Compressively sampled MR image reconstruction using generalized thresholding iterative algorithm
Graphical abstract
Introduction
Magnetic resonance imaging (MRI) is a non-ionizing medical imaging modality that provides excellent visualization of the anatomical structures and functions of human body. It also provides good contrast between different soft tissues of the body. However, the main disadvantage of MRI is that it is a slow imaging technique which causes patient discomfort. During data acquisition it is important to restrain the patient from movement, as motion artifacts degrade the quality of magnetic resonance (MR) image. Many techniques have been used to reduce the amount of data acquired in k-space to decrease the scan time with clinically acceptable reconstruction results [1].
Compressed Sensing (CS) [2] is a unique method that provides accurate image reconstruction even at a sampling rate below the Nyquist criteria with a very little loss of information. In CS, the acquired image must have a sparse representation in a transform domain for a successful reconstruction from the under-sampled k-space data, however certain conditions need to be met for a perfect recovery of images as given below [3]:
- a.
The image must be sparse in some known transform domain, which means that the whole energy of the image must be concentrated in a few significant coefficients.
- b.
The undersampling (in -space) should produce incoherent artifacts.
- c.
Reconstruction should be performed using a non-linear technique that must have spatial attributes which would promote sparsity of representation of the object.
CS is an established technique based on the fact that most of the images are compressible. The CS constrained minimization problem can be solved by using different non-linear optimization algorithms and specific sparsifying transforms. CS was first introduced in MRI by Lustig [4], where Non-linear Conjugate Gradient (NLCG) algorithm with total variation (TV) model was used as the non-linear reconstruction algorithm. In this paper we suggest CS-MRI image reconstruction using -thresholding based iterative algorithm.
The rest of the paper is organized as follows: In Section 2 we discuss some basic theory of CS. Section 3 presents the proposed approach for CS-MRI reconstruction, and in Section 4 the quantifying parameters and datasets used in the experiments are discussed. In Section 5 simulation results obtained from the proposed method and other conventional algorithms are presented. In Section 6 the results are discussed and finally the paper is concluded in Section 7.
Section snippets
Compressed sensing theory
The common model of data acquisition with incomplete measurements for finding the sparsest solution is given by the following formulation [5]:where is the reconstructed image with number of pixels. is the acquired under-sampled k-space data. is the Fourier encoding transform which can be expressed as , where k is the undersampling pattern and is the Fourier transform. Here is the -norm and represents the number of non-zero elements in m. Commonly,
Proposed method for compressed sensing MRI
Many methods have been used for solving CS image reconstruction problem such as Bregman iteration [14], [15], Interior point method [16], RecPF [17], and CoSaMP [18]. A review of CS-MRI reconstruction algorithms can be found in [17], [19]. However, not all of these methods directly solve the -minimization problem. Iterative shrinkage algorithm [20] directly cancels the aliasing caused by undersampling in k-space.
In this paper, a generalized thresholding (-thresholding) based iterative
Data acquisition
The proposed algorithm is tested on phantom and human head data for CS-MRI reconstruction. Fully sampled phantom and human head data is acquired from 1.5 T to 3 T MRI scanners at St. Mary’s Hospital London with eight channel receiver coil and a Gradient Echo sequence with the following parameters: TE = 10 ms, TR = 500 ms, FOV = 20 cm, Bandwidth = 31.25 kHz, Slice Thickness = 3 mm, Flip Angle = 50 degrees, and Matrix Size = 256 × 256. An informed consent was obtained from the volunteers and the
Simulation results
This paper introduces -thresholding based iterative algorithm for CS based MR image reconstruction problem. The results of the proposed method are compared with other iterative thresholding algorithms for CS-MRI image reconstruction. Fig. 4(a) is the variable density sampling mask for AF = 3 which means only 33% of the -space data is acquired. Fig. 4(b) shows the reconstructed image from the fully sampled -space for human head data. Fig. 4(c)–(f) are the reconstructed images using iterative
Discussion
This section provides a quantitative and qualitative analysis of the proposed algorithm for CS-MRI on the three datasets for different AF. In order to demonstrate the effectiveness of the proposed method, a comparison is made with ISTA [5], iterative log thresholding algorithm [23] and iterative hard thresholding algorithm [24].
Fig. 4(a) is the variable density undersampling mask as suggested by Lustig [26]. Fig. 4(c)–(f) are the reconstructed 1.5 T human head images for AF = 3. The results
Conclusion
In this paper an iterative thresholding algorithm with p-thresholding is proposed for CS-MRI reconstruction. We computed PSNR, AP, and SSIM of the reconstructed images with iterative algorithms using different thresholding schemes namely: (1) Log Thresholding; (2) Hard Thresholding; (3) Soft Thresholding and 4) and the proposed method i.e. -thresholding. It is observed that -thresholding based iterative algorithm offers better results in terms of AP, SSIM and PSNR than the conventional soft
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