Convex Optimization in Signal and Image Processing In signal processing, convex optimization plays a useful role in problems such as sparse signal recovery and filter design. What other places does convex optimization appear?

For example, in compressed sensing the Basis pursuit Denoising problem, the LASSO problem and the Dantzig selector can be posed as:

\begin{eqnarray}
\min_{x} \ell(Ax-b)+r(x)
\end{eqnarray}

where $\ell(\cdot)$ is and $r(\cdot)$ are appropriate loss and regularization terms, respectively. Moreover, the design of a filter subject to time and frequency constraints often yields a convex formulation.
There's a whole area of signal processing dedicated to optimal filtering. In pretty much every case I've seen the filtering problem is formulated with a convex cost function. 

Here's a freely available book on the subject - Sophocles J. Orfanidis - Optimum
Signal Processing.