k-space代写 MATLAB、代写MATLAB MRI编程、代写MATLAB程序、MATLAB编程代写、MATLAB作业代写
In MRI, the spatial localisation of internal tissues are realised by gradient encoding,
which innovation was awarded 2013 Nobel Prize. The theory behind it is as simple as Fourier
transformation. The MR signal is received by RF coils and stored in k-space (frequency
domain), which are later transformed into an image by applying inverse Fourier transform.
Namely, the forward Fourier encoding produces k-space data, and the image reconstruction is
realised by using inverse Fourier transform. However, in practice, acquiring full k-space data
are time consuming; therefore, fast MR imaging are always preferable. One common method
is to use phased-array coils acquiring partial k-space and then reconstruct the image with
Sensitivity encoding (SENSE) method or GeneRalized Autocalibrating Partial Parallel
Acquisition (GRAPPA).
Task1:
(1) Please code Fourier transform and inverse Fourier transform with MATLAB in two
methods. You can use “for loops” as one of your method, and the other method of your
own selection, such as “repmat” (exclude MATLAB build-in functions “fft”, “fft2”).
Apply your Fourier transform on the given brain image to generate k-space data in the
form of Ax = b. The brain image is 512x512, resize it to a 64x64 image to satisfy your
computer memory when performing Fourier transform.
(2) Reconstruct the produced k-space data into the brain image with your coded inverse
Fourier transform. Compare the computational time of your methods (two methods in
Task 1-1) with the MATALB given function ‘ifft2’. Use the “fft2” generated k-space
data as the reference and generate k-space error maps of your method to prove your
method is correct. Isthe Fourier transform matrix invertible? What is its inverse matrix?
Task2: From task2, you can use MATLAB functions fft2 and ifft2. In task 2, resize the image
to 256x256.
(1). Generate and display individual coil images with provided sensitivity profiles (from 8-
element phased-array coils) and then produce a combined image (256x256) with using the sumof-square
(SOS) method. Hint: element-wise production; ?
(2). Generate an undersampled k-space with the provided undersampling pattern (reduction
rate = 4, evenly undersampling), and then compare the reconstructed image with the original
image. Why do we see aliasing artefacts when applying inverse Fourier transform? Produce an
undersampling pattern that only sample central 64 lines. Compared the reconstructed image
from this undersampling pattern with the image reconstructed from evenly undersampled kspace,
why two images are differed so much even the same amount of data are sampled?
Task3: In GRAPPA, the central 32 k-space lines are normally fully sampled, which are referred
as to auto-calibration (AC) lines. The reconstruction of missing sample in the i-th coil at an
unacquired position r is simply given by :
r r ( ) (P R y)H
i ri x r g ? (1)
Where gri are sets of reconstruction weights to be calculated. Rr represents the operation of
choosing a block of k-space (from all the coils) out of the entire grid around the k-space
positions indexed by r. The operators Pr represent local sampling patterns that choose only
acquired samples from a block of k-space. Let y be a multi-coil k-space grid concatenated into
a vector in which unacquired data are zero filled. So, the product PrRry is a vector containing
only the acquired k-space neighborhood around the k-space position r.
The gri can be obtained by solving the relation in Eq. (1) at different positions in k-space, where
the xi are known. Typically, this is done in a fully acquired region in the centre of k-space, e.g.,
autocalibration (AC) region. To perform the calibration, it is useful to construct a so-called
calibration matrix, denoted by A, from the AC portion of the acquired data. It is constructed by
sliding a window throughout the AC data, taking each block (Rry)H
inside the AC region to be
a row in the matrix. The columns of A are shifted versions of the AC area, leading to a matrix
structure known as Block-Hankel.
To obtain conditions for the weights gri , Eq. (1) is rewritten using the calibration matrix and
applied to all locations inside the AC region. This yields a set of ideal conditions for the
reconstruction weights:
, where, ei is a vector with “1” in the appropriate
position that chooses the i-th coil data, and “0” elsewhere.
Please try to:
(1). With the given data, generate a k-space undersampling pattern where the reduction rate is
4 but with central 32 lines fully sampled, then form the calibration matrix A.
(2). Explain the relationship between
H P g e r ri i ? and calibration matrix A through SVD
analysis.
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