v2010.10.26 - Convex Optimization

4.7. CONSTRAINING RANK OF INDEFINITE MATRICES 3854.7.0.0.1 Example. Compressed sensing, compressive sampling. [300]As our modern technology-driven civilization acquires and exploitsever-increasing amounts of data, everyone now knows that most of thedata we acquire can be thrown away with almost no perceptual loss − witnessthe broad success of lossy compression formats for sounds, images, andspecialized technical data. The phenomenon of ubiquitous compressibilityraises very natural questions: Why go to so much effort to acquire all thedata when most of what we get will be thrown away? Can’t we just directlymeasure the part that won’t end up being thrown away? −David Donoho [118]Lossy data compression techniques like JPEG are popular, but it isalso well known that compression artifacts become quite perceptible withsignal postprocessing that goes beyond mere playback of a compressedsignal. [220] [245] Spatial or audio frequencies presumed masked by asimultaneity are not encoded, for example, so rendered imperceptible evenwith significant postfiltering (of the compressed signal) that is meant toreveal them; id est, desirable artifacts are forever lost, so highly compresseddata is not amenable to analysis and postprocessing: e.g., sound effects [97][98] [99] or image enhancement (Photoshop). 4.61 Further, there can be nouniversally acceptable unique metric of perception for gauging exactly howmuch data can be tossed. For these reasons, there will always be need forraw (noncompressed) data.In this example we throw out only so much information as to leave perfectreconstruction within reach. Specifically, the MIT logo in Figure 112 isperfectly reconstructed from 700 time-sequential samples {y i } acquired bythe one-pixel camera illustrated in Figure 113. The MIT-logo image in thisexample impinges a 46×81 array micromirror device. This mirror arrayis modulated by a pseudonoise source that independently positions all theindividual mirrors. A single photodiode (one pixel) integrates incidentlight from all mirrors. After stabilizing the mirrors to a fixed butpseudorandom pattern, light so collected is then digitized into one sampley i by analog-to-digital (A/D) conversion. This sampling process is repeatedwith the micromirror array modulated to a new pseudorandom pattern.The most important questions are: How many samples do we need for4.61 As simple a process as upward scaling of signal amplitude or image size will alwaysintroduce noise; even to a noncompressed signal. But scaling-noise is particularlynoticeable in a JPEG-compressed image; e.g., text or any sharp edge.