Reconstruction of a bilevel function such as a bar code signal in a partially blind deconvolution problem is an important task in industrial processes. Existing methods are based on either the local approach or the regularization approach with a total variation penalty. This article reformulated the problem explicitly in terms of change points of the 0-1 step function. The bilevel function is then reconstructed by solving the nonlinear least squares problem subject to linear inequality constraints, with starting values provided by the local extremas of the derivative of the convolved signal from discrete noisy data. Simulation results show a considerable improvement of the quality of the bilevel function using the proposed hybrid approach over the local approach. The hybrid approach extends the workable range of the standard deviation of the Gaussian kernel significantly.
Qu, Leming and Tu, Yi-Cheng. (2006). "Change Point Estimation of Bilevel Functions". Journal of Modern Applied Statistical Methods, 5(2), 347-355.