v2009.01.01 - Convex Optimization

4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES 329
and number of constraints is cut in half by sampling symmetrically.
Convex iteration for cardinality minimization is introduced which allows
perfect reconstruction of a phantom at 4% subsampling rate; 50% Candès’
rate. By making neighboring pixel selection variable, we show image-gradient
sparsity of the Shepp-Logan phantom to be 1.9% ; 33% lower than previously
reported.
We demonstrate application of image-gradient sparsification to the
n×n=256×256 Shepp-Logan phantom, simulating ideal acquisition of MRI
data by radial sampling in the Fourier domain (Figure 87). 4.45 Define a
Nyquist-centric discrete Fourier transform (DFT) matrix
⎡
⎤
1 1 1 1 · · · 1
1 e −j2π/n e −j4π/n e −j6π/n · · · e −j(n−1)2π/n
F =
∆ 1 e −j4π/n e −j8π/n e −j12π/n · · · e −j(n−1)4π/n
1
⎢ 1 e −j6π/n e −j12π/n e −j18π/n · · · e −j(n−1)6π/n
√ ∈ C n×n
⎥ n
⎣ . . . .
... . ⎦
1 e −j(n−1)2π/n e −j(n−1)4π/n e −j(n−1)6π/n · · · e −j(n−1)2 2π/n
(747)
a symmetric (nonHermitian) unitary matrix characterized
F = F T
F −1 = F H (748)
Denoting an unknown image U ∈ R n×n , its two-dimensional discrete Fourier
transform F is
F(U) ∆ = F UF (749)
hence the inverse discrete transform
U = F H F(U)F H (750)
FromA.1.1 no.25 we have a vectorized two-dimensional DFT via Kronecker
product ⊗
vec F(U) ∆ = (F ⊗F ) vec U (751)
and from (750) its inverse [140, p.24]
vec U = (F H ⊗F H )(F ⊗F ) vec U = (F H F ⊗ F H F ) vec U (752)
4.45 k-space is conventional acquisition terminology indicating domain of the continuous
raw data provided by an MRI machine. An image is reconstructed by inverse discrete
Fourier transform of that data sampled on a Cartesian grid in two dimensions.