MR image reconstruction based on framelets and nonlocal total variation using split Bregman method

Abstract

Purpose

An efficient algorithm for magnetic resonance (MR) image reconstruction is needed, especially when sparse sampling is employed to accelerate data acquisition. The aim of this paper is to solve the sparse MRI problem based on nonlocal total variation (NLTV) and framelet sparsity using the split Bregman algorithm. A new method was developed and tested in a variety of MR image acquisitions.

Methods

The proposed method minimizes a linear combination of NLTV, least square data fitting and framelet terms to reconstruct the MR images from undersampled $k$-space data. The NLTV and framelet sparsity are taken as the $L_{1}$-regularization functional and solved by using the split Bregman method. Experiments were conducted to compare the proposed algorithm with several different reconstruction methods, including the operator splitting algorithm, variable splitting method, composite splitting algorithm and its accelerated version called the fast composite splitting algorithm. A detailed evaluation study was done on the reconstruction of MR images which represent varying degrees of object structural complexity. Both qualitative visualization-based and quantitative metric-based evaluations were done.

Results

Numerical results on various data corresponding to different sampling rates showed the advantages of the new method in preserving geometrical features, textures and fine structures. The proposed algorithm was compared with previous methods in terms of the reconstruction accuracy and computational complexity with favorable results.

Conclusion

An efficient new algorithm was developed for compressed MR image reconstruction based on NLTV and framelet sparsity. The algorithm effectively solves a hybrid regularizer based on framelet sparsity and NLTV using the split Bregman method. NLTV makes the recovered image quality sharper by preserving the edges or boundaries more accurately, and framelets often improve image quality. The comparison with alternative method yielded results that demonstrate the superiority of the proposed algorithm for compressed MR image reconstruction.

Appendix

$L_{2}$-Norm

The most popular of all norm is the $L_{2}$-norm. It is used in almost every field of engineering and science as a whole. Following the basic definition, $L_{2}$-norm is defined as

$$\begin{aligned} ||u||_{2}=\sqrt{\sum _{i} u_{i}^{2}} \end{aligned}$$

(47)

$L_{2}$-norm is well known as a Euclidean norm, which is used as a standard quantity for measuring a vector difference. As in $L_{1}$-norm, if the Euclidean norm is computed for a vector difference, it is known as a Euclidean distance:

$$\begin{aligned} ||u_{1}-u_{2}||_{2}=\sqrt{\sum _{i} (u_{1i}-u_{2i})^{2}} \end{aligned}$$

(48)

Image gradient $\nabla$

Image gradient is a directional change in the intensity or color in an image. Image gradients may be used to extract information from images. The gradient of an image is given by the formula

$$\begin{aligned} \nabla f=\frac{\partial f}{\partial x}\widehat{x}+\frac{\partial f}{\partial y}\widehat{y} \end{aligned}$$

(49)

where $\frac{\partial f}{\partial x}\widehat{x}$ is the gradient in the $x$ direction and $\frac{\partial f}{\partial y}\widehat{y}$ is the gradient in the $y$ direction.