^{1}

^{*}

^{2}

^{1}

The Sensitivity Encoding (SENSE) parallel reconstruction scheme for magnetic resonance imaging (MRI) is implemented with non-cartesian sampled k-space trajectories in this paper. SENSE has the special capability to reduce the scanning time for MRI experiments while maintaining the image resolution with under-sampling data sets. In this manner, it has become an increasingly popular technique for multiple MRI data acquisition and image reconstruction schemes. The gridding algorithm is also implemented with SENSE due to its ability in evaluating forward and adjoin operator with non-cartesian sampled data. In this paper, the sensitivity map profile, field map information and the spiral k-space data collected from an array of receiver coils are used to reconstruct unaliased images from under-sampled data. The performance of SENSE with real data set identifies the computational issues to be improved for researched.

Although MRI is a relatively young technology, it has reached a juncture where future advances are limited by the scanning and reconstruction time. Since current MRI scanners already operate at the physical limit of data acquisition speed, mainly due to technical difficulties in producing rapidly switching magnetic field gradients, people resort to parallel imaging techniques that apply phased array coils and parallel reconstruction methods for faster MR imaging. Phased array coils were first exploited in 1987 to reduce phase-encoding lines [

Parallel imaging in MRI is often performed by placing an array of receiver coils around the object to be imaged, with each independent receiver coil lending spatially distinct reception profiles to the collected data sets. The following equation indicates the formation of baseband image in a typical MRI experiment:

ρ ( t ) = ∫ d ( r ) S ( r ) e − i ω ( r ) t e − i 2 π k ( t ) r d r + ε ( t ) (1)

where d(r) is a continuous function of the object’s traverse magnetization at location r; d(r) along with its trajectories can be obtained directly from MRI scanners in k-space, the frequency space; S(r) is the spatial sensitivity profile of the receiver coil; ω(r) is the B0 field inhomogeneity present at location r; k(t) is the input data trajectory at time t; ε(t) is the noise term at time t. Equation (1) shows that the k-space data, the coil sensitivity profile and the field inhomogeneity information are three essential components for parallel image formation. It also suggests that the data acquired from multiple independent coils and the features of those coils can be integrated into a larger system of equations. With each coil receiving its own data, weighted by S_{l}(r), where l = 1 , ⋯ , L , i.e. the complex spatial sensitivity profile of coil l, the parallel imaging model in an integrated matrix form can thus be represented as follows (take L = 3 as an example):

[ ρ 1 ρ 2 ρ 3 ] = [ F ⋅ S 1 F ⋅ S 2 F ⋅ S 3 ] d + [ ε 1 ε 2 ε 3 ] (2)

where ρ_{1} denotes the signal vector received from coil 1 and the image vector ρ is formed by stacking ρ_{l}’s into a single column; S_{l} is the diagonal matrix holding the complex spatial sensitivity profiles on the diagonal entries from the lth coils; F denotes the imaging system matrix.

From the above mathematical model, parallel imaging makes use of signal processing methods to reduce the scanning time and the amount of acquired k-space data while still maintaining the same spatial resolution, i.e. the same area of k-space is still effectively covered in the Nyquist sense with the acquired data from phased array coils. However, according to common signal processing knowledge on sampling theory, if any parallel imaging scheme enables a reduced number of phase encoding lines, the interval between which is 1/Field of View (FOV), aliasing will consequently show up in the reconstructed image when conventional FFT reconstruction is employed.

However, the SENSE parallel imaging scheme [

The SENSE imaging scheme works in this way: To obtain a desired image from the acquired full FOV image data, the image value of pixel (x,y) should be weighted with the coil sensitivity profile S at the corresponding locations (x,y_{l}), where l = 1 , ⋯ , R . And the accurate reconstructed pixel value at the location (x,y) with the k’th coil information is as follows:

I k ( x , y ) = S k ( x , y 1 ) ρ ( x , y 1 ) + S k ( x , y 2 ) ρ ( x , y 2 ) + ⋯ + S k ( x , y R ) ρ ( x , y R ) (3)

This equation can be transformed into matrix notation and illustrated in

I = S ⋅ ρ (4)

To simplify the formulation, in Equation (4) we do not consider the issue of noise correlation.

According to the matrix theory knowledge, we can easily obtain the full FOV image from taking generalized of the sensitivity matrix S:

ρ = ( S H S ) − 1 S H ⋅ I (5)

The gridding algorithm is usually employed to deal with data sets that do not fall on regular cartesian grids in the k-space. The gridding approach allows people to collect and process non-cartesian data sets in a way that previously developed cartesian reconstruction methods, such as FFT based schemes, can be applied without much modification. In this report, the gridding algorithm is performed in a way that the acquired spiral trajectory data is first re-sampled onto cartesian sampled grids before the FFTW library is called for image reconstruction. In the re-sampling process, the gridding algorithm provides an approximation of the adjoint and forward operators in the SENSE imaging scheme. The basic concept of adjoint and forward operators can be expressed as follows:

ρ ( x n ) = ∑ l = 0 l − 1 e + i 2 π f ( x n ) τ l ∑ m = 0 M a l ( t ) d ( m ) e + i 2 π k m x n , (6)

d ( m ) = ∑ l = 0 l − 1 a l ( t ) ∑ n = 0 N ρ ( x n ) e − i 2 π k m x n e − i 2 π f ( x n ) τ l , (7)

where d(m) demotes the k-space sampled data; a(t) denotes the interpolation window function for time segmentation; in this report, the hann window function is calculated and applied to time segmentation; f(x_{n}) denotes the B0 field inhomogeneity map; k_{m} denotes the k-space trajectories. From these two formulations, it is evident that the adjoint operator takes image data value and k-space trajectories as input and returns the k-space data value on the cartesian grid points as output. The forward operator, on the contrary, takes k-space data value and k-space trajectories as input and outputs the image value at each corresponding pixel locations. Although these two operators are distinct from each other, the evaluation of both operators is indispensable for the conjugate gradient method. The following paragraphs will explain the implementation of these two operators in detail.

Finally, the result of the third step is the desired image from the adjoint operator evaluated via gridding algorithm. Although some literatures [

The reconstructed data set in this report is the “Double Vision” data of the “ISMRM Reconstruction Challenge”. The “Double Vision” data originate from 12 axial images in the abdomen collected using a torso phase-array coil, collected at 320 × 320 matrix and resized to four different sizes, 64 × 64 matrix, 128 × 128 matrix, 256 × 256 matrix and 512 × 512 matrix, to test the effectiveness of reconstruction. Field maps were collected using gradient echo images at TE = 3, with a 96 × 96 collected matrix resized to the above mentioned four different sizes. Sensitivity maps are also provided with the dataset. The original data sets were synthesized over 8 spiral trajectories, each with 20,000 points. The k-space trajectories at the first and last interleaves are shown in

The above figures demonstrate that the SENSE parallel imaging scheme with gridding algorithm can effectively handle non-cartesian k-space trajectories. Compare

The implementation of SENSE parallel imaging scheme has shown the effectiveness of reconstructing image from arbitrary k-space trajectories. By incorporating data from multiple receiver coils, the under sampling effect in each coil's data can be effectively compensated. The combination of conjugate gradient algorithm, forward and inverse gridding, together with FFT can reduce the computation complexity as opposed to brute force methods. Still, as can be seen from

Image size | Adjoint Operator (s) | Forward Operator (s) | Estimated Timing (s) | Actual Timing (s) | Estimation/ Actual (%) |
---|---|---|---|---|---|

512 × 512 | 47.53 | 42.34 | 10,992.0 | 12,056.6 | 91.17 |

256 × 256 | 15.99 | 14.15 | 3690.4 | 3827.4 | 96.41 |

128 × 128 | 5.13 | 3.24 | 1080.0 | 1170.9 | 92.24 |

64 × 64 | 1.76 | 0.76 | 358.4 | 372.3 | 96.24 |

Sense parallel imaging based on gridding algorithm can improve the efficiency and speed of image reconstruction from arbitrary k-space trajectory. This term project is a wise investment of time and effort for my future research effort on three dimensional SENSE and it is exciting experience in applying critical thinking skills to the state of the art image reconstruction techniques in my research area. The reconstructed image contains a small number of Additive white Gaussian noise in a given dataset, which can be eliminated by a combination of regularization and SENSE implementations.

The authors declare no conflicts of interest regarding the publication of this paper.

Zhang, L.J., Zhao, L.F. and Liu, G. (2021) Research of Sensitivity Encoding Reconstruction for MRI with Non-Cartesian K-Space Trajectories. Journal of Computer and Communications, 9, 1-9. https://doi.org/10.4236/jcc.2021.91001