v2009.01.01 - Convex Optimization

336 CHAPTER 4. SEMIDEFINITE PROGRAMMING
image-gradient sparsity is actually closer to 1.9% than the 3% reported
elsewhere; e.g., [308,IIB].
2) Numerical precision (≈1E-2) of the fixed point of contraction (771)
is a parameter to the implementation; meaning, direction vector y is
typically updated after fixed-point recursion begins but prior to its
culmination. Impact of this idiosyncrasy tends toward simultaneous
optimization in variables U and y while insuring that vector y settles
on a boundary point of the feasible set (nonnegative hypercube slice)
in (467) at every iteration; for only a boundary point 4.53 can yield the
sum of smallest entries in |Ψ vec U ⋆ | .
Reconstruction of the Shepp-Logan phantom at 103dB image/error is
achieved in a Matlab minute with 4.1% subsampled data; well below an
11% least lower bound predicted by the sparse sampling theorem. Because
reconstruction approaches optimal solution to a 0-norm problem, minimum
number of Fourier-domain samples is bounded below by cardinality of the
image-gradient at 1.9%.
4.6.0.0.11 Example. Compressed sensing, compressive sampling. [260]
As our modern technology-driven civilization acquires and exploits
ever-increasing amounts of data, everyone now knows that most of the data
we acquire can be thrown away with almost no perceptual loss − witness the
broad success of lossy compression formats for sounds, images, and specialized
technical data. The phenomenon of ubiquitous compressibility raises very
natural questions: Why go to so much effort to acquire all the data when
most of what we get will be thrown away? Can’t we just directly measure the
part that won’t end up being thrown away? −David Donoho [100]
Lossy data compression techniques are popular, but it is also well known
that compression artifacts become quite perceptible with signal processing
that goes beyond mere playback of a compressed signal. [191] [211] Spatial or
audio frequencies presumed masked by a simultaneity, for example, become
perceptible with significant post-filtering of the compressed signal. Further,
there can be no universally acceptable and unique metric of perception for
gauging exactly how much data can be tossed. For these reasons, there will
always be need for raw (uncompressed) data.
4.53 Simultaneous optimization of these two variables U and y should never be a pinnacle
of aspiration; for then, optimal y might not attain a boundary point.