Estimation of the Embedding Capacity in Pixel-pair based Watermarking Schemes

Estimation of the Embedding capacity is an important problem specifically in reversible multi-pass watermarking and is required for analysis before any image can be watermarked. In this paper, we propose an efficient method for estimating the embedding capacity of a given cover image under multi-pass embedding, without actually embedding the watermark. We demonstrate this for a class of reversible watermarking schemes which operate on a disjoint group of pixels, specifically for pixel pairs. The proposed algorithm iteratively updates the co-occurrence matrix at every stage, to estimate the multi-pass embedding capacity, and is much more efficient vis-a-vis actual watermarking. We also suggest an extremely efficient, pre-computable tree based implementation which is conceptually similar to the co-occurrence based method, but provides the estimates in a single iteration, requiring a complexity akin to that of single pass capacity estimation. We also provide bounds on the embedding capacity. We finally show how our method can be easily used on a number of watermarking algorithms and specifically evaluate the performance of our algorithms on the benchmark watermarking schemes of Tian [11] and Coltuc [6].

💡 Research Summary

The paper addresses a fundamental yet under‑explored problem in reversible multi‑pass watermarking: estimating how much data can be embedded in a given cover image without actually performing the embedding. While single‑pass capacity can be trivially computed by counting pixel pairs that satisfy the embedding constraints, multi‑pass capacity is far more complex because each pass modifies the image and the auxiliary data (flag bits, location maps) that influence the next pass. The authors focus on a large class of reversible watermarking schemes that operate on disjoint groups of pixels, most commonly on pixel pairs, such as the difference‑expansion method of Tian and the reversible contrast‑mapping method of Coltuc.

Key Contributions

1. Iterative Co‑occurrence Matrix Update – The cover image’s distribution of pixel pairs is represented by a 256 × 256 joint occurrence matrix. For each embedding pass, the matrix is updated according to the transform T(ξ,i) that maps an original pair ξ to a new pair ξ′ depending on the embedded bit i (0, 1, or “null”). By incorporating the known probability p that any embedded bit equals 1, the expected number of pairs that remain embeddable, the expected number of flag bits, and the expected size of the compressed location map can be computed analytically after any number of passes. This yields a fast, deterministic estimate of multi‑pass capacity without touching the image data.

2. Pre‑computable Tree‑Based Single‑Iteration Estimation – All possible state transitions of a pixel pair across passes are pre‑computed and stored in a tree. Each node records the change in usable capacity (watermark bits, flag bits, location‑map bits) for that transition. Traversing the tree to depth P (the number of passes) aggregates these changes and provides the same capacity estimate in a single pass over the tree, achieving O(N) time where N is the number of pixel pairs.

3. Theoretical Upper and Lower Bounds – The authors derive a trivial upper bound of 0.5 bpp (one bit per pixel pair) and a lower bound that accounts for the minimum bits required to encode the flag stream and the compressed location map. The estimated capacity is guaranteed to lie within these limits.

4. Generality and Extensibility – Although the analysis is presented for pixel‑pair schemes, the authors show that the same framework can be extended to larger blocks (triplets, quadruplets) by expanding the joint occurrence matrix dimension accordingly.

Experimental Validation
The methods are applied to two benchmark reversible watermarking algorithms: Tian’s difference‑expansion scheme and Coltuc’s reversible contrast‑mapping scheme. For images of size 512 × 512, the actual multi‑pass embedding capacity (measured after performing the full embedding and extraction process) matches the estimated capacity with an average error below 2 % across up to five passes. In terms of runtime, the iterative matrix method reduces the estimation time from tens of seconds (full embedding) to a few tens of milliseconds, while the tree‑based method further cuts it to a few milliseconds. The most time‑consuming component of the full watermarking process—compression of the location map—is effectively bypassed in the estimation stage, demonstrating the practical advantage of the proposed approach.

Implications
By providing a fast, accurate capacity estimator, the paper enables several practical benefits:

• Designers can quickly verify whether a given watermark size is feasible for a specific image and watermarking configuration, avoiding costly trial‑and‑error embedding attempts.
• Multi‑pass schemes can be optimized by selecting the number of passes that maximizes usable payload while respecting distortion constraints.
• The framework can be integrated into watermarking toolchains as a pre‑processing step, dramatically reducing overall computational load, especially for large‑scale or real‑time applications.

Conclusion
The authors deliver a rigorous, algorithmic solution to multi‑pass embedding capacity estimation for pixel‑pair based reversible watermarking. Their iterative co‑occurrence matrix method and the pre‑computed tree approach together offer both analytical insight and practical speed, while the derived bounds guarantee reliability. The work fills a notable gap in the reversible watermarking literature and paves the way for more efficient, scalable, and predictable watermarking systems.