18
transmission. One such example is THz imaging based on raster scanning. The
raster-scan imaging system obtains each image sample by measuring the THz trans-
mission∕reflection at each spatial position of an object on a uniform grid. This kind
of uniform sampling is governed by the signal’s Nyquist rate. The sampling rate must
be twice the bandwidth in order to accurately represent the signal∕image.
There are many scenarios in which a signal may have a large bandwidth, but does
not contain a lot of information. For instance, a piecewise smooth signal may have
high frequency components necessitating many samples, but can be represented well
by a linear combination of only a few wavelets. In digital photography an image
field may be sampled at 10 million locations, but this information can be effectively
stored with only 100 thousand DCT or wavelet coefficients. Compressive sensing
(CS) takes the logical step of exploiting a signal’s structure to acquire it in fewer
measurements, rather than the observe the whole thing and compress it later. As a
model for structured signals, we first consider compressible signals that are fc-sparse
in some orthogonal basis Φ. This means that when represented in the basis, the signal
has only к non-zero coefficients, where к is much smaller than the signal length N. Let
us now consider an image (or a 2-D signal; this discussion will use the terms, “signal”
and “image” interchangeably). Consider the object a Icngtli-Af signal x of dimension
indexed as æ(n), n E {1, 2,, N}. In this case, x is a 2-D image with pixels ordered in a
N × 1 vector. The image can also be represented in terms of the coefficients {Ai} of an
orthonormal basis expansion x — ɪɪɪ where {t⅛}^=ι are the N × 1 basis vectors.
Forming the coefficient vector a and the N × N basis matrix Φ := [≠ι∣≠2∣ ■ ∙ ■ l'd'ʃv]
by stacking the vectors {ψl} as colums, we can write the samples as x = Φ0. The
image x can be fc-sparse, meaning that the coefficient vector θ is sparse, where only
к <A N coefficients are non-zero. Or x can be r-compressible, meaning that the sorted