A computationally efficient method for reconstructing sequences of MR images from undersampled k-space data
Graphical abstract
Introduction
Magnetic Resonance Imaging (MRI) is an essential medical imaging tool but burdened by its intrinsically slow data acquisition process. Due to the sequential nature of data acquisition in MR imaging modalities, scan times will be reduced if fewer measurements are needed for reconstruction. A fundamental challenge in reduced-data imaging is to recover high-resolution images from undersampled k-space data. When k-space is undersampled, the Nyquist criterion is violated, and conventional Fourier reconstructions exhibit aliasing artifacts. Compressed sensing (CS) (Candes and Tao, 2006, Candes et al., 2006, Candes and Wakin, 2008, Donoho, 2006), on the other hand, aims to reconstruct signals and images from significantly fewer measurements than were traditionally thought necessary. CS has been shown to enable reconstruction of MRI images from much smaller set of measurements than the number of Nyquist-rate samples (Lustig et al., 2007, Lustig et al., 2008, Trzasko and Manduca, 2009, Venkatesh et al., 2010). MRI has two key features that makes it particularly suitable for CS. First, MRI images are naturally compressible by sparse coding in an appropriate transform domain (e.g., by wavelet transform) and secondly, MRI scanners acquire sample images in k-space which is known to be “incoherent” w.r.t. sparse domain, rather than acquiring direct pixel samples (Lustig et al., 2007).
Most reported CS-based works are concerned with reconstruction of static MRI image slices and not sequences of MRI images as in volumetric (3D) and dynamic MR imaging. Dynamic MRI relies on fast MRI sequences and enables direct quantitative visual assessment of organ function over time. In dynamic MRI such as real-time cardiac MRI (rtCMR), functional MRI (fMRI) and joint motion imaging), spatial/temporal resolution is most important. High spatial resolution is needed for visualization of fine details or structures that have diagnostic importance. Simultaneously, the MR image sequence needs to have high temporal resolution to be able to depict changes over time due to motion or intensity variation (Jung et al., 2009). For example in cardiac imaging capturing the temporal variations of heart, which exhibits significant motion (60–200 heart beats per minute), is quite challenging. Currently, dynamic MRI is only possible with a compromise on the achievable spatial/temporal resolution of the reconstructed image sequence due to the time-consuming scanning process (Gamper et al., 2008). Thus, CS-based methods specifically adapted for dealing with dynamic sequences of MRI images, can greatly benefit these applications.
CS-based techniques for reconstruction of sequences of MRI images can broadly be divided into non-causal (or batch-based) and causal methods. In non-causal approaches, the entire T frame needs to be acquired before carrying out the reconstruction, which takes advantage of the temporal sparsity. Examples of such methods include batch-CS (Gamper et al., 2008), which improves upon simple-CS by jointly reconstructing the entire sequence by treating it as a 3D sparse signal, and the following works which treat the underlying image sequence as a group of video frames and carry out the prediction from neighboring frames through motion-adaptive interpolation: kt-FOCUSS (Jung et al., 2009), kt-FOCUSS with motion compensation (MC) (Jung and Ye, 2010) and MASTeR (Asif et al., 2012). The main limitation of such methods is their computational complexity and memory requirement, which for a T-frame acquisition, is roughly times and T times of that of causal methods, respectively. For example in (Otazo et al., 2010), the reconstruction time is reported to be about 2.5 h for a sequence of 10 MR images.
Causal approaches can recover the current frame as soon as its MR data gets acquired, and their memory and computational demand is much lower than that of non-causal (batch) methods. These methods are especially suitable for real-time reconstruction of MRI images. When applied to the reconstruction of fMRI image sequences, the modified-CS-residual (Wei et al., 2011) which is a causal method, has been shown to outperform kt-FOCUSS (Jung et al., 2009), batch-CS (Gamper et al., 2008) and modified-CS (Lu and Vaswani, 2009) in terms of reconstruction quality.
In recent CS literature, a class of greedy reconstruction approaches including the iterative hard thresholding algorithms (Blumensath and Davies, 2009), has been shown to be an appealing alternative to BPDN-based methods due to their low computational complexity, speed and easy implementation. Moreover, their reconstruction performance guarantee is similar to those of BPDN-based methods (Eldar and Kutyniok, 2012).
In this paper, we introduce a fast casual approach which enables detailed and simultaneous reconstruction of dynamic MRI image sequences from a limited number of samples, using an iterative thresholding-based algorithm. Through extensive experimental results, we show that our proposed method achieves superior spatial and temporal reconstruction quality using highly undersampled k-space data, while having a much lower computational complexity and memory requirements than the modified-CS-residual (hereafter referred to as Mod-CS-Res). The paper is organized as follows: next section starts with a description of the notations that precedes the formulation of the problem. Section 3 presents the details of our proposed algorithm as applied to sequences of MRI images. We present and analyze our experimental results in Section 4 before providing some concluding remarks in Section 5.
Section snippets
Reconstruction of dynamic MRI
Notations: Throughout the paper, vectors and matrices are denoted by boldface letters (e.g. ) and we use the notation to denote the sub-matrix containing elements of with indices belonging to set and denotes the (i, j)th element . Scalars are shown by small regular letters (e.g. ) and linear maps are denoted by bold calligraphic uppercase letters (). Superscript added to a matrix refers to that of time/iteration t. Symbol refers to the convolution operator and
Proposed approach
In our approach, we extract some a priori knowledge about the support of the signal of interest in the transform domain that remains valid even when the support is changing and use it to guide the reconstruction process. The idea is based on the observation that in sequences of MRI images, nearby slices are closely related to each other in the sparse domain. Therefore, the conjecture is that we should be able to extract some priori information about sparsity of from . More
Experimental results
The proposed Priori-IHT was tested on six sequences of MRI images (see details in Table 12) some of which are shown in Fig. 3.
Conclusion
In this paper, we presented an iterative thresholding-based algorithm, which enables a detailed and fast reconstruction of dynamic MRI image sequences from highly undersampled k-space. Our proposed method has two major improvements over the state-of-the-art Mod-CS-Res method. Firstly for the extracted priori information, instead of just using the support of the previously reconstructed image as the estimated support of the current frame, a probability matrix is obtained which remains valid even
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