Hawk-Inspired Algorithm With Fractional Calculus Sharpens Multi-Threshold Image Segmentation

Image segmentation is one of those deceptively simple tasks that quietly underpins much of modern computer vision. Before a self-driving car can recognize a pedestrian, before a radiologist’s AI assistant can outline a tumor, and before an inspection robot can spot a hairline crack in a bridge girder, an algorithm must first decide which pixels belong together. One of the most widely used approaches is thresholding: pixels are sorted into classes according to their intensity values, with the dividing lines, or thresholds, chosen to maximize some measure of information content. When an image needs more than two classes, the problem becomes multi-threshold segmentation, and the search space for the best combination of thresholds grows explosively. A team of researchers at Changchun University in China has now introduced a new weapon for that search, described in the journal Cluster Computing, that fuses a nature-inspired optimizer with the mathematics of fractional-order calculus.

The work, led by Guoyuan Ma together with Zeyang Qin, Shuwei Wang, Xin Jin and Jinliang Xing from the School of Mechanical and Vehicle Engineering, tackles a well-known bottleneck. Among the many criteria available for deciding where thresholds should fall, exponential entropy has earned a reputation for producing accurate, information-rich segmentations. The catch is computational: evaluating exponential entropy across every plausible combination of thresholds in a multi-level setting is prohibitively expensive. Exhaustive search is out of the question for anything beyond a handful of thresholds, so researchers have increasingly turned to metaheuristic optimizers, algorithms that explore the search space intelligently rather than exhaustively, borrowing their search logic from swarms, flocks and predators.

The particular predator chosen by the Changchun team is the Harris hawk. The Harris Hawk Optimization algorithm, first introduced in 2019, models the cooperative hunting behavior of these birds of prey, in which a lead hawk chases a quarry while its flockmates take turns in a relay of surprise attacks, or pounce. In algorithmic terms, candidate solutions behave like hawks that alternate between exploratory phases, scanning the space broadly for promising regions, and exploitative phases, closing in on the best solution found so far. The transition between these phases is governed by an energy factor that mimics the quarry’s dwindling escape energy. HHO has proven popular across engineering problems, from photovoltaic parameter extraction to feature selection, but the authors of the new study point out two persistent weaknesses: the initial population tends to lack diversity, which can trap the algorithm near a mediocre solution, and its local search around that solution is often too coarse to polish it to a true optimum on complex landscapes.

Their answer is a multi-strategy enhanced version, which they call IHHO, built from three complementary modifications. The first is a cosine energy factor update strategy. In the original HHO, the energy factor that dictates whether hawks search widely or pounce locally decreases in a largely linear fashion as iterations proceed. By reshaping this decay with a cosine profile, the improved algorithm modulates the exploration-exploitation balance more smoothly, spending more time in broad exploration early on and transitioning gradually to fine-grained local refinement, which the authors argue improves the algorithm’s development capability and stability.

The second and mathematically most distinctive ingredient is a fractional-order calculus strategy, implemented in the Grünwald-Letnikov discrete format. Fractional calculus generalizes ordinary differentiation and integration to non-integer orders, and its discrete Grünwald-Letnikov form expresses a fractional derivative as a weighted sum over past values, with weights that follow a characteristic power-law decay. In the context of an optimization algorithm, this memory-laden update rule lets each hawk’s next position depend not just on its current state but on a fading trail of its history, injecting long-range correlations into the search dynamics. The approach echoes a broader trend: fractional-order operators have recently been used to enhance other optimizers, including a dung beetle optimizer applied to medical image segmentation, and Lévy-flight-like heavy-tailed movement patterns have been observed in physical systems ranging from light transport to foraging animals. By embedding fractional-order dynamics into the hawk position updates, the Changchun team aims to help the swarm escape local optima while retaining fine convergence behavior.

The third strategy is a Lévy-t perturbation. Lévy flights are random walks whose step lengths follow a heavy-tailed distribution, meaning that most steps are short and local but occasional steps are very long. This mixture of dense local sampling with rare long-range jumps is exactly the kind of search pattern that helps metaheuristics balance exploitation and exploration. Applied as a perturbation within IHHO, it gives individual hawks a mechanism to occasionally leap far from the current cluster of solutions, refreshing population diversity when the search stagnates.

To test whether these three strategies genuinely help, the researchers benchmarked IHHO against a battery of competing algorithms on the CEC2017 benchmark function dataset, a standard and demanding test suite used in international optimization competitions to probe how algorithms handle rugged, multimodal and shifted landscapes. IHHO performed well on the key performance indices, and the authors report that it showed good development capability, stability and robustness. Beyond raw benchmark scores, the team also ran ablation experiments, systematically removing individual improvement strategies to verify that each one, the cosine energy factor, the fractional-order update and the Lévy-t perturbation, contributes measurably to the overall performance. This kind of component-level validation is increasingly expected in the metaheuristics literature, where multi-strategy hybrids are sometimes accused of being ad hoc collections of tricks.

The real payoff, however, comes when the optimizer is coupled to the exponential entropy criterion for actual image segmentation. In this framework, the quality of a candidate set of thresholds is scored by the exponential entropy of the resulting segmented image, and the optimizer’s job is to find the threshold combination that maximizes that score without evaluating every possibility. In segmentation experiments on both grayscale and color images, the IHHO-based method outperformed the other algorithms tested, delivering segmentations that better preserved the informational structure of the original images while keeping the computational burden manageable. The combination matters because the two components address different halves of the problem: exponential entropy provides a sensitive, information-theoretic measure of segmentation quality, while IHHO supplies an efficient and reliable search engine capable of locating near-optimal thresholds in a combinatorial space.

The significance of the work extends beyond a single algorithmic recipe. Multi-threshold segmentation sits at the foundation of pipelines in medical imaging, remote sensing, industrial inspection and agricultural machine vision, and the choice of thresholds can determine whether downstream analysis succeeds or fails. Reviews of the field note that nature-inspired optimizers, from particle swarm optimization and grey wolf optimizer to whale optimization and butterfly optimization, have become the dominant tools for multi-level thresholding precisely because exhaustive threshold search scales so badly. Each new optimizer variant is essentially a bet about which search dynamics best navigate the entropy landscape of real images. The Changchun study’s bet, that memory-rich fractional-order dynamics combined with heavy-tailed perturbations and a smoothly scheduled energy factor will outperform simpler schemes, is supported by their benchmark, ablation and segmentation results.

The research, published in Cluster Computing as volume 29, article 805, was funded by the Jilin Provincial Natural Science Foundation under grant number YDZJ202501ZYTS659. The authors state that no datasets were generated or analysed during the current study beyond those used in the reported experiments, and they declare no competing financial interests. For practitioners, the message is that the machinery of fractional calculus, long a fixture of applied mathematics and physics, is proving to be a practical tool for sharpening swarm intelligence. For the vision community, the study adds a strong new contender to the growing family of Harris hawk variants, and suggests that when the next image needs to be carved into a dozen meaningful regions, a flock of mathematically enhanced hawks may be the most efficient way to do it.

Subject of Research: A multi-strategy improved Harris Hawk Optimization algorithm using fractional-order calculus for multi-threshold image segmentation

Article Title: A novel method for image multi-threshold segmentation based on fractional order calculus improved harris hawk optimization algorithm

Article References: Ma, G., Qin, Z., Wang, S., Jin, X., & Xing, J. (2026). A novel method for image multi-threshold segmentation based on fractional order calculus improved harris hawk optimization algorithm. Cluster Computing, 29(14), Article 805. https://doi.org/10.1007/s10586-026-06566-z

Image Credits: AI Generated

DOI: 10.1007/s10586-026-06566-z

Keywords: image segmentation, multi-threshold segmentation, Harris Hawk Optimization, fractional-order calculus, Grünwald-Letnikov, exponential entropy, metaheuristic optimization, Lévy flight, swarm intelligence, CEC2017 benchmarks, computer vision, Cluster Computing