v2009.01.01 - Convex Optimization

338 CHAPTER 4. SEMIDEFINITE PROGRAMMING
Y
y i
Figure 91: One-pixel camera. [299] [323] Compressive imaging camera
block diagram. Incident lightfield (corresponding to the desired image Y )
is reflected off a digital micromirror device (DMD) array whose mirror
orientations are modulated in the pseudorandom pattern supplied by the
random number generators (RNG). Each different mirror pattern produces
a voltage at the single photodiode that corresponds to one measurement y i .
Our approach to reconstruction is to look for low-rank solution to an
underdetermined system:
find X
subject to A vec X = y
rankX ≤ 5
(772)
where vec X is vectorized matrix X ∈ R 46×81 (stacked columns). Each
row of fat matrix A is one realization of a pseudorandom pattern applied
to the micromirrors. Since these patterns are deterministic (known),
then the i th sample y i equals A(i, :) vec Y ; id est, y = A vec Y . Perfect
reconstruction means optimal solution X ⋆ equals scene Y ∈ R 46×81 to within
machine precision.
Because variable matrix X is generally not square or positive semidefinite,
we constrain its rank by rewriting the problem equivalently
find X
subject to A vec
[
X = y
]
W1 X
rank
X T ≤ 5
W 2
(773)