REGULARIZED SENSE RECONSTRUCTION USING ITERATIVELY REFINED
TOTAL VARIATION METHOD
Bo Liu1, Leslie Ying1, Michael Steckner2, Jun Xie3 and Jinhua Sheng1
1
Dept. of Electrical Engineering and Computer Science, University of Wisconsin, Milwaukee, WI
2
Hitachi Medical System America, Twinsburg, OH
3
Dept. of Biophysics, Medical College of Wisconsin, Milwaukee, WI
Index Terms— SENSE, total variation regularization,
Bregman iteration
methods have been proposed for the regularization image
[5,6], but none of them can reconstruct an aliasing-free,
edge-preserving image at high reduction factors.
To address the ill-conditioning problem at high
reduction factors, we propose an improved SENSE
algorithm based on total variation (TV) regularization with
iterated refinement by Bregman iteration. Since the TV
method was first applied to image processing by Rudin and
Osher [8], many successful applications of TV
regularization have been reported including MR image
reconstruction [9-11]. The TV technique uses the bounded
variation norm for the regularization term, and the best
solution is the one that balance the data consistency and
total variation. The advantage of TV regularization lies in
that it penalizes highly oscillatory solution but allows sharp
discontinuities (edges) [10]. With the addition of Bregman
iteration, the TV regularization has been shown to
reconstruct MR images with fine details [12-14]. In this
paper, we applied TV regularization to SENSE
reconstruction and demonstrated its superior performance
when high reduction factors are used.
1. INTRODUCTION
2. TIKHONOV REGULARIZED SENSE
ABSTRACT
SENSE has been widely accepted as one of the standard
reconstruction algorithms for Parallel MRI. When large
acceleration factors are employed, the SENSE
reconstruction becomes very ill-conditioned. For Cartesian
SENSE, Tikhonov regularization has been commonly used.
However, the Tikhonov regularized image usually tends to
be overly smooth, and a high-quality regularization image is
desirable to alleviate this problem but is not available. In
this paper, we propose a new SENSE regularization
technique that is based on total variation with iterated
refinement using Bregman iteration. It penalizes highly
oscillatory noise but allows sharp edges in reconstruction
without the need for prior information. In addition, the
Bregman iteration refines the image details iteratively. The
method is shown to be able to significantly reduce the
artifacts in SENSE reconstruction.
Thanks to the desirable feature of fast data acquisition in kspace without the need of improving the gradient
performance, parallel MRI has been extensively
investigated since its emergence. Several reconstruction
methods have been established in order to unfold the aliased
images caused by sampling at a rate lower than Nyquist rate
during acquisition [1,2]. Among them, SENSE (SENSitivity
Encoding) [1] is known to theoretically be able to give the
exact reconstruction of the imaged object in absent of noise.
However, in practice, a well-known problem of SENSE is
the amplification of data noise due to the ill-conditioned
nature of the inverse problem which is especially serious
when a high reduction factor is employed. General solutions
include optimization of the coil geometry [3, 4] and
application
of
mathematical
regularizations
to
reconstruction [5-7]. A natural choice for regularization is
the Tikhonov scheme primarily due to the existence of a
closed-form solution. A common problem with Tikhonov
regularization is that the reconstruction is overly smooth
unless a high quality regularization image is used. Several
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The parallel imaging equation for the k-space data acquired
at the lth receiver coil can be expressed as
G G
G
G
G
G
D l ( k m ) ³ U ( r ) s l ( r ) e i 2 S k m r dr
(1)
G
G
where U (r ) is the spin density of the desired object, Dl ( k m )
G
is the data measured at the k-space location km by the lth
G
receiver coil whose sensitivity profile is sl (r) . In parallel
imaging, these k-space data from all L channels are acquired
simultaneously, but sampled with a rate R times lower than
the Nyquist sampling rate such that the data acquisition time
is reduced by a factor of R.
After discretization, Eq. (1) can be represented as a
linear equation [1]:
G
G
Ef
d ,
(2)
G
G
where f and d are vectors of the desired image on voxels
and the acquired k-space data, and the element of the
encoding matrix E is given by
ISBI 2007
G G
G
e i 2 S k m ˜ rn s l ( rn ) .
E ^l , m `, n
(3)
The solution to Eq. (2 ) is given by
G
G
f
( E H Ȍ 1E ) 1 E H Ȍ 1 d ,
(4)
where the superscript H indicates transposed complex
conjugate, Ȍ is the receiver noise covariance[1].
When the linear equation in Eq. (2) is ill-conditioned,
the data noise can be amplified which leads to poor
reconstruction. Tikhonov regularization has been used to
address the ill-conditioning problem. In Tiknohov
framework, the reconstruction is given by:
G
G
G 2
G G
f reg arg min
d Ef O ( f f r ) ,
(5)
G
f
^
`
where the regularization parameter O is chosen to balance
the data fitting error and the penalty (or regularization) term
formed from the difference between the expected solution
G
and the prior image f r known as the regularization image.
G
A closed-form solution for f reg exists and is given by
G
G
G
G
(6)
f reg f r (E H ȥ 1E O I ) 1 E H ȥ 1 (d Ef r ) .
Because Tikhonov formulation in Eq. (5) uses L2 norm, the
regularized reconstruction usually suffers from blurring
G
effects unless a high quality regularization image f r is
available. In practice, a low resolution image generated
from several additional lines acquired at the center of the kspace is used as the regularization image. However, its
limited quality cannot remove aliasing artifacts at high
reduction factor R.
Another option to address the ill-conditioning problem
is to use the iterative conjugate gradient (CG) method for
SENSE reconstruction [15]. The CG method has inherent
regularization capability because the low-frequency
components of the solution tend to converge faster than the
high-frequency components. However, in the case of high
reduction factor, the inherent regularization is not sufficient.
3. PROPOSED REGULARIZATION METHOD
3.1. Theory
To better address the ill-conditioning problem at high
reduction factors, we propose to use iteratively refined TV
regularization for SENSE reconstruction. TV regularization
has been proved to be able to recover piecewise smooth
functions without smoothing sharp discontinuities. With TV
regularization, the reconstructed image is given by
G
G
G2
G
,
(7)
f reg arg min d Ef O f
f
^
BV
`
where the regularization term changes from L2 norm to
bounded variance (BV) norm. To keep convexity, we define
the BV norm as
G
2
2
2
2
f
¦ ’x fr ’x fi ’y fr ’y fi , (8)
BV
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where f r and fi are real and imaginary part of the complex
MR image, and ’ x and ’ x denote the gradient along x and
y respectively [13]. Total variation is a characteristic of an
image; the value is usually small for a clean image, but
increases with addition of noise. The use of BV norm in Eq.
(7) allows us to recover images with edges.
To further improve the fine details in reconstruction, we use
an iterative regularization procedure. Instead of stopping at
G
the solution f 0 to Eq. (7), we use it to iteratively refined
TV regularization:
G
G
G
G G
f k arg min{|| d Ef || OD( f , f k 1 )} , for k ! 0 . (9)
f
G
G G
where D( f , f k ) is the Bregman distance between f and
G
f k associated with the BV norm, which is defined as [16]
G G
G
G
G G
G
D ( f , f k ) || f ||BV || f k ||BV f f k , w (|| f k ||BV ) ! (10)
G
where ˜,˜ ! denotes the inner product and w (|| f k ||BV ) is
G
an element of the sub-gradient of the BV norm at point f k .
G G
G
Since the BV norm is convex, D( f , f k ) is convex in f for
G
each f k . The Bregman distance is an indication of the
G
G
increase in || f || BV over || f k ||BV above linear growth with
G
G
slope w (|| f k ||BV ) . It has been shown that the sequence Ef k
G
monotonically converges to the acquired noisy data d , and
for O sufficiently large, the sequence also monotonically
G
gets closer to the noise free data Ef true , whose convergence
is faster than noise [13]. Therefore, if a good stopping rule
is applied, Bregman iteration can recover fine details of the
image before the noise. According to [13], the minimization
in Eq (10) is equivalent to
G
G G
G
G
f k arg min{|| d vk 1 Ef || O || f ||BV } ,
(11)
f
G
G
where vk vk 1 d Ef k for k ! 0, v0 0 . It is seen that the
above formulation is the same as TV regularization except
G
G G
that d is replaced by d v k 1 .
3.2. Optimization
To solve the minimization in Eq (7), we first obtain the
Euler-Lagrange equation:
G
G G
G
’f
0 O’ ˜ ( G ) 2EH (d vk 1 Ef ) ,
(12)
’f
We then can establish the following time dependent partial
differential equation (PDE):
G
G
G G
G
’f
f O’ ˜ ( G ) 2EH (d vk 1 Ef ) .
(13)
’f
In order to solve Eq. (10), we employ the time marching
method [8] which is an iterative numerical method.
Specially, we update the solution iteratively according to
G
Gw
G G
G
’f kw
f k O't ˜ (’ ˜ ( G w ) 2 E H (d vk 1 Ef kw )) (14)
’f k
G
where w is the iteration number starting from zero with f 00
being the initial guess for the desired image, and 't can be
regarded as the time marching step. Suppose W iterations
are needed for solving the minimization in Eq. (11), then the
G
G
G
algorithm proceeds as f 00 , f 01 , …, f 0W (zeroth Bregman),
G
G
G
G
f10 (same as f 0W ), f11 , ..., f1W , etc. The convergence of
minimization is guaranteed [17].
G
f kw1
4. EXPERIMENT RESULTS
The proposal method was tested on a number of data sets.
We show results of phantom data collected on a Hitachi
Airis Elite (Kashiwa, Chiba, Japan) 0.3T permanent magnet
scanner with a four-channel head coil and a single slice spin
echo sequence (TE/TR = 40/1000 ms, 8.4KHz bw,
256 u 256 matrix size, FOV = 220 mm2), and results of in
vivo data acquired on a 3T GE Excite MR system
(Waukesha, WI, USA) with an eight-channel head coil
(MRI Devices, Waukesha, WI, USA) and a 3-D SPGR
pulse sequence (TE = 2.38 ms, TR = 7.32 ms, flip angle =
12°, matrix = 200 u 200, FOV = 187 mm2, slice thickness
= 1.2mm). In both data sets, the full k-space data were
acquired and their sum of square reconstruction is used as
the gold standard for comparison. The data were
downsampled to simulate a 4X reduction factor. The central
32 fully-sampled phase encoding lines were acquired as the
auto-calibration signal (ACS). The ACS was used to
generate a set of low resolution images from multiple coils.
The sensitivity profiles of the coils were estimated from
these images. The sum of the square of the low resolution
images were also used as the initial image for the proposed
regularization method. We used 2 Bregman iterations of Eq.
(11) (k=0,1), with each iteration of minimization solved by
30 iterations of Eq. (14) (w=0, 1, …, 29). The reconstructed
images are shown in Fig. 1 for phantom and Fig. 2 for in
vivo data.
For comparison, reconstructions from the basic SENSE
[1] and Tikhonov regularized SENSE [7] are also shown in
Fig. 1 and 2. in Tikhonov regularized SENSE, the low
resolution image from central k-space is used as the
regularization image. It is seen that all these reconstructions
contain different levels of artifacts at this high reduction
factor of 4. Among all these methods, the proposed method
gives the best reconstruction.
The proposed method converged quickly after 2
Bregman iterations. Each iteration took 18 seconds on a
2.8GHz CPU/512MB RAM PC, which is on slightly longer
than the running time of 5.8 and 16.8 seconds for (c) and (d),
respectively.
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Fig. 1: SENSE reconstruction from phantom acquired with a 4channel coil and a 4X reduction factor. (a) Gold standard
reconstructed from fully sampled data, and (b) SENSE
reconstruction with the proposed regularization method, (c) basic
SENSE, (d) Tikhonov regularized SENSE.
Fig. 2: SENSE reconstruction from in vivo data acquired with an
8- channel head coil and a 4X reduction factor. (a)-(d) are the same
as in Fig. 1.
5. DISCUSSION
Our experiment results demonstrated that TV regularization
is ideal for piecewise smooth functions whose total
variation is small. Because the phantom image in Fig. 1
satisfies such piecewise smooth condition, the proposed
method showed a significant improvement over the existing
techniques. In contrast, the improvement for the in vivo
brain image is not as dramatic; there exists a tradeoff
between least total variation and data consistency.
In addition, we observed that the fine details of image are
gradually reconstructed as the number of Bregman iterations
increases; as the number further increases, noise will
become apparent. As in Fig. 3, the image with two Bregman
iterations recovers the main structure, and an additional
iteration recovers more fine details. But as the iteration
number further increases, noise becomes apparent. It
suggests that the iteration has to stop at some appropriate
stage. The stopping rule similar to [13] was used here to
ensure the best results. In general, the convergence is fast
and computational complexity of each iteration is relatively
low as shown in our experiments.
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Fig. 3: Reconstructions using the proposed method with 1, 2, 3,
and 4 Bregman iterations are shown in (a)-(d).
6. CONCLUSION
A novel regularized SENSE reconstruction method has been
proposed. Based on TV regularization and Bregman
iteration, the method reduces the image artifacts caused by
the ill-conditioned nature of high reduction factors. Results
show this algorithm achieves image quality superior to the
existing methods with little increase in computational time.
Acknowledgments: The authors would like to thank Mr.
Guoqiang Yu for helpful and valuable discussion.
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