Quantitative single point imaging with compressed sensing
Abstract
A novel approach with respect to single point imaging (SPI), compressed sensing, is presented here that is shown to significantly reduce the loss of accuracy of reconstructed images from undersampled acquisition data. SPI complements compressed sensing extremely well as it allows unconstrained selection of sampling trajectories. Dynamic processes featuring short T2* NMR signal can thus be more rapidly imaged, in our case the absorption of moisture by a cerealbased wafer material, with minimal loss of image quantification. The absolute moisture content distribution is recovered via a series of images acquired with variable phase encoding times allowing extrapolation to time zero for each image pixel and the effective removal of T2* contrast.
Quantitative single point imaging with compressed sensing
P. Parasoglou
a
, D. Malioutov
b
, A.J. Sederman
a
, J. Rasburn
c
, H. Powell
c
, L.F. Gladden
a
, A. Blake
b
,
M.L. Johns
a,
*
a
Department of Chemical Engineering and Biotechnology, University of Cambridge, New Museums Site Pembroke Street, Cambridge CB2 3RA, UK
b
Microsoft Research Ltd., 7 J J Thompson Ave., Cambridge CB3 0FB, UK
c
Nestec York Ltd., Nestlé Product Technology Centre, Haxby Road, P.O. Box 204, York YO91 1XY, UK
article info
Article history:
Received 30 March 2009
Revised 23 July 2009
Available online 14 August 2009
Keywords:
Compressed sensing
SPI
Undersampling
kspace
abstract
A novel approach with respect to single point imaging (SPI), compressed sensing, is presented here that is
shown to signiﬁcantly reduce the loss of accuracy of reconstructed images from undersampled acquisi
tion data. SPI complements compressed sensing extremely well as it allows unconstrained selection of
sampling trajectories. Dynamic processes featuring short T
2
NMR signal can thus be more rapidly imaged,
in our case the absorption of moisture by a cerealbased wafer material, with minimal loss of image quan
tiﬁcation. The absolute moisture content distribution is recovered via a series of images acquired with
variable phase encoding times allowing extrapolation to time zero for each image pixel and the effective
removal of T
2
contrast.
Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction
Single point imaging was originally proposed as a ‘solid state’
imaging modality [1,2]. As a pure phaseencoding imaging tech
nique, SPI is relatively immune to artefacts due to chemical shift,
magnetic susceptibility variations and inhomogeneities in the B
0
ﬁeld [3,4] and has thus been extensively used to image samples
with very short transverse relaxation time (T
2
), of the order of
100
l
s [5]. Typically these materials also have short spin–spin
relaxation times (T
2
) and thus frequencyencoding imaging tech
niques are unsuitable due to the minimum limits on the time for
echo formation.
The difﬁculty of slice selection in SPI has conﬁned its use pre
dominantly to one or three dimensional imaging. Since only one
point in kspace (the acquired data which is subsequently Fourier
transformed to produce an image) is typically sampled after each
r.f. signal excitation, total acquisition times are long and in some
cases prohibitive for the time resolution needed for certain appli
cations. Several attempts have been made in the past in order to
speed up acquisition, the most prominent being SPRITE, as devel
oped by Balcom and coworkers [6]; this enables signiﬁcantly fas
ter acquisition as the phase encoding magnetic ﬁeld gradients are
ramped up in discrete steps as opposed to being switched on and
off between each kspace point acquisition.
Conventionally such SPI approaches have been performed by
sampling kspace in a linear raster starting at the extremities of
kspace. Centric scans that result in improved sensitivity, as well
as removal of T
1
contrast, have however almost completely re
placed this original sampling trajectory. Since a better signal to
noise ratio (SNR) is achieved per scan, fewer repeat scans are
needed in total to achieve a required SNR threshold. Centric scans
with different geometrical functions such as spirals and square spi
rals [7,8] have appeared in the literature. As we have shown in a
previous study [9], prior knowledge of the sample shape can en
able the design of near optimum sampling trajectories, where the
SNR is maximised by sampling kspace points with the highest ex
pected value convolved with the largest excited magnetization.
Note that SPI, as a pure phaseencoding imaging pulse sequence,
has the desirable feature of placing no constraints on the sampling
trajectory.
Since imaging speed is important in MRI applications, many re
searches have focused on acquiring only a portion of kspace with
out signiﬁcantly degrading the image quality. Traditionally in
centric scans which employ spiral acquisitions, the extremities of
kspace are undersampled and the values are replaced by zeros
before reconstructing the image with the use of the Fourier trans
form. When all kspace points as dictated by the desired ﬁeld of
view (FOV) and spatial resolution are not sampled, the Nyquist
sampling theorem is theoretically violated. In this case the recon
structed images are expected to show artefacts and diminished
accuracy if reconstructed by a linear transform (such as the Fourier
transform). Compressed sensing (CS) has shown the ability to
reconstruct images which are signiﬁcantly undersampled
10907807/$  see front matter Ó 2009 Elsevier Inc. All rights reserved.
doi:10.1016/j.jmr.2009.08.003
* Corresponding author. Tel.: +44 1223 334767; fax: +44 1223 334796.
Email address: mlj21@cheng.cam.ac.uk (M.L. Johns).
Journal of Magnetic Resonance 201 (2009) 72–80
Contents lists available at ScienceDirect
Journal of Magnetic Resonance
journal homepage: www.elsevier.com/locate/jmr
Quantitative single point imaging with compressed sensing
P. Parasoglou
a
, D. Malioutov
b
, A.J. Sederman
a
, J. Rasburn
c
, H. Powell
c
, L.F. Gladden
a
, A. Blake
b
,
M.L. Johns
a,
*
a
Department of Chemical Engineering and Biotechnology, University of Cambridge, New Museums Site Pembroke Street, Cambridge CB2 3RA, UK
b
Microsoft Research Ltd., 7 J J Thompson Ave., Cambridge CB3 0FB, UK
c
Nestec York Ltd., Nestlé Product Technology Centre, Haxby Road, P.O. Box 204, York YO91 1XY, UK
article info
Article history:
Received 30 March 2009
Revised 23 July 2009
Available online 14 August 2009
Keywords:
Compressed sensing
SPI
Undersampling
kspace
abstract
A novel approach with respect to single point imaging (SPI), compressed sensing, is presented here that is
shown to signiﬁcantly reduce the loss of accuracy of reconstructed images from undersampled acquisi
tion data. SPI complements compressed sensing extremely well as it allows unconstrained selection of
sampling trajectories. Dynamic processes featuring short T
2
NMR signal can thus be more rapidly imaged,
in our case the absorption of moisture by a cerealbased wafer material, with minimal loss of image quan
tiﬁcation. The absolute moisture content distribution is recovered via a series of images acquired with
variable phase encoding times allowing extrapolation to time zero for each image pixel and the effective
removal of T
2
contrast.
Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction
Single point imaging was originally proposed as a ‘solid state’
imaging modality [1,2]. As a pure phaseencoding imaging tech
nique, SPI is relatively immune to artefacts due to chemical shift,
magnetic susceptibility variations and inhomogeneities in the B
0
ﬁeld [3,4] and has thus been extensively used to image samples
with very short transverse relaxation time (T
2
), of the order of
100
l
s [5]. Typically these materials also have short spin–spin
relaxation times (T
2
) and thus frequencyencoding imaging tech
niques are unsuitable due to the minimum limits on the time for
echo formation.
The difﬁculty of slice selection in SPI has conﬁned its use pre
dominantly to one or three dimensional imaging. Since only one
point in kspace (the acquired data which is subsequently Fourier
transformed to produce an image) is typically sampled after each
r.f. signal excitation, total acquisition times are long and in some
cases prohibitive for the time resolution needed for certain appli
cations. Several attempts have been made in the past in order to
speed up acquisition, the most prominent being SPRITE, as devel
oped by Balcom and coworkers [6]; this enables signiﬁcantly fas
ter acquisition as the phase encoding magnetic ﬁeld gradients are
ramped up in discrete steps as opposed to being switched on and
off between each kspace point acquisition.
Conventionally such SPI approaches have been performed by
sampling kspace in a linear raster starting at the extremities of
kspace. Centric scans that result in improved sensitivity, as well
as removal of T
1
contrast, have however almost completely re
placed this original sampling trajectory. Since a better signal to
noise ratio (SNR) is achieved per scan, fewer repeat scans are
needed in total to achieve a required SNR threshold. Centric scans
with different geometrical functions such as spirals and square spi
rals [7,8] have appeared in the literature. As we have shown in a
previous study [9], prior knowledge of the sample shape can en
able the design of near optimum sampling trajectories, where the
SNR is maximised by sampling kspace points with the highest ex
pected value convolved with the largest excited magnetization.
Note that SPI, as a pure phaseencoding imaging pulse sequence,
has the desirable feature of placing no constraints on the sampling
trajectory.
Since imaging speed is important in MRI applications, many re
searches have focused on acquiring only a portion of kspace with
out signiﬁcantly degrading the image quality. Traditionally in
centric scans which employ spiral acquisitions, the extremities of
kspace are undersampled and the values are replaced by zeros
before reconstructing the image with the use of the Fourier trans
form. When all kspace points as dictated by the desired ﬁeld of
view (FOV) and spatial resolution are not sampled, the Nyquist
sampling theorem is theoretically violated. In this case the recon
structed images are expected to show artefacts and diminished
accuracy if reconstructed by a linear transform (such as the Fourier
transform). Compressed sensing (CS) has shown the ability to
reconstruct images which are signiﬁcantly undersampled
10907807/$  see front matter Ó 2009 Elsevier Inc. All rights reserved.
doi:10.1016/j.jmr.2009.08.003
* Corresponding author. Tel.: +44 1223 334767; fax: +44 1223 334796.
Email address: mlj21@cheng.cam.ac.uk (M.L. Johns).
Journal of Magnetic Resonance 201 (2009) 72–80
Contents lists available at ScienceDirect
Journal of Magnetic Resonance
journal homepage: www.elsevier.com/locate/jmr
Quantitative single point imaging with compressed sensing
P. Parasoglou
a
, D. Malioutov
b
, A.J. Sederman
a
, J. Rasburn
c
, H. Powell
c
, L.F. Gladden
a
, A. Blake
b
,
M.L. Johns
a,
*
a
Department of Chemical Engineering and Biotechnology, University of Cambridge, New Museums Site Pembroke Street, Cambridge CB2 3RA, UK
b
Microsoft Research Ltd., 7 J J Thompson Ave., Cambridge CB3 0FB, UK
c
Nestec York Ltd., Nestlé Product Technology Centre, Haxby Road, P.O. Box 204, York YO91 1XY, UK
article info
Article history:
Received 30 March 2009
Revised 23 July 2009
Available online 14 August 2009
Keywords:
Compressed sensing
SPI
Undersampling
kspace
abstract
A novel approach with respect to single point imaging (SPI), compressed sensing, is presented here that is
shown to signiﬁcantly reduce the loss of accuracy of reconstructed images from undersampled acquisi
tion data. SPI complements compressed sensing extremely well as it allows unconstrained selection of
sampling trajectories. Dynamic processes featuring short T
2
NMR signal can thus be more rapidly imaged,
in our case the absorption of moisture by a cerealbased wafer material, with minimal loss of image quan
tiﬁcation. The absolute moisture content distribution is recovered via a series of images acquired with
variable phase encoding times allowing extrapolation to time zero for each image pixel and the effective
removal of T
2
contrast.
Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction
Single point imaging was originally proposed as a ‘solid state’
imaging modality [1,2]. As a pure phaseencoding imaging tech
nique, SPI is relatively immune to artefacts due to chemical shift,
magnetic susceptibility variations and inhomogeneities in the B
0
ﬁeld [3,4] and has thus been extensively used to image samples
with very short transverse relaxation time (T
2
), of the order of
100
l
s [5]. Typically these materials also have short spin–spin
relaxation times (T
2
) and thus frequencyencoding imaging tech
niques are unsuitable due to the minimum limits on the time for
echo formation.
The difﬁculty of slice selection in SPI has conﬁned its use pre
dominantly to one or three dimensional imaging. Since only one
point in kspace (the acquired data which is subsequently Fourier
transformed to produce an image) is typically sampled after each
r.f. signal excitation, total acquisition times are long and in some
cases prohibitive for the time resolution needed for certain appli
cations. Several attempts have been made in the past in order to
speed up acquisition, the most prominent being SPRITE, as devel
oped by Balcom and coworkers [6]; this enables signiﬁcantly fas
ter acquisition as the phase encoding magnetic ﬁeld gradients are
ramped up in discrete steps as opposed to being switched on and
off between each kspace point acquisition.
Conventionally such SPI approaches have been performed by
sampling kspace in a linear raster starting at the extremities of
kspace. Centric scans that result in improved sensitivity, as well
as removal of T
1
contrast, have however almost completely re
placed this original sampling trajectory. Since a better signal to
noise ratio (SNR) is achieved per scan, fewer repeat scans are
needed in total to achieve a required SNR threshold. Centric scans
with different geometrical functions such as spirals and square spi
rals [7,8] have appeared in the literature. As we have shown in a
previous study [9], prior knowledge of the sample shape can en
able the design of near optimum sampling trajectories, where the
SNR is maximised by sampling kspace points with the highest ex
pected value convolved with the largest excited magnetization.
Note that SPI, as a pure phaseencoding imaging pulse sequence,
has the desirable feature of placing no constraints on the sampling
trajectory.
Since imaging speed is important in MRI applications, many re
searches have focused on acquiring only a portion of kspace with
out signiﬁcantly degrading the image quality. Traditionally in
centric scans which employ spiral acquisitions, the extremities of
kspace are undersampled and the values are replaced by zeros
before reconstructing the image with the use of the Fourier trans
form. When all kspace points as dictated by the desired ﬁeld of
view (FOV) and spatial resolution are not sampled, the Nyquist
sampling theorem is theoretically violated. In this case the recon
structed images are expected to show artefacts and diminished
accuracy if reconstructed by a linear transform (such as the Fourier
transform). Compressed sensing (CS) has shown the ability to
reconstruct images which are signiﬁcantly undersampled
10907807/$  see front matter Ó 2009 Elsevier Inc. All rights reserved.
doi:10.1016/j.jmr.2009.08.003
* Corresponding author. Tel.: +44 1223 334767; fax: +44 1223 334796.
Email address: mlj21@cheng.cam.ac.uk (M.L. Johns).
Journal of Magnetic Resonance 201 (2009) 72–80
Contents lists available at ScienceDirect
Journal of Magnetic Resonance
journal homepage: www.elsevier.com/locate/jmr
[10,11] with minimal loss of quantiﬁcation. This image reconstruc
tion method takes advantage of the fact that certain transforma
tions of MR images such as wavelet or spatial differences are
typically very sparse. Sparsity in this transform domain allows
good reconstruction from an undersampled set of measurements
in kspace. Lustig et al. [11], studied the exploitation of sparsity
of MR (conventional frequencyphase encoded) images in a trans
form domain and proposed a nonlinear iterative reconstruction of
the undersampled images based on l
1
optimization which results
in minimal loss of image quality/quantiﬁcation.
In the work presented here, moisture absorption from humidi
ﬁed air by a thin cerealbased wafer material is considered. A 2D
and a 3D binary MR image of the wafer sample is shown in
Fig. 1(a) and (b), respectively. Such moisture absorption (and sub
sequent migration within the wafer foam) occurs during an indus
trial conditioning process. The conditioning process is required in
order to optimise the expansion properties of the wafer for subse
quent fabrication steps during confectionery production. Hence
this process has a signiﬁcant contribution to the ﬁnal quality of
the product. Temporally resolved MRI of such a system is very
challenging. The wafer is highly porous (typically 80–90 vol%)
while the absorbed moisture ranges from 1 to 20 wt% of the solid
content. This low moisture content occurs as ‘bound’ water with
restricted mobility; the porous nature of the wafer also results in
signiﬁcant magnetic susceptibility variations. These collectively re
sult in a short T
2
for the water, typically of the order of 100
l
s,
whilst T
1
remains relatively long (1 s). Imaging the system with
conventional MRI frequencyencode methods is not possible,
hence the use of SPI.
To follow the dynamic moisture absorption process by the wa
fer, the following approach was developed and implemented: SPI
was implemented in 3D with an undersampled acquisition of
33% of kspace. The sampling trajectory was optimised based on
the expected amplitude of the points in kspace as outlined in
our previous publication [9]. Reconstruction of the images from
this acquired kspace data is achieved via a nonlinear iteration
(based on optimizing a convex function involving l
1
norms) as pro
posed by Lustig and coworkers [11]. Each image was also con
structed from four subimages acquired with increasing phase
encoding time, thus allowing quantiﬁcation of the water content
in the image via appropriate signal relaxation analysis. This imag
ing protocol is repeated a number of times during the moisture
absorption process.
2. Method development
2.1. Compressed sensing in MRI
Acquiring all the kspace points or Fourier coefﬁcients of a MR
image is time consuming. Undersampled kspace acquisition is
the method of choice when neither the spatial resolution nor the
number of averages can be compromised for a desired SNR. From
the Nyquist sampling theorem, we expect that when kspace is
strongly undersampled, linear recovery methods will lead to lossy
recovery and exhibit artefacts. It is important to note, however,
that not all of the Fourier coefﬁcients are equally important for
the reconstruction of the image.
Recently compressed sensing [11,12] has attracted interest in
terms of MR imaging. This sampling method takes advantage of
the sparsity of MR images and allows good reconstructions from
signiﬁcantly undersampled kspace. A signal is described as
sparse if it contains only a small number of nonzero values, or if
these values decay very fast. We consider a transform domain
sparsity, where the signal is not sparse but it can be manipulated
to be in some transform domain. In the current context we make
use of the acquired signal in the kspace domain, this is trans
formed into the image domain and we exploit sparsity in a trans
form domain. Note that the image domain can be used as the
transform domain if it presents sufﬁcient sparsity. Lustig et al.
[11,12], studied the exploitation of sparsity of MR images; for
example they considered angiograms which have sparsity in the
image domain. Many other types of ‘natural’ images (e.g., natural
scenes, astronomical images) can exhibit sparsity in terms of their
Fourier or wavelet coefﬁcients, or in terms of discrete gradients;
hence in their transform domain. The idea of taking advantage of
sparsity in order to undersample a signal is motivated by the suc
cess of image compression techniques. ‘Natural’ images and MR
images can thus be compressed signiﬁcantly with minimal loss
of information. Several transforms can be used in order to trans
form an image to a sparse domain – e.g., the discrete cosine trans
form (DCT) as well as the wavelet transform are the basis for
compression tools JPEG and JPEG2000 [13].
Since MR images are compressible, compressed sensing the
ory [10,14] suggests that taking fewer acquisition samples, com
bined with prior knowledge that the image is sparse in the
transform domain, may be sufﬁcient for an accurate image
reconstruction. This is achieved using a nonlinear reconstruction
method based on optimizing a convex function involving l
1

norms, related to the basis pursuit method [10]. In their work
Lustig et al. [11] developed a framework for using CS in MRI,
which has proved to be able to accurately reconstruct MR
images from a small subset of kspace rather than the entire
kspace grid.
2.2. Quantitative single point imaging with optimised sampling
trajectory
A comprehensive study of a near optimum sampling scheme
has been presented in previous work [9], where the sampling tra
jectory is dictated and ranked by the expected magnitude of the
sampled kspace points which we predict based on available prior
knowledge of the sample shape. kSpace points are sorted from
maximum to minimum expected modulus signal intensity and di
vided into interleafs consistent with the centric scan sampling
practise [8,15]. The principle used here is that kspace points with
the highest expected signal intensity are acquired with the highest
value of magnetization. A time interval of 5 T
1
is allowed be
tween each interleaf to allow the magnetization to fully recover.
As has been demonstrated in our previous study [9] a signiﬁcant
a
b
30
mm
y
x
z
17 mm
3.2 mm
y
x
Fig. 1. (a) 2D and (b) 3D binary MR images of the wafer sheet with dimensions
shown. The coordinate system shown is used for all image acquisitions.
P. Parasoglou et al. / Journal of Magnetic Resonance 201 (2009) 72–80
73
[10,11] with minimal loss of quantiﬁcation. This image reconstruc
tion method takes advantage of the fact that certain transforma
tions of MR images such as wavelet or spatial differences are
typically very sparse. Sparsity in this transform domain allows
good reconstruction from an undersampled set of measurements
in kspace. Lustig et al. [11], studied the exploitation of sparsity
of MR (conventional frequencyphase encoded) images in a trans
form domain and proposed a nonlinear iterative reconstruction of
the undersampled images based on l
1
optimization which results
in minimal loss of image quality/quantiﬁcation.
In the work presented here, moisture absorption from humidi
ﬁed air by a thin cerealbased wafer material is considered. A 2D
and a 3D binary MR image of the wafer sample is shown in
Fig. 1(a) and (b), respectively. Such moisture absorption (and sub
sequent migration within the wafer foam) occurs during an indus
trial conditioning process. The conditioning process is required in
order to optimise the expansion properties of the wafer for subse
quent fabrication steps during confectionery production. Hence
this process has a signiﬁcant contribution to the ﬁnal quality of
the product. Temporally resolved MRI of such a system is very
challenging. The wafer is highly porous (typically 80–90 vol%)
while the absorbed moisture ranges from 1 to 20 wt% of the solid
content. This low moisture content occurs as ‘bound’ water with
restricted mobility; the porous nature of the wafer also results in
signiﬁcant magnetic susceptibility variations. These collectively re
sult in a short T
2
for the water, typically of the order of 100
l
s,
whilst T
1
remains relatively long (1 s). Imaging the system with
conventional MRI frequencyencode methods is not possible,
hence the use of SPI.
To follow the dynamic moisture absorption process by the wa
fer, the following approach was developed and implemented: SPI
was implemented in 3D with an undersampled acquisition of
33% of kspace. The sampling trajectory was optimised based on
the expected amplitude of the points in kspace as outlined in
our previous publication [9]. Reconstruction of the images from
this acquired kspace data is achieved via a nonlinear iteration
(based on optimizing a convex function involving l
1
norms) as pro
posed by Lustig and coworkers [11]. Each image was also con
structed from four subimages acquired with increasing phase
encoding time, thus allowing quantiﬁcation of the water content
in the image via appropriate signal relaxation analysis. This imag
ing protocol is repeated a number of times during the moisture
absorption process.
2. Method development
2.1. Compressed sensing in MRI
Acquiring all the kspace points or Fourier coefﬁcients of a MR
image is time consuming. Undersampled kspace acquisition is
the method of choice when neither the spatial resolution nor the
number of averages can be compromised for a desired SNR. From
the Nyquist sampling theorem, we expect that when kspace is
strongly undersampled, linear recovery methods will lead to lossy
recovery and exhibit artefacts. It is important to note, however,
that not all of the Fourier coefﬁcients are equally important for
the reconstruction of the image.
Recently compressed sensing [11,12] has attracted interest in
terms of MR imaging. This sampling method takes advantage of
the sparsity of MR images and allows good reconstructions from
signiﬁcantly undersampled kspace. A signal is described as
sparse if it contains only a small number of nonzero values, or if
these values decay very fast. We consider a transform domain
sparsity, where the signal is not sparse but it can be manipulated
to be in some transform domain. In the current context we make
use of the acquired signal in the kspace domain, this is trans
formed into the image domain and we exploit sparsity in a trans
form domain. Note that the image domain can be used as the
transform domain if it presents sufﬁcient sparsity. Lustig et al.
[11,12], studied the exploitation of sparsity of MR images; for
example they considered angiograms which have sparsity in the
image domain. Many other types of ‘natural’ images (e.g., natural
scenes, astronomical images) can exhibit sparsity in terms of their
Fourier or wavelet coefﬁcients, or in terms of discrete gradients;
hence in their transform domain. The idea of taking advantage of
sparsity in order to undersample a signal is motivated by the suc
cess of image compression techniques. ‘Natural’ images and MR
images can thus be compressed signiﬁcantly with minimal loss
of information. Several transforms can be used in order to trans
form an image to a sparse domain – e.g., the discrete cosine trans
form (DCT) as well as the wavelet transform are the basis for
compression tools JPEG and JPEG2000 [13].
Since MR images are compressible, compressed sensing the
ory [10,14] suggests that taking fewer acquisition samples, com
bined with prior knowledge that the image is sparse in the
transform domain, may be sufﬁcient for an accurate image
reconstruction. This is achieved using a nonlinear reconstruction
method based on optimizing a convex function involving l
1

norms, related to the basis pursuit method [10]. In their work
Lustig et al. [11] developed a framework for using CS in MRI,
which has proved to be able to accurately reconstruct MR
images from a small subset of kspace rather than the entire
kspace grid.
2.2. Quantitative single point imaging with optimised sampling
trajectory
A comprehensive study of a near optimum sampling scheme
has been presented in previous work [9], where the sampling tra
jectory is dictated and ranked by the expected magnitude of the
sampled kspace points which we predict based on available prior
knowledge of the sample shape. kSpace points are sorted from
maximum to minimum expected modulus signal intensity and di
vided into interleafs consistent with the centric scan sampling
practise [8,15]. The principle used here is that kspace points with
the highest expected signal intensity are acquired with the highest
value of magnetization. A time interval of 5 T
1
is allowed be
tween each interleaf to allow the magnetization to fully recover.
As has been demonstrated in our previous study [9] a signiﬁcant
a
b
30
mm
y
x
z
17 mm
3.2 mm
y
x
Fig. 1. (a) 2D and (b) 3D binary MR images of the wafer sheet with dimensions
shown. The coordinate system shown is used for all image acquisitions.
P. Parasoglou et al. / Journal of Magnetic Resonance 201 (2009) 72–80
73
[10,11] with minimal loss of quantiﬁcation. This image reconstruc
tion method takes advantage of the fact that certain transforma
tions of MR images such as wavelet or spatial differences are
typically very sparse. Sparsity in this transform domain allows
good reconstruction from an undersampled set of measurements
in kspace. Lustig et al. [11], studied the exploitation of sparsity
of MR (conventional frequencyphase encoded) images in a trans
form domain and proposed a nonlinear iterative reconstruction of
the undersampled images based on l
1
optimization which results
in minimal loss of image quality/quantiﬁcation.
In the work presented here, moisture absorption from humidi
ﬁed air by a thin cerealbased wafer material is considered. A 2D
and a 3D binary MR image of the wafer sample is shown in
Fig. 1(a) and (b), respectively. Such moisture absorption (and sub
sequent migration within the wafer foam) occurs during an indus
trial conditioning process. The conditioning process is required in
order to optimise the expansion properties of the wafer for subse
quent fabrication steps during confectionery production. Hence
this process has a signiﬁcant contribution to the ﬁnal quality of
the product. Temporally resolved MRI of such a system is very
challenging. The wafer is highly porous (typically 80–90 vol%)
while the absorbed moisture ranges from 1 to 20 wt% of the solid
content. This low moisture content occurs as ‘bound’ water with
restricted mobility; the porous nature of the wafer also results in
signiﬁcant magnetic susceptibility variations. These collectively re
sult in a short T
2
for the water, typically of the order of 100
l
s,
whilst T
1
remains relatively long (1 s). Imaging the system with
conventional MRI frequencyencode methods is not possible,
hence the use of SPI.
To follow the dynamic moisture absorption process by the wa
fer, the following approach was developed and implemented: SPI
was implemented in 3D with an undersampled acquisition of
33% of kspace. The sampling trajectory was optimised based on
the expected amplitude of the points in kspace as outlined in
our previous publication [9]. Reconstruction of the images from
this acquired kspace data is achieved via a nonlinear iteration
(based on optimizing a convex function involving l
1
norms) as pro
posed by Lustig and coworkers [11]. Each image was also con
structed from four subimages acquired with increasing phase
encoding time, thus allowing quantiﬁcation of the water content
in the image via appropriate signal relaxation analysis. This imag
ing protocol is repeated a number of times during the moisture
absorption process.
2. Method development
2.1. Compressed sensing in MRI
Acquiring all the kspace points or Fourier coefﬁcients of a MR
image is time consuming. Undersampled kspace acquisition is
the method of choice when neither the spatial resolution nor the
number of averages can be compromised for a desired SNR. From
the Nyquist sampling theorem, we expect that when kspace is
strongly undersampled, linear recovery methods will lead to lossy
recovery and exhibit artefacts. It is important to note, however,
that not all of the Fourier coefﬁcients are equally important for
the reconstruction of the image.
Recently compressed sensing [11,12] has attracted interest in
terms of MR imaging. This sampling method takes advantage of
the sparsity of MR images and allows good reconstructions from
signiﬁcantly undersampled kspace. A signal is described as
sparse if it contains only a small number of nonzero values, or if
these values decay very fast. We consider a transform domain
sparsity, where the signal is not sparse but it can be manipulated
to be in some transform domain. In the current context we make
use of the acquired signal in the kspace domain, this is trans
formed into the image domain and we exploit sparsity in a trans
form domain. Note that the image domain can be used as the
transform domain if it presents sufﬁcient sparsity. Lustig et al.
[11,12], studied the exploitation of sparsity of MR images; for
example they considered angiograms which have sparsity in the
image domain. Many other types of ‘natural’ images (e.g., natural
scenes, astronomical images) can exhibit sparsity in terms of their
Fourier or wavelet coefﬁcients, or in terms of discrete gradients;
hence in their transform domain. The idea of taking advantage of
sparsity in order to undersample a signal is motivated by the suc
cess of image compression techniques. ‘Natural’ images and MR
images can thus be compressed signiﬁcantly with minimal loss
of information. Several transforms can be used in order to trans
form an image to a sparse domain – e.g., the discrete cosine trans
form (DCT) as well as the wavelet transform are the basis for
compression tools JPEG and JPEG2000 [13].
Since MR images are compressible, compressed sensing the
ory [10,14] suggests that taking fewer acquisition samples, com
bined with prior knowledge that the image is sparse in the
transform domain, may be sufﬁcient for an accurate image
reconstruction. This is achieved using a nonlinear reconstruction
method based on optimizing a convex function involving l
1

norms, related to the basis pursuit method [10]. In their work
Lustig et al. [11] developed a framework for using CS in MRI,
which has proved to be able to accurately reconstruct MR
images from a small subset of kspace rather than the entire
kspace grid.
2.2. Quantitative single point imaging with optimised sampling
trajectory
A comprehensive study of a near optimum sampling scheme
has been presented in previous work [9], where the sampling tra
jectory is dictated and ranked by the expected magnitude of the
sampled kspace points which we predict based on available prior
knowledge of the sample shape. kSpace points are sorted from
maximum to minimum expected modulus signal intensity and di
vided into interleafs consistent with the centric scan sampling
practise [8,15]. The principle used here is that kspace points with
the highest expected signal intensity are acquired with the highest
value of magnetization. A time interval of 5 T
1
is allowed be
tween each interleaf to allow the magnetization to fully recover.
As has been demonstrated in our previous study [9] a signiﬁcant
a
b
30
mm
y
x
z
17 mm
3.2 mm
y
x
Fig. 1. (a) 2D and (b) 3D binary MR images of the wafer sheet with dimensions
shown. The coordinate system shown is used for all image acquisitions.
P. Parasoglou et al. / Journal of Magnetic Resonance 201 (2009) 72–80
73
[10,11] with minimal loss of quantiﬁcation. This image reconstruc
tion method takes advantage of the fact that certain transforma
tions of MR images such as wavelet or spatial differences are
typically very sparse. Sparsity in this transform domain allows
good reconstruction from an undersampled set of measurements
in kspace. Lustig et al. [11], studied the exploitation of sparsity
of MR (conventional frequencyphase encoded) images in a trans
form domain and proposed a nonlinear iterative reconstruction of
the undersampled images based on l
1
optimization which results
in minimal loss of image quality/quantiﬁcation.
In the work presented here, moisture absorption from humidi
ﬁed air by a thin cerealbased wafer material is considered. A 2D
and a 3D binary MR image of the wafer sample is shown in
Fig. 1(a) and (b), respectively. Such moisture absorption (and sub
sequent migration within the wafer foam) occurs during an indus
trial conditioning process. The conditioning process is required in
order to optimise the expansion properties of the wafer for subse
quent fabrication steps during confectionery production. Hence
this process has a signiﬁcant contribution to the ﬁnal quality of
the product. Temporally resolved MRI of such a system is very
challenging. The wafer is highly porous (typically 80–90 vol%)
while the absorbed moisture ranges from 1 to 20 wt% of the solid
content. This low moisture content occurs as ‘bound’ water with
restricted mobility; the porous nature of the wafer also results in
signiﬁcant magnetic susceptibility variations. These collectively re
sult in a short T
2
for the water, typically of the order of 100
l
s,
whilst T
1
remains relatively long (1 s). Imaging the system with
conventional MRI frequencyencode methods is not possible,
hence the use of SPI.
To follow the dynamic moisture absorption process by the wa
fer, the following approach was developed and implemented: SPI
was implemented in 3D with an undersampled acquisition of
33% of kspace. The sampling trajectory was optimised based on
the expected amplitude of the points in kspace as outlined in
our previous publication [9]. Reconstruction of the images from
this acquired kspace data is achieved via a nonlinear iteration
(based on optimizing a convex function involving l
1
norms) as pro
posed by Lustig and coworkers [11]. Each image was also con
structed from four subimages acquired with increasing phase
encoding time, thus allowing quantiﬁcation of the water content
in the image via appropriate signal relaxation analysis. This imag
ing protocol is repeated a number of times during the moisture
absorption process.
2. Method development
2.1. Compressed sensing in MRI
Acquiring all the kspace points or Fourier coefﬁcients of a MR
image is time consuming. Undersampled kspace acquisition is
the method of choice when neither the spatial resolution nor the
number of averages can be compromised for a desired SNR. From
the Nyquist sampling theorem, we expect that when kspace is
strongly undersampled, linear recovery methods will lead to lossy
recovery and exhibit artefacts. It is important to note, however,
that not all of the Fourier coefﬁcients are equally important for
the reconstruction of the image.
Recently compressed sensing [11,12] has attracted interest in
terms of MR imaging. This sampling method takes advantage of
the sparsity of MR images and allows good reconstructions from
signiﬁcantly undersampled kspace. A signal is described as
sparse if it contains only a small number of nonzero values, or if
these values decay very fast. We consider a transform domain
sparsity, where the signal is not sparse but it can be manipulated
to be in some transform domain. In the current context we make
use of the acquired signal in the kspace domain, this is trans
formed into the image domain and we exploit sparsity in a trans
form domain. Note that the image domain can be used as the
transform domain if it presents sufﬁcient sparsity. Lustig et al.
[11,12], studied the exploitation of sparsity of MR images; for
example they considered angiograms which have sparsity in the
image domain. Many other types of ‘natural’ images (e.g., natural
scenes, astronomical images) can exhibit sparsity in terms of their
Fourier or wavelet coefﬁcients, or in terms of discrete gradients;
hence in their transform domain. The idea of taking advantage of
sparsity in order to undersample a signal is motivated by the suc
cess of image compression techniques. ‘Natural’ images and MR
images can thus be compressed signiﬁcantly with minimal loss
of information. Several transforms can be used in order to trans
form an image to a sparse domain – e.g., the discrete cosine trans
form (DCT) as well as the wavelet transform are the basis for
compression tools JPEG and JPEG2000 [13].
Since MR images are compressible, compressed sensing the
ory [10,14] suggests that taking fewer acquisition samples, com
bined with prior knowledge that the image is sparse in the
transform domain, may be sufﬁcient for an accurate image
reconstruction. This is achieved using a nonlinear reconstruction
method based on optimizing a convex function involving l
1

norms, related to the basis pursuit method [10]. In their work
Lustig et al. [11] developed a framework for using CS in MRI,
which has proved to be able to accurately reconstruct MR
images from a small subset of kspace rather than the entire
kspace grid.
2.2. Quantitative single point imaging with optimised sampling
trajectory
A comprehensive study of a near optimum sampling scheme
has been presented in previous work [9], where the sampling tra
jectory is dictated and ranked by the expected magnitude of the
sampled kspace points which we predict based on available prior
knowledge of the sample shape. kSpace points are sorted from
maximum to minimum expected modulus signal intensity and di
vided into interleafs consistent with the centric scan sampling
practise [8,15]. The principle used here is that kspace points with
the highest expected signal intensity are acquired with the highest
value of magnetization. A time interval of 5 T
1
is allowed be
tween each interleaf to allow the magnetization to fully recover.
As has been demonstrated in our previous study [9] a signiﬁcant
a
b
30
mm
y
x
z
17 mm
3.2 mm
y
x
Fig. 1. (a) 2D and (b) 3D binary MR images of the wafer sheet with dimensions
shown. The coordinate system shown is used for all image acquisitions.
P. Parasoglou et al. / Journal of Magnetic Resonance 201 (2009) 72–80
73
[10,11] with minimal loss of quantiﬁcation. This image reconstruc
tion method takes advantage of the fact that certain transforma
tions of MR images such as wavelet or spatial differences are
typically very sparse. Sparsity in this transform domain allows
good reconstruction from an undersampled set of measurements
in kspace. Lustig et al. [11], studied the exploitation of sparsity
of MR (conventional frequencyphase encoded) images in a trans
form domain and proposed a nonlinear iterative reconstruction of
the undersampled images based on l
1
optimization which results
in minimal loss of image quality/quantiﬁcation.
In the work presented here, moisture absorption from humidi
ﬁed air by a thin cerealbased wafer material is considered. A 2D
and a 3D binary MR image of the wafer sample is shown in
Fig. 1(a) and (b), respectively. Such moisture absorption (and sub
sequent migration within the wafer foam) occurs during an indus
trial conditioning process. The conditioning process is required in
order to optimise the expansion properties of the wafer for subse
quent fabrication steps during confectionery production. Hence
this process has a signiﬁcant contribution to the ﬁnal quality of
the product. Temporally resolved MRI of such a system is very
challenging. The wafer is highly porous (typically 80–90 vol%)
while the absorbed moisture ranges from 1 to 20 wt% of the solid
content. This low moisture content occurs as ‘bound’ water with
restricted mobility; the porous nature of the wafer also results in
signiﬁcant magnetic susceptibility variations. These collectively re
sult in a short T
2
for the water, typically of the order of 100
l
s,
whilst T
1
remains relatively long (1 s). Imaging the system with
conventional MRI frequencyencode methods is not possible,
hence the use of SPI.
To follow the dynamic moisture absorption process by the wa
fer, the following approach was developed and implemented: SPI
was implemented in 3D with an undersampled acquisition of
33% of kspace. The sampling trajectory was optimised based on
the expected amplitude of the points in kspace as outlined in
our previous publication [9]. Reconstruction of the images from
this acquired kspace data is achieved via a nonlinear iteration
(based on optimizing a convex function involving l
1
norms) as pro
posed by Lustig and coworkers [11]. Each image was also con
structed from four subimages acquired with increasing phase
encoding time, thus allowing quantiﬁcation of the water content
in the image via appropriate signal relaxation analysis. This imag
ing protocol is repeated a number of times during the moisture
absorption process.
2. Method development
2.1. Compressed sensing in MRI
Acquiring all the kspace points or Fourier coefﬁcients of a MR
image is time consuming. Undersampled kspace acquisition is
the method of choice when neither the spatial resolution nor the
number of averages can be compromised for a desired SNR. From
the Nyquist sampling theorem, we expect that when kspace is
strongly undersampled, linear recovery methods will lead to lossy
recovery and exhibit artefacts. It is important to note, however,
that not all of the Fourier coefﬁcients are equally important for
the reconstruction of the image.
Recently compressed sensing [11,12] has attracted interest in
terms of MR imaging. This sampling method takes advantage of
the sparsity of MR images and allows good reconstructions from
signiﬁcantly undersampled kspace. A signal is described as
sparse if it contains only a small number of nonzero values, or if
these values decay very fast. We consider a transform domain
sparsity, where the signal is not sparse but it can be manipulated
to be in some transform domain. In the current context we make
use of the acquired signal in the kspace domain, this is trans
formed into the image domain and we exploit sparsity in a trans
form domain. Note that the image domain can be used as the
transform domain if it presents sufﬁcient sparsity. Lustig et al.
[11,12], studied the exploitation of sparsity of MR images; for
example they considered angiograms which have sparsity in the
image domain. Many other types of ‘natural’ images (e.g., natural
scenes, astronomical images) can exhibit sparsity in terms of their
Fourier or wavelet coefﬁcients, or in terms of discrete gradients;
hence in their transform domain. The idea of taking advantage of
sparsity in order to undersample a signal is motivated by the suc
cess of image compression techniques. ‘Natural’ images and MR
images can thus be compressed signiﬁcantly with minimal loss
of information. Several transforms can be used in order to trans
form an image to a sparse domain – e.g., the discrete cosine trans
form (DCT) as well as the wavelet transform are the basis for
compression tools JPEG and JPEG2000 [13].
Since MR images are compressible, compressed sensing the
ory [10,14] suggests that taking fewer acquisition samples, com
bined with prior knowledge that the image is sparse in the
transform domain, may be sufﬁcient for an accurate image
reconstruction. This is achieved using a nonlinear reconstruction
method based on optimizing a convex function involving l
1

norms, related to the basis pursuit method [10]. In their work
Lustig et al. [11] developed a framework for using CS in MRI,
which has proved to be able to accurately reconstruct MR
images from a small subset of kspace rather than the entire
kspace grid.
2.2. Quantitative single point imaging with optimised sampling
trajectory
A comprehensive study of a near optimum sampling scheme
has been presented in previous work [9], where the sampling tra
jectory is dictated and ranked by the expected magnitude of the
sampled kspace points which we predict based on available prior
knowledge of the sample shape. kSpace points are sorted from
maximum to minimum expected modulus signal intensity and di
vided into interleafs consistent with the centric scan sampling
practise [8,15]. The principle used here is that kspace points with
the highest expected signal intensity are acquired with the highest
value of magnetization. A time interval of 5 T
1
is allowed be
tween each interleaf to allow the magnetization to fully recover.
As has been demonstrated in our previous study [9] a signiﬁcant
a
b
30
mm
y
x
z
17 mm
3.2 mm
y
x
Fig. 1. (a) 2D and (b) 3D binary MR images of the wafer sheet with dimensions
shown. The coordinate system shown is used for all image acquisitions.
P. Parasoglou et al. / Journal of Magnetic Resonance 201 (2009) 72–80
73
improvement in the SNR and image quality can be achieved by fol
lowing this sampling scheme.
The sampling strategy is based on the fact that SPI is a transient
magnetization imaging method. This means that the magnetiza
tion will reduce from an initial value of M
0
to an equilibrium value
after a certain number of pulseacquire repetitions [16]. Without
any magnetization preparation the magnetization available for
imaging after the n
th
excitation pulseacquire repetition, M
n
, is:
M
n
¼ M
0
ð1 wÞC
n
E
n
þ M
0
w ð1Þ
where E ¼ expðTR=T
1
Þ, C ¼ cosðaÞ and w ¼
1E
1CE
. TR is the time be
tween two successive excitation pulses and
a
is the r.f. pulse tip an
gle. In conventional SPI [5] where kspace is sampled in a rectilinear
trajectory, magnetization is saturated when at the extremities of k
space and the remaining kspace points are sampled with the same
magnetization. This results in a comparatively poor SNR, which is
generally determined by the centre of kspace, hence the preference
for centric sampling trajectories which commence at the centre of
kspace. An additional advantage is that in a centric scan there is
no T
1
weighting of the origin of kspace and hence total image
intensity is directly proportional to the concentration of the sig
nalbearing species [4,8,15,17].
The signal intensity, S(r) at any point, r, in a SPI image will be
given (ignoring T
1
relaxation effects) by [5]:
SðrÞ/
q
ðrÞ exp
t
p
T
2
ðrÞ
sinð
a
Þð2Þ
q
(r) is the local
1
H density (
1
H NMR is used in this paper exclu
sively) and t
p
is the phase encoding time. In cases where t
p
is much
shorter than T
2
, S(r) will be relatively unaffected by signal relaxa
tion; however, if t
p
is comparable or longer than T
2
, the signal inten
sity is attenuated by relaxation effects. Acquisition of S(r) at various
values of t
p
, however, allows us to calculate T
2
ðrÞ using Eq. (2) and
therefore produce a comparatively quantitative image of
q
(r).
With the above sampling method (as presented in [9]), once the
kspace points have been ranked then a subset of them with the
highest intensity can be acquired in a sparse kspace sampling
scheme. In order for this method to be used for compressed sens
ing, it is required that undersampling causes incoherent artefacts,
or more formally the sampling operator must not be easily (spar
sely) represented in the transform domain [14]. It has been shown
[18] that when a good knowledge of the signal exists, then the
sampling of the signal should aim to acquire the coefﬁcients with
the highest expected intensity for optimal results in the CS frame
work. Knowledge of the wafer shape provides such prior knowl
edge. Fig. 2(a) shows a simulated 2D image of a wafer sample,
whilst in Fig. 2(b) the magnitude of its corresponding kspace is
shown. The resultant selected kspace points (20% of the highest
expected intensity) are shown as a binary image in Fig. 2(c). Simu
lations using the sampling trajectory or pattern in Fig. 2(c) reveal
no obvious coherent artefacts which indicate that this sampling
trajectory is sufﬁciently incoherent.
Compressed sensing and SPI constitute a powerful combination
as the use of only phase encoding means that all imaging dimen
sions can be undersampled. By comparison conventional 2D fre
quencyphaseencoding imaging, as used by Lustig et al. [11], can
only be usefully undersampled in the phase dimension. In addi
tion with respect to SPI, there are no constraints on the kspace
sampling trajectory.
2.3. Sparsifying transform and image reconstruction
The sparsifying transform that is used in our case study is spa
tial discrete gradients (i.e., it computes differences of neighbouring
pixels). We compute both the horizontal and the vertical gradients.
Such a linear transform increases the dimension of the transform
space, i.e., the number of coefﬁcients roughly doubles. However,
away from the boundaries of the object in the image these pair
wise differences will be small or close to zero. Only right on the
boundaries will the coefﬁcients be signiﬁcant. Thus spatial ﬁnite
differences is appropriate for our piecewise smooth images and
only a small percentage of the transform coefﬁcients is sufﬁcient
for an accurate, and in our case, sufﬁciently quantitative recon
struction. To conﬁrm this, we considered typical 3D images of
the wafer; both a dry and a 15 wt% moisture sample. Fig. 3 shows
the descending rank order of the coefﬁcients in the Finite Differ
ence transform domain. It is clear that the energy (signal intensity)
of these coefﬁcients is predominately contained in a small subset
and thus that ﬁnite differences sparisfy this particular system well.
The distribution is only slightly broader for the wet sample, indi
cating that the ﬁnite difference approach is ﬁne for all the scenar
ios we considered. It may in future be possible to have an even
better sparsifying transform by designing one directly for the
images of interest [19]. This will potentially reduce the number
of required kspace measurements further.
Typically in SPI, kspace is sampled one point per acquisition
using an appropriate combination of phaseencoding gradients.
To improve the efﬁciency of single point imaging techniques, Bal
com and coworkers have proposed the acquisition of multiple
points of the FID [20–22], which can be coadded to increase the
SNR. These points, at increasing temporal locations along the FID,
however, correspond to increasing kspace coordinates and effec
tively a shrinking ﬁeld of view (FOV). Rescaling to the same FOV is
achieved via use of the chirp ztransform [20]. When the difference
in phase encode time between the successive images is kept to a
minimum (as dictated by hardware restrictions), then the FOV
200
400
600
800
Signal
Intensity (a.u)
0.5
1
1.5
2
2.5
3
x 10
5
Signal
Intensity (a.u)
a
b
c
Fig. 2. (a) Typical 2D image of a wafer sample (b) kspace modulus map (c) our
sampling trajectory employed to acquire 20% of the highest intensity kspace
points.
74 P. Parasoglou et al. / Journal of Magnetic Resonance 201 (2009) 72–80
improvement in the SNR and image quality can be achieved by fol
lowing this sampling scheme.
The sampling strategy is based on the fact that SPI is a transient
magnetization imaging method. This means that the magnetiza
tion will reduce from an initial value of M
0
to an equilibrium value
after a certain number of pulseacquire repetitions [16]. Without
any magnetization preparation the magnetization available for
imaging after the n
th
excitation pulseacquire repetition, M
n
, is:
M
n
¼ M
0
ð1 wÞC
n
E
n
þ M
0
w ð1Þ
where E ¼ expðTR=T
1
Þ, C ¼ cosðaÞ and w ¼
1E
1CE
. TR is the time be
tween two successive excitation pulses and
a
is the r.f. pulse tip an
gle. In conventional SPI [5] where kspace is sampled in a rectilinear
trajectory, magnetization is saturated when at the extremities of k
space and the remaining kspace points are sampled with the same
magnetization. This results in a comparatively poor SNR, which is
generally determined by the centre of kspace, hence the preference
for centric sampling trajectories which commence at the centre of
kspace. An additional advantage is that in a centric scan there is
no T
1
weighting of the origin of kspace and hence total image
intensity is directly proportional to the concentration of the sig
nalbearing species [4,8,15,17].
The signal intensity, S(r) at any point, r, in a SPI image will be
given (ignoring T
1
relaxation effects) by [5]:
SðrÞ/
q
ðrÞ exp
t
p
T
2
ðrÞ
sinð
a
Þð2Þ
q
(r) is the local
1
H density (
1
H NMR is used in this paper exclu
sively) and t
p
is the phase encoding time. In cases where t
p
is much
shorter than T
2
, S(r) will be relatively unaffected by signal relaxa
tion; however, if t
p
is comparable or longer than T
2
, the signal inten
sity is attenuated by relaxation effects. Acquisition of S(r) at various
values of t
p
, however, allows us to calculate T
2
ðrÞ using Eq. (2) and
therefore produce a comparatively quantitative image of
q
(r).
With the above sampling method (as presented in [9]), once the
kspace points have been ranked then a subset of them with the
highest intensity can be acquired in a sparse kspace sampling
scheme. In order for this method to be used for compressed sens
ing, it is required that undersampling causes incoherent artefacts,
or more formally the sampling operator must not be easily (spar
sely) represented in the transform domain [14]. It has been shown
[18] that when a good knowledge of the signal exists, then the
sampling of the signal should aim to acquire the coefﬁcients with
the highest expected intensity for optimal results in the CS frame
work. Knowledge of the wafer shape provides such prior knowl
edge. Fig. 2(a) shows a simulated 2D image of a wafer sample,
whilst in Fig. 2(b) the magnitude of its corresponding kspace is
shown. The resultant selected kspace points (20% of the highest
expected intensity) are shown as a binary image in Fig. 2(c). Simu
lations using the sampling trajectory or pattern in Fig. 2(c) reveal
no obvious coherent artefacts which indicate that this sampling
trajectory is sufﬁciently incoherent.
Compressed sensing and SPI constitute a powerful combination
as the use of only phase encoding means that all imaging dimen
sions can be undersampled. By comparison conventional 2D fre
quencyphaseencoding imaging, as used by Lustig et al. [11], can
only be usefully undersampled in the phase dimension. In addi
tion with respect to SPI, there are no constraints on the kspace
sampling trajectory.
2.3. Sparsifying transform and image reconstruction
The sparsifying transform that is used in our case study is spa
tial discrete gradients (i.e., it computes differences of neighbouring
pixels). We compute both the horizontal and the vertical gradients.
Such a linear transform increases the dimension of the transform
space, i.e., the number of coefﬁcients roughly doubles. However,
away from the boundaries of the object in the image these pair
wise differences will be small or close to zero. Only right on the
boundaries will the coefﬁcients be signiﬁcant. Thus spatial ﬁnite
differences is appropriate for our piecewise smooth images and
only a small percentage of the transform coefﬁcients is sufﬁcient
for an accurate, and in our case, sufﬁciently quantitative recon
struction. To conﬁrm this, we considered typical 3D images of
the wafer; both a dry and a 15 wt% moisture sample. Fig. 3 shows
the descending rank order of the coefﬁcients in the Finite Differ
ence transform domain. It is clear that the energy (signal intensity)
of these coefﬁcients is predominately contained in a small subset
and thus that ﬁnite differences sparisfy this particular system well.
The distribution is only slightly broader for the wet sample, indi
cating that the ﬁnite difference approach is ﬁne for all the scenar
ios we considered. It may in future be possible to have an even
better sparsifying transform by designing one directly for the
images of interest [19]. This will potentially reduce the number
of required kspace measurements further.
Typically in SPI, kspace is sampled one point per acquisition
using an appropriate combination of phaseencoding gradients.
To improve the efﬁciency of single point imaging techniques, Bal
com and coworkers have proposed the acquisition of multiple
points of the FID [20–22], which can be coadded to increase the
SNR. These points, at increasing temporal locations along the FID,
however, correspond to increasing kspace coordinates and effec
tively a shrinking ﬁeld of view (FOV). Rescaling to the same FOV is
achieved via use of the chirp ztransform [20]. When the difference
in phase encode time between the successive images is kept to a
minimum (as dictated by hardware restrictions), then the FOV
200
400
600
800
Signal
Intensity (a.u)
0.5
1
1.5
2
2.5
3
x 10
5
Signal
Intensity (a.u)
a
b
c
Fig. 2. (a) Typical 2D image of a wafer sample (b) kspace modulus map (c) our
sampling trajectory employed to acquire 20% of the highest intensity kspace
points.
74 P. Parasoglou et al. / Journal of Magnetic Resonance 201 (2009) 72–80
improvement in the SNR and image quality can be achieved by fol
lowing this sampling scheme.
The sampling strategy is based on the fact that SPI is a transient
magnetization imaging method. This means that the magnetiza
tion will reduce from an initial value of M
0
to an equilibrium value
after a certain number of pulseacquire repetitions [16]. Without
any magnetization preparation the magnetization available for
imaging after the n
th
excitation pulseacquire repetition, M
n
, is:
M
n
¼ M
0
ð1 wÞC
n
E
n
þ M
0
w ð1Þ
where E ¼ expðTR=T
1
Þ, C ¼ cosðaÞ and w ¼
1E
1CE
. TR is the time be
tween two successive excitation pulses and
a
is the r.f. pulse tip an
gle. In conventional SPI [5] where kspace is sampled in a rectilinear
trajectory, magnetization is saturated when at the extremities of k
space and the remaining kspace points are sampled with the same
magnetization. This results in a comparatively poor SNR, which is
generally determined by the centre of kspace, hence the preference
for centric sampling trajectories which commence at the centre of
kspace. An additional advantage is that in a centric scan there is
no T
1
weighting of the origin of kspace and hence total image
intensity is directly proportional to the concentration of the sig
nalbearing species [4,8,15,17].
The signal intensity, S(r) at any point, r, in a SPI image will be
given (ignoring T
1
relaxation effects) by [5]:
SðrÞ/
q
ðrÞ exp
t
p
T
2
ðrÞ
sinð
a
Þð2Þ
q
(r) is the local
1
H density (
1
H NMR is used in this paper exclu
sively) and t
p
is the phase encoding time. In cases where t
p
is much
shorter than T
2
, S(r) will be relatively unaffected by signal relaxa
tion; however, if t
p
is comparable or longer than T
2
, the signal inten
sity is attenuated by relaxation effects. Acquisition of S(r) at various
values of t
p
, however, allows us to calculate T
2
ðrÞ using Eq. (2) and
therefore produce a comparatively quantitative image of
q
(r).
With the above sampling method (as presented in [9]), once the
kspace points have been ranked then a subset of them with the
highest intensity can be acquired in a sparse kspace sampling
scheme. In order for this method to be used for compressed sens
ing, it is required that undersampling causes incoherent artefacts,
or more formally the sampling operator must not be easily (spar
sely) represented in the transform domain [14]. It has been shown
[18] that when a good knowledge of the signal exists, then the
sampling of the signal should aim to acquire the coefﬁcients with
the highest expected intensity for optimal results in the CS frame
work. Knowledge of the wafer shape provides such prior knowl
edge. Fig. 2(a) shows a simulated 2D image of a wafer sample,
whilst in Fig. 2(b) the magnitude of its corresponding kspace is
shown. The resultant selected kspace points (20% of the highest
expected intensity) are shown as a binary image in Fig. 2(c). Simu
lations using the sampling trajectory or pattern in Fig. 2(c) reveal
no obvious coherent artefacts which indicate that this sampling
trajectory is sufﬁciently incoherent.
Compressed sensing and SPI constitute a powerful combination
as the use of only phase encoding means that all imaging dimen
sions can be undersampled. By comparison conventional 2D fre
quencyphaseencoding imaging, as used by Lustig et al. [11], can
only be usefully undersampled in the phase dimension. In addi
tion with respect to SPI, there are no constraints on the kspace
sampling trajectory.
2.3. Sparsifying transform and image reconstruction
The sparsifying transform that is used in our case study is spa
tial discrete gradients (i.e., it computes differences of neighbouring
pixels). We compute both the horizontal and the vertical gradients.
Such a linear transform increases the dimension of the transform
space, i.e., the number of coefﬁcients roughly doubles. However,
away from the boundaries of the object in the image these pair
wise differences will be small or close to zero. Only right on the
boundaries will the coefﬁcients be signiﬁcant. Thus spatial ﬁnite
differences is appropriate for our piecewise smooth images and
only a small percentage of the transform coefﬁcients is sufﬁcient
for an accurate, and in our case, sufﬁciently quantitative recon
struction. To conﬁrm this, we considered typical 3D images of
the wafer; both a dry and a 15 wt% moisture sample. Fig. 3 shows
the descending rank order of the coefﬁcients in the Finite Differ
ence transform domain. It is clear that the energy (signal intensity)
of these coefﬁcients is predominately contained in a small subset
and thus that ﬁnite differences sparisfy this particular system well.
The distribution is only slightly broader for the wet sample, indi
cating that the ﬁnite difference approach is ﬁne for all the scenar
ios we considered. It may in future be possible to have an even
better sparsifying transform by designing one directly for the
images of interest [19]. This will potentially reduce the number
of required kspace measurements further.
Typically in SPI, kspace is sampled one point per acquisition
using an appropriate combination of phaseencoding gradients.
To improve the efﬁciency of single point imaging techniques, Bal
com and coworkers have proposed the acquisition of multiple
points of the FID [20–22], which can be coadded to increase the
SNR. These points, at increasing temporal locations along the FID,
however, correspond to increasing kspace coordinates and effec
tively a shrinking ﬁeld of view (FOV). Rescaling to the same FOV is
achieved via use of the chirp ztransform [20]. When the difference
in phase encode time between the successive images is kept to a
minimum (as dictated by hardware restrictions), then the FOV
200
400
600
800
Signal
Intensity (a.u)
0.5
1
1.5
2
2.5
3
x 10
5
Signal
Intensity (a.u)
a
b
c
Fig. 2. (a) Typical 2D image of a wafer sample (b) kspace modulus map (c) our
sampling trajectory employed to acquire 20% of the highest intensity kspace
points.
74 P. Parasoglou et al. / Journal of Magnetic Resonance 201 (2009) 72–80
improvement in the SNR and image quality can be achieved by fol
lowing this sampling scheme.
The sampling strategy is based on the fact that SPI is a transient
magnetization imaging method. This means that the magnetiza
tion will reduce from an initial value of M
0
to an equilibrium value
after a certain number of pulseacquire repetitions [16]. Without
any magnetization preparation the magnetization available for
imaging after the n
th
excitation pulseacquire repetition, M
n
, is:
M
n
¼ M
0
ð1 wÞC
n
E
n
þ M
0
w ð1Þ
where E ¼ expðTR=T
1
Þ, C ¼ cosðaÞ and w ¼
1E
1CE
. TR is the time be
tween two successive excitation pulses and
a
is the r.f. pulse tip an
gle. In conventional SPI [5] where kspace is sampled in a rectilinear
trajectory, magnetization is saturated when at the extremities of k
space and the remaining kspace points are sampled with the same
magnetization. This results in a comparatively poor SNR, which is
generally determined by the centre of kspace, hence the preference
for centric sampling trajectories which commence at the centre of
kspace. An additional advantage is that in a centric scan there is
no T
1
weighting of the origin of kspace and hence total image
intensity is directly proportional to the concentration of the sig
nalbearing species [4,8,15,17].
The signal intensity, S(r) at any point, r, in a SPI image will be
given (ignoring T
1
relaxation effects) by [5]:
SðrÞ/
q
ðrÞ exp
t
p
T
2
ðrÞ
sinð
a
Þð2Þ
q
(r) is the local
1
H density (
1
H NMR is used in this paper exclu
sively) and t
p
is the phase encoding time. In cases where t
p
is much
shorter than T
2
, S(r) will be relatively unaffected by signal relaxa
tion; however, if t
p
is comparable or longer than T
2
, the signal inten
sity is attenuated by relaxation effects. Acquisition of S(r) at various
values of t
p
, however, allows us to calculate T
2
ðrÞ using Eq. (2) and
therefore produce a comparatively quantitative image of
q
(r).
With the above sampling method (as presented in [9]), once the
kspace points have been ranked then a subset of them with the
highest intensity can be acquired in a sparse kspace sampling
scheme. In order for this method to be used for compressed sens
ing, it is required that undersampling causes incoherent artefacts,
or more formally the sampling operator must not be easily (spar
sely) represented in the transform domain [14]. It has been shown
[18] that when a good knowledge of the signal exists, then the
sampling of the signal should aim to acquire the coefﬁcients with
the highest expected intensity for optimal results in the CS frame
work. Knowledge of the wafer shape provides such prior knowl
edge. Fig. 2(a) shows a simulated 2D image of a wafer sample,
whilst in Fig. 2(b) the magnitude of its corresponding kspace is
shown. The resultant selected kspace points (20% of the highest
expected intensity) are shown as a binary image in Fig. 2(c). Simu
lations using the sampling trajectory or pattern in Fig. 2(c) reveal
no obvious coherent artefacts which indicate that this sampling
trajectory is sufﬁciently incoherent.
Compressed sensing and SPI constitute a powerful combination
as the use of only phase encoding means that all imaging dimen
sions can be undersampled. By comparison conventional 2D fre
quencyphaseencoding imaging, as used by Lustig et al. [11], can
only be usefully undersampled in the phase dimension. In addi
tion with respect to SPI, there are no constraints on the kspace
sampling trajectory.
2.3. Sparsifying transform and image reconstruction
The sparsifying transform that is used in our case study is spa
tial discrete gradients (i.e., it computes differences of neighbouring
pixels). We compute both the horizontal and the vertical gradients.
Such a linear transform increases the dimension of the transform
space, i.e., the number of coefﬁcients roughly doubles. However,
away from the boundaries of the object in the image these pair
wise differences will be small or close to zero. Only right on the
boundaries will the coefﬁcients be signiﬁcant. Thus spatial ﬁnite
differences is appropriate for our piecewise smooth images and
only a small percentage of the transform coefﬁcients is sufﬁcient
for an accurate, and in our case, sufﬁciently quantitative recon
struction. To conﬁrm this, we considered typical 3D images of
the wafer; both a dry and a 15 wt% moisture sample. Fig. 3 shows
the descending rank order of the coefﬁcients in the Finite Differ
ence transform domain. It is clear that the energy (signal intensity)
of these coefﬁcients is predominately contained in a small subset
and thus that ﬁnite differences sparisfy this particular system well.
The distribution is only slightly broader for the wet sample, indi
cating that the ﬁnite difference approach is ﬁne for all the scenar
ios we considered. It may in future be possible to have an even
better sparsifying transform by designing one directly for the
images of interest [19]. This will potentially reduce the number
of required kspace measurements further.
Typically in SPI, kspace is sampled one point per acquisition
using an appropriate combination of phaseencoding gradients.
To improve the efﬁciency of single point imaging techniques, Bal
com and coworkers have proposed the acquisition of multiple
points of the FID [20–22], which can be coadded to increase the
SNR. These points, at increasing temporal locations along the FID,
however, correspond to increasing kspace coordinates and effec
tively a shrinking ﬁeld of view (FOV). Rescaling to the same FOV is
achieved via use of the chirp ztransform [20]. When the difference
in phase encode time between the successive images is kept to a
minimum (as dictated by hardware restrictions), then the FOV
200
400
600
800
Signal
Intensity (a.u)
0.5
1
1.5
2
2.5
3
x 10
5
Signal
Intensity (a.u)
a
b
c
Fig. 2. (a) Typical 2D image of a wafer sample (b) kspace modulus map (c) our
sampling trajectory employed to acquire 20% of the highest intensity kspace
points.
74 P. Parasoglou et al. / Journal of Magnetic Resonance 201 (2009) 72–80
change between the successive images is minimal and the images
retain all the geometrical characteristics and can hence be sparsi
ﬁed by the same transform.
The image reconstruction method that we use in our study is
based on solving a convex optimization problem involving l
1

norms, a variant of basis pursuit [10,11]. If an image that we want
to reconstruct is stacked as a vector m,
W
is the operator that trans
forms from pixel representation to a sparse representation, F is the
undersampled Fourier transform to k space and y are the kspace
measurements. The reconstruction is then obtained by solving the
following constrained optimization problem:
minjj
W
mjj
1
s tjjFm yjj
2
<
e
ð3Þ
where
e
is a threshold that can be set to the expected noise level.
The l
1
norm acts as a proxy for sparsity – i.e., minimising the above
objective produces an image which has the sparsest representation
in the transform domain while remaining consistent with acquired
measurements. The objective function is minimised using projected
conjugate gradients following the approach of Lustig et al. [11].
Since the objective is convex, the algorithm ﬁnds the global mini
mum in the function. The reconstruction involving the l
1
norm is
known to be a biased estimator for some systems, with the bias
becoming more signiﬁcant at lower SNR [23]. This bias appears in
the sparse domain, so it could affect the reconstructed images in
complex ways. A debiasing scheme such as the one in [23] could
be used. Here we consider the bias and error of the reconstruction
using appropriate simulations. The image of the wafer in Fig. 2(a)
was reconstructed using our methodology applied to 20% of the
highest intensity kspace coefﬁcients. This was done with no mea
surement noise added (SNR = 1) and with the addition of Gaussian
measurement noise to both the real and imaginary components
(SNR = 5, note this ratio corresponds to the lowest SNR of the vari
ous acquired images presented later). The % error between the
reconstructed images and the original images was calculated for
each pixel, the corresponding data is presented as pixel histograms
of % error in Fig. 4. In both cases the error is centred around 0 and
the bias towards a negative error is relatively small. In the case of no
measurement noise (SNR = 1) the mean of the error is only 1.1%,
which is very small. In the case of SNR = 5, the distribution of error
is larger, consistent with the poorer SNR, but the mean of the error
is still only 2.3%, which again is relatively small.
2.4. Experimental setup
All experiments were conducted using a Bruker AV400 spec
trometer equipped with a microimaging r.f. coil of 25 mm inter
nal diameter. Food wafer samples, binary gated 2D crosssectional
and 3D MR images of which are shown in Fig. 1, were placed in a
17 mm internal diameter NMR tube featuring a glass sample
holder and connected to a humidiﬁed air source (18.6 ± 0.5 °C;
relative humidity >95%) at a ﬂow rate of 250 ml min
1
. The 3D
SPI pulse sequence used is schematically shown in Fig. 5 (single
interleaf shown). Each image consisted of four successive sub
images, each acquired with a different range of phase encoding
time (and hence a reduced applied gradient strength to ensure
a consistent FOV), in order to quantify the water content. As
shown in Fig. 5, for each kspace ‘point’, ﬁve points of the FID
were acquired separated by a dwell time of 1
l
s. For the ﬁrst
subimage these corresponded to phase encoding times of 76,
77, 78, 79 and 80
l
s. The second subimage used phase encode
times from 96 to 100
l
s, the third from 116 to 120
l
s and the
fourth from 136 to 140
l
s, respectively. Exceeding this 5
l
s range
resulted in image artefacts in the coadded images reconstructed
with the same FOV via the chirp ztransform, this effect was
made more acute by the presence of sharp edges in our sample.
These images were ﬁrst reconstructed individually through the
optimization scheme that was described earlier and then co
added to improve the SNR.
All images were acquired using a tip angle of 3° corresponding
to a pulse duration of 3
l
s, and a repetition time (T
R
) of 2 ms. The
images were acquired over a ﬁeld of view of 2 1 5cm
3
and
consisted of 64 32 8 pixels. For these undersampled (33% ac
quired) images, followed by compressed sensing reconstruction,
5400 pulseacquire events were required (split into 8 interleafs
of 675 points), which with four signal averages and the acquisi
5000 10000 15000
0
2
4
6
8
10
12
x 10
5
Number of Coefficients
Energy of Coefficient (a.u)
Dry
Wet
Fig. 3. Descending rank order of discrete gradient coefﬁcients of typical 3D images
of both a dry wafer and a wet wafer (15 wt%).
100 75 50 25 0 25 50 75 100
0
10
20
30
40
50
60
70
Pixel Error (%)
Number of Pixels
SNR = 5
SNR =
∞
Fig. 4. Histograms of pixel error (%) resulting from the reconstructed image –
original image for both SNR = 1 and SNR = 5.
P. Parasoglou et al. / Journal of Magnetic Resonance 201 (2009) 72–80
75
change between the successive images is minimal and the images
retain all the geometrical characteristics and can hence be sparsi
ﬁed by the same transform.
The image reconstruction method that we use in our study is
based on solving a convex optimization problem involving l
1

norms, a variant of basis pursuit [10,11]. If an image that we want
to reconstruct is stacked as a vector m,
W
is the operator that trans
forms from pixel representation to a sparse representation, F is the
undersampled Fourier transform to k space and y are the kspace
measurements. The reconstruction is then obtained by solving the
following constrained optimization problem:
minjj
W
mjj
1
s tjjFm yjj
2
<
e
ð3Þ
where
e
is a threshold that can be set to the expected noise level.
The l
1
norm acts as a proxy for sparsity – i.e., minimising the above
objective produces an image which has the sparsest representation
in the transform domain while remaining consistent with acquired
measurements. The objective function is minimised using projected
conjugate gradients following the approach of Lustig et al. [11].
Since the objective is convex, the algorithm ﬁnds the global mini
mum in the function. The reconstruction involving the l
1
norm is
known to be a biased estimator for some systems, with the bias
becoming more signiﬁcant at lower SNR [23]. This bias appears in
the sparse domain, so it could affect the reconstructed images in
complex ways. A debiasing scheme such as the one in [23] could
be used. Here we consider the bias and error of the reconstruction
using appropriate simulations. The image of the wafer in Fig. 2(a)
was reconstructed using our methodology applied to 20% of the
highest intensity kspace coefﬁcients. This was done with no mea
surement noise added (SNR = 1) and with the addition of Gaussian
measurement noise to both the real and imaginary components
(SNR = 5, note this ratio corresponds to the lowest SNR of the vari
ous acquired images presented later). The % error between the
reconstructed images and the original images was calculated for
each pixel, the corresponding data is presented as pixel histograms
of % error in Fig. 4. In both cases the error is centred around 0 and
the bias towards a negative error is relatively small. In the case of no
measurement noise (SNR = 1) the mean of the error is only 1.1%,
which is very small. In the case of SNR = 5, the distribution of error
is larger, consistent with the poorer SNR, but the mean of the error
is still only 2.3%, which again is relatively small.
2.4. Experimental setup
All experiments were conducted using a Bruker AV400 spec
trometer equipped with a microimaging r.f. coil of 25 mm inter
nal diameter. Food wafer samples, binary gated 2D crosssectional
and 3D MR images of which are shown in Fig. 1, were placed in a
17 mm internal diameter NMR tube featuring a glass sample
holder and connected to a humidiﬁed air source (18.6 ± 0.5 °C;
relative humidity >95%) at a ﬂow rate of 250 ml min
1
. The 3D
SPI pulse sequence used is schematically shown in Fig. 5 (single
interleaf shown). Each image consisted of four successive sub
images, each acquired with a different range of phase encoding
time (and hence a reduced applied gradient strength to ensure
a consistent FOV), in order to quantify the water content. As
shown in Fig. 5, for each kspace ‘point’, ﬁve points of the FID
were acquired separated by a dwell time of 1
l
s. For the ﬁrst
subimage these corresponded to phase encoding times of 76,
77, 78, 79 and 80
l
s. The second subimage used phase encode
times from 96 to 100
l
s, the third from 116 to 120
l
s and the
fourth from 136 to 140
l
s, respectively. Exceeding this 5
l
s range
resulted in image artefacts in the coadded images reconstructed
with the same FOV via the chirp ztransform, this effect was
made more acute by the presence of sharp edges in our sample.
These images were ﬁrst reconstructed individually through the
optimization scheme that was described earlier and then co
added to improve the SNR.
All images were acquired using a tip angle of 3° corresponding
to a pulse duration of 3
l
s, and a repetition time (T
R
) of 2 ms. The
images were acquired over a ﬁeld of view of 2 1 5cm
3
and
consisted of 64 32 8 pixels. For these undersampled (33% ac
quired) images, followed by compressed sensing reconstruction,
5400 pulseacquire events were required (split into 8 interleafs
of 675 points), which with four signal averages and the acquisi
5000 10000 15000
0
2
4
6
8
10
12
x 10
5
Number of Coefficients
Energy of Coefficient (a.u)
Dry
Wet
Fig. 3. Descending rank order of discrete gradient coefﬁcients of typical 3D images
of both a dry wafer and a wet wafer (15 wt%).
100 75 50 25 0 25 50 75 100
0
10
20
30
40
50
60
70
Pixel Error (%)
Number of Pixels
SNR = 5
SNR =
∞
Fig. 4. Histograms of pixel error (%) resulting from the reconstructed image –
original image for both SNR = 1 and SNR = 5.
P. Parasoglou et al. / Journal of Magnetic Resonance 201 (2009) 72–80
75
change between the successive images is minimal and the images
retain all the geometrical characteristics and can hence be sparsi
ﬁed by the same transform.
The image reconstruction method that we use in our study is
based on solving a convex optimization problem involving l
1

norms, a variant of basis pursuit [10,11]. If an image that we want
to reconstruct is stacked as a vector m,
W
is the operator that trans
forms from pixel representation to a sparse representation, F is the
undersampled Fourier transform to k space and y are the kspace
measurements. The reconstruction is then obtained by solving the
following constrained optimization problem:
minjj
W
mjj
1
s tjjFm yjj
2
<
e
ð3Þ
where
e
is a threshold that can be set to the expected noise level.
The l
1
norm acts as a proxy for sparsity – i.e., minimising the above
objective produces an image which has the sparsest representation
in the transform domain while remaining consistent with acquired
measurements. The objective function is minimised using projected
conjugate gradients following the approach of Lustig et al. [11].
Since the objective is convex, the algorithm ﬁnds the global mini
mum in the function. The reconstruction involving the l
1
norm is
known to be a biased estimator for some systems, with the bias
becoming more signiﬁcant at lower SNR [23]. This bias appears in
the sparse domain, so it could affect the reconstructed images in
complex ways. A debiasing scheme such as the one in [23] could
be used. Here we consider the bias and error of the reconstruction
using appropriate simulations. The image of the wafer in Fig. 2(a)
was reconstructed using our methodology applied to 20% of the
highest intensity kspace coefﬁcients. This was done with no mea
surement noise added (SNR = 1) and with the addition of Gaussian
measurement noise to both the real and imaginary components
(SNR = 5, note this ratio corresponds to the lowest SNR of the vari
ous acquired images presented later). The % error between the
reconstructed images and the original images was calculated for
each pixel, the corresponding data is presented as pixel histograms
of % error in Fig. 4. In both cases the error is centred around 0 and
the bias towards a negative error is relatively small. In the case of no
measurement noise (SNR = 1) the mean of the error is only 1.1%,
which is very small. In the case of SNR = 5, the distribution of error
is larger, consistent with the poorer SNR, but the mean of the error
is still only 2.3%, which again is relatively small.
2.4. Experimental setup
All experiments were conducted using a Bruker AV400 spec
trometer equipped with a microimaging r.f. coil of 25 mm inter
nal diameter. Food wafer samples, binary gated 2D crosssectional
and 3D MR images of which are shown in Fig. 1, were placed in a
17 mm internal diameter NMR tube featuring a glass sample
holder and connected to a humidiﬁed air source (18.6 ± 0.5 °C;
relative humidity >95%) at a ﬂow rate of 250 ml min
1
. The 3D
SPI pulse sequence used is schematically shown in Fig. 5 (single
interleaf shown). Each image consisted of four successive sub
images, each acquired with a different range of phase encoding
time (and hence a reduced applied gradient strength to ensure
a consistent FOV), in order to quantify the water content. As
shown in Fig. 5, for each kspace ‘point’, ﬁve points of the FID
were acquired separated by a dwell time of 1
l
s. For the ﬁrst
subimage these corresponded to phase encoding times of 76,
77, 78, 79 and 80
l
s. The second subimage used phase encode
times from 96 to 100
l
s, the third from 116 to 120
l
s and the
fourth from 136 to 140
l
s, respectively. Exceeding this 5
l
s range
resulted in image artefacts in the coadded images reconstructed
with the same FOV via the chirp ztransform, this effect was
made more acute by the presence of sharp edges in our sample.
These images were ﬁrst reconstructed individually through the
optimization scheme that was described earlier and then co
added to improve the SNR.
All images were acquired using a tip angle of 3° corresponding
to a pulse duration of 3
l
s, and a repetition time (T
R
) of 2 ms. The
images were acquired over a ﬁeld of view of 2 1 5cm
3
and
consisted of 64 32 8 pixels. For these undersampled (33% ac
quired) images, followed by compressed sensing reconstruction,
5400 pulseacquire events were required (split into 8 interleafs
of 675 points), which with four signal averages and the acquisi
5000 10000 15000
0
2
4
6
8
10
12
x 10
5
Number of Coefficients
Energy of Coefficient (a.u)
Dry
Wet
Fig. 3. Descending rank order of discrete gradient coefﬁcients of typical 3D images
of both a dry wafer and a wet wafer (15 wt%).
100 75 50 25 0 25 50 75 100
0
10
20
30
40
50
60
70
Pixel Error (%)
Number of Pixels
SNR = 5
SNR =
∞
Fig. 4. Histograms of pixel error (%) resulting from the reconstructed image –
original image for both SNR = 1 and SNR = 5.
P. Parasoglou et al. / Journal of Magnetic Resonance 201 (2009) 72–80
75
change between the successive images is minimal and the images
retain all the geometrical characteristics and can hence be sparsi
ﬁed by the same transform.
The image reconstruction method that we use in our study is
based on solving a convex optimization problem involving l
1

norms, a variant of basis pursuit [10,11]. If an image that we want
to reconstruct is stacked as a vector m,
W
is the operator that trans
forms from pixel representation to a sparse representation, F is the
undersampled Fourier transform to k space and y are the kspace
measurements. The reconstruction is then obtained by solving the
following constrained optimization problem:
minjj
W
mjj
1
s tjjFm yjj
2
<
e
ð3Þ
where
e
is a threshold that can be set to the expected noise level.
The l
1
norm acts as a proxy for sparsity – i.e., minimising the above
objective produces an image which has the sparsest representation
in the transform domain while remaining consistent with acquired
measurements. The objective function is minimised using projected
conjugate gradients following the approach of Lustig et al. [11].
Since the objective is convex, the algorithm ﬁnds the global mini
mum in the function. The reconstruction involving the l
1
norm is
known to be a biased estimator for some systems, with the bias
becoming more signiﬁcant at lower SNR [23]. This bias appears in
the sparse domain, so it could affect the reconstructed images in
complex ways. A debiasing scheme such as the one in [23] could
be used. Here we consider the bias and error of the reconstruction
using appropriate simulations. The image of the wafer in Fig. 2(a)
was reconstructed using our methodology applied to 20% of the
highest intensity kspace coefﬁcients. This was done with no mea
surement noise added (SNR = 1) and with the addition of Gaussian
measurement noise to both the real and imaginary components
(SNR = 5, note this ratio corresponds to the lowest SNR of the vari
ous acquired images presented later). The % error between the
reconstructed images and the original images was calculated for
each pixel, the corresponding data is presented as pixel histograms
of % error in Fig. 4. In both cases the error is centred around 0 and
the bias towards a negative error is relatively small. In the case of no
measurement noise (SNR = 1) the mean of the error is only 1.1%,
which is very small. In the case of SNR = 5, the distribution of error
is larger, consistent with the poorer SNR, but the mean of the error
is still only 2.3%, which again is relatively small.
2.4. Experimental setup
All experiments were conducted using a Bruker AV400 spec
trometer equipped with a microimaging r.f. coil of 25 mm inter
nal diameter. Food wafer samples, binary gated 2D crosssectional
and 3D MR images of which are shown in Fig. 1, were placed in a
17 mm internal diameter NMR tube featuring a glass sample
holder and connected to a humidiﬁed air source (18.6 ± 0.5 °C;
relative humidity >95%) at a ﬂow rate of 250 ml min
1
. The 3D
SPI pulse sequence used is schematically shown in Fig. 5 (single
interleaf shown). Each image consisted of four successive sub
images, each acquired with a different range of phase encoding
time (and hence a reduced applied gradient strength to ensure
a consistent FOV), in order to quantify the water content. As
shown in Fig. 5, for each kspace ‘point’, ﬁve points of the FID
were acquired separated by a dwell time of 1
l
s. For the ﬁrst
subimage these corresponded to phase encoding times of 76,
77, 78, 79 and 80
l
s. The second subimage used phase encode
times from 96 to 100
l
s, the third from 116 to 120
l
s and the
fourth from 136 to 140
l
s, respectively. Exceeding this 5
l
s range
resulted in image artefacts in the coadded images reconstructed
with the same FOV via the chirp ztransform, this effect was
made more acute by the presence of sharp edges in our sample.
These images were ﬁrst reconstructed individually through the
optimization scheme that was described earlier and then co
added to improve the SNR.
All images were acquired using a tip angle of 3° corresponding
to a pulse duration of 3
l
s, and a repetition time (T
R
) of 2 ms. The
images were acquired over a ﬁeld of view of 2 1 5cm
3
and
consisted of 64 32 8 pixels. For these undersampled (33% ac
quired) images, followed by compressed sensing reconstruction,
5400 pulseacquire events were required (split into 8 interleafs
of 675 points), which with four signal averages and the acquisi
5000 10000 15000
0
2
4
6
8
10
12
x 10
5
Number of Coefficients
Energy of Coefficient (a.u)
Dry
Wet
Fig. 3. Descending rank order of discrete gradient coefﬁcients of typical 3D images
of both a dry wafer and a wet wafer (15 wt%).
100 75 50 25 0 25 50 75 100
0
10
20
30