Deblurring of One Dimensional Bar Codes via Total Variation Energy Minimisation

Using total variation based energy minimisation we address the recovery of a blurred (convoluted) one dimensional (1D) barcode. We consider functionals defined over all possible barcodes with fidelity to a convoluted signal of a barcode, and regularised by total variation. Our fidelity terms consist of the L^2 distance either directly to the measured signal or preceded by deconvolution. Key length scales and parameters are the X-dimension of the underlying barcode, the size of the supports of the convolution and deconvolution kernels, and the fidelity parameter. For all functionals, we establish regimes (sufficient conditions) wherein the underlying barcode is the unique minimiser. We also present some numerical experiments suggesting that these sufficient conditions are not optimal and the energy methods are quite robust for significant blurring.
Comment: version 1: 44 pages, 9 figures; version 2: 36 pages, 8 figures, revised version in which the results are presented in a bit more generality and the appendix has been shrunk significantly; submitted to SIAM Journal on Imaging Sciences; version 3: after final revisions, accepted by SIAM Journal on Imaging Sciences