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where and are the extracted real part and imaginary part of the complex watermark signal sample at time instant.
If we decide to use a more complex watermark signal such as the multicomponent signal displayed in Figure 4, the WVD would not be appropriate as it would have cross-terms which might hide the actual feature of our signature.
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In the second application, we study the quality assessment of the watermarked content by investigating the extracted complex chirp watermark in the absence of the original watermark.
The detection does not require the original image, instead it uses the special feature of the extracted complex chirp watermark signal for content authentication.
The watermark complex signal length is chosen equal to 512 samples.
The detection is performed by simply extracting the watermark complex chirp signal and, then, evaluating its magnitude.
Every value of the real part,, and every value of the imaginary part,, of the unitary amplitude watermark complex sample,, is quantized into an integer value from 0 to 127.
The second proposed fragile watermarking method, based on wavelet analysis, uses complex chirp signals as watermarks.
Check 'Include watermark' in the 'Watermark' section.
The advantages of using complex chirp signals as watermarks are manyfold, among these one can cite (i) the wide frequency range of such signals making the watermarking capacity very high and (ii) the easiness in adjusting the FM/AM parameters to generate different watermarks.
For size 4 × 4 sub-block image, in order to obtain a higher visual quality, it is assumed that a smooth texture block can embed 1-bit watermark, a medium complex texture block can embed 2-bit watermark, a complex texture block can be embedded 3-bit watermark, a low gray block can embed 2-bit watermark, a middle gray block can embed 1-bit watermark, and a high gray block can embed 2-bit watermark.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com