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A novel, rapid algorithm to speed up and improve the reconstruction of sensitivity encoding (SENSE) MRI was proposed in this paper. The essence of the algorithm was that it iteratively solved the model of simple SENSE on a pixel-by-pixel basis in the region of support (ROS). The ROS was obtained from scout images of eight channels by morphological operations such as opening and filling. All the pixels in the FOV were paired and classified into four types, according to their spatial locations with respect to the ROS, and each with corresponding procedures of solving the inverse problem for image reconstruction. The sensitivity maps, used for the image reconstruction and covering only the ROS, were obtained by a polynomial regression model without extrapolation to keep the estimation errors small. The experiments demonstrate that the proposed method improves the reconstruction of SENSE in terms of speed and accuracy. The mean square errors (MSE) of our reconstruction is reduced by 16.05% for a 2D brain MR image and the mean MSE over the whole slices in a 3D brain MRI is reduced by 30.44% compared to those of the traditional methods. The computation time is only 25%, 45%, and 70% of the traditional method for images with numbers of pixels in the orders of 10^{3}, 10^{4}, and 10^{5}–10^{7}, respectively.

Parallel imaging (PI) is one of the most important applications of the phased-array surface coils introduced to the field of MRI more than two decades ago [

Consider a simple 2D SENSE with an acceleration rate of

The traditional method for solving ^{5} , making the computation time of direct image reconstruction impractically long in clinical settings; (ii) It solved the equations for the entire FOV and did not take into account the region of support [

An iterative method was introduced recently for the fast reconstruction of the SENSE-based MRI [

For the model of simple 2D SENSE with an acceleration factor

This is a simplified version of the direct problem previously given in

There are three approaches to SENSE reconstruction [

We briefly analyze the three approaches to recovering an 8 × 8 matrix from two aliased 4 × 8 images acquired with an acceleration factor of 2 (^{st} iteration, and then we unfold the second lines in the aliased images to obtain the second and sixth row at the 2^{nd} iteration, and so on. With the pixel-by-pixel iteration, we obtain the ^{st} iteration, the ^{nd} iteration, until the ^{th} iteration; Then the scan turns to the next row, and we obtain the ^{th} iteration, and so on. In this paper, we chose the pixel-by-pixel iteration since it only requires least memory.

In matrix notation, equations in

Here ^{1} and ^{2} are two column vectors of eight elements. In essence, the solution of

The ROS is detected from the scout images of eight channels [_{i}

Calculate the power image defined as:

Find the support area. Here we choose the threshold as the 1% of maximum intensity value:

Perform the opening operation to eliminate the noise artifacts:

Use the filling method of mathematical morphologic operations to fill the holes:

The mean essence of the algorithm is thresholding based on the signal-to-noise ratio (SNR), because the brain area usually contains higher signal than the background does. But other morphological operations are also necessary to ensure an example of these procedures is illustrated in

In the literature [

We used a second-order polynomial regression model to approximate the realistic sensitivity map and to reduce noises. The model is as follows:

Let ^{2}, ^{2}, _{20}, _{11}, _{02}, _{10}, _{01}, _{00}]^{T} denote the predictor vector (regressor), and then

Suppose there are

Here model(ROS)={;model(P)|P ∈ ROS}.

Finally, the regressor

Here S(full)= {S(P)|P ∈ full}, and model(full)= {model(P)|P ∈ full}. However, the regressor

Here S(ROS)= {S(P)|P ∈ROS}. An example of the estimation of sensitivity maps is illustrated in

For the ROS-based rapid algorithm of SENSE reconstruction, we classified the pair of aliasing pixels into four groups as follows:

Both pixels fall into the ROS. We solve

The point of (^{1} of

The point of (

Neither of the pair falls into the ROS. We assign a value of 0 to each of the pixels. As seen from

We carried out experiments to assess the performance of our method and, in particular, to compare our method with the traditional methods. The experiments were carried out on the platform of Windows XP on a desktop PC rquipped with an Intel Pentium 4, 3 GHz processor and 2 GB memory. The programs were developed via Matlab 2010b.

The traditional method used ROS as a correction tool, viz., to multiply final reconstruction results with the ROS to reduce the background noises. We call it “ROS-based correction” for short. In our method, we used the ROS at the reconstruction stage to group the pairs into different types, which is referred to as “ROS-based reconstruction”. The main differences of ROS-based correction and ROS-based reconstruction lie in the following three points (shown in the red font in

ROS-based correction calculates the sensitivity map of the full image, which involves extrapolation; however, ROS-based reconstruction only needs the sensitivity map within the ROS, which only involves interpolation.

ROS-based correction directly solves

ROS-based correction needs to correct the final result by multiplying it with the ROS, while the ROS-based reconstruction is free from this procedure.

We compared our proposed ROS-based reconstruction method with traditional ROS-based correction method. We first used a fully sampled T2 weighted brain MRI image as a reference data set, undersampled the data with an acceleration rate of 2, applied the aforementioned two methods to reconstruct the images from the undersampled data, and compared the results of the two methods using mean absolute error (MAE) and mean square errors (MSE), which were calculated against the ground truth. To facilitate fair comparison, we calculated MAEs and MSEs on the ROS instead of the whole FOV. In order to assess the performance of sensitivity map estimation, we added Gaussian noise of zero mean and 10^{−4} variance, which indicates the pixels in the image will have a fluctuation of

The reason why our ROS-based reconstruction method can achieve less error leans on the H-shape area in

The advantages of our ROS-based reconstruction method over the conventional method are more significant for 3D MRI than for 2D MRI. We used a 128 × 128 × 64 MRI data and added Gaussian noise with zero mean and 10^{−4} variance. The curves of MAE and MSE of two methods

We compared the computation times of our method and the conventional method using datasets of 2D/3D MRI, 3D MRSI, 3D DTI and 3D fMRI. The 1^{st} 3 dimensions are spatial dimensions and the 4^{th} for 3D MRSI, 3D DTI and fMRI are spectral, angular and temporal respectively. For images with small size, e.g., a 2D MRI of 256 × 256, the computation time of the proposed method is less than half of the conventional method (

In this study, we have proposed an ROS-based method for the reconstruction of SENSE MRI. The method involves an ROS-based accurate estimation of sensitivity maps and an ROS-based pixel-by-pixel iterative algorithm for the reconstruction. The experiments show that the method is fast and significantly improve the quality of reconstruction of SENSE MRI.

The MSE of our reconstruction is reduced by 16.05% for a 2D brain MR image and the mean MSE over the whole slices in a 3D brain MRI is reduced by 30.44% compared to those of the traditional methods. The computation time is only 25%, 45%, and 70% of the traditional method for images with numbers of pixels in the orders of 10^{3}, 10^{4}, and 10^{5}–10^{7}, respectively.

However, the computation advantage of our method depends on the support size of the brain in the FOV. If the brain occupies most of the FOV, the computational advantage will be compromised. Therefore, our method is suitable for images which contain large background area.

One of an interesting future work will be on the combination of ROS and other techniques, such as the 3D wavelet representation [

A pictorial illustration of the SENSE principles. The k-space is scanned every other row. The rows are for the first, the second and the eighth channels, respectively. Column 1 & 3 are the sensitivity maps, column 2 & 4 are the brain or the images to be reconstructed, and final column is the aliased brain images acquired by the corresponding channels. The red square marker denotes the pixel at location (

Schematic illustration of the three approaches to the SENSE reconstruction. The acceleration rate is 2. The upper 4 × 8 (pink) and the lower 4 × 8 (blue) matrices in the above represent the voxels in the aliased images. The full image is an 8 × 8 matrix. (^{st} and the 5^{th} rows in the first iteration, and the 2^{nd} and the 6^{th} rows in the 2^{nd} iteration, and so on; (

Illustration of obtaining ROS: (

Estimation of sensitivity maps. From left to right columns: eight scout images (O_{1} to O_{8}) from eight channels; energy ratios (D_{1} to D_{8}); Ds within ROS; regressor parameters by _{i}_{i}

Flowchart of (

Comparison of the quality of the reconstructions: (_{2}-weighted brain MR image; (

3D Brain Reconstruction Results: (

Comparison of the computation times between the proposed method and the traditional method. The x-axis denotes the total number of image voxels, and the y-axis denotes the ratio of computation times of the proposed method to traditional method. We used logarithm scale for clearance.

Comparison of computation times image reconstruction by the conventional and the proposed methods for different MR modalities (

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2D MRI | 256 × 256 | 65,536 | 5.7051 | 2.5774 |

3D MRI | 128 × 128 × 64 | 1,048,576 | 91.4645 | 62.2536 |

3D MRSI | 32 × 32 × 8 × 512 | 4,194,304 | 365.9453 | 254.6059 |

3D DTI | 128 × 128 × 16 × 24 | 6,291,456 | 549.7143 | 383.4347 |

3D fMRI | 64 × 64 × 32 × 64 | 8,388,608 | 735.9560 | 512.3298 |