v2009.01.01 - Convex Optimization

4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES 325
phantom(256)
Figure 86: Shepp-Logan phantom from Matlab image processing toolbox.
4.6.0.0.10 Example. Compressive sampling of a phantom.
In summer of 2004, Candès, Romberg, & Tao [61] and Donoho [100] released
papers on perfect signal reconstruction from samples that stand in violation
of Shannon’s classical sampling theorem. The only condition on these defiant
signals is that they be sparse or some affine transformation of them be
sparse; essentially, they proposed sparse sampling theorems asserting average
sample rate independent of signal bandwidth and less than Shannon’s rate.
Minimum sampling rate:
of Ω-bandlimited signal: 2Ω (Shannon)
of k-sparse length-n signal: k log 2 (1+n/k)
(Candès/Donoho)
(confer Figure 80). Certainly, much was already known about nonuniform
or random sampling [32] [180] and about subsampling or multirate
systems [80] [310]. Vetterli, Marziliano, & Blu [319] had congealed a
theory of noiseless signal reconstruction, in May 2001, from samples
that violate the Shannon rate. They anticipated the Candès/Donoho
sparsifying transform by recognizing: it is the innovation (onset) of
functions constituting a (not necessarily bandlimited) signal that determines
minimum sampling rate for perfect reconstruction. Average onset
(sparsity) Vetterli et alii call rate of innovation. The vector inner-products
that Candès/Donoho call measurements, Vetterli calls projections. From
those projections Vetterli demonstrates reconstruction (by digital signal