Redundancy Encoding for Fast Dynamic MR Imaging using Structured Sparsity

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1 Redundancy Encoding for Fast Dynamic MR Imaging using Structured Sparsity Vimal Singh and Ahmed H. Tewfik Electrical and Computer Engineering Dept., The University of Texas at Austin, USA Abstract. For dynamic magnetic resonance imaging (MRI) applications, a critical requirement is to reduce image acquisition times while maintaining high resolutions and signal-to-noise ratios. In this paper, a fast MRI technique based on encoding of the k-space redundancy using data-driven dictionaries learned for sparse representation of signals is presented. The novelty of the technique lies in separating the redundancy encoding into multiple dictionaries in appropriate transform domains. Specifically, the method proposes to learn: (1) a dictionary in the spatial- Fourier space to capture the central k-space redundancy and, (2) a second dictionary in the image-patch space to capture the redundancy in the k- space periphery. Redundancy encoding through sparse representations in multiple dictionaries allows for recovery of MR images at high undersampling rates. Experimental validation results for fast cardiac cine and myocardial perfusion imaging on in vivo datasets are better than the stateof-the-art and images are recovered with peak-signal-to-noise-ratios 29 db for accelerations up to Keywords: Sparse Representations, Dictionary Learning, Structured Sparsity, Magnetic Resonance Imaging, Image Reconstruction 1 Introduction Currently, many medical diagnostic applications rely on magnetic resonance imaging (MRI) of physiological functions. For e.g., first-pass myocardial perfusion MRI is used to detect and evaluate ischemic heart disease [1]. Regional perfusion defects can be detected by analyzing the signal variability in an image time-series. In addition, a high spatial resolution is required to accurately localize the defected tissue [1]. Thus, a critical requirement for dynamic MRI is to reduce image acquisition times while maintaining high spatial resolutions to capture the underlying process with high-information rates, which is difficult to achieve in MRI. Due to the physical and physiological constraints data acquisition in MRI is sequential in time. The sequential acquisition process leads to a trade-off between: 1) the acquisition time, 2) the signal-to-noise ratio (SNR), and 3) the spatial resolution of the image. Consequently, in dynamic MRI due to limited acquisition times, recovered MR images suffer from low SNRs and poor

2 2 Singh et. al. resolutions. In this paper, a fast dynamic MR imaging method based on encoding of spatio-temporal redundancy existing in physiological functions through sparse representation of signals in multiple learned dictionaries is presented. Recently, few approaches have relied on combining redundancy encoding techniques and Compressed Sensing type sparse recovery methods to improve imaging speeds of dynamic MRI [1, 2]. The highly constrained back projection (HYPR) technique of [1] captures the temporal redundancy in a time-series and recovers sparse MR angiography images with acceleration factors up to 100. However, for not-so-sparse myocardial perfusion images, acceleration gains of 4 only, have been reported [1]. Similarly, the k-t focal under-determined system solver (k-t FOCUSS) approach of [2] encodes the redundancy in the temporal dimension using prediction schemes and in a subsequent step, uses the sparsity of the residual time-series to recover images using the FOCUSS method. In the prediction step of the k-t FOCUSS approach, the near-static image locations are approximated either as temporal average over the acquired time-series or merely as the closest image-patch from a(the) reference image(s) (MEMC:motion-estimation-motion-compensation). In both prediction schemes, the sparsity of residual signals and consequently the SNRs are adversely affected at high acceleration factors. In this paper, a fast MRI technique for dynamic MR applications is presented. The proposed technique relies on sparse representation of signals in learned dictionaries to encode the redundancy in the spatial and temporal dimensions of the k-space data for dynamic MR applications. The novelty of the technique lies in identifying multiple dictionaries in appropriate transform domains to capture redundancy in both (1) the central and, (2) the peripheral k-space data. The central k-space data identifies high energy terms (HETs), which vary slowly in the spatial and/or temporal dimensions for physiological functions. In contrast, the peripheral k-space data identifies low energy terms (LETs), which are responsible for adding details in images and show significant variations. However, the added details are localized in the image space. Consequently, the proposed method relies on learning dictionaries in both (1) the spatial-fourier space and, (2) the image-patch space. The sparse representations of the HETs allow for its recovery from under-sampled central k-space data. Thus, at high acceleration factors the under-sampling occurs in both the central and the peripheral k-space. This is unlike any of the fast MRI methods previously published, where the central k-space region has always been sampled densely. The preliminary results of using the proposed approach for experimental fast cardiac cine and myocardial perfusion imaging on in-vivo datasets are better than the state-of-the-art method and images are recovered with Peak-SNRs 29dB for accelerations up to The rest of this paper is organized as follows. In sections 2 and 3, dictionary learning techniques and the image recovery algorithm are presented, respectively. Experimental results using the presented fast MRI method on in-vivo cardiac cine and perfusion datasets are summarized in section 4. Finally, the section 5 concludes the paper.

3 Fast Dynamic MRI using Structured Sparsity 3 2 Redundancy Encoding using Structured Sparsity Separation of redundancy encoding in multiple dictionaries is based on the assumption that physiological functions at coarse scale approximations vary slowly over space and time. These slow variations can be captured using sparse representations of the HETs. Consequently, the proposed technique relies on encoding the redundancy in HETs for dynamic MR applications in the spatial-fourier space. Furthermore, since the LETs are responsible for adding localized image details such as edges and fine structures, a dictionary for sparse representation of image-patches is proposed to encode the redundancy in LETs. To encode the redundancy prevalent in HETs, a dictionary learning technique based on identification of multiple low-dimensional subspaces from the HETs of reference images is used. Very low resolution reference images can be acquired either in a rest-phase prior to or intermittently during the imaging procedure [1, 2]. To identify the dominant subspaces in the HETs of the reference images, the iterative subspace identification (ISI) [3] method is used. The ISI method identifies the subspaces where the data lives by finding the sparsest representation of each training vector in terms of the other vectors in the training data For more details on the ISI method, the reader is referred to [3]. Let, {A i } J i=1 denote the J subspaces identified by the ISI algorithm from the HETs of the reference images. Let, D denote a structured dictionary obtained by concatenating all subspaces {A i } J i=1. Then, a central k-space region Y, residing in the ith HETs variation model A i, can be represented using a unit-block sparse vector p as : Y = A i c i = D p, p = [ 0 T... c i T... 0 T ] T (1) where, c i are its non-zero description coefficients in A i. Using the sparsity of representations p in the learned dictionary D, under-sampling of central k-space region is possible at high acceleration factors. More details on under-sampling of the central k-space follow in sections 3 and 4. To encode the redundancy in LETs, a dictionary for sparse representations of image-patches is proposed. Previously, patch-based dictionaries have been used for regularizing CS-based-MRI recovery. However, initializations of such methods using zero-filled reconstructions have shown improvements at low accelerations only. To learn a patch-based dictionary from a given image u, the following problem is solved: min G,Γ Gα j R j u 2 2 s.t. α j 0 T 0 j (2) j where, R j is a patch extraction operator, the l 0 quasi-norm encodes the sparsity of representation and T 0 is the degree of sparsity. Γ is used to denote the set {α j } j of sparse representation of all patches and G is the patch-based dictionary. The formulation of (2), minimizes the total fitting error of all image patches with respect to the dictionary G, subject to sparsity constraints. For all results reported in this paper, the K-SVD algorithm of [4] has been used to learn the patch-based dictionary G. The K-SVD is an iterative technique which alternates

4 4 Singh et. al. between sparse coding and dictionary-atom update steps to learn a dictionary for sparse representation of training data[3, 4]. 3 The Image Recovery Algorithm This section presents the image recovery algorithm from under-sampled k-space data for the proposed approach. For easier understanding, the complete recovery algorithm is divided into two parts: (1) SSMRI, structured sparse MRI and (2) SDMRI, structured dictionary MRI. The SSMRI recovery corresponds to the Fourier projected image obtained after restoring all recovered HETs to the under-sampled central k-space data. To recover a candidate HETs vector Y, from its under-sampled observation Ỹ using the structured dictionary D in (1), the following problem is solved: min p Ỹ D p 2, s.t. p 0,B 1 (3) where, p 0,B 1 represents the unit-block sparsity constraint. Problem (3) is solved by reformulating it as J least square problems and picking the solution yielding the minimum error at the sampled HETs. For more details on recovery of sparse signals from under-sampled data using a structured dictionary, the reader is referred to [5]. The SSMRI recovery procedure simply decodes the HETs using its under-sampled version based on the redundancy encoded in the dictionary D. The second part of the image recovery algorithm restores the image details using the patch-based dictionary G. For restoration of details the following problem is solved: min u,α TV (u) + µ j ( α j 1 + ν ) 2 Gα j R j u λ F ss u f ss 2 2 (4) where, T V (u) denotes the total-variational norm [6] of the recovered image u, f ss are the sampled k-space locations along with the restored HETs by the SSMRI step, and F ss is the Fourier sampling operator which selects locations in the Fourier transform of u corresponding to those of f ss. Problem (4) reduces the TV-norm of the image being recovered with constraints on l 1 -norm sparsity of image-patch representations and the data fidelity measure with respect to SSMRI recovery. The parameters µ and λ control the weighting between the sparsity of patch-representations and the data-fidelity. Solving (4) is difficult due to the non-differentiability of the T V and l 1 terms. The algorithm used to solve (4) introduces auxiliary variables and uses classical quadratic penalty terms to yield an alternate minimization procedure. For more details on the algorithm and its convergence analysis, the reader is referred to [6]. The image recovered after restoring the details will be termed as the SDMRI image. 4 Results The proposed fast MRI technique is experimentally validated using in-vivo cardiac cine and perfusion datasets (complex raw data). To evaluate the quality

5 Fast Dynamic MRI using Structured Sparsity 5 of MR image recovery following metrics are used: (1) PSNR: peak-signal-tonoise-ratio and (2) HFEN: high frequency error norm. PSNR in decibels (db) is computed as the ratio of the peak reference intensity to the root mean square of the reconstruction error. HFEN quantifies the quality of reconstruction of edges and fine features, i.e., the information carried by the LETs. To calculate the HFEN, a 5x5 rotationally symmetric Laplacian of Gaussian (LoG) filter is used to capture the edges in the absolute reconstruction error image and the HFEN is calculated as the Forbenius norm of the error edge image. (a) (b) (c) Fig. 1: Image recovery using the SDMRI method under the ideal under-sampling scheme (a) original image (b) recovered image (PSNR: db, HFEN: 0.83) (c) sampling mask at R = (see text for details). The cine data was acquired from 1.5T Philips scanner with a k-t measurement size of 256x256x25. The field of view (FOV) was 345mm x 270mm and the slice thickness was 10 mm. The acquisition sequence was steady-state free precession (SSFP) with a flip angle of 50 degree and TR= 3.45 msec. The heart frequency was 66 bpm and the retrospective cardiac gating was used. The dictionary D to capture the redundancy in HETs is trained for the central 32x32 k-space and the patch-based dictionary with 256 atoms, to capture the redundancy in LETs is trained for 8x8 patches with a sparsity of T 0 = 15. Five planes uniformly spaced in time are randomly selected for training the HETs dictionary. Due to the lack of data from a different subject using same imaging sequence and hardware to train the LETs dictionary G, fully referenced images for 2 of the selected HETs planes are used. The remaining 20 planes are used as test images. Fig. 1 shows the SDMRI recovered image for a test image using the ideal sampling scheme. Ideally, the k-space should be under-sampled using uniformly and Gaussian distributed samples in the central and peripheral regions, respectively as shown in 1c. Uniform samples in HETs (red in green region) ensure a well-conditioned inverse problem (3) and Gaussian samples in the periphery (red in white region) provide tighter sampling for mid-spatial frequencies than that for high frequencies, thus gradually increasing the aliasing energy towards high-frequencies. Fig. 2 compares the performance of SSMRI, SDMRI and k- t FOCUSS (with MEMC) techniques by showing recovered images and HFEN maps for the test image Fig. 1a. Clearly, the SDMRI technique outperforms

6 6 Singh et. al. (a) (b) (c) (d) (e) (f) (g) (h) Fig. 2: MR image recovery for the in-vivo cardiac cine data. (a) & (d) SSMRI image and its HFEN map (PSNR: 30.8 db, HFEN: 2.0). (b) & (e) SDMRI image and its HFEN map (PSNR: 35.0 db, HFEN: 1.6). (c) & (f) kt-focuss image and its HFEN map (PSNR: 29.9 db, HFEN: 4.9). (g) under-sampling masks at acceleration factor R=7.11 for the SSMRI/SDMRI. (h) (mean ±0.25 std. deviation) PSNR vs. R plots for the SSMRI (green- ), SDMRI (blue- )and kt-focuss (red- ) techniques. the other techniques at a moderately high acceleration factor of R= Fig. 2g shows the Cartesian under-sampling used for the SSMRI/SDMRI techniques where phase encodes in the central region are not acquired (white lines in the gray region) as shown in the zoomed in-set. The patch based-evolution step adds details to the smooth approximation of the SSMRI step which results in lower HFEN as seen in the HFEN maps Figs. 2d & 2e. The k-t FOCUSS prediction step on zero-filled reconstructions from under-sampled data leads to addition of noise. In contrast, the proposed method relies on regularized sequential addition

7 Fast Dynamic MRI using Structured Sparsity 7 of details. Specifically, the use of patch-based dictionary as detail restoration step leverages the benefits of the SSMRI step (artifact free smooth approximations), yielding higher accuracy and graceful performance degradation with increasing R as seen in the Fig. 2h. (a) (b) (c) (d) (e) (f) (g) (h) Fig. 3: Performance comparison of techniques on the in-vivo perfusion data. Zoomedin images are shown. (a) & (d) fully-sampled reference images at peak LV uptake and at post-contrast (marked blood pool (red) and myocardium (green) regions), respectively. (b) & (e) reconstructed peak LV uptake (PSNR: db, HFEN: 1.61) & post-contrast (PSNR: 31.20dB, HFEN: 1.18) images using SDMRI, respectively. (c) & (f) reconstructed peak LV uptake (PSNR: db, HFEN: 23.94) & postcontrast (PSNR: db, HFEN: 10.97) images using ktfocuss (with MEMC), respectively. (g) & (h) time-series plots of averaged signal intensity of the blood pool and myocardium regions (with zoomed in-set)for the SDMRI and ktfocuss recovered images, respectively. Fig. 3 shows the performance of the proposed method on an in-vivo perfusion data at an acceleration of R = The perfusion data was acquired on a 3T Siemens scanner with a saturation-recovery sequence ( TR\TE = 25.1ms,

8 8 Singh et. al. saturation recovery time= 100 ms) at the University of Utah. The data for each of the 70 temporal slices was acquired on a Cartesian grid of 90x190 (phaseencodes x frequency encodes). The data has significant motion artifacts as the subject was not capable of holding the breath for the entire imaging duration. For simulation, every 5 th temporal slice is selected for HETs dictionary D training and fully sampled images corresponding to 2 of the selected (for HETs) temporal slices are used for patch-based dictionary training. The HETs dictionary D is trained for the 48x48 central k-space and the patch-based dictionary with 256 atoms is trained for 4x4 patches with a sparsity of T 0 = 5. The SDMRI recovers the edges and preserves the homogeneity of localized regions much better than the k-t FOCUSS technique. In addition, time series plots of averaged signal intensity in selected blood pool and myocardium regions shown in Figs. 3g & 3h are more accurate for the SDMRI technique. 5 Conclusion This paper presents a fast MRI technique based on encoding of the k-space redundancy in dynamic MR applications using multiple learned dictionaries. The novelty of the technique lies in identifying multiple dictionaries through sparse representation of signals in (1) the spatial-fourier space and, (2) the image-patch space. Experimental results on in-vivo cardiac cine and myocardial perfusion datasets empirically verify that the separation of redundancy encoding in multiple dictionaries achieves higher redundancy encoding along with sparsity, thus allow for accurate MR image recovery at high under-sampling rates. The preliminary results using the presented technique recovers images at peak-snrs 29dB for accelerations up to References 1. L. Ge, A. Kino, M. Griswold, C. Mistretta, J. Carr, and D. Li, Myocardial perfusion MRI with sliding-window conjugate-gradient HYPR, Magnetic Resonance in Medicine, vol. 62, no. 4, pp , H. Jung, K. Sung, K. Nayak, E. Kim, and J. Ye, k-t FOCUSS: A general compressed sensing framework for high resolution dynamic MRI, Magnetic Resonance in Medicine, vol. 61, no. 1, pp , B. Gowreesunker and A. Tewfik, Learning Sparse Representation Using Iterative Subspace Identification, Signal Processing, IEEE Transactions on, vol. 58, pp , june M. Aharon, M. Elad, and A. Bruckstein, K-SVD: An Algorithm for Designing Overcomplete Dictionaries for Sparse Representation, Signal Processing, IEEE Transactions on, vol. 54, no. 11, pp , D. Wang and A. Tewfik, Real Time 3D Visualization of Intraoperative Organ Deformations Using Structured Dictionary, Medical Imaging, IEEE Transactions on, vol. 31, pp , april J. Yang, Y. Zhang, and W. Yin, A fast TVL1-L2 minimization algorithm for signal reconstruction from partial Fourier data, IEEE J. Special Topics Signal Processing, to appear, 2008.