v2007.11.26 - Convex Optimization

296 CHAPTER 4. SEMIDEFINITE PROGRAMMINGFigure 77: Massachusetts Institute of Technology (MIT) logo, including itswhite boundary, may be interpreted as a rank-5 matrix. (Stanford Universitylogo rank is much higher;) This constitutes Scene Y observed by theone-pixel camera in Figure 78 for Example 4.6.0.0.8.4.6.0.0.8 Example. Compressed sensing, compressive sampling. [225]As our modern technology-driven civilization acquires and exploitsever-increasing amounts of data, everyone now knows that most of the datawe acquire can be thrown away with almost no perceptual loss − witness thebroad success of lossy compression formats for sounds, images, and specializedtechnical data. The phenomenon of ubiquitous compressibility raises verynatural questions: Why go to so much effort to acquire all the data whenmost of what we get will be thrown away? Can’t we just directly measure thepart that won’t end up being thrown away? −David Donoho [81]Lossy data compression techniques are popular, but it is also well knownthat losses become quite perceptible with signal processing that goes beyondmere playback of a compressed signal. Spatial or audible frequencies maskedby a simultaneity become perceptible with significant post-filtering of thecompressed signal, for example. Further, there can be no universallyacceptable and unique metric of perception for gauging exactly how muchdata can be tossed. For these reasons, there will always be need for raw(uncompressed) data.In this example we throw out only so much information as to leaveperfect reconstruction within reach. Specifically, the MIT logo in Figure 77is perfectly reconstructed from 700 time-sequential samples {y i } acquiredby the one-pixel camera illustrated in Figure 78. The MIT-logo imagein this example effectively impinges a 46×81 array micromirror DMD.