v2010.10.26 - Convex Optimization

378 CHAPTER 4. SEMIDEFINITE PROGRAMMINGvec −1 fFigure 109: Aliasing of Shepp-Logan phantom in Figure 107 resulting fromk-space subsampling pattern in Figure 108. This image is real becausebinary mask Φ is vertically and horizontally symmetric. It is remarkablethat the phantom can be reconstructed, by convex iteration, given onlyU 0 = vec −1 f .transform [284, p.53]) demands vertical symmetry about row n +1 and2horizontal symmetry 4.53 about column n+1.2Define ⎡⎤1 0 0⎢−1 1 0 ⎥∆ ⎢⎣−1 1 . .............. 1 00 T −1 1Express an image-gradient estimate⎡U ∆∇U ⎢ U ∆ T⎣ ∆ U∆ T U⎤∈ R n×n (851)⎥⎦⎥⎦ ∈ R4n×n (852)4.52 (848) is a diagonalization of matrix P whose binary eigenvalues are δ(vec ΘΦΘ) whilethe corresponding eigenvectors constitute the columns of unitary matrix F H ⊗F H .4.53 This condition on Φ applies to both DC- and Nyquist-centric DFT matrices.