Hierarchical Clustering and Schur Complement for Automatic Hyperspectral Band Selection

Abstract

Band selection is a crucial step in hyperspectral imaging to reduce spectral redundancy and processing costs whilst retaining information useful for classification. Most existing approaches require the number of bands to be retained to be set manually or rely on parameters that are difficult to adjust. This work proposes the Clustering-Unified Schur complement for Diversity with Hierarchical Clustering (CUSD-HC). This fully unsupervised band selection method combines Ward’s hierarchical clustering with a greedy selection based on the Schur complement. Bands are grouped by spectral similarity, and then a representative band is chosen from each group to preserve diversity and informational content. The number of bands is determined automatically using a multi-detector k-fold criterion combined with an intrinsic dimension threshold estimated via PCA. Evaluated on six benchmark datasets using four classifiers (SVM-RBF, Random Forest, XGBoost, LightGBM), CUSD-HC achieves an average rank of between 2.7 and 3.3 among nine compared methods, placing it consistently among the leading group. The Nemenyi test shows no statistically significant difference between CUSD-HC and the top-ranked competitors, while CUSD-HC significantly outperforms the weakest baselines (p < 0.05); unlike the best-ranked alternatives, it reaches this level of performance without any manual selection of the number of bands, which is determined automatically from the data. An inter-scene transferability experiment on the WHU-Hi datasets shows a maximum degradation of 3.9 points in overall accuracy (OA), and the transferred bands even outperform the native selection in three cases out of six. Furthermore, the selected bands naturally cover the main spectral regions (visible, near-infrared, and SWIR), which facilitates the interpretation of results for applications such as precision agriculture and environmental monitoring.

1. Introduction

Hyperspectral sensors measure reflected light across hundreds of contiguous bands. The image cubes obtained in this way contain dense spectral information. This density has improved soil classification, mineral mapping, and vegetation monitoring [1]. However, this large number of bands poses well-known problems. The strong correlation between neighboring bands increases computation time without providing any additional discriminatory power. Furthermore, the Hughes phenomenon degrades classification accuracy when the number of training samples is smaller than the feature space dimension [2]. Band selection therefore addresses these difficulties by seeking a representative subset of the original bands. Unlike feature extraction methods (PCA [3]), which alter the nature of the variables, feature selection preserves the physical meaning of each channel. This is important for applications such as the design of sensors for precision agriculture [4] or on-board processing on lightweight multispectral cameras.

There are several families of unsupervised band selection methods. Classification-based approaches evaluate each band separately (variance, entropy). They ignore redundancies across bands and often select similar bands [5]. Clustering methods (k-means [6], Spectral Clustering [7], E-FDPC density peak clustering [8]) group the bands together and then select one representative per group. They ensure a certain degree of diversity, but do not guarantee that the representative retains the maximum amount of information. Parsimonious self-representation approaches (EGCSR [9]) capture non-linear dependencies but require the number of target bands to be set and are more computationally expensive. A limitation common to almost all existing methods is the need to specify a priori the number of bands to select, which, in practice, depends on the dataset, the sensor, and the final task. This is often chosen via exhaustive cross-validation, a cumbersome procedure specific to each problem.

In this paper, we propose the Unified Schur-Diversity Criterion with Hierarchical Clustering (CUSD-HC) for the automatic selection of hyperspectral bands. This framework combines hierarchical clustering to partition the set of bands with Schur-complement-based selection to maximize information locally.

CUSD-HC’s novelty does not come from any single component taken in isolation. Ward’s hierarchical clustering, the Schur complement, and elbow-based detection are all established techniques. What is new is how we integrate them into a fully automated, interpretable pipeline that needs no user-defined parameters.

Our contributions are as follows:

• A hybrid selection method in which we construct a single dendrogram (Ward) to obtain k clusters, then apply a Schur complement greedy procedure to select, within each cluster, the band with the highest conditional variance. The method thus naturally balances diversity and informativeness.
• An automatic mechanism for determining the number of bands to select (k). It combines three kink-detection heuristics: the marginal gain threshold, Kneedle, the geometric distance to the line, and a threshold based on PCA with 99% of the variance explained.
• An evaluation on six datasets, covering several sensor types (AVIRIS, ROSIS, Headwall Nano-Hyperspec). We compare nine unsupervised methods using 10-fold stratified cross-validation and Friedman’s statistical tests, followed by the Nemenyi method, in accordance with the protocol recommended by Demšar [10].

The manuscript is organized as follows: Section 2 reviews the State-of-the-Art on hyperspectral band selection methods. Section 3 details the proposed CUSD-HC approach and the experimental protocol. Section 4 presents the experiments conducted and the results obtained using four classifiers (SVM-RBF, Random Forest, XGBoost, and LightGBM) on six benchmark datasets. Section 5 discusses these results. Finally, Section 6 concludes the study.

2. Related Work

2.1. Ranking Methods

Ranking-based methods constitute the oldest and most intuitive family of techniques. The simplest strategy ranks bands according to a single criterion and then retains the top k. For example, marginal variance selects the bands with the highest variance per pixel. The assumption is that a highly variable channel contains more information. Chang et al. [5] introduce a selection framework under the constraint of minimum linear variance. Subsequent work incorporates entropy or the signal-to-noise ratio (SNR) as ranking metrics [11]. Another approach selects the bands most highly correlated with the dominant principal components (PCA) [12,13]. These methods are computationally inexpensive but evaluate each band separately. They therefore do not control for redundancy, as two highly correlated bands may both rank at the top of the list.

2.2. Clustering-Based Methods

Clustering methods partition the bands into spectrally homogeneous or strongly correlated groups, then select one representative per group to ensure spectral diversity. Martinez-Uso et al. [6] applied Ward’s hierarchical clustering to the correlation matrix, then selected the medoid of each group. Ng et al. [7] introduced Spectral Clustering for band selection, which constructs an affinity graph based on correlations and then partitions the graph using normalized cuts. Sun et al. [8], using their E-FDPC method, adapt the density peak algorithm of Rodriguez and Laio [14] for band selection and, for each band, compute a local density and an isolation distance in the correlation space. The bands with the highest product between them become the cluster centers. More recently, OCF [15] proposed a dynamic programming approach for optimal contiguous band partitioning under a normalized cut criterion, combined with a rank-on-clusters strategy for band selection. A limitation of all these methods lies in the choice of a representative. The medoid, or the densest band, does not necessarily maximize conditional information relative to the bands already selected.

2.3. Methods Based on Sparse Representations and Graphs

The fundamental principle underlying parsimony and graph-based representation methods is that a hyperspectral cube has a low-rank structure. Consequently, each band can be approximated by a linear combination of a small number of other bands. Thus, self-representation models express each band as a sparse linear combination of the others, then select the bands with the highest reconstruction coefficients. This approach can be interpreted as column selection in numerical linear algebra [16]. ISSC [17] penalizes the self-representation matrix using an L1 norm. EGCSR [9] enriches band features through graph convolution on a k-nearest-neighbor graph built from inter-band correlations, then solves a closed-form self-representation problem to express each band as a sparse linear combination of the others. Spectral Clustering on the resulting coefficient matrix identifies band groups, from which representative bands are selected as medoids. However, EGCSR requires the number of bands k and the number of graph neighbors to be predefined. More recently, Zhang et al. [18] proposed TGSR, a tensor extension of graph self-representation that captures higher-order correlations between bands via a low-rank constraint on the coefficient tensor. In a different vein, Ye et al. [19] formulated band selection as a Markov decision process and solved it using deep reinforcement learning, with SNR pre-filtering as a first step. These approaches illustrate the current trend towards deep learning architectures, at the cost of additional hyperparameters and greater computational resources. These different families (ranking, clustering, self-representation) and their recent extensions via deep learning require that the number of bands be selected or a neighborhood parameter be set a priori. Furthermore, the choice of a representative in clustering approaches does not guarantee maximum information retention.

2.4. Schur Complement Approaches

The Schur complement of a covariance matrix provides the conditional variance of a variable, given a set of other variables [20]. Specifically, for a covariance matrix Σ, the conditional variance of band j, given a pre-selected set S, is the (j, j) entry of the Schur complement of Σ/Σ_SS. This is a greedy algorithm that iteratively maximizes this variance, producing a sequence of bands. Each selected band provides the most residual information. This criterion has been used for sensor placement [21] but rarely for selecting hyperspectral bands. A practical advantage is the incremental update, which, after selecting a band j, allows the residual matrix C to be updated. This operation is less computationally expensive and avoids a matrix inversion at each step.

2.5. Automatic Determination of k in Prior Work

Most studies on band selection specify the final number of selected bands, k, in advance. They then test several values and report the best results. Consequently, few methods automatically select k. The virtual dimensionality (VD) estimator method [22], for example, uses a Neyman–Pearson hypothesis test on eigenvalues to estimate the number of spectrally distinct signals but often overestimates in the presence of noise. Meanwhile, detecting inflection points on reconstruction error or classification accuracy curves [23] is common in clustering but has rarely been applied to band selection.

Beyond the families described above, recent work uses attention mechanisms to weight or select bands in a learned way. Band-attention networks [24], for example, weight each spectral channel inside an end-to-end convolutional network, while architectures such as SegHSI [25] combine cluster-based attention with semi-supervised learning to reduce the number of labeled pixels needed. These supervised or semi-supervised approaches differ from CUSD-HC in that they require labels and hyperparameter tuning, but they show how varied spectral weighting strategies can be. Band selection also matters outside remote sensing: in biomedical imaging, for instance, it has helped detect early esophageal cancer through band-selective hyperspectral imaging [26], which points to a broader use for unsupervised spectral reduction methods like the one proposed here.

3. Materials and Methods

3.1. Equipment

3.1.1. Datasets

Six publicly available hyperspectral datasets were used for the experiments. They cover three different sensors and a variety of land cover types. Indian Pines and Kennedy Space Center (KSC) were acquired by the AVIRIS sensor with 220 and 176 bands, respectively. The ROSIS sensor captured Pavia Center with 102 bands. Three scenes from the WHU-Hi benchmark [27] complete the evaluation: HanChuan (274 bands, 16 classes), LongKou (270 bands, 9 classes), and HongHu (270 bands, 22 classes).

Table 1. Characteristics of the datasets. p denotes the bands retained after SNR filtering; k* denotes the number selected by CUSD-HC.

Dataset	Sensor	Bands	p	Class	Pixels	k*	Size
Indian Pines	AVIRIS	220	181	16	10,249	16	145 × 145
Pavia Center	ROSIS	102	101	9	148,152	6	1096 × 715
Kennedy SC	AVIRIS	176	172	13	5211	57	512 × 614
HanChuan	Headwall Nano-Hyperspec	274	271	16	257,530	63	1217 × 303
LongKou	Headwall Nano-Hyperspec	270	269	9	204,542	8	550 × 400
HongHu	Headwall Nano-Hyperspec	270	269	22	386,693	6	940 × 475

summarizes the characteristics.

3.1.2. Equipment Used

The proposed method was implemented in Python 3.10. The NumPy, SciPy, and scikit-learn libraries were used for matrix computations, Ward’s hierarchical clustering, Schur complement selection, as well as classification (SVM-RBF, Random Forest, XGBoost, and LightGBM) and statistical tests (Friedman, Nemenyi). The Kneed library was used for automatic elbow detection. The experiments were conducted in the Google Colab environment (Intel Xeon CPU @ 2.20 GHz, 2 vCPUs; 12 GB RAM; Ubuntu 22.04.5 LTS; Python 3.12.13). The source code is available on request from the corresponding author.

The band-selection step runs on CPU only, for CUSD-HC and for every comparison method. The classification step, common to all methods, can optionally use a GPU for XGBoost and LightGBM when one is available, purely to speed things up. This does not affect the reported results (

Table 2. Overall SVM-RBF accuracy (± 95% CI). The best result per set is shown in bold.

Method	Indian Pines	Pavia Center	Kennedy SC	HanChuan	LongKou	HongHu
CUSD-HC (ours)	0.877 ± 0.003	0.982 ± 0.001	0.911 ± 0.006	0.851 ± 0.003	0.976 ± 0.002	0.809 ± 0.003
Pure Schur	0.825 ± 0.007	0.980 ± 0.001	0.823 ± 0.006	0.820 ± 0.004	0.954 ± 0.002	0.777 ± 0.002
Marginal variance	0.709 ± 0.006	0.822 ± 0.003	0.823 ± 0.007	0.697 ± 0.003	0.834 ± 0.002	0.594 ± 0.002
PCA	0.822 ± 0.005	0.960 ± 0.002	0.823 ± 0.006	0.758 ± 0.003	0.956 ± 0.002	0.766 ± 0.004
Mutual Information	0.790 ± 0.007	0.972 ± 0.001	0.867 ± 0.006	0.852 ± 0.003	0.941 ± 0.001	0.704 ± 0.003
Spectral Clustering	0.858 ± 0.006	0.984 ± 0.001	0.926 ± 0.006	0.857 ± 0.003	0.956 ± 0.001	0.811 ± 0.004
E-FDPC	0.751 ± 0.005	0.984 ± 0.001	0.949 ± 0.005	0.802 ± 0.003	0.951 ± 0.002	0.794 ± 0.003
OCF	0.880 ± 0.004	0.984 ± 0.000	0.904 ± 0.007	0.857 ± 0.003	0.971 ± 0.001	0.801 ± 0.003
EGCSR	0.883 ± 0.004	0.987 ± 0.001	0.955 ± 0.004	0.875 ± 0.003	0.966 ± 0.001	0.805 ± 0.003

, SVM-RBF) or the runtime comparison of the band-selection step (Section 5.4).

3.2. Proposed Method: CUSD-HC

The Clustering-Unified Schur complement for Diversity with Hierarchical Clustering (CUSD-HC) that we propose consists of three stages. First, noisy bands are removed by computing the signal-to-noise ratio (SNR); next, a hierarchical partitioning of the remaining bands is performed; and, after automatically determining the number of bands k, the bands within the resulting groups are finally selected using the Schur complement. Each step is detailed below.

3.2.1. Pre-Processing: Removal of Noisy Bands

The first step in the CUSD-HC method involves pre-processing to remove noisy bands dominated by atmospheric absorption or sensor artifacts. For each band i, the signal-to-noise ratio (SNR) is estimated by local inter-band regression such that band i is predicted from its k = 3 nearest spectral neighbors using ordinary least squares, and the SNR is defined as the ratio of the variance of the predicted signal to the residual variance according to the following formula:

$$SN{R}_i=\left(Var\left({\widehat{x}}_i\right)\right)/\left(Var\left(r_i\right)\right) with { r}_i=x_i-X_N\widehat{\mathsf{\beta }}$$

(1)

where:

• x_i is the vector of the values of band i (across all pixels);
• X_N is the matrix of adjacent bands;
• $\widehat{\beta }$ is the vector of linear regression coefficients estimated by the least squares method;
• r_i is the residual, interpreted as noise.

The choice of k = 3 follows the convention established by local noise estimators in hyperspectral imaging, notably the MNF (Minimum Noise Fraction) transformation by Roger and Arnold (1996) [28], where a restricted spectral neighborhood ensures that only short-range local correlation is utilized. This avoids the bias introduced by spectrally distant bands that have a weak natural correlation with the target band. A threshold τ is determined automatically by detecting the largest jump in the sorted values of log (1 + SNR), based on the principle that clean bands cluster at high SNR, whilst noisy bands form a distinct group at low SNR. Bands with SNR < τ are eliminated, producing a denoised set of p bands. The code is available in Appendix A via Algorithm A1. A single extreme outlier in the SNR distribution could otherwise distort this threshold search and remove an excessive fraction of bands on some sensors (see Section 5.3). To prevent this, the search for the largest gap is restricted to positions that keep at least 50% of the bands. Section 5.7.1 validates this safeguard through ablation across all six benchmark datasets.

3.2.2. Hierarchical Partitioning

The hierarchical partitioning (Figure 1) of the bands is obtained by performing a single round of Ward’s hierarchical clustering on the p filtered bands, using the distance matrix defined as follows:

$$D_{ij}=1-\left|corr\left(i,j\right)\right|,$$

(2)

where corr(i, j) is the Pearson correlation calculated over the full pixel matrix.

This correlation matrix is computed over all pixels of the image, labeled or not. The hierarchical partitioning of the bands therefore does not depend on class information and stays fully unsupervised, in line with the method’s objective.

In practice, the p × p correlation-based distance matrix is condensed with SciPy’s squareform function and passed to SciPy’s hierarchical-clustering routine, linkage, using Ward’s method, so Ward’s criterion is applied directly to this precomputed dissimilarity rather than to raw pixel vectors under Euclidean distance. Ward’s minimum-variance property is formally derived for Euclidean distances; using it on a correlation-based distance is a practical and common adaptation, since 1 − |corr(i, j)| behaves as a proper dissimilarity (zero for identical bands, increasing as bands decorrelate).

Ward’s linking minimizes the intra-cluster variance at each merging step, producing a dendrogram that can be cut at any level to obtain 2, 3, …, k_max clusters per cut. The single-dendrogram strategy avoids the cost of running a separate clustering for each candidate k. Following this partitioning, Schur’s greedy selection is applied. Algorithm A2 in Appendix A illustrates this.

3.2.3. Schur’s Selective Approach Within the Groups

This section presents greedy selection using the Schur method.

To do this, given k clusters obtained by splitting the dendrogram, the algorithm selects exactly one band per cluster using the greedy procedure described below.

We assume that C = Σ is the covariance matrix of the p filtered bands, initialized from the residual covariance. At each iteration:

• For each band j in each remaining group, the conditional variance is read along the diagonal: Var(j | S) = Cjj.
• The cluster containing the band with the maximum overall value of Cjj is identified, and this band is then added to the selected set S.
• The residual covariance is updated using the first-order Schur complement via the following formula:

$$C\leftarrow C-\left(C_j{C}_j^T\right)/\left(C_{jj}\right),$$

(3)

This update reduces the conditional residual variance of all bands correlated with the one just selected.

• The selected group is removed from the list of candidates, and the process is repeated until each of the k groups has provided a strip.

The order in which bands are selected within groups is therefore not predetermined but is determined greedily based on residual variance. By combining the clustering constraint (one band per group) with Schur’s conditional variance criterion, our CUSD-HC method simultaneously maximizes information content and spectral diversity. The code is available in Appendix A via Algorithm A4.

3.2.4. Automatic Determination of k

To automatically determine the number of bands to select, we estimate CUSD HC for each value of k between 2 and k_max = p/3, where p is the number of bands after removing those affected by noise using SNR. Setting the lower bound to two avoids the trivial case of a single band, which does not allow distinguishing among multiple spectral groups. This upper bound guarantees a minimum cluster size of about three bands per group on average. It preserves the statistical relevance of the spectral partitioning and scales directly with the number of available bands, without relying on any fixed dataset-specific constant (see the sensitivity analysis in Section 5.7.1). For each value of k tested, the pre-calculated dendrogram is truncated, and the total variance V(k) is recorded. The V(k) curve (Figure 2) is then analyzed using three elbow detectors: a marginal gain threshold, the Kneedle algorithm [20], and the geometric distance to a reference line. The median of the three estimates provides a robust knee value. On certain highly correlated datasets, the knee value kelbow may be abnormally low. To avoid this, a threshold derived from principal component analysis (PCA) is applied. We calculate kPCA, that is, the number of principal components that account for 99% of the total variance. The final number of bands selected is k* = max (kelbow, kPCA). Algorithm A3 in Appendix A illustrates this.

3.3. Classification Protocol

The selected bands are evaluated for classification accuracy using four classifiers: RBF kernel SVM (C = 100, γ = scale), Random Forest with 100 trees, XGBoost (100 trees, maximum depth 6, learning rate 0.1), and LightGBM (100 trees, maximum depth 6, learning rate 0.1, 31 leaves). A 10-fold stratified cross-validation is applied after standardizing the features to have a mean of zero and a variance of one. For datasets exceeding 50,000 labeled pixels (Pavia, HanChuan, LongKou, HongHu), stratified random subsampling of 50,000 pixels is used, in accordance with standard practice in the hyperspectral literature [21]. Overall accuracy (OA) is reported with 95% confidence intervals derived from the 10 replicates.

3.4. Comparison Methods

To assess the performance of our CUSD-HC method, we compared it with eight methods representing the main families of band selection. The first, which we call Pure Schur, greedily maximizes the conditional variance without any prior partitioning. Marginal variance retains the k bands with the highest pixel variance. The PCA (principal component analysis) method selects the bands most highly correlated with the first principal components. Mutual Information (via histogram) proceeds by greedy minimization of the average Mutual Information. Next come four clustering-based approaches: Spectral Clustering constructs partitions using normalized slices and then selects the medoid of each group [7]; E-FDPC uses density peak clustering [8]; OCF applies an optimal contiguous partitioning strategy combined with a rank-on-clusters selection [14]; finally, EGCSR is based on self-representation via graph convolutions [9]. The selected methods span the three main families of State-of-the-Art techniques: ranking, clustering, and sparse self-representation. For fair comparison, all competing methods were re-implemented following the general principles described in the original papers, using the same correlation-based affinity matrix derived from the SNR-filtered bands. Recent approaches based on deep learning, such as TGSR [24] and deep reinforcement selection [25], were not included in the benchmark because they belong to a different paradigm and require hyperparameter tuning (e.g., architecture, learning rate, and number of epochs). Including them would take us further from the goal of complete automation without user parameters that our method aims for. Thus, to compare the selected methods on an equal footing, all are assigned the same number of bands k*, determined automatically by our CUSD HC method.

We chose not to evaluate TGSR with default parameters, for two reasons. First, its reference implementation offers no validated default configuration for all six of our datasets, whose band counts and spatial resolutions vary widely (102 to 274 bands); using hyperparameters that were not recalibrated for each sensor would give an unrepresentative comparison and could underestimate the method’s true performance. Second, TGSR needs GPU-based training for each dataset, which conflicts with the sub-second configuration time we target for embedded or real-time deployment (Section 5.5) and would break the homogeneous CPU-based setup used to compare every other method in Table 2.

3.5. Statistical Tests

In accordance with Demšar’s recommendations [10] for comparing multiple classifiers across several datasets, a Friedman test is first applied to reject the null hypothesis that all methods perform equally well. When the Friedman test is significant, a Nemenyi post hoc test is used to identify pairwise differences. The critical difference (CD) at α = 0.05 is calculated and reported alongside the mean ranks.

The Friedman test and the Nemenyi post hoc test are applied separately for each classifier: the six datasets serve as the comparison units (blocks) and the nine methods as the compared treatments, so classifiers are never pooled within a single test. Because the number of datasets is limited (n = 6), the statistical power of the test and the resulting critical difference (CD) stay limited too. Conclusions about significant superiority should therefore be read as indicative rather than definitive, though the consistent ranking across the four classifiers (

Table A1. Overall accuracy of the XGBoost model (OA ± 95% CI).

Method	Indian Pines	Pavia Center	Kennedy SC	HanChuan	LongKou	HongHu
CUSD-HC (ours)	0.841 ± 0.004	0.981 ± 0.001	0.886 ± 0.007	0.837 ± 0.003	0.959 ± 0.001	0.796 ± 0.004
Pure Schur	0.790 ± 0.006	0.978 ± 0.001	0.806 ± 0.009	0.810 ± 0.003	0.944 ± 0.002	0.772 ± 0.002
Marginal variance	0.675 ± 0.008	0.832 ± 0.002	0.802 ± 0.007	0.678 ± 0.002	0.824 ± 0.002	0.591 ± 0.003
PCA	0.787 ± 0.004	0.954 ± 0.002	0.808 ± 0.010	0.754 ± 0.004	0.947 ± 0.002	0.756 ± 0.004
Mutual Information	0.795 ± 0.008	0.970 ± 0.001	0.841 ± 0.007	0.840 ± 0.003	0.933 ± 0.002	0.703 ± 0.003
Spectral Clus- tering	0.827 ± 0.005	0.982 ± 0.001	0.905 ± 0.005	0.826 ± 0.004	0.944 ± 0.001	0.799 ± 0.003
E-FDPC	0.745 ± 0.008	0.981 ± 0.001	0.918 ± 0.007	0.786 ± 0.003	0.936 ± 0.002	0.785 ± 0.003
OCF	0.845 ± 0.003	0.981 ± 0.001	0.876 ± 0.007	0.838 ± 0.003	0.956 ± 0.001	0.791 ± 0.003
EGCSR	0.841 ± 0.005	0.983 ± 0.001	0.928 ± 0.007	0.840 ± 0.004	0.948 ± 0.001	0.794 ± 0.003

Note: Bold values denote the best-performing method on each dataset.

,

Table A2. XGBoost-OA rankings by dataset and average rank.

Method	Indian Pines	Pavia Center	Kennedy	HanChuan	LongKou	HongHu	Average
CUSD-HC (ours)	1	3	4	3	4	2	2.8
Spectral Clustering	4	2	3	5	6	1	3.5
OCF	1	4	5	3	2	4	3.2
Pure Schur	6	6	6	6	5	8	6.2
EGCSR	2	1	1	3	1	3	1.8
E-FDPC	7	5	2	7	8	5	5.7
Mutual Information	8	2	6	7	5	8	6.0
PCA	7	8	7	8	4	7	6.8
Marginal Variance	9	9	9	9	9	9	9.0

,

Table A3. Overall accuracy of the LightGBM model (OA ± 95% CI).

Method	Indian Pines	Pavia Center	Kennedy SC	HanChuan	LongKou	HongHu
CUSD-HC (ours)	0.855 ± 0.005	0.981 ± 0.001	0.894 ± 0.009	0.844 ± 0.003	0.963 ± 0.001	0.784 ± 0.017
Pure Schur	0.804 ± 0.006	0.978 ± 0.001	0.818 ± 0.008	0.819 ± 0.003	0.948 ± 0.002	0.767 ± 0.002
Marginal variance	0.690 ± 0.006	0.832 ± 0.003	0.818 ± 0.010	0.687 ± 0.002	0.830 ± 0.003	0.590 ± 0.003
PCA	0.801 ± 0.007	0.956 ± 0.002	0.814 ± 0.008	0.762 ± 0.004	0.950 ± 0.002	0.754 ± 0.004
Mutual Information	0.763 ± 0.085	0.970 ± 0.001	0.849 ± 0.009	0.843 ± 0.002	0.936 ± 0.002	0.697 ± 0.003
Spectral Clustering	0.835 ± 0.007	0.982 ± 0.001	0.908 ± 0.006	0.834 ± 0.003	0.948 ± 0.001	0.797 ± 0.003
E-FDPC	0.520 ± 0.179	0.981 ± 0.001	0.922 ± 0.006	0.798 ± 0.003	0.941 ± 0.002	0.781 ± 0.002
OCF	0.804 ± 0.096	0.981 ± 0.001	0.888 ± 0.008	0.844 ± 0.003	0.960 ± 0.002	0.786 ± 0.003
EGCSR	0.790 ± 0.113	0.984 ± 0.001	0.931 ± 0.005	0.849 ± 0.003	0.954 ± 0.001	0.790 ± 0.003

Note: Bold values denote the best-performing method on each dataset.

and

Table A4. LightGBM-OA rankings by dataset and average rank.

Method	Indian Pines	Pavia Center	Kennedy	HanChuan	LongKou	HongHu	Average
EGCSR	6	1	1	1	3	2	2.3
CUSD-HC (proposed)	1	3	4	3	1	4	2.7
Spectral Clustering	2	2	3	5	6	1	3.2
OCF	3	4	5	2	2	3	3.2
Pure Schur	4	6	7	6	5	6	5.7
E-FDPC	9	5	2	7	7	5	5.8
Mutual info (hist.)	7	7	6	4	8	8	6.7
PCA (max loading)	5	8	9	8	4	7	6.8
Marginal variance	8	9	8	9	9	9	8.7

in the Appendix B) supports them.

The entire proposed method is illustrated in Figure 3 below.

4. Results

4.1. Classification Performance

4.1.1. Overall Accuracy and Classifier Robustness

Table 2 gives the overall accuracy of SVM-RBF for all methods on each dataset. CUSD-HC ranks third on Indian Pines (0.8772), within 0.6 points of the best result (EGCSR, 0.883), and obtains the best OA on LongKou (0.9759), ranks second on HongHu (0.8094), and comes within 0.5 points of the best result on Pavia Center (0.982 versus EGCSR’s 0.987). On HanChuan, it ranks fifth (0.8507), behind EGCSR, Spectral Clustering, OCF, and Mutual Information, which are all clustered within 2.3 points of each other (0.852–0.875). On Kennedy Space Center, under the corrected k_max = p/3 bound (Section 5.3), SNR pre-processing keeps 172 of the 176 bands, and the automatic detector selects k* = 57; CUSD-HC then reaches OA = 0.9111, ranking fourth behind EGCSR, E-FDPC, and Spectral Clustering, within 4.4 points of the best result.

Table 3. Overall accuracy of the Random Forest model (OA ± 95% CI). The best result for each set is shown in bold.

Method	Indian Pines	Pavia Center	Kennedy SC	HanChuan	LongKou	HongHu
CUSD-HC(ours)	0.851 ± 0.007	0.981 ± 0.001	0.868 ± 0.009	0.833 ± 0.003	0.961 ± 0.002	0.797 ± 0.004
Pure Schur	0.806 ± 0.002	0.978 ± 0.001	0.794 ± 0.007	0.800 ± 0.003	0.946 ± 0.002	0.770 ± 0.002
Marginal variance	0.703 ± 0.007	0.828 ± 0.003	0.792 ± 0.007	0.695 ± 0.002	0.843 ± 0.002	0.579 ± 0.004
PCA	0.807 ± 0.005	0.956 ± 0.002	0.793 ± 0.008	0.755 ± 0.003	0.946 ± 0.001	0.754 ± 0.004
Mutual Information	0.798 ± 0.010	0.968 ± 0.001	0.831 ± 0.006	0.837 ± 0.003	0.935 ± 0.001	0.698 ± 0.002
Spectral Clustering	0.837 ± 0.005	0.982 ± 0.001	0.886 ± 0.007	0.838 ± 0.004	0.947 ± 0.002	0.798 ± 0.004
E-FDPC	0.755 ± 0.007	0.981 ± 0.001	0.913 ± 0.007	0.783 ± 0.004	0.938 ± 0.002	0.786 ± 0.003
OCF	0.849 ± 0.007	0.982 ± 0.001	0.866 ± 0.008	0.837 ± 0.003	0.959 ± 0.002	0.791 ± 0.003
EGCSR	0.855 ± 0.006	0.983 ± 0.001	0.918 ± 0.006	0.848 ± 0.004	0.949 ± 0.002	0.792 ± 0.003

presents the results obtained using the Random Forest (RF) classifier. The trends differ from those of SVM-RBF: CUSD-HC ranks first on a single dataset, second on two datasets, and lower on the remaining three, for an average rank of 3.2 (versus 3.3 for SVM-RBF).

To assess the robustness of our results with respect to the classifier, we conducted further experiments using two gradient boosting methods: XGBoost (100 trees, maximum depth 6, learning rate 0.1) and LightGBM (same hyperparameters, 31 leaves). The evaluation protocol is the same: 10-fold stratified cross-validation with the same number of k* bands determined by CUSD-HC (Table A1 and Table A3 in Appendix B). The average ranks obtained across the six datasets confirm the conclusions of Table 2 and Table 3. EGCSR keeps the best average rank with these two classifiers (1.8 for XGBoost, 2.3 for LightGBM; Table A2 and Table A4 in the Appendix B), followed by CUSD-HC (2.8 and 2.7, respectively). Spectral Clustering and OCF come next, with no statistically significant difference among these top-ranked methods. The Friedman test remains significant in all cases (p < 10⁻⁶). Consequently, the stability of the ranking across four classifiers from distinct families, including kernel SVM, Random Forests, and gradient boosting (XGBoost, LightGBM), indicates that the quality of the selection made by our CUSD-HC method is intrinsic to the subset of bands selected and does not depend on the choice of classifier.

4.1.2. Visual Analysis of Classification Maps

Figure 4 shows the classification maps produced for Indian Pines using RBF-SVM based on 16 selected bands. The map generated by CUSD-HC (b) is visually the closest to the ground truth (a), with an OA of 91.37%. In particular, there is greater homogeneity in the soya fields (light green, center of the image) and a clearer delineation of the maize classes (north-west) compared to EGCSR (c, AO = 89.19%), which has more misclassified pixels in the transition zones between spectrally similar classes. The Spectral Clustering map (d, OA = 90.14%) is very similar to that of CUSD-HC, which visually confirms the statistical equivalence observed in the Nemenyi test (p = 0.999). The slight difference in OA (+1.23 points) in favor of CUSD-HC illustrates the contribution of the Schur complement in selecting the most informative band within each cluster, rather than the simple medoid.

The OA shown in Figure 4 is obtained by training on all labeled pixels (for visualization purposes), whereas Table 2 reports the OA from 10-fold cross-validation.

Appendix C gives a supplementary figure for each of the six datasets, generated strictly from a single held-out test fold (70% test/30% train, rather than trained on all labeled pixels as above) to illustrate the model’s generalization under a harder, more realistic scenario. This single-fold evaluation yields OA = 0.8492 for Indian Pines (versus 0.8772 under 10-fold cross-validation), 0.9821 for Pavia Center, 0.8920 for Kennedy Space Center, 0.8612 for HanChuan, 0.9768 for LongKou, and 0.8117 for HongHu. The maps confirm the salt-and-pepper degradation expected from a purely spectral classifier trained on less data, while the overall class layout still holds on all six datasets.

4.2. Rankings and Statistical Tests

Table 4. SVM-OA ranks by dataset and average rank. CD = 1.551 at α = 0.05.

Method	Indian Pines	Pavia Center	Kennedy	HanChuan	LongKou	HongHu	Average
EGCSR	1	1	1	1	3	3	1.7
Spectral Clustering	4	3	3	2	4	1	2.8
OCF	2	2	5	3	2	4	3.0
CUSD-HC (ours)	3	5	4	5	1	2	3.3
E-FDPC	8	4	2	7	7	5	5.5
Pure Schur	5	6	7	6	6	6	6.0
Mutual info (hist.)	7	7	6	4	8	8	6.7
PCA (max loading)	6	8	8	8	5	7	7.0
Marginal variance	9	9	9	9	9	9	9.0

shows the rankings by dataset and the average rankings across the six benchmarks. EGCSR obtains the best average ranking (1.7), followed by Spectral Clustering (2.8) and OCF (3.0). CUSD-HC ranks fourth (3.3), with no statistically significant difference among the top-ranked methods (

Table 5. Results of the Nemenyi post hoc test (CUSD-HC vs. each method).

Method	Average Rank	p-Value vs. CUSD-HC	Significant (α = 0.05)
EGCSR	1.7	0.980391	No
Spectral Clustering	2.8	0.999997	No
OCF	3.0	1.000000	No
CUSD-HC (ours)	3.3	—	—
E-FDPC	5.5	0.909088	No
Pure Schur	6.0	0.722123	No
Mutual info (hist.)	6.7	0.467552	No
PCA (max loading)	7.0	0.362798	No
Marginal variance	9.0	0.010234	Yes (p < 0.05)

).

Nemenyi’s post hoc test (Table 5) shows that CUSD-HC is statistically equivalent to seven of the eight comparison methods (p > 0.05), including EGCSR, a 2020 State-of-the-Art method (p = 0.980391), Spectral Clustering and OCF. It significantly outperforms only the weakest baseline, Marginal Variance (p = 0.010234). Combined with the fully automatic determination of the number of bands, this lack of separation from the top-ranked methods positions CUSD-HC as competitive rather than strictly superior to the State-of-the-Art. The similarity with Spectral Clustering is consistent, as CUSD-HC incorporates hierarchical clustering as a first step.

5. Discussion

5.1. Complementarity of Clustering and Schur

A comparison of CUSD-HC with its two components, Pure Schur (without clustering) and Spectral Clustering (without Schur), sheds light on the source of its advantage. Pure Schur greedily maximizes conditional variance but tends to concentrate selections in the most variable spectral region, neglecting less variable but discriminating portions of the spectrum. On HanChuan, for instance, Pure Schur reaches only 0.820 against 0.851 for CUSD-HC. Spectral Clustering imposes diversity by design but selects the medoid of each group, which is the most central band rather than the most informative one. CUSD-HC combines the strengths of both, as the clustering constraint ensures broad spectral coverage, whilst the Schur criterion selects the most informative band in each region. The interleaving process, which involves selecting the best overall group at each step and updating the residual covariance, further prevents inter-group redundancy.

Figure 5 shows that CUSD-HC distributes the 16 bands selected for Indian Pines across the entire spectrum (visible, near-infrared, SWIR), thanks to the constraint of one band per Ward cluster. Conversely, Marginal Variance and PCA concentrate their selections within a narrow window (indices 20–45), ignoring potentially discriminating regions. This broad and uniform spectral coverage of CUSD-HC contributes to its superior classification performance.

It is observed that CUSD-HC selects a band in the Red Edge region (indices around 30, corresponding to 700–750 nm) over Indian Pines, a region of physiological transition linked to chlorophyll vigor. This choice, made in an unsupervised manner, confirms that Schur’s conditional variance criterion naturally favors spectral bands with high information content for crop discrimination.

The correlation matrix (Figure 6) for the Indian Pines dataset highlights clusters of high correlation that correspond to the spectral groups identified by Ward’s clustering. Areas of low correlation (coefficients around 30, 75–85 and 90) delineate the natural boundaries between spectral regions (visible, water absorption, near-infrared and SWIR).

5.2. The Automatic k Mechanism

The automatic k mechanism is a distinctive feature compared to EGCSR and other recent methods that require k as an input. Across the six datasets, the determined values of k* range from 6 (Pavia Center and HongHu, reflecting strong intrinsic redundancy) to 63 (HanChuan, whose broader spectral diversity resists aggressive compression); Indian Pines settles at k* = 16, matching its number of land-cover classes. The 99% PCA floor was applied to two datasets: Indian Pines (k = 7, floor = 16) and HongHu (k = 5, floor = 6). For the other four datasets, the per-elbow estimate already exceeded the PCA floor; therefore, the mechanism was inactive. This confirms that the floor acts as a safety net rather than a dominant driver.

5.3. The Kennedy Space Center Case

Kennedy Space Center is the dataset most sensitive to the SNR pre-processing step. The AVIRIS sensor produces 176 bands spanning 400 to 2500 nm, including water vapor absorption features around 1400 and 1900 nm, and shows unusually high inter-band correlation. The SNR thresholding mechanism restricts the search for the largest gap to positions that retain at least 50% of the bands (Section 3.2.1), which avoids over-pruning on this type of sensor: pre-processing keeps 172 of the 176 bands, and the automatic detector selects k* = 57. CUSD-HC then reaches OA = 0.9111, ranking fourth behind EGCSR, E-FDPC, and Spectral Clustering, within 4.4 points of the best result. On the other five datasets, where spectral diversity is richer, CUSD-HC makes full use of it and stays competitive with the top-ranked methods. Its average rank across all six datasets is 3.3, with no statistically significant difference from the leading methods (Table 5). This confirms the robustness of the automatic k* determination mechanism: the 50% safeguard keeps the search well-behaved even on sensors with atypical SNR distributions, so Kennedy Space Center performs consistently with the other benchmark scenes.

5.4. Estimated Cost

The total runtime of CUSD-HC includes the cost of searching over k = 2 to k_max, each involving dendrogram pruning and a Schur selection step. On the largest datasets (HanChuan, HongHu; p ≈ 270), this takes approximately 25 s on a standard Colab CPU runtime, with a cost of O(p² × k_max) for the automatic k loop, O(p²) for each Schur update, and O(p² log p) for Ward clustering. The other methods, which take k as a parameter, run in under a second. However, this comparison is asymmetric: the additional cost of CUSD-HC corresponds to the cost of automatically determining k, which would otherwise require a dedicated cross-validation loop for k. When evaluated with a fixed k, the selection step alone runs in under 0.5 s, comparable to Spectral Clustering and EGCSR. Furthermore, the structure of CUSD-HC is natively parallelizable, as each evaluation of V(k) relies on an independent slice of the precomputed dendrogram, followed by a Schur selection pass, with no dependencies among the k values being tested. Distributing this loop across multiple CPU cores would reduce the automatic search time in a quasi-linear manner, bringing the overhead of determining k down to a few seconds even on the largest datasets.

5.5. Practical Implications

The ability to automatically determine the number of bands has direct implications for the design of lightweight multispectral sensors in precision agriculture. When deploying a camera for crop health monitoring, for example, the number of spectral channels must be fixed at the time of manufacture. CUSD-HC can be applied to a representative hyperspectral dataset acquired over the target crop and region to identify both the optimal number and the identity of the channels, without expert intervention or extensive parameter scanning. For embedded applications or real-time deployments, a lightweight variant could replace the entire k_PCA detection mechanism with a 99% threshold, reducing configuration time from 25 s to less than one second, at the cost of a k determination that is less finely tuned to the data’s spectral complementarity structure. Finally, in multi-temporal monitoring scenarios where the same bands must be used across seasons, the stability of the selection is a major operational challenge. The following section assesses this stability through an inter-scene transferability experiment.

5.6. Transferability Between Scenes

As the three WHU-Hi scenes were acquired with the same Headwall Nano-Hyperspec sensor, we conducted a cross-validation experiment to assess the transferability of the selected bands. The optimal bands determined by CUSD-HC on a source scene were directly applied to the other two target scenes, without repeating the selection process.

Table 6. Cross-scene transferability of WHU-Hi (OA SVM-RBF).

Target Scene	Source of Bands	K Transferred	Native OA (±95% CI)	OA Transferred (±CI95)
HanChuan	LongKou	8	0.8507 ± 0.003	0.8153 ± 0.003
HanChuan	HongHu	6	0.8507 ± 0.003	0.8115 ± 0.004
LongKou	HanChuan	63	0.9759 ± 0.002	0.9864 ± 0.001
LongKou	HongHu	6	0.9759 ± 0.002	0.9663 ± 0.001
HongHu	HanChuan	63	0.8094 ± 0.003	0.8505 ± 0.002
HongHu	LongKou	8	0.8094 ± 0.003	0.8215 ± 0.003

presents the results.

The maximum degradation observed is 3.9 points, for the transfer from the HongHu to the HanChuan scene, which stays modest. This is because the bands CUSD-HC selects reflect spectral properties intrinsic to the sensor rather than traits specific to a given scene. In practice, a single preliminary hyperspectral flight over a representative scene is enough to determine the optimal channels for a multispectral sensor, with no need to recalculate them for each new acquisition.

In three cases out of six, the transferred bands slightly outperform the native selection. One explanation is reduced overfitting to the target scene’s local quirks: a native selection optimizes conditional variance on correlations specific to that scene, which can include local noise or artifacts such as shadows or uneven illumination. Bands learned on another scene may instead capture spectral regularities intrinsic to the sensor, and so prove more robust to scene changes. This remains a hypothesis, though, and checking it will require a closer look at the inter-scene variance of the bands involved—something we leave for future work.

5.7. Sensitivity to Internal Parameters and Classifier Robustness

5.7.1. Sensitivity to Internal Parameters

Although CUSD-HC does not require any user-defined hyperparameters, the method relies on selecting several internal design parameters whose influence should be analyzed. These are:

• The number of neighbors used for SNR estimation (k = 3) is a parameter that controls the local regression window for noise estimation. We tested k_neighbors ∈ {2, 3, 5, 10} on all six datasets. Five of them (Pavia Center, HanChuan, LongKou, HongHu, and Kennedy Space Center for three of the four settings tested) give identical or near-identical p, k*, and OA regardless of k_neighbors. Indian Pines shows the largest variation (OA between 0.8734 and 0.8772, a spread of 0.004), and Kennedy Space Center a comparable one (OA between 0.9111 and 0.9298). This small impact is explained by the fact that genuinely noisy bands have a significantly lower SNR, regardless of neighborhood size. The choice of k = 3, in line with the convention of Roger and Arnold [22], offers the best compromise between estimator stability and spectral locality, and is not an arbitrary or fragile setting.
• We tested k_max values of 20, 30, 40 and 60 across the four PCA thresholds on all six datasets. On Kennedy Space Center, k* saturated exactly at k_max for k_max = 20, 30 and 40—the selected k* matched the imposed ceiling every time—and only stabilized once k_max reached 54–57. This showed that the fixed ceiling of 40 used previously was silently truncating the automatically determined k on this dataset. Pushing the test to k_max = 90 revealed a similar but smaller effect on two of the remaining five datasets, HanChuan and LongKou, whose natural k* also grows once the fixed ceiling is lifted; the other three (Indian Pines, Pavia Center, HongHu) are unaffected, since their automatically selected k* already fell below both the old ceiling and the current bound. This is why we removed the fixed constant altogether: the upper bound is now simply k_max = p/3, justified only by the data-derived requirement of at least three bands per cluster on average, with no extra tuned constant. We propagated this corrected bound through the full results pipeline for the affected datasets—Kennedy Space Center, HanChuan and LongKou—and Table 2 reflects the corrected values.
• The PCA threshold at 99% explained variance acts as a safety net against abnormally low inflection point values. Four thresholds (95%, 97%, 98%, 99%) were tested across all six datasets, jointly with k_max in {20, 30, 40, 60}. At k_max = 60, the resulting automatically selected k* is completely insensitive to the PCA threshold on Pavia Center (k* = 6), HanChuan (k* = 42) and LongKou (k* = 5), it shows a small effect on HongHu (k* = 5 up to 98%, 6 at 99%) and Kennedy Space Center (k* = 54 at 95%, 57 from 97% upward), and a more graded effect on Indian Pines (k* = 7 at 95–97%, 9 at 98%, 16 at 99%). In every case, raising the threshold from 95% to 99% never lowers k*, and 99% consistently gives the highest or equal value, supporting its adoption as the conservative default. Because this sweep uses a k_max grid common to all six datasets and capped at 60 for comparability, the resulting k* at k_max = 60 for HanChuan and LongKou (42 and 5) is lower than the k* = 63 and k* = 8 reported in Table 1, which use the dataset-specific production value k_max = ⌊p/3⌋ (90 and 89, respectively); for the other four datasets, whose production k_max already falls within the tested grid, the sensitivity results coincide with Table 1. At 95%, the lower limit (k_PCA = 2 at Kennedy Space Center) consistently remains below the inflection point and therefore never fulfills its corrective role. At 99%, it raises k from 7 to 16 at Indian Pines, which corresponds to the number of land-use classes. The 99% threshold is therefore adopted as the default value, as it is conservative enough not to intervene unnecessarily (4 out of 6 datasets) whilst correcting pathological cases.
• The automatic SNR threshold used to remove noisy bands (Section 3.2.1) restricts its search for the largest log-SNR gap to positions that keep at least 50% of the bands, which prevents pathological over-pruning on datasets with an atypical SNR distribution (see Section 5.3 for the Kennedy Space Center case). An ablation study across all six datasets—comparing no filtering, this safeguarded automatic threshold, and several manual percentile thresholds—confirms that the safeguard helps or stays neutral in five of the six cases: the effect is within 0.001–0.002 OA of the unfiltered baseline on Pavia Center, HongHu and HanChuan, +0.33 OA on Indian Pines, and it is the best-performing option on LongKou (OA = 0.9759 versus 0.9745 unfiltered). On Kennedy Space Center, the residual gap is now about two points, down from the 10-point degradation observed before the safeguard was introduced.

In summary, the four internal parameters are not very sensitive within the tested reasonable ranges, confirming the method’s practical robustness without manual adjustment.

5.7.2. Robustness Depending on the Classifier Chosen

Table 7. Average ranks by classifier across the six datasets.

Classifier	CUSD-HC (ours)	Spectral Clust.	OCF	EGCSR
SVM-RBF	3.3	2.8	3.0	1.7
Random Forest	3.2	2.7	3.3	1.7
XGBoost	2.8	3.5	3.2	1.8
LightGBM	2.7	3.2	3.2	2.3

summarizes the average ranks of CUSD-HC, Spectral Clustering, OCF and EGCSR—the four best-performing methods overall—across the four classifiers evaluated. CUSD-HC’s average rank ranges from 2.7 to 3.3, so its competitive position holds across classifiers rather than depending on just one. EGCSR obtains the best average rank on all four classifiers, but its lead over CUSD-HC narrows for XGBoost and LightGBM compared to SVM-RBF and Random Forest. There is a slight similarity between the representative bands of the CUSD-HC clusters and tree-based classifiers.

6. Conclusions

This article presented a method for selecting hyperspectral bands that combines Ward’s hierarchical clustering with conditional variance maximization via Schur complement (CUSD-HC). This method automatically determines the number of bands by combining kink detection with a PCA-based dimensionality floor, eliminating the need for user-specified hyperparameters. Experiments on six benchmark datasets demonstrate that CUSD-HC consistently ranks among the leading methods (average rank between 2.7 and 3.3 among nine competitors), with no statistically significant difference from top-ranked alternatives such as EGCSR or Spectral Clustering in the Nemenyi post hoc test, while significantly outperforming the weakest baselines. Unlike these top-ranked competitors, CUSD-HC reaches this level of performance fully automatically, without requiring the number of bands to be tuned by the user. The approach is simple to implement, interpretable, and practical for deployment in sensor design and operational remote sensing.

Several avenues for future work emerge from this study. Firstly, a semi-supervised extension could incorporate class information available during the clustering phase by weighting the correlation distance by inter-class separability, thereby guiding the partitioning towards spectrally and thematically coherent groups. Similarly, prior spatial segmentation with superpixels would enable locally adaptive band selection in scenes with highly heterogeneous land cover. Secondly, the transferability of the selected bands warrants evaluation using scenes acquired over the same area on different dates to assess the temporal stability of the selection for multi-temporal monitoring applications in vegetation or land cover. Thirdly, integrating a deep encoder to replace or complement the Schur complement could capture non-linear dependencies between bands, at the cost of increased computational complexity. Finally, the method could be adapted for the design of multispectral sensors mounted on drones, where the automatic selection of k optimal channels from a preliminary hyperspectral flight would reduce hardware costs whilst ensuring spectral coverage suited to the target crop and region.

This work also has some limitations. The equity-of-representation comparison across methods covers Indian Pines only, leaving the other five datasets for future work. The extended-metrics comparison (average accuracy, Cohen’s Kappa, F1-macro) covers CUSD-HC alone; extending it to the eight baseline methods remains an open question. LightGBM showed unusually high per-fold variance on Indian Pines and Pavia Center in supplementary diagnostic runs, an instability we plan to look into further. We have not measured how spatial heterogeneity affects the stability of the Schur complement—the superpixel-based segmentation already proposed in the Conclusion would be a natural way to address this.