v2009.01.01 - Convex Optimization

4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES 327
June 2004: Candès, Romberg, & Tao publish in cyberspace: 5481 complex samples
(22 radial lines, ≈256 complex samples per) to reconstruct noiseless
256×256-pixel Shepp-Logan phantom by 1-norm minimization of an
image-gradient integral estimate called total variation; id est, 8.4%
subsampling of 65536 data. [61,1.1] [60,3.2]
March 2006: Chartrand discovers 4.40 that reconstruction of the Shepp-Logan
phantom is possible with only 2521 complex samples (10 radial lines,
Figure 87); 3.8% subsampled data input to a (nonconvex) 1 2 -norm
total-variation minimization. [67,IIIA]
April 2007: Trzasko & Manduca recover the phantom with same subsampling
as Chartrand, but they realize an accurate 0-norm total-variation
minimization algorithm that is 10× faster 4.41 than previous methods.
[308] [309]
Passage of these few years witnessed an algorithmic speedup and dramatic
reduction in minimum number of samples required for perfect reconstruction
of the noiseless Shepp-Logan phantom. But minimization of total variation
is ideally suited to recovery of any piecewise-constant image, like a phantom,
because the gradient of such images is highly sparse by design.
There is no inherent characteristic of real-life MRI images that would
make reasonable an expectation of sparse gradient. Sparsification of an
image-gradient tends to preserve edges. Then minimization of total variation
seeks an image having fewest edges. There is no deeper theoretical foundation
than that. When applied to human brain scan or angiogram, with as
much as 20% of 256×256 Fourier samples, we have observed 4.42 a 30dB
4.40 I try using p = 1 2
in place of p =1[-norm] to reconstruct the Shepp-Logan phantom,
and am very surprised to see how well it works. This is by alternating gradient descent
with projection onto the constraint. In hindsight, one aspect appears to have been crucial
to its success: regularization of the cusp with a relatively large epsilon, then decreasing
epsilon after convergence. At the time I did this solely to try to speed up convergence,
based on experience with TV minimization algorithms (like lagged diffusivity). But it now
seems to be what allows the algorithm to avoid local minima, as these are filled in by the
larger epsilon.
−Rick Chartrand
4.41 Their Matlab code runs in a minute, on an Intel-based dual-processor 2006 vintage
laptop computer, capable of recovery at 50dB image/error from 3.2% subsampled data.
4.42 Work on real-life images was performed in conjunction with Christine Law at Lucas
Center for Imaging, Stanford University.