v2009.01.01 - Convex Optimization

328 CHAPTER 4. SEMIDEFINITE PROGRAMMING
image/(reconstruction)error ratio 4.43 barrier that seems impenetrable by
the total-variation objective. Total-variation minimization has met with
moderate success, in retrospect, only because some medical images are
moderately piecewise-constant signals. One simply hopes a reconstruction,
that is in some sense equal to a known subset of samples and whose gradient
is most sparse, is that unique image we seek.
The total-variation objective, operating on an image, is expressible
as norm of a linear transformation (765). It is natural to ask
whether there exist other sparsifying transforms that might break the
real-life 30dB barrier (@20% 256×256 data) in MRI. There has been much
research into application of wavelets, discrete cosine transform (DCT),
randomized orthogonal bases, splines, etcetera, but with suspiciously little
focus on objective measures like image/error or illustration of difference
images; the predominant basis of comparison instead being subjectively
visual (Duensing & Huang, ISMRM Toronto 2008). 4.44 Despite choice of
transform, there seems yet to have been a breakthrough of the 30dB barrier.
Application of compressed sensing to MRI, therefore, remains fertile in 2008
for continued research.
Lagrangian form of compressed sensing in imaging
We now repeat Candès’ image reconstruction experiment from 2004 [61]
which led to discovery of sparse sampling theorems. But we achieve
perfect reconstruction with an algorithm based on vanishing gradient of the
compressed sensing problem’s Lagrangian; our contraction method (p.334)
[308,IIIA] is computationally 10× faster than contemporary methods
because matrix multiplications are replaced by fast Fourier transforms
4.43 Noise considered here is due only to the reconstruction process itself; id est, noise
in excess of that produced by the best reconstruction of an image from a complete set
of samples in the sense of Shannon. At less than 30dB image/error, artifacts generally
remain visible to the naked eye. We estimate about 50dB is required to eliminate noticeable
distortion in a visual A/B comparison.
4.44 I have never calculated the PSNR of these reconstructed images [of Barbara].
−Jean-Luc Starck
The sparsity of the image is the percentage of transform coefficients sufficient for
diagnostic-quality reconstruction. Of course the term “diagnostic quality” is subjective.
...I have yet to see an “objective” measure of image quality. Difference images, in my
experience, definitely do not tell the whole story. Often I would show people some of my
results and get mixed responses, but when I add artificial Gaussian noise to an image,
often people say that it looks better.
−Michael Lustig