Ant Colony Algorithm Hides Invisible Watermarks in Color Images Using Quaternion Transforms

Every day, billions of color images travel across open networks — photographs shared on social media, medical scans transmitted between hospitals, satellite frames beamed down from orbit, and digital artwork sold on marketplaces. With that traffic comes an urgent question: how can the owner of an image prove ownership when anyone can copy, compress, crop, or otherwise tamper with the file? A research team at Fujian Normal University in Fuzhou, China, has now unveiled an answer that blends nineteenth-century mathematics, modern signal processing, and an optimization algorithm inspired by the foraging behavior of ants. Writing in the journal Cluster Computing, Shiyue Tang, Huangzhi Xia, and Yifen Ke describe a color image watermarking framework that hides an invisible mark inside a picture so robustly that it survives most common attacks, yet so subtly that the human eye cannot detect it.

The heart of the new method is an exotic mathematical setting known as the Discrete Quaternionic Quadratic Phase Fourier Transform, abbreviated DQQPFT. To understand why the authors reached for such a formidable-sounding tool, it helps to consider how color images are usually handled. Most watermarking schemes treat the red, green, and blue channels of an image as three separate grayscale images, processing each one independently with a fixed transform kernel. That approach is computationally convenient, but it throws away something valuable: the natural correlation between the color channels. A color image is not simply three monochrome images stacked together; the channels are deeply intertwined, and any method that severs those ties loses information that could be used to make a watermark stronger and more secure.

Quaternions offer a way out. Invented by William Rowan Hamilton in 1843, quaternions extend complex numbers to a four-dimensional number system with one real part and three imaginary parts. That structure turns out to be a remarkably natural fit for color images: the three imaginary components can encode the three color channels simultaneously, while the real part can carry additional information. When a color image is represented as a matrix of quaternions, transforms built on quaternion algebra can operate on all three channels at once, preserving the inter-channel relationships that separable schemes discard. The DQQPFT goes a step further than earlier quaternion transforms by introducing a quadratic phase function with controllable parameters, giving the transform a flexible, non-rigid kernel that can be tuned rather than fixed.

That tunability is not just a mathematical curiosity — it is the security backbone of the scheme. The parameters that shape the DQQPFT kernel are incorporated directly into the system as cryptographic keys. An attacker who somehow extracts a suspicious signal from a watermarked image still cannot recover the hidden watermark without knowing the exact kernel parameters, and because those parameters are continuous values with high sensitivity, the space of possible keys is far too large to search by brute force. In effect, the transform itself becomes part of the lock: even someone with full knowledge of the embedding algorithm would face a key space that renders guessing computationally infeasible.

Of course, elegance in theory must survive the realities of computation. Quaternion arithmetic is inherently more expensive than real-number arithmetic, and naive implementations of quaternion transforms can impose a heavy computational burden. The authors sidestepped this obstacle with a clever algebraic trick: they implemented the DQQPFT through an equivalent real-matrix representation of quaternions. By mapping quaternion operations onto ordinary real matrices, the transform can be computed with standard, highly optimized numerical routines while remaining mathematically consistent with the original quaternion formulation. The result is a scheme that keeps the theoretical advantages of the quaternion domain without paying the full performance penalty that quaternion arithmetic would otherwise demand.

With the transform in place, the next challenge is deciding exactly how and where to embed the watermark. The team turned to Quaternion Singular Value Decomposition, or QSVD, a technique that breaks the transformed image into components whose stability under common image-processing operations makes them ideal hiding places. But within QSVD there is a fundamental tension that every watermarking designer must confront. Embed the watermark too strongly, and it will resist attacks — compression, noise, filtering, geometric distortions — but the host image will visibly degrade. Embed it too gently, and the image looks perfect while the watermark washes away at the first sign of trouble. Imperceptibility and robustness pull in opposite directions, and there is no single setting that maximizes both.

The researchers framed this tension as a formal bi-objective optimization problem and solved it with an Enhanced Multi-Objective Ant Colony Optimization algorithm, which they call E-MOACO. Ant colony optimization is a swarm intelligence technique modeled on the way real ants find efficient paths to food by laying and following pheromone trails: solutions that perform well reinforce the trails that led to them, gradually steering the colony toward the best regions of the search space. The enhanced version introduced by the team adds two key innovations. Dynamic weight scheduling continuously rebalances how much each objective — visual quality versus attack resistance — influences the search as it progresses, while adaptive Gaussian perturbation injects controlled randomness that helps the colony escape local optima where it might otherwise stagnate. A knee-point selection strategy then picks, from the family of balanced trade-off solutions the algorithm produces, the single solution that offers the best overall compromise between the competing goals.

The experimental results are striking. Across extensive tests on standard image databases, the watermarked images achieved an average peak signal-to-noise ratio, or PSNR, exceeding 39 decibels without any attacks applied — a level of fidelity at which the embedded mark is genuinely invisible to human observers. More importantly, the watermark remained reliably detectable under duress. The average normalized correlation, a measure of how faithfully the extracted watermark matches the original, stayed above 0.99 under common single attacks such as noise addition, filtering, and compression. Even under mixed attacks, where multiple distortions are combined, the correlation remained above 0.90, indicating that the watermark survives the kind of compound abuse that real-world images frequently endure.

Those numbers matter because they address the central weakness of many existing watermarking schemes. Fixed-kernel transforms offer an attacker a predictable target, and separable channel-by-channel processing leaves color correlations unexploited. By combining a parameterized quaternion transform with adaptive, optimization-driven embedding, the new method raises the bar on both fronts simultaneously: the key-dependent kernel complicates unauthorized extraction, while the bi-objective optimization ensures the embedding strength is tailored to each image rather than dictated by a one-size-fits-all rule. The work also reflects a broader trend in the field, as researchers increasingly deploy evolutionary and swarm-based optimizers — genetic algorithms, particle swarms, and now enhanced ant colonies — to navigate the trade-offs that classical watermarking designs handled with hand-tuned heuristics.

The implications extend well beyond academic curiosity. As generative artificial intelligence floods the internet with synthetic imagery and copyright disputes grow more contentious, techniques that can embed verifiable, attack-resistant ownership marks into color images are becoming critical infrastructure for the digital economy. The Fujian Normal University team’s framework, supported by the Natural Science Foundation of Fujian Province, points toward watermarking systems that are simultaneously invisible, resilient, and cryptographically protected — a combination that could prove decisive for medical imaging, remote sensing, digital forensics, and content authentication alike. The research was published in Cluster Computing, volume 29, article 723, on 5 September 2026, with the DOI 10.1007/s10586-026-06491-1.

Subject of Research: Robust color image watermarking in the DQQPFT domain using enhanced multi-objective ant colony optimization

Article Title: A novel color image watermarking method in DQQPFT domain via enhanced multi-objective ant colony optimization

Article References: Tang, S., Xia, H., & Ke, Y. (2026). A novel color image watermarking method in DQQPFT domain via enhanced multi-objective ant colony optimization. Cluster Computing, 29(12), Article 723. https://doi.org/10.1007/s10586-026-06491-1

Image Credits: AI Generated

DOI: 10.1007/s10586-026-06491-1

Keywords: color image watermarking, DQQPFT, quaternion transforms, QSVD, ant colony optimization, multi-objective optimization, digital watermarking, image security, swarm intelligence, PSNR, normalized correlation, cryptographic keys