QUALITY OF SENSE AND GRAPPA RECONSTRUCTIONS IN
PARALLEL IMAGING.
A.Iles, A.Bessaid
Université de Tlemcen, FT, Département GEE. B.P. 230
Tlemcen 13000, Algérie
amel_iles@yahoo.fr
Abstract:
Many techniques for the reconstruction of parallel MRI images are performed which can be used to accelerate image
acquisition using an RF coil array. Recently, parallel imaging methods have become commercially available, and the
SENSE technique is very useful in clinical applications. In this study, we compare this technique with GRAPPA
technique which can provide high resolution. Algorithms can be illustrated for these methods to compare MR imaging
performance with limited receiver channels as well as analysis of SNR in the resulting images, and normalized mean
squared error (NMSE) for different acceleration factor.
Key words : Parallel MRI, SENSE, GRAPPA, SNR, NMSE.
1.
INTRODUCTION :
In the last years, MRI became an important consideration in
clinical application because it does not use any radiation
also it produces high-resolution images with a high signalto-noise ratio (SNR); the major limitation is that it operate at
the limits of potential imaging speed, high speed acquisition
is necessary.
Parallel MRI is one method to reduce MRI scan time.
However, several partially parallel acquisition strategies
have been proposed [1,2,3,4] which work either in the
frequency domain [1,3] or in the spatial domain [2,4]. These
rapid MRI techniques use spatial information in a multiple
receiver coil array; multiple phase-encoded data are derived
in parallel from a single phase-encoded NMR signal.
In 1997 did Sodickson and al. [1] report the first successful
experiments using parallel receivers for the purpose of scan
time reduction, introducing the SMASH method
(SiMultaneous Acquisition of Spatial Harmonics). In 1999
Klaas P. Pruessmann and al [2] reformulate the problem of
image reconstruction from multiple receiver data. This
concept has been named SENSE (SENSitivity Encoding).
After, a GeneRalized Autocalibrating Partially Parallel
Acquisitions (GRAPPA) technique is published in 2002 by
Mark A. Griswold and al [3]. A more generalized view of
the variable density AUTO-SMASH (VD-AUTO-SMASH)
technique [5] is used to generate uncombined coil images
from each coil in the array.
In 2004, M. Blaimer and al compare in their work the
several partially parallel acquisitions (PPA) as comparison
of the image quality of SENSE and GRAPPA
reconstructions; they provide that the results were nearly
identical reconstruction quality in the head human region.
GRAPPA reconstruction is particularly beneficial in areas
where accurate coil sensitivity maps may be difficult to
obtain. In inhomogeneous regions with low spin density
such as the lung and the abdomen, it can be difficult to
determine precise spatial coil sensitivity information. In
these regions, the image quality of SENSE reconstructions
might therefore suffer from inaccurate sensitivity maps [6].
In contrast, the GRAPPA algorithm provides good quality
image reconstructions, since the sensitivity information is
extracted from the k-space. In GRAPPA, central k-space
lines are fit to calculate the reconstruction parameters. This
fitting procedure involves global information and is
therefore not affected by localized inhomogeneities. The use
of lines near the center of k-space also ensures that there is
sufficient information to achieve a good reconstruction
quality.
In this study, we compare the both GRAPPA and SENSE
techniques. However, a brief technical overview of these
pMRI methods is given. The SNR, and normalized mean
squared error (NMSE) parameters are treated, they are used
to quantify the quality of reconstruction. The effects of
different acceleration factors on these parameters are also
discussed.
2.
THEORY :
In parallel MRI methods, the sensitivity can be estimated by
a sum of squares (SOS) image of all coil images.
The SENSE method requires explicit channel sensitivity
maps, against GRAPPA is an autocalibrated method that
does not require explicit sensitivity maps but nevertheless
needs channel calibration data.
2.1. SENSE:
This technique utilizes the information provided by multiple
receiver coils to skip portions of k-space that would
normally be phase encoded by a gradient field. The factor
that describes the reduced number of k-space samples
compared to the fully sampled on is referred to as reduction
factor R [2]. This results in a decreased sampling density
along the phase-encoded direction of k-space, which in turn
causes the problem of aliasing. This aliasing cannot be
undone, the coil sensitivity information is necessary.
Figure 1. SENSE reconstruction.
For a single aliased pixel, if nc denote the number of
receiver coils. The complex coil sensitivities at the np
superimposed positions form an nc x np sensitivity matrix
S:
𝑆𝑐,𝑝 = 𝑠𝑐 (𝑟𝑝 )
(1)
Where the subscripts c and p count the coils and the
superimposed pixels, respectively, rp denotes the position
of the pixel p, and Sc is the spatial sensitivity of the coil c.
For a single pixel in the reduced FOV and n c many
receiver coils, the sensitivity matrix S can then be used to
represent the aliasing process as:
𝑎⃗ = 𝑆𝑣⃗
(2)
Where 𝑣⃗ is an np×1 vector containing the correct pixel
values for the np unique points in the full FOV. After
aliasing, 𝑎⃗ contains nc×1 many different linear
combinations of the R pixel values in 𝑣⃗ .
The unfolding matrix U is given by using Moore-Penrose
pseudo-inverse as:
𝑈 = (𝑆 𝐻 𝑆)−1 𝑆 𝐻
(3)
Using the unfolding matrix, signal separation is
performed by:
𝑣⃗ = 𝑈𝑎⃗
(4)
The GRAPPA method uses the following equation to
reconstruct the missing k-space lines of the j-th coil at a
line (ky _ mΔky) offset from the normally acquired data
using a blockwise reconstruction:
𝑁 −1
𝑏
𝑆𝑗 (𝑘𝑦 − 𝑚𝛥𝑘𝑦 ) = ∑𝐿𝑙=1 ∑𝑏=0
𝑛(𝑗, 𝑏, 𝑙, 𝑚)𝑆𝑙 (𝑘𝑦 − 𝑏𝑅𝛥𝑘𝑦 ) (5)
Where the variable b specifies the reconstruction block, R
represents the acceleration factor. Nb is the number of
blocks used in the reconstruction. Block is defined as a
single acquired line and R - 1 missing lines (see Figure 2,
right side). The index l counts through the individual
coils. RΔky is the step-size in k-space between each of the
sampled lines, where each value of b provides the location
in k-space of a line to be used in the reconstruction. For a
line offset by mΔky, the value of the weights n(j,b,l,m) is
found by solving equation (5) separately for each ky in the
k-space center.
This process is repeated for each coil in the array,
resulting in L uncombined single coil images which can
then be combined using a conventional sum of squares
reconstruction.
2.3. Quality of image reconstruction:
In order to analyze the quality of reconstruction image,
two quantification parameters would be calculated,
signal-to-noise ratio (SNR) and the normalized mean
squared error (NMSE) of image
Where the resulting vector 𝑣⃗ has length np and lists
separated pixel values for the originally superimposed
positions. By repeating this procedure for each pixel in
the reduced FOV a non-aliased full-FOV image is
obtained (see figure 1.).
2.3.1. Signal to Noise Ratio (SNR):
2. 2. GRAPPA:
2.3.2. Normalized mean squared error (NMSE):
Griswold and al. proposed a further extension named
GeneRalized
Autocalibrating
Partially
Parallel
Acquisitions (GRAPPA). This technique utilizes inferred
information about the spatial sensitivities of the receiver
coils by applying multiple blockwise reconstructions to
directly calculate missing lines of k-space for each coil
(This process is shown in Figure 2.). Data acquired in
each coil of the array (black circles) are fit to the autocalibration signal (ACS) line (gray circles). Sampled lines
from all coils are used to fit a single ACS line from a
single coil, (an ACS line from coil 4 in this case). [3]
The normalized mean squared error (NMSE) is the total
power in the complex difference image, found by
subtracting the reference and the reconstructed images
and squaring the magnitude of the result, divided by the
total power in the reference image, found by squaring the
magnitude of the reference image [8]:
The SNR is determined as the ratio of a region of interest
for signal (ROS) and standard derivation (SD) of the
region of interest for noise (RON) [7]:
SNR(db) = 20log10
NMSE =
Mean ROS
(6)
Std.deviation of RON
∑||Ireference (x,y)|−|Ireconstructed (x,y)||
∑|Ireference (x,y)|2
2
(7)
This definition is equivalent to the artifact power (AP)
[7]. If Ireference= Ireconstructed, no artefact in the
reconstructed image and the reconstructed image is
identical to the reference image. Similarly, NMSE will be
a higher value, if the reconstructed image is significantly
different than the reference image, and it represents
reduced image quality, which suggests both increased
image artifacts and noise.
neighboring undersampled kspace phase-encoding lines
(blocks) and channels as input. A set of eight
autocalibration signal (ACS) lines in the central k-space
were acquired [3]. The reconstruction algorithms are
applied on simulated data set.
4.1. SNR:
For SNR estimation, the mean intensity in a region of
signal (boxed regions labeled ROS in the Figure 3.) was
divided by the standard deviation of the noise region
intensity (boxed regions labeled RON). The results of the
computer simulations are shown in Figure 4. It indicates
the relationship between acceleration factor and SNR, The
reconstruction was performed for various acceleration
factors ranging between 2 and 8. It is shown that the
GRAPPA reconstruction results in higher SNR than the
SENSE reconstruction at all accelerations tested. The
GRAPPA reconstruction using four blocks present a
higher SNR compared to the GRAPPA reconstruction
using two blocks for low acceleration factors.
Figure 2. Four acquired lines are used to fit a single ACS line in
coil 4. One block on the right is a single acquired line plus the
missing lines adjacent for an acceleration factor of two.
3.
The GRAPPA reconstruction achieves nearly perfect SNR
efficiency, so that nearly no loss in performance should be
expected, especially for low acceleration factors.
MATERIAL AND METHODS:
For this study, we have used computer simulations to
establish the SNR and the NMSE performance at different
acceleration factors. Simulated SENSE and GRAPPA
imaging were performed using image of standard
resolution phantoms type Shepp-Logan and brain image
with matrix size 256 x 256. Simulated images were
generated using eight element array coil. Images were
reconstructed at various acceleration factors from two to
eight; these calculations were performed using a sum of
squares reconstruction to demonstrate the benefits of the
uncombined-coil approaches. In our experience, two and
four blocks were used to minimize computational
complexity of each reconstruction; as mentioned by
Griswold and al the numbers of blocks in the range of
four to eight showed similar good results [3].
The images acquired at integer acceleration factors were
then subjected to quantitative evaluation of both SNR and
artifact power. They were evaluated using multiple
reconstructed of the same images.
4.
RESULTS AND DISCUSSION :
In this Section, we present the experimental results. The
Shepp-Logan phantom and brain image with matrix size:
256 x 256 were considered to demonstrate the
performance of the both SENSE and GRAPPA
reconstruction. Reference images were reconstructed
using the sum of square (SOS) algorithm.
For SENSE reconstruction, the number of k-space lines
were skipped to produce aliased images, depending upon
the acceleration factor, for an acceleration factor of two,
one out of every two phase encode steps were removed.
The inverse Fourier transform of the sub-sampled k-space
gave us aliased images.
GRAPPA, recovers the missing k-space data. However, it
reproduces a set of k-space data for each channel. It uses
Figure 3. Brain image reconstruction, using eight-element
phased array coil. Signal measurements were taken from the
boxed regions labeled ROS, and noise estimates were taken
from the boxed regions labeled with an RON, used for
measuring signal-to-noise ratios.
4.2. NMSE:
Similarly, the results of the NMSE simulations are shown
in Figure 5. It indicates the relationship between
acceleration factor and normalized mean squared error.
The reconstruction was performed for various
acceleration factors ranging between 2 and 8. There is a
sharp increase in the error after acceleration factor of 4.
After inverse DFT, the analytical data are transformed in
space domain; they are shown together with the difference
to the direct pixelization of the phantoms.
Space domain data and differences are presented in Figure
6. They exhibit ringing artifacts due to insufficient highfrequency sampling.
radiofrequency coil
1997;38:591–603.
SNR fct R
35
4 block
2 block
SENSE
30
SNR
Magn
Reson
Med
2. Pruessmann KP, Weiger M, Scheidegger MB, Boesiger
P. 1999. SENSE: sensitivity encoding for fast MRI. Magn
Reson Med 42(5):952–962.
25
20
3. Griswold MA, Jakob PM, Heidemann RM, Nittka M,
Jellus V, Wang J, and al. 2002. Generalized
autocalibrating partially parallel acquisitions (GRAPPA).
Magn Reson Med 47(6):1202–1210.
15
10
5
arrays.
2
3
4
5
acceleration factor
6
7
8
4. Griswold MA, Jakob PM, Nittka M, Goldfarb JW,
Haase A. Partially parallel imaging with localized
sensitivities (PILS). Magn Reson Med 2000;44:602–609.
Figure 4. SNR for different acceleration factors
5. Heidemann R, and al. VD-AUTO-SMASH IMAGING.
Magn Reson Med. 2001; 45:1066-1074.
NMSE fct R
0.7
4 block
2 block
SENSE
0.6
6. Blaimer M, and al. SMASH, SENSE, PILS, GRAPPA
How to Choose the Optimal Method. Top Magn Reson
Imaging. 2004; 15:223-236.
NMSE
0.5
0.4
7. Ji JX, Son JB, Rane SD. 2007. PULSAR: A Matlab
toolbox for parallel magnetic resonance imaging using
array coils and multiple channel receivers. Concepts
Magn Reson B: Magn Reson Eng 31B:24–36.
0.3
0.2
0.1
0
2
3
4
5
acceleration factor
6
7
8
Figure 5. NMSE for different acceleration factors.
The images in Figure 6, demonstrate shepp logan and
brain images implementations of the GRAPPA and
SENSE imaging techniques using the eight element
phased array coil. Figure 6a shows the GRAPPA
reconstruction image obtained using four blocks, Figure
6b shows the equivalent image obtained using two blocks
and Figure 6c shows the equivalent image with the
SENSE technique. Nearly similar results are obtained in
GRAPPA method while the SENSE method contains
bigger error compared to the both GRAPPA 2 blocks and
GRAPPA 4 blocks reconstruction.
CONCLUSION :
We based on the Blaimer article, which find no
differences between the both SENSE and GRAPPA
images. We decide evaluates the SNR and NMSE
performances for comparison between these two parallel
imaging methods. We have found that SENSE method
acquire low SNR and high NMSE compared to GRAPPA
method. Also, we have proved that images with more
blocks showed essentially the same image quality. So,
GRAPPA imaging includes a potential for increased SNR
and decreased blurring artifacts, especially for low
acceleration factors.
REFERENCES :
1. Sodickson DK, Manning WJ. Simultaneous acquisition
of spatial harmonics (SMASH): fast imaging with
8. LIU Xiao-fang, YE Xiu-zi, ZHANG San-yuan, LIU
Feng. Regularized Least Squares Estimating Sensitivity
for Self-calibrating Parallel Imaging. JOURNAL OF
COMPUTERS. 2011; VOL. 6, NO. 5
image reference
image reference
image reconstruction
image reconstruction
difference images
difference images
a)
difference images
image reference
image reconstruction
image reference
image reconstruction
image reference
image reconstruction
difference images
b)
difference images
image reference
image reconstruction
difference images
c)
Figure 6. Comparison of GRAPPA and SENSE reconstruction using eight-channel array. Reconstructed image of Brain and Shepp
Logan phantom; and error maps. Left: phantom reference image. Middle: image reconstruction with acceleration factor of two.
Right: error maps between the both reference and reconstructed images. a) GRAPPA reconstruction with eight ACS lines and four
blocks. b) GRAPPA reconstruction with eight ACS lines and two blocks. c) SENSE reconstruction.