Private Watermarking

What are 'public watermarking', 'blind watermarking', 'semi-blind watermarking', 'private watermarking', 'non-blind watermarking' and 'asymmetric watermarking'?

Has been imported from the related field of steganography.
Originally, public watermarking and blind watermarking mean the same, but the wording was confusing with public-key watermarking software. 'Signal processing people' took over the field, so only the later tends to remain. In these schemes, the cover signal (the original signal) is not needed during the detection process to detect the mark. Solely the key, which is typically used to generate some random sequence used during the embedding process, is required. These types of schemes can be used easily in mass market electronic equipment or software.

In some cases you may need extra information to help your detector (in particular to synchronise its random sequence on the possibly distorted test signal). In particular some watermarking schemes require access to the 'published' watermarked signal, that is the original signal just after adding the watermark. People refer to these schemes as semi-blind watermarking schemes.

Private watermarking and non-blind-watermarking mean the same: the original cover signal is required during the detection process.

At last, by asymmetric watermarking images or public-key watermarking, people refer to photo watermarking schemes with properties reminding asymmetric cryptosystem (or public key cryptosystem). No such system really exists yet although some possible suggestions have been made. In this case, the detection process (and in particular the detection key) is fully known to anyone as opposed to blind watermarking where a secret key is required. So here, only a 'public key' is needed for verification and a 'private key' (secret) is used for the embedding though. Knowledge of the public key does not help to compute the private key (at least in a reasonable time), it does not either allow removal of the mark nor it allows an attacker to forge a mark.