To read a watermark from printed images requires that the watermarking system read correctly after affine distortions. One way to recover from affine distortions is to add a synchronization signal in the Fourier frequency domain and use this synchronization signal to estimate the applied affine distortion. Using the Fourier Magnitudes one can estimate the linear portion of the affine distortion. To estimate the translation one must first estimate the phase of the synchronization signal and then use phase correlation to estimate the translation. In this paper we provide a new method to measure the phase of the synchronization signal using only the data from the complex Fourier domain. This data is used to compute the linear portion, so it is quite convenient to estimate the phase without further data manipulation. The phase estimation proposed in this paper is computationally simple and provides a significant computational advantage over previous methods while maintaining similar accuracy. In addition, the phase estimation formula gives a general way to interpolate images in the complex frequency domain.
Robert G. Lyons,
John D. Lord,
"Estimating synchronization signal phase", Proc. SPIE 9409, Media Watermarking, Security, and Forensics 2015, 94090P (4 March 2015); doi: 10.1117/12.2083390; https://doi.org/10.1117/12.2083390