  This paper proposes a general framework to design a sparse sensing matrix
$\ensuremath{\mathbf{A}}\in \mathbb{R}^{m\times n}$, in a linear measurement
system $\ensuremath{\mathbf{y}} = \ensuremath{\mathbf{Ax}}^{\natural} +
\ensuremath{\mathbf{w}}$, where $\ensuremath{\mathbf{y}} \in \mathbb{R}^m$,
$\ensuremath{\mathbf{x}}^{\natural}\in \RR^n$, and $\ensuremath{\mathbf{w}}$
denote the measurements, the signal with certain structures, and the
measurement noise, respectively. By viewing the signal reconstruction from the
measurements as a message passing algorithm over a graphical model, we leverage
tools from coding theory in the design of low density parity check codes,
namely the density evolution, and provide a framework for the design of matrix
$\ensuremath{\mathbf{A}}$. Particularly, compared to the previous methods, our
proposed framework enjoys the following desirable properties: ($i$)
Universality: the design supports both regular sensing and preferential
sensing, and incorporates them in a single framework; ($ii$) Flexibility: the
framework can easily adapt the design of $\bA$ to a signal
$\ensuremath{\mathbf{x}}^{\natural}$ with different underlying structures. As
an illustration, we consider the $\ell_1$ regularizer, which correspond to
Lasso, for both the regular sensing and preferential sensing scheme.
Noteworthy, our framework can reproduce the classical result of Lasso, i.e.,
$m\geq c_0 k\log(n/k)$ (the regular sensing) with regular design after proper
distribution approximation, where $c_0 > 0$ is some fixed constant. We also
provide numerical experiments to confirm the analytical results and demonstrate
the superiority of our framework whenever a preferential treatment of a
sub-block of vector $\bx^{\natural}$ is required.
