliased images, (2) sensitivity map, (3) original
image(for comparison), (4) phase encode direction, i.e.,
horizontal or vertical, (5) acceleration factor between
two and eight, (6) regularization type: a: Polynomial
(order 1), b: Polynomial (order2), c: Tikhonov Regularization, d: Wavelet based regularization, (7) Undersampling: User can select if the loaded data is already
aliased and there is no need for under-sampling or
loaded data needs to be under-sampled to simulate
aliased data. Output: (1) reconstructed image, (2) g-factor map, (3) SNR value, (4) artefact power. The flowchart for this generalized SENSE reconstruction is
given in Fig. 2. For ease of use, this algorithm has been
linked with a GUI interface. Matlab(version 7.6) has
been used for this GUI platform and all the functions
needed during reconstruction have been associated with
different fields in the GUI for an efficient working.
In MRI, the raw data we acquire are complex
(having a magnitude and a phase component). Normally, the magnitude is enough to visualize an MRI
image and the phase component is ignored. However,
phase plays a crucial role in the SENSE reconstruction
process. The coil sensitivity varies in space in both
phase and magnitude and the phase information is
essential for an accurate reconstruction. So, all the steps
shown in the algorithm involve complex arithmetic. (2)
V. EVALUATION OF RECONSTRUCTION
The performance of the parallel image reconstruction
algorithm can be evaluated by two quantification parameters: (1) signal to noise ratio (2) artefact power
1. Signal to Noise Ratio (SNR): During the process of reconstruction, the user is asked to
select a region of interest for signal (ROS)
and a region of interest for noise (RON), normally the background. Then SNR is calculated by using the following formula (26):
SNRðdBÞ ¼ 20 log10
Mean ROS
Std:Deviation of RON
[7]
2. Artefact Power (AP): The concept of AP has
been derived from ‘‘Square Difference Error.’’
Here, it is presumed that a reference image
(full FOV) is available and the AP in the
reconstructed image will be evaluated on the
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OMER AND DICKINSON
Figure 2 Algorithm for SENSE Reconstruction.
basis of this reference image. AP can be calculated using the following formula (26):
AP ¼
2
I reference ðx; yÞjjI reconstruded ðx; yÞ
jI reference ðx; yÞj2
[8]
It is clear from the above formula that if Ireference
¼ Ireconstructed, the AP will be zero meaning that there
is no artefact in the reconstructed image and the
reconstructed image is identical to the reference
image. Similarly, AP will be a bigger value (i.e.,
closer to 1) if the reconstructed image is significantly
different than the reference image.
VI. RESULTS AND DISCUSSION
To demonstrate the performance of this implementation, the reconstruction algorithm is first applied on
simulated data and then on the experimental data
sets. We used a 1.5 Tesla GE scanner at St. Mary’s
Hospital London with an eight channel head coil and
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IMPLEMENTATION OF SENSE RECONSTRUCTION
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Figure 3 Images acquired by eight separate coils of an eight array head coil showing differing
spatial localization of the signals.
a Gradient Echo sequence with the following parameters: TE ¼ 10 m sec, TR ¼ 500 m sec, FOV ¼ 20
cm, Bandwidth ¼ 31.25 KHz, Slice Thickness ¼ 3
mm, Flip Angle ¼ 908, Matrix Size ¼ 256 256.
For both datasets, the full k-space data were
acquired (Fig. 3) and their ‘‘sum of squares’’ reconstruction was used as a reference image. Then, the
specified number of k-space lines were skipped to
produce aliased images, depending upon the acceleration factor, e.g., for an acceleration factor of two,
one out of every two phase encode steps were
removed, for an acceleration factor of three, two out
of every three phase encode steps were removed. The
inverse Fourier transform of the sub-sampled k-space
gave us aliased images. To have the sensitivity map,
the central lines of k-space of the full FOV data were
truncated by using cosine taper window (25). In this
way, low resolution images of the coils were
obtained and then these values were normalized by
dividing by sum of square image, thus giving us the
sensitivity maps (Fig. 4). Information about the noise
captured by the coils during imaging process is very
useful in the reconstruction process. The noise
images are obtained by switching off the RF signal
and capturing the images. Thus, the signals obtained
are just the noise images and not the signals from the
Figure 4 Sensitivity maps obtained by dividing low resolution coil images (of Fig. 3) by sum
of squares image.
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OMER AND DICKINSON
Figure 5 Aliased images (acceleration factor of 2) obtained by skipping phase encode steps.
object being imaged. These aliased images (Fig. 5)
along with sensitivity information (Fig. 4) are used to
reconstruct the image [Fig. 6(b)].
Figure 7 indicates the relationship between acceleration factor and SNR, acceleration factor and artefact power. The reconstruction was performed using
the above algorithm for various acceleration factors
ranging between 2 and 8. It is noticed that the SNR
deteriorates abruptly after acceleration factor of 4.
Similarly, there is a sharp increase in the artefact
power after acceleration factor of 5. The possible reason for this may be the fact that the receiver coils
must have independent and distinct sensitivity profiles to give a good reconstruction. It means that we
have to see how many distinct receiver channels are
there in the direction of phase encoding. It is quite
probable that some of the eight receiver coil channels
may not have distinct sensitivity profiles particularly
in the phase encoding direction. Although, theoretically we say that the maximum achievable accelera-
tion factor from a receiver coil array is less than
maximum number of coils present in the array, but it
is valid only if each coil has a distinct profile. If this
condition is not fulfilled then reconstruction will not
be of good quality for higher acceleration factors.
Another important point to consider is that the coil
array system used here is circular. A circular array
has sensitivity variations not only in the direction of
phase encoding but along both x and y axis. It gives a
complex profile of coil sensitivity as compared to a
Linear Array (in which maximum sensitivity change
is only in one direction either x or y). This two directional change in sensitivity may add some inconsistency to the reconstruction because it becomes difficult to have accurate estimation of the sensitivity
changes in both directions.
These results validate the accuracy of the reconstruction algorithm. It is important to note that these
results are obtained without any regularization. Better results may be obtained with regularization and
Figure 6 (a) Sum of square image, (b) reconstructed image, (c) g-map.
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IMPLEMENTATION OF SENSE RECONSTRUCTION
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Figure 7 (a) SNR for different acceleration factors and (b) A.P. for different acceleration factors.
better coil design especially for higher acceleration
factors because these measures can help improve the
SNR as well as decrease the artefact power. This will
be the subject of our future work.
VII. CONCLUSIONS
A detailed account of a GUI based implementation
of SENSE reconstruction is presented. There are several parameters to be defined during the process of
image reconstruction, e.g., phase encode direction
(horizontal or vertical), acceleration factor, regularization type etc. This algorithm presents a generalized
SENSE reconstruction method giving a flexibility to
select different parameters needed for the reconstruction. The GUI interface provides a more manageable
way to load the aliased images, sensitivity maps and
other related data as well as facilitates the selection
of different reconstruction parameters in an interactive way. The results of reconstruction and the g-factor map are also displayed on the same window thus
making it easy to analyze the reconstruction. SNR,
g-factor, and artefact power are the parameters which
are used to quantify the quality of reconstruction.
The user can under-sample the k-space, if there is a
need to produce aliased images. It is noticed that the
g-factor deteriorates badly as soon as the acceleration
factor exceeds 4 (Fig. 7). This is due to a significant
loss in the k-space data because for an acceleration
factor of 5, just one out of five phase encode lines are
acquired thus causing a loss in SNR. Higher acceleration factors result in more phase encode steps being
skipped causing even more degradation in SNR. This
tool is available freely on request to the corresponding author.
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BIOGRAPHIES
Hammad Omer is a PhD candidate at
Department of Bioengineering, Imperial
College London. He received his B.Eng.
degree in Electronic Engineering from
Dawood College of Engineering & Technology, Pakistan in 2002 followed by Masters of Computer Science from University
of Karachi, Pakistan in 2003. Then he did
MSc in Bioengineering from Imperial College London in 2006. His current research
is focussed on image reconstruction using
parallel MRI.
Dr. Robert J. Dickinson, M.A., PhD, MBA
is a lecturer at the Department of Bio-engineering, Imperial College. He graduated
with a degree in physics from Cambridge
University, and then obtained a PhD in
Biophysics from the University of London
in ultrasound signal processing. Dr Dickinson has extensive experience in medical
imaging, in both hospital and industrial
environments. He worked on MRI coil
development and system integration at Picker International Ltd
and ultrasound imaging in a small start-up company where he
developed a sub-1mm intravascular ultrasound imaging catheter
for imaging coronary arteries. He has substantial experience in
the biocompatibility and other patient compatibility issues of
invasive medical devices, together with commercialization and IP
transfer. He has filed over a number of patents and published
papers, and has CE marked a number of medical devices. He has
worked with Emcision Ltd on their range of electrosurgical devices with over 3000 patients treated to date. His current research
interests include imaging in surgery, interventional imaging, miniaturising medical devices and electro surgery.
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a