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Chapter 9:
Compressive Sensing and Compressive Holography
It is well known from communication theory that for a sampled signal, the sampling rate must be greater than twice the signal bandwidth for faithful reproduction of the original signal. The concept of sampling at the Nyquist rate was postulated by Shannon in 1949; in the same year, Golay introduced the idea of artificial discrete multiplex coding in optical measurements. More than 50 years later, Candes, Tao, and Romberg and Donoho have demonstrated that signals that are sparse in a certain basis and sampled by multiplex encodings may be accurately inferred with high probability using many fewer measurements than suggested by Shannon’s sampling theorem in a process referred to as compressive sensing (CS). This section summarizes the basic concept of CS and provides an example of its application to holography. As stated by Brady et al., holography can be considered as a complex encoding of a signal (recording both the amplitude and phase) to which CS may be applied. In conventional optical imaging, only the intensities (no phase information) can be recorded, which results in rather poor measurement conditioning.
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