Elsevier

Journal of Magnetic Resonance

Volume 286, January 2018, Pages 91-98
Journal of Magnetic Resonance

Compressively sampled MR image reconstruction using generalized thresholding iterative algorithm

https://doi.org/10.1016/j.jmr.2017.11.008Get rights and content

Highlights

  • A generalized thresholding (p-thresholding) based iterative algorithm is proposed for CS-MRI reconstruction.

  • The proposed algorithm is tested on phantom and human head data for CS-MRI reconstruction.

  • The results of the proposed method are compared with other iterative thresholding algorithms for CS image reconstruction.

  • The proposed approach outperforms the soft and hard thresholding based iterative algorithms for CS image reconstruction.

Abstract

Compressed sensing (CS) is an emerging area of interest in Magnetic Resonance Imaging (MRI). CS is used for the reconstruction of the images from a very limited number of samples in k-space. This significantly reduces the MRI data acquisition time. One important requirement for signal recovery in CS is the use of an appropriate non-linear reconstruction algorithm. It is a challenging task to choose a reconstruction algorithm that would accurately reconstruct the MR images from the under-sampled k-space data. Various algorithms have been used to solve the system of non-linear equations for better image quality and reconstruction speed in CS. In the recent past, iterative soft thresholding algorithm (ISTA) has been introduced in CS-MRI. This algorithm directly cancels the incoherent artifacts produced because of the undersampling in k-space. This paper introduces an improved iterative algorithm based on p-thresholding technique for CS-MRI image reconstruction. The use of p-thresholding function promotes sparsity in the image which is a key factor for CS based image reconstruction. The p-thresholding based iterative algorithm is a modification of ISTA, and minimizes non-convex functions. It has been shown that the proposed p-thresholding iterative algorithm can be used effectively to recover fully sampled image from the under-sampled data in MRI. The performance of the proposed method is verified using simulated and actual MRI data taken at St. Mary’s Hospital, London. The quality of the reconstructed images is measured in terms of peak signal-to-noise ratio (PSNR), artifact power (AP), and structural similarity index measure (SSIM). The proposed approach shows improved performance when compared to other iterative algorithms based on log thresholding, soft thresholding and hard thresholding techniques at different reduction factors.

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 k-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 p-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]:minmm0s.t.y=Fum.where mCN is the reconstructed image with N number of pixels. yCM is the acquired under-sampled k-space data. Fu is the Fourier encoding transform which can be expressed as Fu=kF, where k is the undersampling pattern and F is the Fourier transform. Here m0 is the 0-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 1-minimization problem. Iterative shrinkage algorithm [20] directly cancels the aliasing caused by undersampling in k-space.

In this paper, a generalized thresholding (p-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 p-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 k-space data is acquired. Fig. 4(b) shows the reconstructed image from the fully sampled k-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. p-thresholding. It is observed that p-thresholding based iterative algorithm offers better results in terms of AP, SSIM and PSNR than the conventional soft

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