A nonlinear inverse filter is proposed for restoring signals degraded by a linear system and additive Gaussian noise. The proposed filter consists of combination of a linear high pass filter and an ε-filter, which is modified from the cascaded linear filter. The nonlinear property of the ε-filter is utilized to suppress pre-enhanced additive random noise and to restore sharp edges. It is demonstrated that the filter can be reduced to a multi-layered neural network model, and the optimal design is described by using the back propagation algorithm. The nonlinear function is approximated by a piecewise linear function, which results in simple and robust training algorithm. An application to image restoration is also presented, illustrating the effectiveness over the linear filter, especially when the amplitude of additive noise is small.