project1.md

Project 1: SENSE Reconstruction

[TOC]

1. Theory

1.1 Mathematical description

• According to the theory of Fourier transform, downsampling in frequency domain by factor $R$, will cause the image to be cyclically extended by period $L/R$ ($L$ is the length of data) in spatial domain.

• Then, the alised voxels are as below

$$ \begin{aligned} \begin{bmatrix} I_0(y) \ I_1(y) \ \vdots \ I_{N_c-1} \end{bmatrix} &= \begin{bmatrix} C_0(y) & \cdots & C_0(y + (N_A-1)L/R) \\ C_1(y) & \cdots & C_1(y + (N_A-1)L/R) \\ \vdots & \vdots & \vdots \\ C_{N_c-1}(y) & \cdots & C_{N_c-1}(y + (N_A-1)L/R) \end{bmatrix} \begin{bmatrix} m(y) \ m(y+L/R) \ \vdots \ m(y+(N_A-1)L/R) \end{bmatrix} \\ I &= Cm \end{aligned} \tag{1} $$

   In which, $C$ and $I$ are sensitivity maps and alised image of each channel respectively, and $N_c$ is the number of channels, $N_A = L/R$. Then the unfolded voxels are

$$ m = \left(C^{H}\Phi^{-1}C \right)^{-1} C^{H}\Phi^{-1}I \tag{2} $$

   $\Phi$ is the $N_c \times N_c$ coil noise correlation matrix. The coil sensitivity images are

$$ \begin{aligned} C_j = \frac{I_j}{I} = \frac{I_j}{\sqrt{\sum_i (\vert I_i \vert^2 / \sigma_i^2)}} ;,; j = 1,2,\cdots,N_c \end{aligned}\tag{3} $$

   The factor $\mathrm{g}$ is

$$ g_i = \sqrt{[\left(C^{H}\Phi^{-1}C \right)^{-1}]{ii} \left(C^{H}\Phi^{-1}C \right){ii}} \tag{4} $$

1.2. Flow chart

• Using Fourier transform (FT) to reconstruct the full-sampled k-space, and then combine all channel through sum of square
• Compute the sensitivity map according to the equation (3)
• Undersample in k-space by acceleration factor $R$, and then obtain the sub-sampled image by FT
• Perform SENSE reconstruction according to equation (1).

Figure 1. SENSE reconstruction flow chart

2. Results

• The program execution begins and ends in file project1.m. Function senseKernel() realizes the SENSE algorithm.

Figure 2. Sensitivity maps

Figure 3. Acceleration factor R = 2

Figure 4. Acceleration factor R = 2

Figure 5. Acceleration factor R = 2