By Avella P., Boccia M., Sforza A.
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23) The detection criterion ✥ ✒ is the maximum value of ✥ ✼ among all ✠ possible shifts , for a given realization of ✠ . 22 is the distribution of ✥ ✼ realizations for a fixed shift or for independent re☞ alizations. In our scenario, the different detection values ✥ ✼ are derived combining all realizations from the fixed set of✠ realizations in . For this✞ reason, distribution of ✏ ✼ for varying shifts ✞ will not exactly fit the ✔ distribution. The way that it will differ from ✔ distribution is described below.
Terpolation. 12 shows the effect of a fractional cropping (maintaining constant image size) followed by cubic spline interpolation  on bit-error rate. 12: Crop of a fraction of pixel and interpolation. This property also permits to boost the detection performances using 46 Chapter 3. 2-D periodic patterns for image watermarking side information at the embedding stage. This strategy is also called performing informed embedding procedure [75, 76]. Since the cover media is known to the encoder, one can optimize the coding process in order to adapt the watermark to the channel characteristics.
We will explain below which combinations of keys should not be used. The pattern is filled with the elements of the sequence in such a way that there exists in the pattern a constant relation between elements in adjacent positions. If the position ✄✄✟✏☎✞✠✫✞ in the pattern is filled with the✠ ✍ symbol of the sequence, then we will take the symbol ✹✵✞ in position ✄ ✍✏✎ in the sequence to fill position ✄✄✟ ✎ ☎✞✠✫✞ of the pattern. The construction is illustrated in fig. 3. In the same way, ✠ ✞ the ✞ are symbol at position ✄✄✟✆☎✞✠☎✞ and the one at position ✄✄✟✏☎✞✠ ✎ cyclic positions away in the sequence.
A Branch-and-Cut Algorithm for the Median-Path Problem by Avella P., Boccia M., Sforza A.