# 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 under-sampled 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 cereal-based 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.

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**ABSTRACT:**We demonstrate a method to manipulate magnetic resonance data such that the moments of the signal spatial distribution are readily accessible. Usually, magnetic resonance imaging relies on data acquired in so-called k-space which is subsequently Fourier transformed to render an image. Here, via analysis of the complex signal in the vicinity of the centre of k-space we are able to access the first three moments of the signal spatial distribution, ultimately in multiple directions. This is demonstrated for biofouling of a reverse osmosis (RO) membrane module, rendering unique information and an early warning of the onset of fouling. The analysis is particularly applicable for the use of mobile magnetic resonance spectrometers; here we demonstrate it using an Earth's magnetic field system. Copyright © 2015 Elsevier Inc. All rights reserved.Journal of Magnetic Resonance 02/2015; 252(252):145-150. · 2.30 Impact Factor - SourceAvailable from: J.S. (Hans) Vrouwenvelder
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**ABSTRACT:**Compressed sensing (CS) is considered as a promising signal processing technique, and successful applications of the CS theory depend mainly on the accuracy and speed of the reconstruction algorithms. In this paper, a generalized objective functional, which has been developed using the combinational estimation and an extended stabilizing functional, is proposed. An efficient iterative scheme, which integrates the beneficial advantages of the homotopy method, the shuffled frog-leaping (SFL) algorithm and the harmony search (HS) algorithm, is designed for searching a possible global optimal solution. Numerical simulations are implemented to evaluate the numerical performances and effectiveness of the proposed algorithm. Excellent numerical performances and encouraging results are observed. For the cases considered in this paper, a dramatic improvement in the reconstruction accuracy is achieved, which indicates that the proposed algorithm is a promising candidate for solving CS inverse problem.Computers & Electrical Engineering 07/2011; 37(4). · 0.99 Impact Factor

Page 1

Quantitative single point imaging with compressed sensing

P. Parasogloua, D. Malioutovb, A.J. Sedermana, J. Rasburnc, H. Powellc, L.F. Gladdena, A. Blakeb,

M.L. Johnsa,*

aDepartment of Chemical Engineering and Biotechnology, University of Cambridge, New Museums Site Pembroke Street, Cambridge CB2 3RA, UK

bMicrosoft Research Ltd., 7 J J Thompson Ave., Cambridge CB3 0FB, UK

cNestec York Ltd., Nestlé Product Technology Centre, Haxby Road, P.O. Box 204, York YO91 1XY, UK

a r t i c l ei n f o

Article history:

Received 30 March 2009

Revised 23 July 2009

Available online 14 August 2009

Keywords:

Compressed sensing

SPI

Under-sampling

k-space

a b s t r a c t

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 under-sampled acquisi-

tion data. SPI complements compressed sensing extremely well as it allows unconstrained selection of

sampling trajectories. Dynamic processes featuring short T?

in our case the absorption of moisture by a cereal-based 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?

2NMR signal can thus be more rapidly imaged,

2contrast.

? 2009 Elsevier Inc. All rights reserved.

Journal of Magnetic Resonance 201 (2009) 72–80

Contents lists available at ScienceDirect

Journal of Magnetic Resonance

journal homepage: www.elsevier.com/locate/jmr

Page 2

1. Introduction

Single point imaging was originally proposed as a ‘solid state’

imaging modality [1,2]. As a pure phase-encoding imaging tech-

nique, SPI is relatively immune to artefacts due to chemical shift,

magnetic susceptibility variations and inhomogeneities in the B0

field [3,4] and has thus been extensively used to image samples

with very short transverse relaxation time (T?

100 ls [5]. Typically these materials also have short spin–spin

relaxation times (T2) and thus frequency-encoding 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 k-space (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 co-workers [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 k-space point acquisition.

2), of the order of

1090-7807/$ - see front matter ? 2009 Elsevier Inc. All rights reserved.

doi:10.1016/j.jmr.2009.08.003

* Corresponding author. Tel.: +44 1223 334767; fax: +44 1223 334796.

E-mail address: mlj21@cheng.cam.ac.uk (M.L. Johns).

Page 3

Conventionally such SPI approaches have been performed by

sampling k-space in a linear raster starting at the extremities of

k-space. Centric scans that result in improved sensitivity, as well

as removal of T1contrast, 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 k-space points with the highest ex-

pected value convolved with the largest excited magnetization.

Note that SPI, as a pure phase-encoding 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 k-space with-

out significantly degrading the image quality. Traditionally in

centric scans which employ spiral acquisitions, the extremities of

k-space are under-sampled and the values are replaced by zeros

before reconstructing the image with the use of the Fourier trans-

form. When all k-space 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 imageswhicharesignificantlyunder-sampled

Page 4

[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 under-sampled set of measurements

in k-space. Lustig et al. [11], studied the exploitation of sparsity

of MR (conventional frequency-phase encoded) images in a trans-

form domain and proposed a non-linear iterative reconstruction of

the under-sampled images based on l1-optimization which results

in minimal loss of image quality/quantification.

In the work presented here, moisture absorption from humidi-

fied air by a thin cereal-based 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?

whilst T1remains relatively long (?1 s). Imaging the system with

conventional MRI frequency-encode 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 under-sampled acquisition of

33% of k-space. The sampling trajectory was optimised based on

the expected amplitude of the points in k-space as outlined in

our previous publication [9]. Reconstruction of the images from

this acquired k-space data is achieved via a non-linear iteration

(based on optimizing a convex function involving l1-norms) as pro-

posed by Lustig and co-workers [11]. Each image was also con-

2for the water, typically of the order of 100 ls,

Page 5

this acquired k-space data is achieved via a non-linear iteration

(based on optimizing a convex function involving l1-norms) as pro-

posed by Lustig and co-workers [11]. Each image was also con-

structed from four sub-images 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.

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 co-ordinate system shown is used for all image acquisitions.

Page 6

2. Method development

2.1. Compressed sensing in MRI

Acquiring all the k-space points or Fourier coefficients of a MR

image is time consuming. Under-sampled k-space 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 k-space is

strongly under-sampled, 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 under-sampled k-space. A signal is described as

sparse if it contains only a small number of non-zero 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 k-space domain, this is trans-

formed into the image domain and we exploit sparsity in a trans-

Page 7

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 under-sample 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 JPEG-2000 [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 non-linear reconstruction

method based on optimizing a convex function involving l1-

norms, related to the basis pursuit method [10]. In their work

Page 8

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 k-space rather than the entire

k-space 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 k-space points which we predict based on available prior

knowledge of the sample shape. k-Space 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 k-space points with

the highest expected signal intensity are acquired with the highest

value of magnetization. A time interval of 5 ? T1is allowed be-

tween each interleaf to allow the magnetization to fully recover.

As has been demonstrated in our previous study [9] a significant

Page 9

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 M0to an equilibrium value

after a certain number of pulse-acquire repetitions [16]. Without

any magnetization preparation the magnetization available for

imaging after the nthexcitation pulse-acquire repetition, Mn, is:

Mn¼ M0ð1 ? wÞCnEnþ M0w

where E ¼ expð?TR=T1Þ, C ¼ cosðaÞ and w ¼

tween two successive excitation pulses and a is the r.f. pulse tip an-

gle. In conventional SPI [5] where k-space is sampled in a rectilinear

trajectory, magnetization is saturated when at the extremities of k-

space and the remaining k-space points are sampled with the same

magnetization. This results in a comparatively poor SNR, which is

generally determined by the centre of k-space, hence the preference

for centric sampling trajectories which commence at the centre of

k-space. An additional advantage is that in a centric scan there is

no T1weighting of the origin of k-space and hence total image

intensity is directly proportional to the concentration of the sig-

nal-bearing species [4,8,15,17].

The signal intensity, S(r) at any point, r, in a SPI image will be

given (ignoring T1relaxation effects) by [5]:

?

q(r) is the local1H density (1H NMR is used in this paper exclu-

sively) and tpis the phase encoding time. In cases where tpis much

shorter than T?

tion; however, if tpis comparable or longer than T?

sity is attenuated by relaxation effects. Acquisition of S(r) at various

ð1Þ

1?E

1?CE. TR is the time be-

SðrÞ / qðrÞexp ?

tp

T2?ðrÞ

?

sinðaÞð2Þ

2, S(r) will be relatively unaffected by signal relaxa-

2, the signal inten-

74

P. Parasoglou et al./Journal of Magn

Page 10

tion; however, if tpis comparable or longer than T?

sity is attenuated by relaxation effects. Acquisition of S(r) at various

values of tp, however, allows us to calculate T?

therefore produce a comparatively quantitative image of q(r).

With the above sampling method (as presented in [9]), once the

k-space points have been ranked then a subset of them with the

highest intensity can be acquired in a sparse k-space sampling

scheme. In order for this method to be used for compressed sens-

ing, it is required that under-sampling 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 k-space is

shown. The resultant selected k-space 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 under-sampled. By comparison conventional 2D fre-

quency-phase-encoding imaging, as used by Lustig et al. [11], can

only be usefully under-sampled in the phase dimension. In addi-

tion with respect to SPI, there are no constraints on the k-space

sampling trajectory.

2, the signal inten-

2ðrÞ using Eq. (2) and

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.

Page 11

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) k-space modulus map (c) our

sampling trajectory employed to acquire 20% of the highest intensity k-space

points.

gnetic Resonance 201 (2009) 72–80

Page 12

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 k-space measurements further.

Typically in SPI, k-space is sampled one point per acquisition

using an appropriate combination of phase-encoding gradients.

To improve the efficiency of single point imaging techniques, Bal-

com and co-workers have proposed the acquisition of multiple

points of the FID [20–22], which can be co-added to increase the

SNR. These points, at increasing temporal locations along the FID,

however, correspond to increasing k-space co-ordinates and effec-

tively a shrinking field of view (FOV). Rescaling to the same FOV is

achieved via use of the chirp z-transform [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

Page 13

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 l1-

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

under-sampled Fourier transform to k-space and y are the k-space

measurements. The reconstruction is then obtained by solving the

following constrained optimization problem:

5000

Number of Coefficients

1000015000

0

2

4

6

8

10

12

x 10

5

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%).

Page 14

under-sampled Fourier transform to k-space and y are the k-space

measurements. The reconstruction is then obtained by solving the

following constrained optimization problem:

minjjWmjj1

s ? tjjFm ? yjj2< e

ð3Þ

where e is a threshold that can be set to the expected noise level.

The l1-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 l1-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 k-space 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

Page 15

-100-75 -50 -25

Pixel Error (%)

025 50 75100

0

10

20

30

40

50

60

70

Number of Pixels

SNR = 5

SNR = ∞