v2010.10.26 - Convex Optimization

4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES 375Lagrangian form of compressed sensing in imagingWe now repeat Candès’ image reconstruction experiment from 2004 whichled to discovery of sparse sampling theorems. [70,1.2] But we achieveperfect reconstruction with an algorithm based on vanishing gradient of acompressed sensing problem’s Lagrangian, which is computationally efficient.Our contraction method (p.381) is fast also because matrix multiplications arereplaced by fast Fourier transforms and number of constraints is cut in halfby sampling symmetrically. Convex iteration for cardinality minimization(4.5) is incorporated which allows perfect reconstruction of a phantom at4.1% subsampling rate; 50% Candès’ rate. By making neighboring-pixelselection adaptive, convex iteration reduces discrete image-gradient sparsityof the Shepp-Logan phantom to 1.9% ; 33% lower than previously reported.We demonstrate application of discrete image-gradient sparsification tothe n×n=256×256 Shepp-Logan phantom, simulating idealized acquisitionof MRI data by radial sampling in the Fourier domain (Figure 108). 4.50Define a Nyquist-centric discrete Fourier transform (DFT) matrix⎡⎤1 1 1 1 · · · 11 e −j2π/n e −j4π/n e −j6π/n · · · e −j(n−1)2π/nF 1 e −j4π/n e −j8π/n e −j12π/n · · · e −j(n−1)4π/n1⎢ 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(836)a symmetric (nonHermitian) unitary matrix characterizedF = F TF −1 = F H (837)Denoting an unknown image U ∈ R n×n , its two-dimensional discrete Fouriertransform F isF(U) F UF (838)hence the inverse discrete transformU = F H F(U)F H (839)4.50 k-space is conventional acquisition terminology indicating domain of the continuousraw data provided by an MRI machine. An image is reconstructed by inverse discreteFourier transform of that data interpolated on a Cartesian grid in two dimensions.