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Compressed sensing in k-space: from magnetic resonanceimaging and synthetic aperture radar

Mike E. Davies, Chaoran Du, Shaun I. Kelly, Ian W. Marshall, Gabriel Rilling and Yuehui Tao

University of Edinburgh, Edinburgh EH9 3JL, U.K

ABSTRACT

We consider two imaging applications of compressed sensing where the acquired data corresponds to samplesin the Fourier domain (aka k- space). The first one is magnetic resonance imaging (MRI), which has beenone of the standard examples in the compressed sensing literature. The second one is synthetic aperture radar(SAR). We consider the practical issues of applying compressed sensing ideas in these two applications notingthat the physical prossesses involved in these two sensing modalities are very different. We consider the issuesof: appropriate image models and sampling strategies, dealing with noise, and the need for calibration.

Keywords: compressed sensing, k-space sampling, MRI, SAR

1. INTRODUCTION

To digitally acquire images it is necessary to reduce the information into a finite set of numbers. This sampling hastraditionally been done by taking equally spaced signal readings in the appropriate acquisition domain, where thenumber of measurements taken is directly related to the essential bandwidth of the image of interest. However,the recent theory of compressed sensing (CS) offers the potential to reduce the number of measurements requiredbased around prior knowledge of the signals/images of interest.1,2 The theory is potentially applicable to a widerange of signal acquisition tasks where it is feasible to take measurements that are in some way incoherent to theassumed signal model.3 For example, a key motivating problem for CS has been magnetic-resonance imaging(MRI), where the aim is to reconstruct detailed anatomical images of a patient from acquired lines in k-space (thespatial Fourier Transform of the image). The acquisition system achieves this by exciting the spins of protonsthrough a sophisticated arangement of magnetic fields. The acquisition of each line in k-space takes the orderof 5 msec. Therefore reducing the number of lines that are needed will potentially reduce the acquisition time.Since MRI images are often sparse in the spatial (e.g. angiography) or wavelet (e.g. brain images) domains CSoffers a possible solution. CS also has potential applications in a wide range of other imaging modalities thatbroadly sample in k-space such as synthetic aperture radar imaging and X-ray tomography.

In order to realize these ideas in practice it is necessary to consider the various engineering challenges as-sociated with a specific modality. The nature of the image must first be considered to define the appropriateform of regularization. The different physical acquisition processes result will inevitably result in different issuesassociated with the sampling mechanism, the treatment of noise and the issue of calibration. Finally the endgoal of the imaging will also influence the design choices that need to be made. In this paper we illustrate howthese ideas impact on two very specific applications of CS that are currently being explored at the Universityof Edinburgh to: (1) carotid blood flow measurement using phase contrast MRI;4,5 and (2) target location andclassification in spotlight SAR imaging.6–9 Although the underlying k-space acquisition appears similar theactual solutions to these problems are very different.

Further author information: (Send correspondence to Mike E. Davies)M.E.D: e-mail: [email protected], Telephone: +44 131 650 5795

Wavelets and Sparsity XIV, edited by Manos Papadakis, Dimitri Van De Ville, Vivek K. Goyal, Proc. of SPIE Vol. 8138, 81381G · © 2011 SPIE · CCC code: 0277-786X/11/$18

doi: 10.1117/12.893446

Proc. of SPIE Vol. 8138 81381G-1

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2. CS IN MRI

CS offers the opportunity of reduced scan times in MRI and therefore opens up the potential for an increaseduse of more advanced imaging modalities. It should be noted that, in practice, CS will not offer significant gainsto static 2D imaging since the time taken to fully sample such an image is not prohibitive (approximately 1sec. for 256 × 256 image). The scenarios where CS offers most potential are therefore the more advanced MRImodalities such as dynamic MRI, perfusion MRI, 3D MRI, etc.

For dynamic (2D) MRI, multiple time frames are recorded to form a movie. Assuming that each line issampled at the Nyquist rate (Cartesian sampling) in, say, the x-direction, the dynamic MRI reconstruction canbe performed for each (y, t) slice at each x-location independently. Classical undersampling strategies for dynamicMRI are generally based on tiling of the signal support in the 2D Fourier domain (y; f), where f is the frequencywith respect to time (t).10,11 This exploits a 2D equivalent of the classical 1D Nyquist sampling theory wherethe achievable acceleration is limited by how packable the (y; f) support is using an optimal lattice structure.Other strategies such as kt-BLAST12 allow some aliasing but minimize its impact through a judicious latticesampling and a Wiener-filter based (linear) reconstruction. In contrast, application of CS techniques to dynamicMRI13adopt random sampling patterns, e.g. the use of a random ordering of k-space samples, followed by one ofthe many nonlinear sparse reconstruction algorithms. Here, we describe an attempt to obtain the best of bothworlds. We consider the specific dynamic MRI example of phase contrast carotid blood flow measurement. Forfull details see.5

2.1 Accelerating phase contrast carotid blood flow MRI

Measurement of carotid blood flow can be achieved via MRI using a technique referred to as phase constrastvelocity encoding. This involves acquiring two sets of time frames, applying a velocity encoding to one and usingthe other one as a reference. The velocity flow perpendicular to the image can then be obtained from the phasedifference between the two sets of frames. As the artery is localized a single coil may be used and the usualcalibration issues associated with coil sensitivity can be avoided.

For a typical (y, t) plane at a given x location going through the right common carotid artery (RCCA), thesupport, S of the signal ux in the (y, f) domain can be modelled as a cross (see Fig. 1). The support can beviewed as the combination of a static part with content only at f = 0 (i.e. no blood flow) and a dynamic partthat is localized in a small region of interest corresponding to the RCCA. Let us consider a single x location.The task is therefore to design an appropriate undersampling operator, Φ, that samples in the ky-t space (wehave supressed any indexing with respect to x) such that the signal u can be recovered from the observedmeasurements v = Φu. Note that the selected Φ acts on each location of x as we are actually fully samplinglines in kx.

Since the signal, u, is sparse in the (y, f) domain we can apply conventional CS strategies using randomsampling in k-t space and `1 reconstruction. Although CS theory tells us that random sampling is good, inthe general sparse setting it is common practice to densely sample the centre of k-space in order to capture theregion of dominant energy for the image sequence. However, CS theory gives us little guidance as to how best todo this. Applying such CS techniques to the carotid artery blood flow measurement problem was done in4 andacceleration factors of about 3− 4 were achieved while not incurring significant errors in the velocity estimates.Studies of different sampling patterns in k-t space have indicated that the process is reasonably insensitive tothe precise density of the pattern as long as the centre of k space is fully sampled.

Achieving accelerations beyond 3−4 using standard CS techniques appears challenging. However, we actuallyknow much more about the signal support set in (y, f). Specifically, we know the location of the static supportand that the dynamic support forms a localized contiguous block associated with the artery location (see Fig. 1).Note it is only the location of the artery that is unknown. Intuitively this means we have a much smaller searchspace to consider in the signal reconstruction. We can also exploit the support structure to design samplingpatterns that have good properties without recourse to random sampling. To do this we split the support intotwo parts which are much more amenable to traditional lattice sampling.



x

y

f (Fourier domain of t)

y

Figure 1. Left: Carotid slice with the RCCA marked in yellow. An x slice going through the RCCA is marked in blue.Right: signal support model: static signal (dark gray) and dynamic ROI (light gray).

2.2 Multi-lattice sampling and support splitting

Random sampling in CS is preferred because it can be guaranteed (with high probability) to provide a nearoptimal acceleration and conditioning in the worst case over all sparse support sets. However, when the spaceof possible support sets is highly restricted then random sampling may well be non-optimal. Indeed, we alreadyknow from classical sampling theory that for a packable support set in 2D a lattice sampling strategy is optimaland provides a reconstruction operator that has an optimal condition number (equal to 1).

If we divide the support set in Fig. 1 into two packable subsets: the static and the dynamic components,we can consider a sampling strategy that is the composition of optimal sampling lattices for each componentindividually. We term this multi-lattice sampling. It can be seen as a generalization of the multi-coset samplingframework investigated by Feng and Bresler.14,15 An example of ’individual’ optimal sampling patterns isillustrated in Fig. 2. Let us consider these support sets separately. Assuming the width of the dynamic bandis at most B pixels, then B parallel lines in a dynamic lattice are required for its reconstruction, provided itslocation is known. For the static component it is sufficient to sample each ky location once, resulting in standardNyquist sampling of the static part.

The composition of two such lattices is sufficient to enable reconstruction of the combined support if therows of the two sampling operators are linearly independent. Furthermore, additional redundancy can be addedto one or both sampling lattices to make the location of the dynamic support identifiable. Given the limitednumber of options in generating such lattice pairs it is possible to numerically optimize the composite samplingpattern through an exhaustive search in order to achieve an optimal conditioning and maximal acceleration.

In a similar manner the location of the dynamic support can also be found through exhaustive seach. Assum-ing an upper bound B on the size of the dynamic region is known, then the detection of the dynamic support canbe achieved by an exhaustive search for the support that is most consistent with the measurements. As thereare only N − B + 1 possible dynamic supports sets this procedure has linear time complexity. This should becompared with the usually intractable `0 exhaustive search in standard CS.

Once the full support, S has been determined the signal can be reconstructed by applying the pseudo-inverseof the sampling operator restricted to this support: u = Φ†Sv,

2.3 Further refinements

The above strategy enables acceleration rates that are competitive with the standard CS strategy to be achieved.To further increase acceleration we can make two simple modifications. First, we borrow an idea from kt-



t

ky

t

ky

f

y

f

y

Figure 2. Example of ’individually’ optimal lattices (top) and their associated (y, f) packing (bottom) for the static (left)and dynamic (right) components.

BLAST12 and use a regularized Wiener filter based estimator restricted to the detected support, which in thiscase takes the form:

uS = WΦHS

(ΦSWΦH

S + σ2I)−1

v. (1)

where W is a diagonal matrix representing the covariance of the signal and σ2 is the measurement noise variance.In k-t BLAST W is estimated through a training sequence. For the blood flow measurement application we areable to use a simple parametric model for W .5

For the second refinement we observe that the blood velocity profile generally varies smoothly across the artery.therefore the high spatial frequencies of the dynamic component may be ignored. The same is not true for thestatic part of the image which typically exhibits edge discontinuities. The benefits of this observation are two fold.Setting the high spatial frequencies for the dynamic component to zero further regularizes the reconstruction,reducing the reconstruction error. However, more importantly, ignoring the high spatial frequencies for thedynamic component means that we no longer need to sample the dynamic lattice near the edges of k-space. Thisallows us to achieve even higher acceleration factors.

An example of the reconstructed flow possible using this technique is shown in Fig. 3. The plot shows thevelocity maps for a fully sampled scan and for an 8× acceleration using multi-lattice sampling of the same data.The quality of the reconstructed flow is comparable to that achieved using standard CS reconstruction withrandom sampling at 4× acceleration.

We finally mention it should be possible generalize the multi-lattice sampling to scenarios with multipledynamic components. Here the support could also be found through exhaustive search as long as the numberremained small. Alternatively a model based CS recontruction technique could be adopted.16

3. CS IN SAR IMAGING

Synthetic aperture radar (SAR) is an active ground imaging system based on coherent processing of multipleradar echoes acquired along the path of a moving platform (aircraft or satellite). Assuming free space propagationof the radar waves, scalar wavefields, no mobile targets and single bounce scattering by ground reflectors, themap from the SAR image, u, to the received radar echoes, v, can be modelled as linear, v = Φu. Under furtherassumptions such as constant terrain elevation and flat wavefronts (far-field scenario) the SAR sensing operator



fully sampled 8× acceleration

Figure 3. The velocity maps from a fully sampled scan (left) and an 8× acceleration (right) based on multi-lattice samplingof the same raw data. The plots show the velocity time histories (blue) for each pixel of the reconstructed artery.

Φ can be viewed as sampling k-space through the Fourier slice theorem.17 Based upon this assumption the fullbackprojection algorithm is usually replaced by the popular Polar Format Algorithm (PFA) that works directlyin k-space. Note, however, that with the advent of fast back and re-projection operators18,19 fast reconstructionis not actually constrained by these simlifying assumptions. This provides enhanced image quality and allowsscenarios such as general flight paths and non flat terrain models to be dealt with.

The application of Compressed Sensing to SAR offers the possibility to reconstruct images from partial SARdata. This is of operational interest for a number of reasons, such as:

• missing radar echoes at some locations along the synthetic aperture. This typically results from the needto use the radar antenna for other tasks, e.g. multi-function radar.

• missing frequency bands in the radar signal. This occurs in ultra wide band SAR where the desired band ofthe radar contains sub-bands that are saturated by other communications systems or in which transmissionis not allowed.20

• arbitrary missing data. This may occur if one throws away samples from the acquired data to reduce theamount of data stored on the acquisition platform.

However, in order to apply the principles of CS to SAR it is first necessary to identify the exploitable structurewithin the image data.

3.1 SAR image model

In contrast to the MRI example considered in the previous section raw SAR data has significantly differentstatistical properties. Like all coherent imaging systems SAR suffers greatly from speckle noise. Speckle noise isgenerated when there are a large number of independent scatterers within a range cell. The complex reflectionsthen combine in an incoherent manner and the reconstructed pixel can be modelled as the sum of complex-valuedreflectivities of the sub-pixel objects. Moreover, for two adjacent pixels, the sub-pixel objects can be assumedindependent. This leads to a multiplicative i.i.d. complex Gaussian noise model for the complex-valued imageat such locations, that is often referred to as speckle noise. In constrast to this corner and edge objects aredominated by specular reflection effects. Such objects are common in man-made structures or vehicles and leadto very high magnitudes for the corresponding pixels of the SAR image, typically 103 times larger than themagnitude of the background clutter.



Viewed as a whole, most of the SAR image contains low level multiplicative speckle noise which behaves likenonstationary complex Gaussian white noise. In terms of information content, this means that the complex-valued SAR image has a very high entropy and therefore a very low compressibility. In particular it cannot bemodelled as sparse in any dictionary which precludes the use of compressed sensing ideas to recover the full SARimage from partial SAR data. However, the dominant reflections off man-made structures do exhibit a strongsparsity in the native pixel domain. We can therefore split the image into these two parts

u = us + ubg, (2)

with us corresponding to the few very bright pixels and ubg to the “background” lower reflectivity clutter pixelscontaminated by speckle noise. The success of CS applied to SAR therefore depends heavily on which partsof the image are of interest. For surveillance and military applications imaging of the man-made objects is aprimary interest and are good candidate applications for CS.

3.2 mixed `1/`2 reconstructionGiven the image decomposition proposed in (2) it is fairly straight forward to modify classical compressed sensingreconstruction to tackle this problem. Here we use a mixed `1 and `2 regularisation, motivated by the standard CStheory.1,2 The framework allows the accurate reconstruction of coherent point targets while still reconstructingthe background albeit at a reduced resolution and constrast. Furthermore, when used as a preprocessing step forAutomatic Target Recognition (ATR) good classification rates are still achievable as significant undersamplingrates, e.g. 93% from 25% of the data. See7,9 for further details.

The sparse component us is first reconstructed using an `1 penalized Least Squares reconstruction:

us = argminu

‖v − Φu‖22 + λ ‖u‖1 (3)

where λ is a constant which controls the level of sparsity in the reconstructed image and must be set based onthe expected target-to-clutter ratio. Similar reconstruction techniques has already been used for fully-sampleddata in the context of superresolution.21,22

Although the background clutter will not be recovered using (3) the residual error, (v − Φus), still containsinformation on the background image. The simplest means to make the inversion well-posed is to use an `2regularisation on the SAR image. We therefore reconstruct ubg as follows:

ubg = argminu

‖v − Φ(us + u)‖22 + λ ‖u‖2 (4)

This reconstruction will clearly suffer from the effects of undersampling, however, recall that even with fullysampled data the background image is heavily corrupted by multiplicative noise. The undersampled backgroundreconstruction typically suffers from a loss of contrast and resolution. However, by first extracting the coherentcomponent, the background is no longer swamped by the sidelobes of bright point targets. Figures 4(a) and 4(b)demonstrate the visual improvement achievable at modest (50%) undersampling ratios.

3.3 Auto-focusWe finally consider an additional complication in spotlight-mode SAR systems, which is that the round trippropagation delay to the scene centre must be accurately estimated. Otherwise phase errors can occur. Errorsin this estimate introduce unknown phase errors to the acquired data.23 These phase errors can degrade andproduce distortions in the reconstructed image. Under mild assumptions the phase error can be treated as anunknown constant phase error in the range Fourier transformed phase history.23 In fully sampled SAR thephase errors can be corrected using classical autofocus methods such as the Phase Gradient Autofocus (PGA)method,24 which implicitly exploits the sparse point like nature of SAR images to correct phase errors. Phasecorrection can therefore be naturally incorporated explicitly into the sparsity framework.25 We can thereforemodify the reconstruction cost function to generate a CS-autofocus as follows

(φ,us) = argminφ,u

∥∥v − diag(ejφ)Φu∥∥2

2+ λ ‖u‖1 (5)



where, φ ∈ [−π, π]m is a vector containing the estimated phase errors and diag(ejφ) is a diagonal matrixcontaining the elements ejφ. We now jointly estimate the sparse image component and the phase errors togetherin an iterative alternating minimization approach. Fixing φ reduces the problem back to (3), while with u fixed adirect solution for the program for φ is obtained with almost negligible computation. Although the cost functionis no longer convex we have found this strategy empirically to gives good results. An example of CS performancewith and without autofocus is shown in figures 4(c) and 4(d).

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Figure 4. Image formation using 4◦ of the Gotcha data set, 50% uniform randomly under-sampled in cross-range. (a) `2regularised LS reconstruction. (b) Mixed `1/`2 regularised LS reconstruction. (c) Mixed `1/`2 regularised LS reconstruc-tion with phase errors. (d) Mixed `1/`2 regularised LS reconstruction with phase errors using auto-focus.



4. CONCLUSIONS

In this paper we have discussed some practical issues that arise when attempting to apply the clean mathematicaltheory of CS in real applications. Issues such as the appropriate signal model, sampling strategy and calibrationneed to be considered. We have also found that often it is the noise (including modelling inaccuracies) thatdefines the limiting factor on our ability to undersample.

ACKNOWLEDGMENTS

This work was supported in part by: the MOD Competition of Ideas contract, grant number RT/COM/5/028,EPSRC grants [EP/F039697/1, EP/H012370/1], the MOD University Defence Research Centre on Signal Pro-cessing and the European Commission through the SMALL project under FET-Open, grant number 225913.

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