v2009.01.01 - Convex Optimization

4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES 335
Observe that P (759), in the equality constraint from problem (765), is
not a fat matrix. 4.52 Although number of Fourier samples taken is equal to
the number of nonzero entries in binary mask Φ , matrix P is square but
never actually formed during computation. Rather, a two-dimensional fast
Fourier transform of U is computed followed by masking with ΘΦΘ and
then an inverse fast Fourier transform. This technique significantly reduces
memory requirements and, together with contraction method of solution,
is the principal reason for relatively fast computation when compared with
previous methods.
convex iteration
Direction vector y is initialized to 1 until the first fixed point is found; which
means, the first sequence of contractions begins calculating the (1-norm)
solution U ⋆ to (765) via problem (767). Once U ⋆ is found, vector y is
updated according to an estimate of image-gradient cardinality c : Sum
of the 4n 2 −c smallest entries of |Ψ vec U ⋆ |∈ R 4n2 is the optimal objective
value from a linear program, for 0≤c≤4n 2 − 1
4n ∑
2
π(|Ψ vec U ⋆ |) i = minimize 〈|Ψ vec U ⋆ | , y〉
y∈R 4n2
subject to 0 ≼ y ≼ 1
i=c+1
y T 1 = 4n 2 −c
(467)
where π is the nonlinear permutation-operator sorting its vector argument
into nonincreasing order. An optimal solution y to (467), that is an
extreme point of its feasible set, is known in closed form: it has 1 in each
entry corresponding to the 4n 2 −c smallest entries of |Ψ vec U ⋆ | and has 0
elsewhere (page 295). The updated image U ⋆ is assigned to U t , the
contraction sequence is recomputed solving (767), direction vector y is
updated again, and so on until convergence.
There are two features that distinguish problem formulation (767) and
our particular implementation of it (available on Wıκımization):
1) An image-gradient estimate may engage any combination of four
adjacent pixels. In other words, the algorithm is not locked into a
four-point gradient estimate (Figure 89); number of points constituting
an estimate is determined by direction vector y . Indeed, we find
only 5092 zero entries in y for the Shepp-Logan phantom; meaning,
4.52 Fat is typical of compressed sensing problems; e.g., [60] [67].