Truncation effects in SENSE reconstruction

Abstract

Finite sampling is an important practical issue in Fourier imaging systems. Although data truncation effects are well understood in conventional Fourier imaging where a single uniform receiver channel is used for data acquisition, this issue is not yet fully addressed in parallel imaging where an array of nonuniform receiver channels is used for sensitivity encoding to enable sub-Nyquist sampling of k-space. This article presents a systematic analysis of the problem by comparing the truncation effects in parallel imaging with those in conventional Fourier imaging. Specifically, it derives a convolution kernel function to characterize the truncation effects, which is shown to be approximately equal to that associated with the conventional Fourier imaging scheme. This article also describes a set of conditions under which significant differences between the truncation effects in parallel imaging and conventional Fourier imaging occur. The results should provide useful insight into interpreting and reducing data truncation effects in parallel imaging.

Introduction

Parallel imaging using multiple receiver coils has emerged as an efficient tool to reduce MRI data acquisition time [1], [2], [3], [4], [5], [6], [7], [8], [9], with a wide range of applications [10], [11], [12], [13], [14], [15], [16]. The k-space data collected in a sensitivity-encoded (SENSE) imaging experiment using an array of L receiver coils can be expressed, in general, as [2]

$$D_l\left({\mathbf{k}}_n\right)=\int \rho \left(\mathbf{r}\right)s_l\left(\mathbf{r}\right)e^{i2\pi {\mathbf{k}}_n·\mathbf{r}}d\mathbf{r}\text{,}$$

where s_l(r) is the sensitivity function of the lth coil, l=1, 2, …, L, ρ(r) is the desired image function and D_l(k_n) is the data sample measured at k-space location k_n by the lth coil. This article focuses on Cartesian k-space sampling since nonuniform sampling of k-space can introduce additional image artifacts, analysis of which is beyond the scope of this article. Invoking the separability of the Fourier transform in Cartesian sampling, Eq. (1) can be rewritten as sequential 1D Fourier transforms along each spatial direction. Therefore, without loss of generality, we will focus our discussion on 1D sensitivity encoding (assumed to be along the x direction) by rewriting Eq. (1) as

$$D_l\left(n\Delta \hat{k}\right)=\underset{-B/2}{\overset{B/2}{\int }}\rho \left(x\right)s_l\left(x\right)e^{-i2\pi n\Delta \hat{k}x}dx\text{,}$$

where ρ(x) is assumed to be support limited to |x|<B/2, k is used for k_x for simplicity and the sampling interval Δkˆ is chosen to be Δkˆ=RΔk with Δk=1/B and R≥1. Notice that when R=1, the Nyquist sampling criterion is satisfied; otherwise, the k-space data is undersampled by a factor of R≤L, thereby increasing the imaging speed by a factor of R. When there is no data truncation, the Fourier image produced by the lth coil from the infinite Fourier series with coefficients D_l(nΔkˆ) [17] can be written as [9]

$$d_l\left(x\right)=\sum _{m=0}^{R-1}\rho (x-m\hat{B})s_l(x-m\hat{B})\text{,}$$

for B/2−Bˆ<x<B/2 and l=1, 2, …, L, where Bˆ=B/R is known as the reduced field of view [2]. Because the sensitivity-weighted images, ρ(x)s_l(x), have a spatial support of B, ρ(x−mBˆ)s_l(x−mBˆ) for different m=0, 1, …, R−1 will overlap, producing the well-known aliasing effect in d_l(x). Based on Eq. (3), we can remove the aliasing error in d_l(x) to obtain the desired image function ρ(x). Specifically, rewriting Eq. (3) in matrix form yields

$$\mathbf{S}\mathbf{\rho }=\mathbf{d}\text{,}$$

where

$$\mathbf{S}=\left[\begin{array}{cccc}{s}_1\left(x\right)& s_1(x-\hat{B})& \dots & s_1(x-(R-1\left)\hat{B}\right)\\ s_2\left(x\right)& s_2(x-\hat{B})& \dots & s_2(x-(R-1\left)\hat{B}\right)\\ ⋮& ⋮& \ddots & ⋮\\ s_L\left(x\right)& s_L(x-\hat{B})& \dots & s_L(x-(R-1\left)\hat{B}\right)\end{array}\right]\text{,}$$

$$\mathbf{\rho }=\left[\begin{array}{c}\rho \left(x\right)\\ \rho (x-\hat{B})\\ ⋮\\ \rho (x-(R-1\left)\hat{B}\right)\end{array}\right]\text{and}\mathbf{d}=\left[\begin{array}{c}{d}_1\left(x\right)\\ d_2\left(x\right)\\ ⋮\\ d_L\left(x\right)\end{array}\right]\text{.}$$

The least-squares solution for ρ is given by¹

$$\mathbf{\rho }=({\mathbf{S}}^H\mathbf{S}{)}^{-1}{\mathbf{S}}^H\mathbf{d}\text{,}$$

which is the basic SENSE reconstruction formula. Equation (5) implicitly assumes that an infinite number of samples D_l(nΔkˆ) is available, which is not valid in practice. This article analyzes the truncation effect in SENSE reconstruction in comparison with those seen in conventional Fourier imaging using a single, uniform data acquisition channel with Nyquist sampling.

The rest of the article is organized as follows. Section 2 presents a general theoretical analysis of the truncation effects in the reconstruction given by Eq. (5). Section 3 discusses several specific issues and suggests some methods that can mitigate the truncation effects, with some representative simulation and experimental results. The conclusion of the study is given in Section 4.