v2010.10.26 - Convex Optimization

374 CHAPTER 4. SEMIDEFINITE PROGRAMMINGwith as much as 20% of 256×256 Fourier samples, we have observed 4.46 a30dB image/reconstruction-error ratio 4.47 barrier that seems impenetrableby the total-variation objective. Total-variation minimization has met withmoderate success, in retrospect, only because some medical images aremoderately piecewise-constant signals. One simply hopes a reconstruction,that is in some sense equal to a known subset of samples and whose gradientis most sparse, is that unique image we seek. 4.48The total-variation objective, operating on an image, is expressible asnorm of a linear transformation (854). It is natural to ask whether thereexist other sparsifying transforms that might break the real-life 30dB barrier(any sampling pattern @20% 256×256 data) in MRI. There has beenmuch research into application of wavelets, discrete cosine transform (DCT),randomized orthogonal bases, splines, etcetera, but with suspiciously littlefocus on objective measures like image/error or illustration of differenceimages; the predominant basis of comparison instead being subjectivelyvisual (Duensing & Huang, ISMRM Toronto 2008). 4.49 Despite choice oftransform, there seems yet to have been a breakthrough of the 30dB barrier.Application of compressed sensing to MRI, therefore, remains fertile in 2008for continued research.4.46 Experiments with real-life images were performed by Christine Law at Lucas Centerfor Imaging, Stanford University.4.47 Noise considered here is due only to the reconstruction process itself; id est, noisein excess of that produced by the best reconstruction of an image from a complete setof samples in the sense of Shannon. At less than 30dB image/error, artifacts generallyremain visible to the naked eye. We estimate about 50dB is required to eliminate noticeabledistortion in a visual A/B comparison.4.48 In vascular radiology, diagnoses are almost exclusively based on morphology of vesselsand, in particular, presence of stenoses. There is a compelling argument for total-variationreconstruction of magnetic resonance angiogram because it helps isolate structures ofparticular interest.4.49 I have never calculated the PSNR of these reconstructed images [of Barbara].−Jean-Luc StarckThe sparsity of the image is the percentage of transform coefficients sufficient fordiagnostic-quality reconstruction. Of course the term “diagnostic quality” is subjective....I have yet to see an “objective” measure of image quality. Difference images, in myexperience, definitely do not tell the whole story. Often I would show people some of myresults and get mixed responses, but when I add artificial Gaussian noise to an image,often people say that it looks better.−Michael Lustig