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. Author manuscript; available in PMC: 2011 Jun 23.
Published in final edited form as: Conf Proc IEEE Eng Med Biol Soc. 2010;2010:3390–3393. doi: 10.1109/IEMBS.2010.5627934

PSF Model-Based Reconstruction with Sparsity Constraint: Algorithm and Application to Real-Time Cardiac MRI

Bo Zhao 1, Justin P Haldar 1, Zhi-Pei Liang 1
PMCID: PMC3121182  NIHMSID: NIHMS300530  PMID: 21097243

Abstract

The partially separable function (PSF) model has been successfully used to reconstruct cardiac MR images with high spatiotemporal resolution from sparsely sampled (k,t)-space data. However, the underlying model fitting problem is often ill-conditioned due to temporal undersampling, and image artifacts can result if reconstruction is based solely on the data consistency constraints. This paper proposes a new method to regularize the inverse problem using sparsity constraints. The method enables both partial separability (or low-rankness) and sparsity constraints to be used simultaneously for high-quality image reconstruction from undersampled (k,t)-space data. The proposed method is described and reconstruction results with cardiac imaging data are presented to illustrate its performance.

I. INTRODUCTION

Several model-based methods have been proposed to enable reconstruction from highly undersampled (k, t)-space data [1]–[5]. The partially separable function (PSF) model assumes that the (k, t)-space MR signal s(k, t) is partially separable to the Lth order [4] such that

s(k,t)==1Lu(k)v(t), (1)

where {u(k)}=1L and {v(t)}=1L can be treated as spatial and temporal basis functions, respectively. Due to strong spatiotemporal correlations in cardiac MR signals, L can usually be chosen to be very small [6], which significantly reduces the number of degrees of freedom of the model.

Consider the following Casorati matrix formed from the samples of s(k, t),

S=[s(k1,t1)s(k1,tM)s(kN,t1)s(kN,tM)], (2)

where N represents the number of k-space points at each time instant for a total of M instants in an ‘ideal’ data set {s(kn,tm)}n,m=1N,M. If s(k, t) is Lth-order partially separable, it is easy to show that the rank of S is upper bounded by L [4] and S can be decomposed as

S=UkVt, (3)

where Vt ∈ ℂL×M is a basis for the temporal subspace of S, and Uk ∈ ℂN×L is a matrix of k-space coefficients. With sparse sampling, many entries of S are missing. The actual measured data can be expressed as

d=Ω(S)+ξ, (4)

where d ∈ ℂp is the measured data, Ω: ℂN×M → ℂp is the sparse sampling operator, and ξ ∈ ℂp represents measurement noise. In this paper, we assume that Vt is known (i.e., predetermined from navigator signals as is done in [4], [6]) and our goal is to determine Uk from d. The basic PSF method solves the following unconstrained problem:

U^k=argminUkCN×L||dΩ(UkVt)||2. (5)

A necessary condition for (5) to be well-posed is that each k-space location is sampled at least L times. In practice, we need much more measurements to avoid a poorly conditioned inverse problem. In the previous work, spatial-spectral support information has been exploited to improve the conditioning of the problem and reduce reconstruction artifacts [7], [8].

In this paper, we use an (x, f)-domain sparsity constraint to regularize the inverse problem. An (x, f)-domain sparsity constraint has been used in previous work [9]–[11]. Our method is the first attempt to use both explicit partial separability and (x, f )-domain sparsity for image reconstruction from highly undersampled (k, t)-space data. The rest of paper is organized as follows: Section II describes the proposed formulation and image reconstruction algorithm; Section III presents an application of the proposed method to real-time cardiac MRI and discussion of related work, followed by a brief conclusion in Section IV.

II. PSF solution with sparsity constraint

Incorporating an (x, f)-domain sparsity constraint, the PSF reconstruction problem can be formulated as

graphic file with name nihms300530e1.jpg (6)

where ℱs represents a spatial Fourier transform operator, Ft is a temporal Fourier transform matrix, graphic file with name nihms300530u1.jpg, and λ is a regularization parameter. In the above expression, both ℱs and Ft are assumed to be orthonormal, and ||·||1 of a matrix is defined as the sum of the magnitudes of all entries in the matrix. The ℓ1 penalty (or ℓ1 regularization) in (6) has been widely used for finding sparse solutions to ill-posed inverse problems [12].

We propose to solve (6) using an additive half-quadratic regularization algorithm [11], [13], [14] and a continuation procedure [11], [15], [16]. First, the non-differentiable ℓ1 norm is replaced by a differentiable approximation

||UsVf||1n=1Nm=1Mϕ((UsVf)n,m), (7)

where ϕ(t): ℝ → ℝ is the Huber function defined as

ϕ(t)={t2/2α,iftα,tα/2,ift>α. (8)

Note that ϕ(t) is a continuously differentiable function, and the accuracy of the approximation in (7) is controlled by the parameter α. As α → 0, the Huber function approaches |t|. Furthermore, it can be shown that ϕ(t)=mingR{12α(tg)2+g}. Therefore, (7) can be rewritten as

n=1Nm=1Mϕ(UsVfn,m)=minGCN×M12α||UsVfG||F2+||G||1, (9)

where G is an auxiliary matrix and ||·||F denotes the Frobenius norm. Using (7) and (9), the problem in (6) can be approximated as

graphic file with name nihms300530e2.jpg (10)

We solve (10) using an alternating optimization procedure, which first optimizes Us with a fixed value of G, and then optimizes G with a fixed value of Us. This procedure is repeated until Us and G converge. At the ℓth iteration and for a fixed Us(1), minimizing (10) over G is accomplished using the following soft-shresholding operation

Gn,m()={0,ifQn,m<α,Qn,mQn,m(Qnmα),ifQn,mα, (11)

where Q=Us(1)Vf. With a fixed G(ℓ), minimizing (10) with respect to Us is equivalent to

graphic file with name nihms300530e3.jpg (12)

By changing variables Uk = ℱsUs and B(ℓ) = FsG(ℓ)Ft, we can convert (12) into

U^k=argminUkCN×L||Ω(UkVt)d||22+λ2αUkVtB()F2. (13)

Though (13) is a large-scale quadratic optimization problem, it has a simple decoupled structure, which enables very efficient evaluation. Specifically,

Ω(UkVt)=[Ω1(uk1Vt)ΩN(ukNVt)], (14)

where uknC1×L represents the nth row of Uk, and Ωn is the sampling operator that takes samples from the nth row of UkVt. Thus, (13) is equivalent to solving

u^kn=argminuknC1×L||Ωn(uknVt)dn||22+λ2αuknVtbn()F2 (15)

for 1≤nN, where dn is the measured data from the nth row of S, and bn() is the nth row of B(ℓ). In (15), each quadratic optimization problem is equivalent to

graphic file with name nihms300530e4.jpg (16)

for 1 ≤ nN, where Inline graphic(·) = ΩnVt), ℬn(·) = ·Vt, Inline graphic and Inline graphic are the adjoint operators of Inline graphic and ℬn, respectively. Since each normal equations in (16) only has L unknowns, it can be solved very efficiently. For example, ideally the conjugate gradient method can solve (15) or (16) in only L iterations if the exact arithmetic is assumed [17]. Moreover, the completely decoupled structure is quite suitable for parallel computation for further computational accelerations.

The above alternating optimization procedure gives an optimal solution to (10). To obtain a good approximate to (6), a continuation scheme is used. The continuation procedure starts with a large value of α, which causes the alternating algorithm to converge rapidly [14]. Then we decrease the value of α in (10) and use the previous solution as a ‘warm start’ for the new alternating optimization. The above procedure is repeated until the cost function of (10) well approximates the cost function in (6).

III. Results and Discussion

A. Simulation Results

The proposed method has been validated systematically using both human and rat cardiac phantoms. These phantoms were used to simulate free-breathing cardiac imaging experiments with variable heartbeat periods. Snapshot images from two phantoms in the diastolic phase of the cardiac cycle are shown in Fig. 1. In the experiments, 200 phase encodings and 256 samples per readout were used with the human phantom, while 256 phase encodings and 256 samples per readout were used with the rat phantom.

Fig. 1.

Fig. 1

Snapshot images in the human and rat phantoms.

The data acquisition scheme shown in Fig. 2 was used to collect measurements. This sampling pattern consists of acquiring training data in the center of k-space and imaging data in the other region of k-space (See Fig. 2). The training data is used for estimating Vt, while the imaging data is used to solve Us. For the simplicity of discussion, we define the number of data frames as M/Np, where Np represents the number of phase encodings at each time instant. In the simulation, 11 data frames of data were of data were generated for both phantoms.

Fig. 2.

Fig. 2

Data acquisition scheme in the simulation. Blue dots represent the training data, while black dots represent the imaging data.

In addition to the proposed method (order-16 PSF with sparsity constraint), two other reconstruction methods were applied to process this data: order-6 PSF (the best-performing basic PSF method for the data set tested) and ℓ1-regularized reconstruction without PSF modeling [10]. For both methods including ℓ1-regularization, the regularization parameters were chosen manually to optimize the performance.

Some representative snapshot images reconstructed from the human and rat data sets (zoomed-in to the heart) are shown in Fig. 3 and 4, respectively. It is clear that the proposed method has a much better spatial resolution than the other two methods, particularly in the myocardium and the blood pool, where the other two methods produced significant artifacts. Relative reconstruction error defined as, i.e.,

Fig. 3.

Fig. 3

Reconstructions of the human phantom (zoomed-in to the heart) using 11 data frames.

Fig. 4.

Fig. 4

Reconstructions of the rat phantom (zoomed-in to the heart) using 11 data frames.

error=||SU^kVt||F||S||F, (17)

are shown in Table I, which are consistent with the improved performance of the proposed method.

TABLE I.

Relative reconstruction errors.

Phantom Order-6 PSF ℓ1 recon. Proposed
Human 6.45% 5.98% 4.01%
Rat 4.94% 6.90% 2.10%

Tests were also done to compare the amount of data required for (a) ℓ1 regularized reconstruction without PSF modeling, (b) basic PSF with L = 16, and (c) the proposed method with L = 16. Reconstruction errors as a function of the amount of acquired data are shown in Fig. 5, and reconstructed images are shown in Fig. 6. As expected, reconstruction performance improves with the increasing data for both basic PSF and the proposed method. However, the proposed method can reconstruct higher quality images especially when the number of data frames is small compared with the basic PSF with the same model order. For ℓ1 regularized reconstruction without PSF modeling, the reconstructed images suffer from significant blurring artifacts although the reconstructions are stable with different amount of measurements.

Fig. 5.

Fig. 5

Reconstruction errors versus the number of data frames for the human phantom. The errors for ℓ1 regularized reconstruction almost do not change with the increasing amount of data. Basic PSF reconstruction needs a large amount of measured data for accurate reconstruction, while the proposed method can yield high-quality reconstructions with much less data.

Fig. 6.

Fig. 6

Reconstructions of the human phantom (zoomed-in to the heart). (a)–(d) show ℓ1 regularized reconstructions without PSF modeling using 16.5, 22, 27.5, 33 data frames, respectively; (e)–(h) show basic order-16 basic PSF reconstructions using 16.5, 22, 27.5, and 33 data frames, respectively; (i)–(l) show reconstructions with the proposed method (order-16 PSF with sparsity constraint) using 16.5, 22, 27.5, and 33 data frames, respectively.

B. Discussion

Simultaneous use of rank and sparsity constraints was also considered in [11]. Instead of imposing an explicit rank constraint, the algorithm in [11] minimizes the nuclear norm. We have observed that the explicit rank constraint can be more effective than the nuclear norm rank constraint [18]. In terms of computation, the proposed algorithm has the following advantages over the ℓ1 and nuclear norm minimization algorithm. The algorithm in [11] requires one SVD (or partial SVD) at each iteration, which is much more time-consuming than solving the decoupled least squares problems in (16). In addtion, for the proposed method both spatial and temporal Fourier transforms can be computed on the spatial and temporal basis, instead of the whole data matrix, which is more efficient.

IV. Conclusion

A new method has been described for reconstruction of highly undersampled (k, t)-space data. This method simultaneously enforces partial separability (low-rankness) and encourages sparsity in the (x, f)-domain. The resulting reconstruction problem is efficiently solved using an additive half-quadratic optimization procedure. The proposed method has been validated with simulated data, and shows significantly better performance than the existing methods that use rank or sparsity constraints alone.

Acknowledgments

This work was supported in part by the following research grants: NIHP41-EB-00197, NIH-P41-RR-02395, and NSF-CBET-07-30623.

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