v2010.10.26 - Convex Optimization

4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES 379that is a simple first-order difference of neighboring pixels (Figure 110) tothe right, left, above, and below. 4.54 ByA.1.1 no.31, its vectorization: forΨ i ∈ R n2 ×n 2vec ∇U =⎡⎢⎣⎤∆ T ⊗ I∆ ⊗ I⎥I ⊗ ∆I ⊗ ∆ T⎡⎦ vec U ⎢⎣Ψ 1Ψ T 1Ψ 2Ψ T 2⎤⎥⎦ vec U Ψ vec U ∈ R4n2 (853)where Ψ∈ R 4n2 ×n 2 . A total-variation minimization for reconstructing MRIimage U , that is known suboptimal [207] [71], may be concisely posedwhereminimize ‖Ψ vec U‖ 1Usubject to P vec U = f(854)f = (F H ⊗F H ) vec K ∈ C n2 (855)is the known inverse subsampled Fourier data (a vectorized aliased image,Figure 109), and where a norm of discrete image-gradient ∇U is equivalentlyexpressed as norm of a linear transformation Ψ vec U .Although this simple problem statement (854) is equivalent to a linearprogram (3.2), its numerical solution is beyond the capability of even themost highly regarded of contemporary commercial solvers. 4.55 Our onlyrecourse is to recast the problem in Lagrangian form (3.1.2.1.2) and writecustomized code to solve it:minimizeUminimize 〈|Ψ vec U| , y〉Usubject to P vec U = f≡〈|Ψ vec U| , y〉 + 1λ‖P vec U − 2 f‖2 2(a)(b)(856)4.54 There is significant improvement in reconstruction quality by augmentation of anominally two-point discrete image-gradient estimate to four points per pixel by inclusionof two polar directions. Improvement is due to centering; symmetry of discrete differencesabout a central pixel. We find small improvement on real-life images, ≈1dB empirically,by further augmentation with diagonally adjacent pixel differences.4.55 for images as small as 128×128 pixels. Obstacle to numerical solution is not acomputer resource: e.g., execution time, memory. The obstacle is, in fact, inadequatenumerical precision. Even when all dependent equality constraints are manually removed,the best commercial solvers fail simply because computer numerics become nonsense;id est, numerical errors enter significant digits and the algorithm exits prematurely, loopsindefinitely, or produces an infeasible solution.