MRI reconstruction from partial k-space data by iterative stationary wavelet transform thresholding Mohammad H. Kayvanrad 1,2, Charles A. McKenzie 1,2,3, Terry M. Peters 1,2,3 1 Robarts Research Institute, Western University, Canada 2 Biomedical Engineerin, Western University, Canada 3 Medical Biophysics, Western University, Canada Abstract. One approach to accelerating data acquisition in magnetic resonance imaging is acquisition of partial k-space data and recovery of missing data based on the sparsity of the image in the wavelet transform domain. We hypothesize that the application of stationary wavelet transform (SWT) thresholding as a sparsity-promoting operation results in improved artifact removal performance in comparison with the regular decimated wavelet transform (DWT). On this grounds, we develop an iterative SWT thresholding reconstruction. We demonstrate that an acceleration factor of 3 can be achieved using this approach on a human brain scan while only suffering a 9.61% penalty in signal to noise ratio without compromising the spatial resolution of the image. 1 Introduction Patient comfort and cost considerations limit the number of different examinations that can be performed on a patient in an MR scanner to those that can be implemented in a maximum 30-45 min time period. There is, therefore, a strong desire to reduce the acquisition time without compromising the quality of the acquired images. In addition to hardware-based parallel imaging techniques [10], which demonstrate impressive reduction in the acquisition time, a complementary approach to reducing the MRI acquisition time is acquisition of partial k-space data and use of the sparsity of the image in a transform domain, e.g., wavelet, as an a priori reconstruction constraint to interpolate the missing data [11]. This is commonly referred to as compressed sensing or compressive sampling (CS) [3]. In this paper, we consider the same problem of reconstruction of MR images from under-sampled k-space data based on their sparsity in the wavelet transform domain. Nevertheless, while this work can be categorized under CS in the sense that it involves reconstruction of an object from under-sampled observations based on a sparsity constraint, our reconstruction approach is not based on the conventional norm-minimization approaches commonly used for CS. In fact, our iterative reconstruction algorithm is similar to the Papoulis-Gerchberg reconstruction algorithm [12]. However, for the sake of completeness and due to its relevance, let us briefly review the literature of CS:
In this paper we assume any point on the k-space grid is either sampled or replaced by zero, and denote under-sampling by a linear operation, U F, defined on the Fourier space, F. The relationship between the fully-sampled k-spaced data, F F, and the under-sampled k-space data, F u F, can be expressed as: F u = U F F (1) Assume an under-sampled set of k-space data, F u, corresponding to an unknown fully-sampled k-space dataset, F. Equation (1) describes the relationship between F u and F. The objective of CS is to reconstruct F, or equivalently in the spatial domain, f, from the under-sampled k-space data, F u, based on an a priori sparsity constraint. f = F F is the spatial domain representation of F, where F is the Fourier transform operation and denotes the adjoint operation. Assume there exists an orthogonal sparsifying transform, Ψ, to the sparse domain. It is very common to assume Ψ is a wavelet transform since MR images have a sparse representation in this domain. Wavelet sparsity, often measured in terms of an l p -norm (1 p 2), is often incorporated as a regularizing penalty term in order to solve the above problem [6]. Also, It has been shown that the addition of a finite difference penalty term, usually referred to as total variation (TV), improves the reconstruction quality [2]. The problem can be cast in a constrained optimization framework: min Ψf lp + αt V (f ) s.t. U F Ff F u l2 < ɛ (2) f where f denotes the solution and α is a constant. Although the solution to this problem may be found using a general-purpose optimizer [2], [11], use of iterative thresholding algorithms is also a common approach to computing the solution [6], [8], [1]. We note once more that while, our objective is to reconstruct MR images from under-sampled k-space data based on the wavelet sparsity of the images, we don t follow the conventional CS reconstruction by optimizing norm penalty terms (e.g., l 1 or TV). Our reconstruction algorithm is similar to the Papoulis-Gerchberg reconstruction, and is not intended to optimize any particular norm. This is important to note for a clear understanding of the methods and the results described in this paper. The results are, however, compared with those obtained by conventional CS methods, as one may naturally expect. For the sake of clarity, especially since it was mentioned that our reconstruction is similar to the Papoulis-Gerchberg (P-G) reconstruction algorithm, it will be helpful to also have a few words on the relation between our reconstruction and P-G before proceeding to a detailed explanation of the reconstruction algorithm and the rationale behind it. The P-G algorithm was originally developed for reconstruction from partial spatial or frequency domain data with a finite support constraint in the other domain. The signal is reconstructed by alternating between these domains to re-enforce the data and support constraints in the corresponding domains. This algorithm has been also used for MRI reconstruction from limited k-space observations with the assumption of a finite spatial
support constraint on the image [4], [9], [13]. However, as we describe in section 3, a similar reconstruction can be obtained with the assumption of a wavelet sparsity constraint (equivalent to the support constraint in P-G). Similar to the P-G algorithm, the image is reconstructed by alternating between the frequency domain and the wavelet domain to re-enforce the known k-space data and sparsity constraints, respectively. Besides the domain on which these constraints are defined (wavelet vs. spatial), they also differ in the sense that while a known finite support can be considered a hard constraint, wavelet sparsity is a soft constraint. We hypothesize that the application of stationary wavelet transform (SWT) thresholding as a sparsity-promoting operation results in improved artifact removal performance in comparison with the regular decimated wavelet transform (DWT) (section 2). The rationale is based on the observation that better de-noising/artifact-removal performance can be achieved by SWT thresholding compared to DWT due to the translation-invariance of SWT [5]. On this grounds we develop a SWT iterative thresholding algorithm by alternating between the frequency domain, in which the k-space data constraint is re-enforced, and the SWT domain, in which the sparsity constraint is re-enforced (section 3). 2 SWT thresholding reconstruction Thresholding is a well-established technique for denoising [7] and sparse recovery [6]. It has been shown that denoising with the regular decimated wavelet transform (DWT) sometimes results in visual artifacts. Some of these artifacts are attributed to the lack of translation invariance of the wavelet basis and can be eliminated by the application of stationary wavelet transform (SWT) [5]. Although use of SWT thresholding in place of DWT thresholding is very well established for denoising applications, it seems to have been neglected in sparse recovery applications. In the discussion that follows we demonstrate the advantages of SWT thresholding over DWT thresholding as a sparsity-promoting operation for sparse recovery by some point spread function (PSF) analysis. With many pulse sequences one may not achieve further time savings by under-sampling in the readout (k x ) direction since the samples in the readout direction are usually acquired within a single repetition time, T R. Therefore, assuming full sampling in the readout direction, the problem reduces to a 1D (for 2D MRI) or 2D (for 3D MRI) interpolation problem in the phase encode (k y and k z ) directions. In this paper, we consider the 2D MRI case and assume k-space is under-sampled in the phase-encode (k y ) direction only. Therefore, it is sufficient to consider a delta input in the y direction only for the sake of our PSF analysis. We assume a pre-allocated grid of size M N and let f(m, n) = δ(n) be a discrete delta function in the y direction in the spatial domain, where m M/2 and n N/2. Note that, for simplicity, we assume the { origin is 1; n = 0 located at the center of the grid. δ is the Dirac delta function, δ(n) = 0; n 0.
Transforming f(m, n) to the Fourier domain, under-sampling, and transforming back to the spatial domain result in the under-sampling PSF, f u, in the spatial domain (figure 1(a)): f u = F U F Ff (3) Consider the wavelet decomposition of f u using decimated and stationary wavelet transforms: C DW T = Ψ DW T f u, and C SW T = Ψ SW T f u, where Ψ DW T and Ψ SW T are the (orthogonal) decimated and stationary wavelet transforms and C DW T and C SW T are the corresponding wavelet decomposition coefficients, respectively. Assume a level-dependent thresholding operation, γ, acting on the decomposition coefficients: CDW T = γ{c DW T }, and C SW T = γ{c SW T }. The corresponding PSFs are computed by wavelet reconstruction of the thresholded coefficients: fdw T = ΨDW C T DW T, and f SW T = ΨSW C T SW T (figures 1(b) and 1(c)). The signal-to-alias ratio, defined as the energy of the peak (signal) to the energy of the side-lobe (alias), provides a quantitative means of comparing these functions with each other and with the original aliasing PSF, f u. Apparently, a higher signal-to-alias ratio, achieved by SWT thresholding, indicates less aliasing interference (SWT signal-to-alias ratio > DWT signal-to-alias ratio > Aliased signal-to-alias ratio). Since we are mainly working with stationary wavelet transforms, from now on we drop the subscripts for stationary wavelet transform. Also, in order to simplify our notations, let us define a SWT thresholding operation, Γ, such that f = Γ {f u } (4) 3 Methods In section 2 we showed that the signal-to-alias ratio can be increased by the application of a thresholding operation to the SWT coefficients of the aliased image. Assume an under-sampled image f u corresponding to an under-sampled k-space dataset F u. Starting with f u as the initial estimate to the solution, a better estimate is achieved by removing some of the aliasing artifacts by the thresholding (sparsity-promoting) operation: g (1) = Γ {f u }. The superscript denotes the iteration number, i.e., the first iteration. We employ the same notation to denote the iteration number throughout this section. Nonetheless, both under-sampling (U F ) and thresholding (Γ ) operations reduce the energy of the image. Consequently, g (1) has reduced energy compared to f u and f u has reduced energy compared to f. 4 In addition, while thresholding should have revealed more features of the image by removing some of the aliasing artifacts, it may as well have affected the known k-space samples. 4 In fact, f u = F F U has the minimum energy among all the solutions consistent with the k-space data since we assume the unobserved k-space samples are simply replaced by zero in F U. This is usually called a minimum-energy reconstruction.
(a) Aliasing PSF (signal-to-alias ratio = 0.505) (b) DWT (signal-to-alias ratio = 0.585) (c) SWT (signal-to-alias ratio = 0.685) Fig. 1. Point spread functions (PSF) resulting from k-space under-sampling (1(a) ) followed by the application of DWT (1(b)) and SWT (1(c)) thresholding on the aliased impulse input image. Mathematically, F u = U F F U F G (1), where G (1) = Fg (1) is the Fourier transform of g (1). In words, if G (1) is under-sampled in the same way k-space was, i.e., U F G (1), it will not necessarily be consistent with the original k-space data. Therefore, before further progress, the known k-space samples are recovered: F (1) = G (1) U F G (1) + F u. Note that F (1) has higher energy than F u, since some of the unknown coefficients, which are replaced by zeros in F u, take an estimated value in F (1). f (1) is a better estimate to f than f u. This estimate can be improved by repeating the above procedure in a an iterative manner. That is, at the nth iteration starting with the latest estimate at the previous iteration, f (n 1), the next estimate is achieved by a sparsity-promoting operation, g (n) = Γ {f (n 1) } (5)
followed by recovery of the known k-space samples, F (n) = G (n) U F G (n) + F u (6) Combining these two operations, and noting f (n) = F F (n), the iterations can be expressed as f (n) = Γ {f (n 1) } F U F FΓ {f (n 1) } + F F u (7) The iterations are initialized with the minimum-energy reconstruction, f (0) = f u, and continue until a convergence criterion, e.g., changes lass than a certain threshold, f (n) f (n 1) / f (n) < ɛ, or a maximum number of iterations is reached. A whole-brain scan of a volunteer was acquired at 3T, with a 32-channel head coil, using fast spin echo (FSE), with the following parameters: TR/TE = 3600ms/80ms, ETL = 15, Grid size = 256x256 (resolution = 1mm), Number slices = 68 (slice thickness = 2mm). Human data used in this work were acquired using a protocol approved by the institutional Office of Research Ethics. The fully-sampled dataset is used as the gold standard. Each slice is randomly under-sampled with a Gaussian pdf with a zero mean (corresponding to the center of k-space, i.e., the zero frequency) along the phase-encode (K y ) direction (see section 1). The under-sampling pattern is chosen by following the Monte Carlo experiments described in [11] to make the experiments repeatable. The under-sampled set of data in each slice is reconstructed by iterative SWT thresholding and by direct l 1 + TV optimization. 4 Results The results indicate that an acceleration factor of 3 can be achieved while only suffering 9.61% in the signal-to-noise ratio (SNR) (Figures 2(a), 2(c), and 3(b)). Note that although the aliasing artifacts are due to k-space under-sampling and are not noise, we include these artifacts in noise and use SNR as a quantitative measure of reconstruction quality in addition to RMSE. The results also show noticeable improvement over reconstruction by l 1 + TV optimization, performed using the code supplement to [11] (figures 2(d) and 3(c)). Reconstruction by SWT thresholing also results in better reconstruction quality compared to low-resolution sampling with the same under-sampling factor (figures 2(b) and 3(a)). Nevertheless, while in [11] it is shown that reconstruction by l 1 + TV optimization results in higher reconstruction quality compared to low-resolution sampling at an acceleration of 2.4, it seems the reconstruction quality degrades compare to low-resolution sampling at an acceleration of 3. 5 Discussion While MRI data used in this paper were acquired using a fast spin echo pulse sequence, this does not affect the generality of the described methods and they can
(a) Fully-sampled (b) Low resolution (c) SWT thresholding (d) l1 and TV (SNR=28.4 a.u.) (SNR=18.9 a.u.) (SNR=25.6 a.u.) optimization (SNR=6.48 a.u.) Fig. 2. Reconstruction from under-sampled (under-sampling factor = 3) k-space data. 2(a) Fully-sampled k-space (Gold standard) 2(b) Low resolution image 2(c) Reconstruction by iterative SWT thresholding 2(d) Reconstruction by l1 and TV optimization. be used to accelerate acquisition with other pulse sequences as well. Although a whole brain scan with the described resolution can be acquired in about 6 minutes using fast spin echo, a time that is quite reasonable for a clinical MRI, an acceleration ratio of 3, illustrated in the results, will reduce the acquisition time to 2 minutes without noticeably reducing the reconstruction quality. Once additional acquisitions are added to the scan session, the time savings will become noticeable not only from a patient-comfort perspective, also because the time savings will enable us to run additional acquisitions to acquire more information, which is valuable for diagnosis/treatment purposes. 6 Conclusion We hypothesize that the aliasing artifacts due to random k-space under-sampling show noise-like characteristics. Since stationary wavelet transform (SWT) thresholding is commonly used for denoising to eliminate the pseudo-gibbs artifacts associated with the translation-variance of the decimated wavelet transform, and based on the above hypothesis, we propose to use iterative SWT thresholding for reconstruction of under-sampled MRI data. The results indicate that an acceleration factor of 3 can be achieved while only suffering a 9.61% penalty in signal-to-noise ratio (SNR), also exhibiting a noticeable improvement compared to low-resolution sampling and reconstruction by l1 + TV optimization. References 1. Thomas Blumensath and Mike E. Davies. Iterative hard thresholding for compressed sensing. Applied and Computational Harmonic Analysis, 27(3):265 274, November 2009.
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