Automatic Cluster Detection in Hyperspectral Images using Quantum Walrus Optimizers

Abstract

The task of determining the correct number of clusters in hyperspectral images is not a straightforward procedure as the images feature complex spectral data, redundancy, and lack of validation images. Hence, automatic clustering methods are often preferred in practical applications, as they can efectively handle the complexity of spectral data and the absence of validation images. This study introduces two novel algorithms viz., the Qubit Walrus Optimizer (QbWaO) and Qutrit Walrus Optimizer (QtWaO) to leverage quantum principles to address the limitations of classical optimization techniques, which are inspired by the breeding behavior of walruses. QbWaO and QtWaO are designed to automatically detect clusters in hyperspectral images (HSI) by efectively balancing exploration and exploitation. This makes them particularly suited for high-dimensional clustering tasks in a real time environment. The concept of quantum Hadamard gates is used to initialize the population and induce diversity. The optimal clusters are then determined using the Adjusted Rand Index as the fitness function. score and the ′ score are used to determine the quality of the grouping. The comparative results suggest that the proposed methods outperform classical approaches in most scenarios, demonstrating their efectiveness in automatically clustering hyperspectral images.

Keywords

• eol>walrus optimizer
• qutrit
• qubit
• Hadamard gate
• hyperspectral image

1. Introduction

The emergence of spectral cameras and sensors has significantly improved the ability to collect detailed information about ground data in various fields, such as agriculture, law enforcement, geography, and military applications [ 1 ]. Hyperspectral sensors capture data across hundreds of contiguous spectral bands. This broader spectral and wide area coverage helps represent a combination of multiple materials, each contributing to the pixel’s overall spectral signature with varying abundance levels. Hyperspectral imaging (HSI) is a powerful technology, but the vast amount of data in contiguous spectral bands presents a major computational challenge. Band Selection (BS) techniques are commonly employed as a preprocessing step to identify unique spectral bands. A key challenge is designing efective criteria for informative bands while retaining important spectral information. Recently, information theory-based methods [ 1 ] have demonstrated great potential in HSI band selection.

One critical task in HSI processing is automatic cluster detection [ 2 ]. This aims to identify underlying patterns, classes, or regions without ground-truth images. Traditional clustering algorithms, such as k-means [ 3 ] or fuzzy c-means [ 4 ], often struggle with HSI’s high-dimensional, nonlinear nature. Optimization-based clustering techniques have gained popularity to overcome these challenges. Nature-inspired metaheuristic optimization techniques, such as Genetic Algorithms [ 5 ], Particle Swarm Optimization [ 6 ], and Tabu Search [ 7 ], have emerged as powerful tools in HSI clustering. These methods efectively explore the vast solution space, adapt to HSI data’s complex spectral signatures and high-dimensional structure, and avoid local minima, thus ofering superior performance over traditional clustering algorithms. The Walrus Optimizer (WaO) [ 8 ] is a relatively new optimization algorithm ofering eficient global search capability while balancing navigating and leveraging the search space. WaO [ 8 ] can sufer from slow and premature convergence when solving complex problems. The No Free Lunch Theorem afirms that no one-size-fits-all algorithm can efectively address all NPhard problems [ 9 ]. Researchers are creating improved metaheuristic algorithms to efectively address the diverse array of real-world challenges. In recent years, quantum-based metaheuristic methods have transpired as potential solutions to these problems [ 2 ]. These algorithms enhance search eficiency by employing quantum computing principles such as superposition and entanglement. In recent years, qubit or bi-level quantum metaheuristics have drawn substantial research attention, but developing higher-order quantum metaheuristics remains a complex challenge [ 10 ]. However, applying quantumbased approaches to complex clustering problems requires innovative designs and frameworks. In this work, qubit and qutrit versions of the Walrus Optimizer algorithm viz., Qubit Walrus Optimizer (QbWaO) and the Qutrit Walrus Optimizer (QtWaO) are introduced. The breeding behavior of walruses has inspired these, and quantum principles have enhanced their limitations. QbWaO and QtWaO are designed to automatically detect clusters in hyperspectral images by leveraging quantum-inspired mechanisms to balance exploration and exploitation efectively. This makes it ideal for complex highdimensional HSI clustering. Furthermore, QbWaO and QtWaO incorporate breeding-inspired dynamics and foraging behavior to maintain population diversity.

The primary contributions are as follows.

• emphQubit and Qutrit-Based WaO: This is the first instance of developing and applying multilevel quantum versions of the Walrus Optimizer Algorithm for clustering hyperspectral images. • Enhanced Exploration Phase with Global Best Selection: Instead of selecting a random walrus, the global best solution or the best walrus is utilized during the exploration phase. This efectively guides the entire population towards the leader. • Band Selection Based on Spectral Variability (BSSV) [ 11 ]: The proposed method selects the most dissimilar bands by utilizing spectral variability similarity, ensuring the removal of redundant information. • Population Diversity Enhancement with qubit and qutrit Hadamard Gate: The population is initialized using either qubit or qutrit Hadamard gate, which equally distributes individuals across the search space, increasing diversity in the population.

The paper is organized as follows: Section 2 presents a brief review of the literature. Relevant concepts are discussed in Section 3. Section 4 contains the details of the main proposed work. Section 5 thoroughly describes the dataset used, the experimental design, and the result anyalysis. Finally, Section 6 ofers a concise conclusion of the proposed methodology.

2. Brief Survey of Related Works

Hyperspectral imaging has revolutionized the way detail information about ground targets is captured and analyzed in various fields such as agriculture, geographical monitoring, and defense-based applications. However, this vast amount of data leads to significant computational challenges, necessitating efective band selection techniques to identify informative bands while preserving essential spectral information [ 11 ]. Recent studies have shown that information theory-based methods ofer promising results in HSI band selection, allowing for the extraction of critical features from complex datasets [ 1 ]. In [ 12 ], superpixel segmentation is performed to find the heterogeneous area in region-specific hypergraphs, and then a consensus matrix is built to select bands eficiently for hyperspectral imagery. Clustering assessment metrics are employed to determine the optimal number of clusters. Adjusted Rand Index (AIndex) [ 13 ], and PBM index [ 14 ] are some well-known metrics.

Estimating optimal clusters in HSI is important as ground truth data is often absent [ 15 ]. A lot of HSI clustering methods have been reviewed in [ 15 ]. Although an efective process, unsupervised Artificial DNA Spectral Matching proposed in [ 16 ] takes a huge amount of time for execution. Optimization-based clustering techniques have gained traction in addressing these challenges. The Walrus Optimization Algorithm (WaO)[ 8 ] is a recent addition to this suite of optimization algorithms, distinguished by its eficient global search capabilities and its balanced approach to exploration and exploitation of the search space, but it sufers from slow and premature convergence. As no individual optimization algorithm can universally solve all NP-hard problems[ 9 ], developing enhanced metaheuristic algorithms that can adapt to a wide range of real-world challenges is needed.

In light of these considerations, quantum-based metaheuristic methods have emerged as potential solutions, harnessing the quantum computing principles, such as superposition, coherence and decoherence—to improve search eficiency [ 2 ]. While qubit-based quantum metaheuristics have garnered significant attention, advancing higher-order quantum metaheuristics, such as those utilizing qutrits, remains a complex challenge [ 10 ]. In [17], a qubit-based Grey Wolf Optimizer (QBGWO) is introduced, incorporating quantum rotation and NOT gates to enhance solution quality. A memetic quantuminspired evolutionary algorithm was proposed in [18] which combines quantum genetic algorithms with tabu search. It balances global exploration via quantum rotation gates and local exploitation through directional mutations, achieving faster convergence and improved performance on benchmark functions. In [19], a qubit based Diferential Evolution algorithm is developed for eficiently identifying optimal clusters without prior knowledge and outperforming competitive algorithms in accuracy and convergence speed.

While qubit-based algorithms generally outperform classical metaheuristics, they remain constrained by the No Free Lunch Theorem [ 9 ]. Consequently, there is significant research interest in developing higher-order quantum metaheuristics, specifically qudit-based methods. A three-valued quantum or ( qutrit) based Genetic Algorithm was developed in [20]. In [ 2 ], the Diferential Evolution algorithm has been introduced for the automated clustering of hyperspectral images. Additionally, six quantum methodologies based on the Artificial Hummingbird Algorithm, Particle Swarm Optimization, and Genetic Algorithms have been presented in [ 10 ], utilizing both bi-level and tri-level quantum logic for unsupervised clustering of HSI data.

3. Significant Related Concepts

This section discusses several key concepts, viz., band selection method used, foundational principles of quantum computing, and the WaO[ 8 ] algorithm.

3.1. Band Selection Based on Spectral Variability [11]

Spectral Information Divergence (SID) measures the diference between two spectral bands by evaluating their probability distributions. To compute the SID first, the probability vectors are defined as follows. For two spectral bands and , we first define their probability vectors as - For band : - For band : = =

∑︀=1

∑︀=1 , = 1, 2, . . . , , = 1, 2, . . . , where, is the spectral dimension of the HSI dataset. The SID between the two bands and are defined using the Kullback-Leibler divergence (relative entropy) as

( , ) = ( || ) + ( || ), where, the Kullback-Leibler divergence ( || ) is given by The channel transition probability is defined as ( || ) = ∑︁ log =1 .

( , ) | = ∑︀ =1 ( , ) , The bands with the lowest SID values are selected as the distinct bands because they exhibit the least similarity in spectral information.

3.2. Quantum Computing Principles

Quantum-based machines represent a powerful computational paradigm that merges quantum mechanics with fundamentals of computing [ 10 ]. The basic unit of quantum machines are the qubit. Qubits exist in two distinct states: |0⟩ and |1⟩ [ 10 ]. These states can be expressed mathematically in the following column vector notation [21].

|0⟩ = ︂( 1)︂ 0 , |1⟩ = ︂( 0)︂ 1 Qubits exist in a superposition of its basis states (|0⟩ and |1⟩). The number of states that can be represented by n qubits in superposition grows exponentially (2) and thus provides increased computational eficacy with fewer resources. The superposition state of a qubit, denoted as |⟩ (where = 2 for qubit), is given by

|⟩ = 0|0⟩ + 1|1⟩ The probabilities that the qubits are in states |0⟩ or |1⟩ are represented by the complex coeficients 0 and 1, respectively. These probabilities also satisfy the following condition.

The normalization condition for a qubit is expressed as follows [21].

0 ⩽ ||2 ⩽ 1, for = 0, 1 02 + 12 = 1 (1) (2) (3) (1) (2) (3) (4) (5) (6) Quantum systems can exhibit a wider range of discrete energy levels and exist in more than two states, known as qudits [ 2 ]. The smallest multilevel quantum system is called a qutrit [ 2 ]. A qutrit can similarly be represented as ⎛1⎞ ⎛0⎞

⎛0⎞ |0⟩ = ⎝0⎠ , 0 |1⟩ = ⎝1⎠ , 0 |2⟩ = ⎝0⎠ 1 Qudits ofer better performance than qubits [ 10 ]. They require fewer units to achieve superior results than a larger number of qubits, thus reducing computation time. Mathematically, to represent the same information as qubits, only log2(3) qutrits are needed, resulting in an eficiency increase of approximately log2(3) ≈ 1.6 [ 2 ]. This reduction significantly lowers decoherence in qudit-based systems, enhancing their computational power. A qutrit can be expressed in a superposition state as follows.

|⟩ = 0|0⟩ + 1|1⟩ + 2|2⟩

The normalization condition for a qutrit is given by

︂[ 01 02 03 . . . 0]︂

11 12 13 . . . 1 ⎡01 02 03 . . . 0⎤ ⎣11 12 13 . . . 1⎦

21 22 23 . . . 2 Furthermore, the probabilities associated with the states of a qutrit must satisfy the following condition.

3.3. Walrus Optimizer Algorithm [8]

The WaO [ 8 ] algorithm is based on the behavior of walruses. Their ways of migrating, breeding, roosting, and foraging are considered by Han et al.. This approach models walruses social structures and role divisions, assuming populations interpret behavior through danger and safety signals. The steps of WaO [ 8 ] are briefly described in Algorithm 1.

The WaO [ 8 ] uses a population of "walruses" (agents) to explore and exploit the search space for optimal solutions to optimization problems. The algorithm divides the population into adults (90%) and juveniles (10%), with diferent roles for males, females, and juveniles.

Key features of WaO [ 8 ] include danger and safety signals, which encompasses the phases of exploration and exploitation in the algorithm. In high-risk conditions, walruses migrate causing enhanced searching in the entire solution space, while in safer conditions, they reproduce causing exploitation of the search space. Migration is achieved by adjusting positions based on other walruses’ locations, while reproduction involves updating positions through male influence (using a Halton sequence) and female influence from both the male and the best solution. Juveniles update their positions to avoid predators.

4. Proposed Work

The proposed methodology can be divided into two sections viz., BSSV based Band Depletion and Quantum Walrus Optimizer Algorithms. Band minimization is done as stated in Section 3.1.

4.1. Quantum Walrus Optimizers (QbWaO and QtWaO)

The quantum versions viz., Qubit Walrus Optimizer and Qutrit Walrus Optimizer algorithms follow the same steps as mentioned in Algorithm 1. The modifications introduced are as follows: • Qubit or Qutrit Encoding and Observation [20] : The qubit walrus population is represented as follows. 0 ⩽ ||2 ⩽ 1, for = 0, 1, 2 (qutrit) (7) (8) Similarily, the qutrit walrus population is represented in the following manner.

As Hadamard gate initializes all the states with equal amplitude values, qubit states are initialized in the following manner.

, = 1/(2) + 1/(2) (23) where, 01 = 1/(2) and 11 = 1/(2) for each individual qubit. Similarly, the following method is used for a qutrit.

, = 1/(3) + 1/(3) + 1/(3) (24)

Algorithm 1 Walrus Optimizer Algorithm [8]

1: Input: Population consisting of Walrus of dimension, Maximum iterations , 1, 2, 3,4 and 5 are random numbers between (0,1), is Migration step, is a distress coeficient designated by random number between (0,1). 2: is divided into 45% Male Walrus (MW) with top fitness values, 45% Female Walrus (FW) and 10% Young Walrus (YW) in

Exploitation Phase. Best Male Walrus (BW1) has the highest fitness, and (BW2) has the second highest fitness value. 3: Initialize the population using , = random(0, 1) = · = 1 −

,

= 2 × , = 2 × 1 − 1 4: Calculate fitness values for each walrus 5: while ≤ do 6: Calculate danger signal where, , , and are given by if | | ≥ 1 then

Exploration phase Update positions of each walrus where, the Migration step is calculated as where, is a randomly selected position, and

= 1 − else

Exploitation phase if Safety signal ≥ 0.5 then

Breeding behavior for each MW do

Use the Halton sequence to update the position end for for each FW do

Update

,+1 = , + = , − , · · 3

1 1 + exp (︁ − − ×1/02 )︁ 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18: 19: 20: 21: 22: 23: 24: 25: 26: 27: 28: 29: 30: (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) endforis generated using Lévy distribution to imitate Lévy movement else

Foraging behavior if | | ≥ 0.5 then

Gathering behavior Update positions It,+1 = It, + · ( It1,1 − It,)

+ (1 − ) · ( 1It − It,) ,+1 = ( − ,) · = 1 + , · ,+1 = 1, + 2,

2 1,+ 1 = 1, − 1 · 1 · | 1, − ,| 2,+ 1 = 2, − 2 · 2 · | 2, − ,|

= · 5 − , = tan( ), = 1, 2 ,+1 = , · − | 1 − ,| · 4 end for for each YW do

Update position where: where,

Calculate and else

Fleeing behavior

Update positions else if 0, ≤ < 1, then population is equally distributed in the solution space.

Equations (23) and (24) sufice the normalization principles stated in Section 3.2. The following Algorithms 2 and 3 are used for finding the probable classical state observations [ 20]. The notations of , and are same as specified in Algorithm 1. The resulting classical representation in binary and ternary form after observations for each walrus are • Equqation (13) is replaced by the equation given below.

︀[ 0 1 0 1 1 1 1 0 0 · · ·

︀] ︀[ 0 2 0 1 1 2 1 0 0 · · ·

︀] = − , · · 3 (25) (26) (27) Instead of considering any random Walrus, the best Walrus with the highest fitness value enhances the exploration phase. • At the end of every iteration, it is checked whether the quantum normalization principles are adhered to. When it does not stand true Equations (23) and (24) are used to reinitialize those walruses. • In every iteration, a random count of zeros is incorporated into the population for which is true. The non-zero values designate the cluster centers using k-mean++ [ 3 ] algorithm, the appropriate clusters are constructed. The AIndex [ 13 ] is used as the Cluster Validity Index to identify the optimal cluster numbers.

The complexity of the band minimization is as follows:

( × × ) Here bands are present each of dimensions × pixels. For the QbWaO, if is a randomly chosen pixel intensity value, represnts the length of individual population, is the total number of iterations then the worst-case time complexity is:

For QtWaO, the time complexity is as follows:

(2 × × × ) (3 × × × )

5. Findings and Analysis

This section presents the various parameters used, the dataset employed, the statistical tests conducted, and the analysis of the experimental results.

5.1. HSI Dataset

The WHU-Hi-LongKou dataset (WHU) [22] [23] was collected in 2018, featuring scenes of Longkou Town in Hubei province, China. It has 9 classes spread over an agricultural landscape with six types of crops. It has a spatial dimension of 550 × 400 pixels and a spectral dimension of 270 spectral bands. The UAV-borne hyperspectral imagery has a spatial resolution of approximately 0.463 meters.

5.2. Experimental Configurations and Analysis

QbWaO and QtWaO are compared with their classical version WaO [ 8 ] using the WHU-Hi-LongKou dataset [22] [23]. To conduct an impartial study, all algorithms are executed for 50 times over 100 iterations. An Intel Core i7-8700 processor and a Windows 11 machine were used. All simulations were conducted on MATLAB 2023b. In WaO [ 8 ], each Walrus has dimensions of 20 and 20 walruses were considered. The dimensions were kept the same for both QbWaO and QtWaO. The number of walruses taken was 16 for QbWaO and 14 for QtWaO, respectively. For QbWaO, 8 male and female walruses and 2 young walruses were taken. In QtWaO, 7 male and female walruses and 2 young walruses were taken. The proposed methods are also compared with Kmeans [ 3 ] algorithm using taking the predefined cluster number as 6.

Various statistical tests like mean and standard deviation are recorded in Table 1 for all three algorithms. The time required for the convergence of each algorithm is noted in Table 1. From Table 1 and Figure 1 it can be observed that QtWaO takes far lesser time to converge than the other two algorithms. Table 2 contains the optimal AIndex [ 13 ] values, automatically detected cluster numbers, score [24] and ′ score [24] are presented. The score [24] and ′ score [24] help determine the clustering quality. QtWaO produces 10 clusters nearest to the classes in the WHU Dataset [22] [23]. The clustered images, along with the minimized image produced by using BSSV [ 11 ] and the ground truth image of WHU Dataset [22] [23] are presented in Figure 2.

The null hypothesis is evaluated using the One-Way ANOVA test [25], which assesses whether the results originate from the same probability distribution. The null hypothesis is rejected if the value is below the 1% significance level, indicating support for the alternative hypothesis. The results are summarized in Table 3. Additionally, Tukey’s post hoc test [25] is conducted. The box plots for both tests are shown in Figure 3.

Based on the various tests conducted and the parameters evaluated, QtWaO generally outperforms QbWaO in most cases. Also, the population size and convergence speed of QtWaO are far better than those of both QbWaO and WaO.

6. Conclusion

Qubit and Qutrit Walrus Optimizer algorithms significantly advance unsupervised clustering for hyperspectral imagery. By integrating quantum principles and biological inspirations, QbWaO and QtWaO efectively address the challenges associated with the slow and premature convergence of the Walrus Optimizer algorithm. The enhanced exploration phase integrated with the Hadamard gate and band selection strategies improves clustering performance and ensures the robustness of the algorithms in high-dimensional spaces. The qutrit version performs better in almost all aspects, producing a near-optimal number of clusters in less time. Future work could explore enhancements to these quantum algorithms, including implementing more advanced quantum techniques and their application to various domains requiring sophisticated data clustering solutions. Additionally, parameter reduction can be implemented in the Walrus Optimizer.

Acknowledgments

Thanks to the developers of ACM consolidated LaTeX styles https://github.com/borisveytsman/acmart and to the developers of Elsevier updated LATEX templates https://www.ctan.org/tex-archive/macros/ latex/contrib/els-cas-templates.

Declaration on Generative AI

The author(s) have not employed any Generative AI tools. [17] Y. Liu, C. Li, J. Xiao, Z. Li, W. Chen, X. Qu, J. Zhou, Qegwo: Energy-eficient clustering approach for industrial wireless sensor networks using quantum-related bioinspired optimization, IEEE Internet of Things Journal 9 (2022) 23691–23704. [18] A. S. Hesar, M. Houshmand, A memetic quantum-inspired genetic algorithm based on tabu search,

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(a) (b) Figure 3: (a) One-Way ANOVA test [25], (b) Tukey’s post hoc test [25] - results for WHU Dataset [22] [23]

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Figure 2: (a) Ground Truth image of WHU Dataset [22] [23], (b) Resultant image using BSSV [11], (c) Clustered image using WaO [8], (d) Clustered image using QbWaO, (e) Clustered image using QtWaO with AIndex [13], (f) results using -means [3] (cluster number = 6) on WHU Dataset [ 22 ] [23]