v2009.01.01 - Convex Optimization

334 CHAPTER 4. SEMIDEFINITE PROGRAMMING
Figure 89: Neighboring-pixel candidates from [309] for estimating
image-gradient. Our implementation selects adaptively from darkest four •
about central.
Because of P idempotence and Hermitian symmetry and sgn(κ)= κ/|κ| ,
this is equivalent to
lim
ǫ→0
(
Ψ T δ(y)δ(|Ψ vec U| + ǫ1) −1 Ψ + λP ) vec U = λPf (770)
where small positive constant ǫ ∈ R + has been introduced for invertibility.
When small enough for practical purposes 4.51 (ǫ≈1E-3), we may ignore the
limiting operation. Then the mapping, for λ ≫ 1/ǫ and 0 ≼ y ≼ 1
vec U t+1 = ( Ψ T δ(y)δ(|Ψ vec U t | + ǫ1) −1 Ψ + λP ) −1
λPf (771)
is a contraction in U t that can be solved recursively in t for its unique
fixed point; id est, until U t+1 → U t . [197, p.300] Calculating this inversion
directly is not possible for large matrices on contemporary computers because
of numerical precision, so we employ the conjugate gradient method of
solution [128,4.8.3.2] at each recursion in the Matlab program.
4.51 We are looking for at least 50dB image/error ratio from only 4.1% subsampled data
(10 radial lines in k-space). With this setting of ǫ, we actually attain in excess of 100dB
from a very simple Matlab program in about a minute on a 2006 vintage laptop. By
trading execution time and treating image-gradient cardinality as a known quantity for
this phantom, over 160dB is achievable.