v2010.10.26 - Convex Optimization

376 CHAPTER 4. SEMIDEFINITE PROGRAMMINGΦFigure 108: MRI radial sampling pattern, in DC-centric Fourier domain,representing 4.1% (10 lines) subsampled data. Only half of these complexsamples, in any halfspace about the origin in theory, need be acquiredfor a real image because of conjugate symmetry. Due to MRI machineimperfections, samples are generally taken over full extent of each radialline segment. MRI acquisition time is proportional to number of lines.FromA.1.1 no.31 we have a vectorized two-dimensional DFT via Kroneckerproduct ⊗vecF(U) (F ⊗F ) vec U (840)and from (839) its inverse [166, p.24]vec U = (F H ⊗F H )(F ⊗F ) vec U = (F H F ⊗ F H F ) vec U (841)Idealized radial sampling in the Fourier domain can be simulated byHadamard product ◦ with a binary mask Φ∈ R n×n whose nonzero entriescould, for example, correspond with the radial line segments in Figure 108.To make the mask Nyquist-centric, like DFT matrix F , define a circulant[168] symmetric permutation matrix 4.51Θ [ 0 II 0]∈ S n (842)Then given subsampled Fourier domain (MRI k-space) measurements in4.51 Matlab fftshift()