The signal reconstruction model with noise in compressed sensing can be expressed as an l₁-norm problem. To reconstruct high-precision image with a small number of observations, restricted isometric property (RIP) and non-coherent are required to design the observation matrix. However, it is very difficult to judge the RIP of matrix. In view of the uncertainty of the observation matrix, the  l₁-norm problem is transformed into a stochastic optimization model with probability constraint, which is usually non-smooth. A smooth conservative approximation function of the characteristic function is constructed. The properties of the approximation function are discussed. It is proved that proposed smooth approximation function is equivalent to probability constrained function when the parameter is small enough. The corresponding smooth approximation problem is established.

compressed sensing, probability constraint, characteristic function, smooth approximation.