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Sparse signal and image recovery from Compressive Samples
In this paper we present an introduction to Compressive Sampling
(CS), an emerging model-based framework for data acquisition
and signal recovery based on the premise that a signal
having a sparse representation in one basis can be reconstructed
from a small number of measurements collected in a
second basis that is incoherent with the first. Interestingly, a
random noise-like basis will suffice for the measurement process.
We will overview the basic CS theory, discuss efficient
methods for signal reconstruction, and highlight applications
in medical imaging
Sparsity and Incoherence in Compressive Sampling
We consider the problem of reconstructing a sparse signal from a
limited number of linear measurements. Given randomly selected samples of
, where is an orthonormal matrix, we show that minimization
recovers exactly when the number of measurements exceeds where is the number of
nonzero components in , and is the largest entry in properly
normalized: . The smaller ,
the fewer samples needed.
The result holds for ``most'' sparse signals supported on a fixed (but
arbitrary) set . Given , if the sign of for each nonzero entry on
and the observed values of are drawn at random, the signal is
recovered with overwhelming probability. Moreover, there is a sense in which
this is nearly optimal since any method succeeding with the same probability
would require just about this many samples
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