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Reconstruction Project for Principle of MRI
Deadline: January 5 2025 at 12:00 pm.
1. Your lab write-up should include answers to all the questions. Please comment and clearly justify your answers. The MATLAB or Python codes and report of this lab need to be uploaded to the Blackboard system before the deadline. You need to submit: 1) All the code files that you have generated and 2) A report, in pdf with the obtained figures, answers and detailed comments. Failure to submit code will result in a 40% reduction of the assignment mark.
2. AI tools (such as ChatGPT) can be employed for studying and programming purposes.
1. Iterative SENSE (50 marks)
Provided Materials
1. C (.npy/.mat) File contains coil sensitivity map of each channel, symbol: C, shape: [Nc, Npix, Npix], Nc = 12, Npix = 256.
Assignment
1. (10 marks) Apply the provided coil sensitivity map C to the shepp-logan phantom image m with matrix size 256x256 and plot the simulated multi-coil images. Employ the root-sum-of-square approach and the weighted coil sensitivity approach to combine the data from the individual coils. Depict and compare the combined images with respect to the original fully sampled image m.
Scoring points:
3 marks for the multi-coil images.
3 marks for the root-sum-of-square approach.
3 marks for the weighted coil sensitivity approach.
1 mark for the comparison.
2. (10 marks) Generate 2 uniform under-sampling patterns with acceleration factors of 3 and 7 (U3
and U7
) and 2 random under-sampling patterns (UR3
and UR7
). Each sampling pattern must be a matrix with 1 s in the sampled positions and 0 s in the remaining entries. Obtain the corresponding point spread functions (PSFs) and comment about the expected aliasing generated by these under-sampling patterns.
Scoring points:
1 mark for U3 .
1 mark for U7 .
1 mark for UR3.
1 mark for UR7.
1 marks for the PSF of U3.
1 marks for the PSF of U7.
1 marks for the PSF of UR3.
1 marks for the PSF of UR7.
2 marks for the comparison.
3. (10 marks) Obtain the aliased images for each coil as a result of under-sampling with the generated patterns. For this you should use:
where F is the Fourier transform, F
−1
is the inversed Fourier transform and bi are the aliased images for each coil i = 1,...,Nc. Depict and compare the aliased images for the different under-sampling factors and patterns.
Scoring points:
2 marks for the image sampled by U3.
2 marks for the image sampled by U7.
2 marks for the image sampled by UR3.
2 marks for the image sampled by UR7.
2 marks for the comparison.
4. (15 marks) The SENSE under-sampled reconstruction can be written as a linear problem:
Em = b
where m is the image to be reconstructed and the encoding matrix E = UFC corresponds to the forward sampling operator, with U the under-sampling operator, F the Fourier transform operator, C the coil sensitivity maps and b the acquired k-space data.
Iterative SENSE reconstruction is obtained by solving the linear problem Em = b as a least square minimization m^ = arg minm ∥Em − b∥
2
2
. Implement the Gradient Descent method to solve the above problem.
Show and compare your results for the under-sampling patterns generated in previous part. What can you conclude from them? How many iterations are needed to reconstruct acquisitions with the different sampling patterns?
Scoring points:
10 marks for the intermediate results during the optimization.
3 marks for the final result of the reconstruction.
2 marks for the comparison and the conclusion.
(Optimization is not necessarily done by Gradient Decent, other optimizers from2. Non-Cartesian Reconstruction (30 + 5 marks) pytorch (such as Adam) is also allowed. Regularization term can be employed to enhance the convergence)
5. (5 marks) Define the reconstruction error as the difference between the fully sampled image m and your reconstructions m^ as:
e = m − m^
and the mean square error (MSE) of the reconstruction as
where i=1,...,N indicate each pixel in the image.
For all reconstructions (i.e. uniform and random 3x and 7x) plot the MSE w.r.t the number of iterations for the Gradient descent iterative SENSE method implemented. Comment about the convergence of the method.
Scoring points:
3 marks for the MSE curve.
2 marks for the comments.
1. K (.npy/.mat) File contains the spiral sampling trajectory, K with shape of [NPE, NRO, Ndim], NPE refers to the number of spiral interleaves, NRO refers to the number of points in each spiral interleaf, Ndim refers to the number of dimensions.
2. Om (.npy/.mat) File contains the off-resonance frequency of every pixel, symbol: ω(x→), shape: [Npix, Npix], Npix = 256
1. (10 marks) Show the provided k-space trajectory. Then, use the Non-Uniform Fast Fourier Transform (NUFFT) toolbox provided in the following to simulate the k-space data sampled with the provided Spiral trajectory K of the shepp-logan phantom. The NUFFT toolboxes are: PyNUFFT[1] (https://pypi.org/project/pynufft/) for Python; MIRT[2] (https://github.com/JeffFessler/mirt/tree/main) for MATLAB.
Scoring points:
5 marks for the figure of the trajectory.
5 marks for the plot of the simulated k-space data, including real part andimaginary part. If image is correctly reconstructed in (2), no points deduced.
2. (10 marks) Reconstruct the phantom image from the simulated k-space data. You can use inverse NUFFT or inverse Discrete Fourier transform. For this task, you may need to design the appropriate density compensation function using method such as the Voronoi diagram or any other method. In the answer, you should clarify the method you are using and show the reconstructed image.
Scoring points:
8 marks for the final reconstructed image.
2 marks for the explanation to the employed method.
3. (10 marks) Perform 4-fold undersampling on the simulated k-space data in (1), and reconstruct the image from the undersampled data. The under-sampling artifacts should be reduced or removed. For this task you can use BART (https://mrirecon.github.io/bart/) or any other software/toolbox. Please show the undersampled image and reconstructed image. You also need to briefly describe the method you use such as parallel imaging or compressed sensing. Note that BART is a command line tool runs in Linux or WSL. For help, please refers to webinars of BART, or use bart or bart <sub-command> -h .
Scoring points:
4 marks for the reconstructed image.
4 marks for the aliased image (for comparison).
2 marks for the description of the employed method.
4. (5 marks, bonus question) Simulate the k-space data sampled by the provided Spiral trajectory K with additional off-resonance effect, and then reconstruct the image from the k-space with off-resonance. You can use the following formula to simulate off-resonance:
ω(x→) denotes the off-resonance angular frequency at position x→.
Please show the reconstructed image I(x→) affected by off-resonance, and describe the off-resonance artifacts in Spiral MRI.
Scoring points:
4 marks for the reconstructed image affected by off-resonance.
1 mark for the description.
(bonus can be used to offset the previously lost points)
Reference
[1] J.-M. Lin, “Python Non-Uniform Fast Fourier Transform (PyNUFFT): An Accelerated Non-Cartesian MRI Package on a Heterogeneous Platform (CPU/GPU),” Journal of Imaging, vol. 4, no. 3, Art. no. 3, Mar. 2018, doi: 10.3390/jimaging4030051.
[2] J. A. Fessler and B. P. Sutton, “Nonuniform fast Fourier transforms using min-max interpolation,” IEEE Transactions on Signal Processing, vol. 51, no. 2, pp. 560–574, Feb. 2003, doi: 10.1109/TSP.2002.807005.
