v2010.10.26 - Convex Optimization

4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES 373Reconstruction of the Shepp-Logan phantom (Figure 107), from aseverely aliased image (Figure 109) obtained by Magnetic ResonanceImaging (MRI), was the impetus driving Candès’ quest for a sparse samplingtheorem. He realized that line segments appearing in the aliased imagewere regions of high total variation. There is great motivation, in themedical community, to apply compressed sensing to MRI because it translatesto reduced scan-time which brings great technological and physiologicalbenefits. MRI is now about 35 years old, beginning in 1973 with Nobellaureate Paul Lauterbur from Stony Brook USA. There has been muchprogress in MRI and compressed sensing since 2004, but there have alsobeen indications of 1-norm abandonment (indigenous to reconstruction bycompressed sensing) in favor of criteria closer to 0-norm because of acorrespondingly smaller number of measurements required to accuratelyreconstruct a sparse signal: 4.455481 complex samples (22 radial lines, ≈256 complex samples per) wererequired in June 2004 to reconstruct a noiseless 256×256-pixel Shepp-Loganphantom by 1-norm minimization of an image-gradient integral estimatecalled total variation; id est, 8.4% subsampling of 65536 data. [70,1.1][69,3.2] It was soon discovered that reconstruction of the Shepp-Loganphantom were possible with only 2521 complex samples (10 radial lines,1Figure 108); 3.8% subsampled data input to a (nonconvex) -norm 2total-variation minimization. [76,IIIA] The closer to 0-norm, the fewer thesamples required for perfect reconstruction.Passage of a few years witnessed an algorithmic speedup and dramaticreduction in minimum number of samples required for perfect reconstructionof the noiseless Shepp-Logan phantom. But minimization of total variationis 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 wouldmake reasonable an expectation of sparse gradient. Sparsification of adiscrete image-gradient tends to preserve edges. Then minimization of totalvariation seeks an image having fewest edges. There is no deeper theoreticalfoundation than that. When applied to human brain scan or angiogram,4.45 Efficient techniques continually emerge urging 1-norm criteria abandonment; [80] [356][355,IID] e.g., five techniques for compressed sensing are compared in [37] demonstratingthat 1-norm performance limits for cardinality minimization can be reliably exceeded.