=== VQ watermarking techniques ===

=== VQ watermarking techniques ===

The technique presented next has been published in [7] and is based on the method introduced in [3] and extended in [4]. The algorithm introduced in [3] is described next:

The technique presented next has been published in <ref name="Charalampidis2005">Charalampidis, D., “Improved Robust VQ-based Watermarking,” Electronics Letters, 41 (23), 1272-1273, 10 November 2005</ref> and is based on the method introduced in <ref name="Huang2001"/> and extended in <ref name="Lu2005"/>. The algorithm introduced in <ref name="Huang2001"/> is described next:

:1. Consider an image <math>X</math> of <math>M \times N</math>, and the binary watermark <math>\mathbf{W}</math> of size <math>M_W \times N_W</math>.  

:1. Consider an image <math>X</math> of <math>M \times N</math>, and the binary watermark <math>\mathbf{W}</math> of size <math>M_W \times N_W</math>.  

:10. Then, <math>key_1</math> is used to obtain the estimated watermark <math>\mathbf{W}'</math> from permuted watermark <math>\mathbf{W}_{P'}</math>.

:10. Then, <math>key_1</math> is used to obtain the estimated watermark <math>\mathbf{W}'</math> from permuted watermark <math>\mathbf{W}_{P'}</math>.