v2009.01.01 - Convex Optimization

330 CHAPTER 4. SEMIDEFINITE PROGRAMMING
Φ
Figure 87: MRI radial sampling pattern in DC-centric Fourier domain
representing 4.1% (10 lines) subsampled data. Only half of these complex
samples, in theory, need be acquired for a real image because of conjugate
symmetry. Due to MRI machine imperfections, samples are generally taken
over full extent of each radial line segment. MRI acquisition time is
proportional to number of lines.
Radial sampling in the Fourier domain can be simulated by Hadamard
product ◦ with a binary mask Φ∈ R n×n whose nonzero entries could, for
example, correspond with the radial line segments in Figure 87. To make
the mask Nyquist-centric, like DFT matrix F , define a permutation matrix
for circular shift 4.46 Θ ∆ =
[ 0 I
I 0
]
∈ S n (753)
Then given subsampled Fourier domain (MRI k-space) measurements in
incomplete K ∈ C n×n , we might constrain F(U) thus:
and in vector form, (35) (1664)
ΘΦΘ ◦ F UF = K (754)
δ(vec ΘΦΘ)(F ⊗F ) vec U = vec K (755)
Because measurements K are complex, there are actually twice the number
of equality constraints as there are measurements.
4.46 Matlab fftshift().