v2010.10.26 - Convex Optimization

4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES 383There are two features that distinguish problem formulation (856b) andour particular implementation of it [Matlab on Wıκımization]:1) An image-gradient estimate may engage any combination of fouradjacent pixels. In other words, the algorithm is not lockedinto a four-point gradient estimate (Figure 110); number of pointsconstituting an estimate is directly determined by direction vectory . 4.59 Indeed, we find only c = 5092 zero entries in y ⋆ for theShepp-Logan phantom; meaning, discrete image-gradient sparsity isactually closer to 1.9% than the 3% reported elsewhere; e.g., [355,IIB].2) Numerical precision of the fixed point of contraction (860) (≈1E-2for almost perfect reconstruction @−103dB error) is a parameter tothe implementation; meaning, direction vector y is updated aftercontraction begins but prior to its culmination. Impact of thisidiosyncrasy tends toward simultaneous optimization in variables Uand y while insuring y settles on a boundary point of its feasible set(nonnegative hypercube slice) in (862) at every iteration; for only aboundary point 4.60 can yield the sum of smallest entries in |Ψ vec U ⋆ |.Almost perfect reconstruction of the Shepp-Logan phantom at 103dBimage/error is achieved in a Matlab minute with 4.1% subsampled data(2671 complex samples); well below an 11% least lower bound predicted bythe sparse sampling theorem. Because reconstruction approaches optimalsolution to a 0-norm problem, minimum number of Fourier-domain samplesis bounded below by cardinality of discrete image-gradient at 1.9%. 4.6.0.0.13 Exercise. Contraction operator.Determine conditions on λ and ǫ under which (860) is a contraction andΨ T δ(y)δ(|Ψ vec U t | + ǫ1) −1 Ψ + λP from (861) is positive definite. 4.59 This adaptive gradient was not contrived. It is an artifact of the convex iterationmethod for minimal cardinality solution; in this case, cardinality minimization of a discreteimage-gradient.4.60 Simultaneous optimization of these two variables U and y should never be a pinnacleof aspiration; for then, optimal y might not attain a boundary point.