Restricted Isometry Property (RIP) in Compressive Sensing What is the meaning of Restricted Isometry Property (RIP) condition in Compressive Sensing for Sparse Signal Analysis?
How can we define Restricted Isometry Constant (RIC) for the RIP condition?

Thanks in Advance!
The restricted isometry property states that:
$$\begin{equation}
  (1-\delta_S)||x||_2^2 \le ||A x||_2^2 \le (1+\delta_S)||x||_2^2
\end{equation}$$
for any $S$-sparse vector $x$. The restricted isometry constant is $\delta_S$, $0 &lt; \delta_S &lt; 1$.

This means that the matrix $A$ is guaranteed to only change the length of any vector $x$ "very little" as long as the vector $x$ is at least $S$-sparse (has at most $S$ non-zero coefficients).

Suppose we have arbitrary $\frac{S}{2}$-sparse vectors $x$. In order to be able to reconstruct such vectors in general, from measurements taken as $y = A x$, we need to be sure that it is possible to distinguish between measurements $y_1 = A x_1$ and $y_2 = A x_2$ of any two such vectors. If $y_1 = y_2$ for any two such vectors $x_1$ and $x_2$, we would not be able to distinguish them and reconstruct them unambigously. Therefore we need to make sure that measurements of any two $\frac{S}{2}$-sparse vectors are "sufficiently different".

If we calculate the difference between any two $\frac{S}{2}$-sparse vectors, their difference can be at most $S$-sparse. So for reconstructing any $\frac{S}{2}$-sparse vector correctly from measurements taken with $A$, the restricted isometry property quantifies how well $A$ lets us do that (the smaller $\delta_S$, the better).

For an early introduction to compressed sensing and the restricted isometry property (and other concepts), see Candès &amp; Wakin, 2008.