A Graphical Generalized Implementation of SENSE Reconstruction … · 2011-06-28 · A Graphical Generalized Implementation of SENSE Reconstruction Using Matlab HAMMAD OMER, ROBERT

A Graphical GeneralizedImplementation of SENSEReconstruction UsingMatlabHAMMAD OMER, ROBERT DICKINSON

Department of Bioengineering, Imperial College London, United Kingdom

ABSTRACT: Parallel acquisition of Magnetic Resonance Imaging (MRI) has the poten-

tial 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 recon-

struction 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. Sig-

nal-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. � 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

Magnetic Resonance Imaging (MRI) has been of

great value in the medical diagnostics for the past

several years and has provided a tremendous poten-

tial to identify different pathological conditions in

the human body. One major limitation of current

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 recon-

struction of Parallel MRI which is based on the fact

that receiver sensitivity generally has an encoding

Received 18 September 2009; revised 16 April2010; accepted 27 April 2010

Correspondence to: Hammad Omer; E-mail: [email protected]

Concepts inMagnetic Resonance Part A, Vol. 36A(3) 178–186 (2010)

Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/cmr.a.20160

� 2010 Wiley Periodicals, Inc.

178

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 loca-

tion of the actual image, an important step of recon-

struction is to separate the signal contribution from

each pixel location of the aliased image and to allo-

cate 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 com-

ponents will be different for each coil (2) thus ensur-ing 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). Owingto 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 in-

formation. Further details on this can be found in (9).

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

from ‘‘R’’ locations, equally spaced along the sub-

sampled direction, overlap in the image. Field of view

(FOV) reduction can be stated mathematically by say-

ing 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 sig-

nal and displaced replicates (11):

IjðyÞ ¼XNA�1

n¼0

Cjðyþ nL=RÞMðyþ nL=RÞ

where j ¼ 0; 1; . . . . . .Nc � 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 equa-

tion, 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 nota-

tion. 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 formu-

lated as solving a set of linear equations defined in

(2) where:

I ¼

I0ðyÞI1ðyÞ�

INc�1ðyÞ

26664

37775M ¼

MðyÞMðyþ L=RÞ

�Mðyþ ðNA � 1ÞL=RÞ

26664

37775

C ¼

C0ðyÞ � � C0ðyþ ðNA � 1ÞL=RÞ� � ��

CNc�1ðyÞ CNc�1ðyþ ðNA � 1ÞL=RÞ

26664

37775

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 theencoding matrix (also named as Sensitivity Matrix)

which contains spatial information about each coil

IMPLEMENTATION OF SENSE RECONSTRUCTION 179

Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a

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 C�1I [3]

The inverse of matrix C in (3) can be imple-

mented by using Moore-Penrose pseudo-inverse

given by:

M ¼ ðCt CÞ�1Cth i

I [4]

where ‘‘M’’ is the unfolded image.

III. QUALITY OF RECONSTRUCTION

One issue with parallel acquisition is the loss of Sig-

nal-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 tra-

jectories or coil designs giving higher SNR as com-

pared 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-condi-

tioned. The standard method of reconstruction given

a poorly conditioned matrix can amplify the noise in

reconstructed SENSE images. The noise amplifica-

tion 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 dif-

ferent 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

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 autocorrela-

tion effect (24). The quantification of this noise

amplification factor is done with ‘‘g-factor.’’ The ‘‘g-factor’’ describes how well the coil array encodes the

magnetization distribution of the object. A smaller g-factor 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)

SNRSENSE ¼ SNRNormal

gffiffiffiR

p [5]

Here, the factor ‘‘R’’ implies the expected loss in

SNR that results by reducing the scan time by accel-

eration factor ‘‘R’’ and ‘‘g’’ is the geometry factor

which represents noise magnification that occurs

when aliasing is reconstructed. The g-factor is deter-mined by (1):

gi ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðCtc�1CÞ�1h i

iiCtc�1C� �

ii

r[6]

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 alsorelated to a property of the matrix ‘‘CtC�1C’’ 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 magnifi-

cation 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 isquite useful when considering how to design a coil

which is to be used for SENSE.

180 OMER AND DICKINSON


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 Regula-

rization, d: Wavelet based regularization, (7) Under-

sampling: 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-fac-tor map, (3) SNR value, (4) artefact power. The flow-

chart 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). Nor-

mally, 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 pa-

rameters: (1) signal to noise ratio (2) artefact power

1. Signal to Noise Ratio (SNR): During the pro-

cess of reconstruction, the user is asked to

select a region of interest for signal (ROS)

and a region of interest for noise (RON), nor-

mally the background. Then SNR is calcu-

lated by using the following formula (26):

SNRðdBÞ ¼ 20 log10MeanROS

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

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.]



basis of this reference image. AP can be cal-

culated using the following formula (26):

AP ¼ � Ireferenceðx; yÞ �j jIreconstrudedðx; yÞ�� 2� Ireferenceðx; yÞj j2 [8]

It is clear from the above formula that if Ireference¼ 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 implementa-

tion, 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

Figure 2 Algorithm for SENSE Reconstruction.



a Gradient Echo sequence with the following param-

eters: 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’’ recon-

struction was used as a reference image. Then, the

specified number of k-space lines were skipped to

produce aliased images, depending upon the acceler-

ation 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.

Figure 3 Images acquired by eight separate coils of an eight array head coil showing differing

spatial localization of the signals.



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 accel-

eration factor and SNR, acceleration factor and arte-

fact 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 rea-

son for this may be the fact that the receiver coils

must have independent and distinct sensitivity pro-

files 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, theoreti-

cally 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 acomplex 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 direc-

tional change in sensitivity may add some inconsis-

tency to the reconstruction because it becomes diffi-

cult to have accurate estimation of the sensitivity

changes in both directions.

These results validate the accuracy of the recon-

struction algorithm. It is important to note that these

results are obtained without any regularization. Bet-

ter results may be obtained with regularization and

Figure 6 (a) Sum of square image, (b) reconstructed image, (c) g-map.

Figure 5 Aliased images (acceleration factor of 2) obtained by skipping phase encode steps.



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 sev-

eral parameters to be defined during the process of

image reconstruction, e.g., phase encode direction

(horizontal or vertical), acceleration factor, regulari-

zation type etc. This algorithm presents a generalized

SENSE reconstruction method giving a flexibility to

select different parameters needed for the reconstruc-

tion. 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 interac-

tive way. The results of reconstruction and the g-fac-tor 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 accelera-

tion factors result in more phase encode steps being

skipped causing even more degradation in SNR. This

tool is available freely on request to the correspond-

ing 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 & Tech-

nology, Pakistan in 2002 followed by Mas-

ters of Computer Science from University

of Karachi, Pakistan in 2003. Then he did

MSc in Bioengineering from Imperial Col-

lege London in 2006. His current research

is focussed on image reconstruction using

parallel MRI.

Dr. Robert J. Dickinson, M.A., PhD, MBAis a lecturer at the Department of Bio-engi-

neering, 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 Dickin-

son 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 devi-

ces with over 3000 patients treated to date. His current research

interests include imaging in surgery, interventional imaging, mini-

aturising medical devices and electro surgery.



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A Graphical Generalized Implementation of SENSE Reconstruction … · 2011-06-28 · A Graphical Generalized Implementation of SENSE Reconstruction Using Matlab HAMMAD OMER, ROBERT