Table 1 Summary of the models used in relevant studies

Jian and Koch et al. [5]

To describe digital watermarking schemes

Original data

Embedding (bit-carrier selector, bit-noise generator, bit-carrier modifier)

Steganography

Limited consideration of the inputs, outputs, component-functions, and watermarking properties for image applications

Watermarked data

Extraction (bit-carrier selector, bit-pattern matching)

Spread-spectrum communications

Limited to spread-spectrum communication-based watermarking schemes

Degraded data (as a copy of watermarked data)

Signal processing

Identification code (as watermark)

O’Sullivan et al. [8]

To determine the optimal hiding strategy, where watermarking is considered as a game between an attacker and information hider

Input and output data (e.g., images, audio, etc as a vector)

Encoder

Information theory

Limited consideration of the image application scenarios, inputs, outputs, component-functions, and watermarking properties

Message (as watermark)

Decoder

Steganography

Cox et al. [11]

To examine the similarities between watermarking and traditional communication models

Cover data (as a vector)

Perceptual distance function

Spread-spectrum communications

Limited consideration of image application scenarios (e.g., that use only spread-spectrum-based schemes), inputs, outputs, component functions, and watermarking properties

Watermark message

Encoding function

Watermarked cover data

Extraction function

Mixing function

Petitcolas et al. [4]

To illustrate a simplified case of watermarking concept

Mark (as fingerpring or watermark)

Embedding

Information hiding

Limited consideration of inputs, outputs, and components

Stego image

Recovery

May not be useful to study image watermarking schemes rigorously

Marked image

Cohen and Lapidoth [9]

To compute the coding capacity of the watermarking game for a Gaussian cover text and squared mean error distortions

Cover text

Encoder

Game theory

Limited consideration of inputs, outputs, and components

Message

Decoder

Information theory

Watermarking is considered as a game in a copyright application scenario

Stego-text

Secret key

Adelsbach et al. [12]

To analyse security of watermarking schemes against protocol attacks (e.g., copy, ambiguity attacks)

Unwatermarked object

Key generation

Cryptography

Limited consideration of inputs, outputs, and components

Watermarked object

Embedding

Application scenarios are limited to dispute resolving protocols

Watermark

Detection

Key

Barni et al. [13]

To provide a general security framework for robust watermark

Original content

Embedding (feature extraction and mixing, watermark generation)

Information theory

The concept of fair and unfair attacks may not be realistic

Watermark

Decoding

Cryptography

Limited consideration of inputs, outputs, and components (e.g., what original content includes)

Watermarked content

Signal processing

Key (for embedding and detection)

Li et al. [1]

To illustrate the formulation of the security definitions and the attacker models

Original and watermarked work (as a vector)

Watermark generation

Data hiding

Limited consideration of inputs and outputs

Watermark (as bit sequence)

Watermark embedding

Cryptography

The model represents only a simplified case of watermarking

Watermark detector

Signal processing

Perceptual distance function

Moulin and O’Sullivan [15]

To evaluate hiding capacity in an optimal attack context (as a data-hiding game)

Host-data (image, audio, video, etc.)

Encoder

Information theory

Limited consideration of inputs, outputs, and component functions (e.g., inputs and outputs are not conventional for watermarking)

Message

Decoder

Data-hiding

Side information

Game theory

Composite data (contains hidden message)

Mittelholzer [6]

To characterize embedding process and attacked stego image (for analyzing secrecy and robustness in terms of mutual information)

Cover data

Stego encoder

Information theory

Limited consideration of inputs, outputs, and component functions

Key

Stego channel

Steganography

More related to steganography schemes

Secret message

Stego decoder

Cachin [10]

To quantify steganographic security

Cover text

Key generation

Information theory

Limited consideration of inputs, outputs, and component functions

Stego text

Embedding

Steganography

More related to steganography schemes

Secret key

Extraction

Limited to the passive attack scenarios

Adelsbach et al. [7]

To formalize robustness considered as a core security property, of watermarking

Cover data

Key generation

Cryptography

Limited consideration of inputs, outputs, and component functions

Stego data

Embedding

Limited to robust watermarking schemes

Watermark

Detection

Key (for embedding and detection)

Secret parameter (used as key generation input)