v2009.01.01 - Convex Optimization

4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES 337
Figure 90: Massachusetts Institute of Technology (MIT) logo, including its
white boundary, may be interpreted as a rank-5 matrix. (Stanford University
logo rank is much higher;) This constitutes Scene Y observed by the
one-pixel camera in Figure 91 for Example 4.6.0.0.11.
In this example we throw out only so much information as to leave
perfect reconstruction within reach. Specifically, the MIT logo in Figure 90
is perfectly reconstructed from 700 time-sequential samples {y i } acquired
by the one-pixel camera illustrated in Figure 91. The MIT-logo image
in this example effectively impinges a 46×81 array micromirror device.
This mirror array is modulated by a pseudonoise source that independently
positions all the individual mirrors. A single photodiode (one pixel)
integrates incident light from all mirrors. After stabilizing the mirrors to
a fixed but pseudorandom pattern, light so collected is then digitized into
one sample y i by analog-to-digital (A/D) conversion. This sampling process
is repeated with the micromirror array modulated to a new pseudorandom
pattern.
The most important questions are: How many samples do we need for
perfect reconstruction? Does that number of samples represent compression
of the original data?
We claim that perfect reconstruction of the MIT logo can be reliably
achieved with as few as 700 samples y=[y i ]∈ R 700 from this one-pixel
camera. That number represents only 19% of information obtainable from
3726 micromirrors. 4.54
4.54 That number (700 samples) is difficult to achieve, as reported in [260,6]. If a minimal
basis for the MIT logo were instead constructed, only five rows or columns worth of
data (from a 46×81 matrix) are independent. This means a lower bound on achievable
compression is about 230 samples; which corresponds to 6% of the original information.