Average Band Correlation in Hyperspectral Analysis

• ABC is a statistic that computes the average absolute Pearson correlation of each spectral band with every other, serving as a clear redundancy indicator.
• It is applied in hyperspectral band selection by favoring bands with low ABC scores to reduce multicollinearity and computational overhead without sacrificing classification quality.
• ABC can be combined with measures like VIF and MI in clustering frameworks to boost the performance of hyperspectral image classification.

Average Band Correlation (ABC) is a correlation-based statistic for hyperspectral band analysis that quantifies how strongly a spectral band co-varies, in magnitude, with the rest of the spectrum. In hyperspectral band selection, ABC is used primarily as a per-band redundancy measure: bands with smaller average absolute Pearson correlation to other bands are treated as less redundant and therefore more desirable for retention in compact subsets (Deb et al., 24 Jan 2025). A related but distinct usage appears in correlation-matrix theory, where ABC denotes the arithmetic mean of off-diagonal correlations of a correlation matrix and serves as a global descriptor linked to eigenvalue structure and eigenvector alignment (Stepanov et al., 2021). Subsequent hyperspectral work integrates ABC with Variance Inflation Factor (VIF) and Mutual Information (MI) in a multicollinearity-aware, parameter-free selection pipeline (Deb et al., 26 Sep 2025).

1. Definition and terminological scope

In the band-selection formulation, ABC is defined per band. For a hyperspectral cube $X \in \mathbb{R}^{W \times H \times N}$, let $N$ be the number of spectral bands after pre-processing, and let $n = W \cdot H'$ be the number of pixels used after excluding any removed background pixels. For each pair of bands $i$ and $j$, a Pearson correlation coefficient $r_{ij}$ is computed across pixels. The ABC score of band $i$ is then

$$\text{ABC}_i = \frac{1}{N-1}\sum_{j=1,\, j\ne i}^{N} |r_{ij}|.$$

This produces $N$ scalar scores, $\text{ABC}_1,\dots,\text{ABC}_N$, one per band. In "Correlation-Based Band Selection for Hyperspectral Image Classification" (Deb et al., 24 Jan 2025), selection decisions are made solely from these per-band values.

A terminological divergence arises in "Generic Features in the Spectral Decomposition of Correlation Matrices" (Stepanov et al., 2021). There, for a correlation matrix $N$0, the quantity

$N$1

is identified as ABC in spectral-band applications. This is a single global average over off-diagonal correlations, not a per-band score. The distinction is substantive rather than cosmetic: the hyperspectral selector in (Deb et al., 24 Jan 2025) uses $N$2 for bandwise retention, whereas the spectral-decomposition analysis in (Stepanov et al., 2021) studies the matrix-level mean correlation $N$3 and its dispersion $N$4.

Usage	Definition	Role
Per-band ABC	$N$5	Bandwise redundancy score
Global ABC	$N$6	Matrix-level mean correlation

A common misconception is that ABC is necessarily a single scalar for an entire dataset. The literature represented here does not support a unique convention. Instead, it shows two related usages: a per-band statistic for selection (Deb et al., 24 Jan 2025, Deb et al., 26 Sep 2025) and a global off-diagonal mean for spectral analysis (Stepanov et al., 2021).

2. Data model, correlation estimation, and preprocessing

The hyperspectral formulations compute ABC from inter-band correlations estimated across pixels, not across classes. In (Deb et al., 24 Jan 2025), each band is treated as a random variable observed over $N$7 pixels after pre-processing; no class labels are used in correlation or ABC computation, so the method is fully unsupervised. In (Deb et al., 26 Sep 2025), after masking out background pixels, the retained data are arranged as $N$8 with columns $N$9, and Pearson correlations are computed between band columns across the $n = W \cdot H'$0 retained pixels.

For two bands $n = W \cdot H'$1 and $n = W \cdot H'$2 represented as vectors of length $n = W \cdot H'$3, the Pearson correlation used in (Deb et al., 24 Jan 2025) is

$n = W \cdot H'$4

ABC uses $n = W \cdot H'$5, so strong positive and strong negative linear relationships are treated as equally redundant (Deb et al., 24 Jan 2025, Deb et al., 26 Sep 2025).

The pre-processing conventions are also explicit. In (Deb et al., 24 Jan 2025), Salinas originally contains 224 bands, of which 20 were removed due to water absorption and low SNR; Pavia University retained 103 bands after removing broken bands. Background pixels were excluded from both correlation and classification analyses. The paper states that “all pixels were standardized” prior to correlation; this typically means per-band $n = W \cdot H'$6-score standardization. In (Deb et al., 26 Sep 2025), background masking and bandwise standardization are likewise performed before CC and ABC computation.

The computational profile is quadratic in the number of bands. In (Deb et al., 24 Jan 2025), naive computation of the full correlation matrix $n = W \cdot H'$7 requires $n = W \cdot H'$8 time and $n = W \cdot H'$9 memory, with an additional $i$0 cost to average the off-diagonal absolute correlations into $i$1. Practical implementations can vectorize the computation or exploit matrix symmetry to halve operations. The multicollinearity-aware pipeline in (Deb et al., 26 Sep 2025) has the same dominant correlation cost, stated as $i$2 time and $i$3 memory in that paper’s notation.

3. ABC as a band-selection mechanism

The simplest ABC-based selector uses a fixed threshold on the per-band scores. In (Deb et al., 24 Jan 2025), the algorithm begins by removing low-SNR, water-absorption, or broken bands as appropriate to the dataset, removing background pixels, standardizing the remaining data per band, and flattening the image into a matrix $i$4. It then computes the Pearson correlation matrix $i$5, evaluates $i$6 for each band, and retains all bands satisfying $i$7 with a fixed threshold $i$8. No tie-breaking or ordering is required, though the paper notes that bands may be sorted in ascending ABC for reporting (Deb et al., 24 Jan 2025).

This thresholded procedure is explicitly unsupervised at the selection stage. Labels enter only in the downstream evaluation phase, where an SVM classifier is trained on the selected bands using a 70/30 train/test pixel split, and class-wise precision, recall, F1, overall accuracy (OA), and Kappa are reported (Deb et al., 24 Jan 2025).

A more elaborate use of ABC appears in the dependence-measures pipeline of (Deb et al., 26 Sep 2025). There, ABC remains the per-band average absolute Pearson correlation,

$i$9

but it is not itself thresholded for final selection. Instead, pairwise VIF is computed via

$j$0

and a candidate pre-selection set $j$1 is formed from band pairs satisfying $j$2, where

$j$3

With $j$4, the method becomes threshold-free in the sense emphasized by the paper, with $j$5 (Deb et al., 26 Sep 2025). MI with the ground-truth labels is then computed for bands in $j$6, a two-dimensional $j$7 feature space is constructed, and k-means clustering is applied. The selected subset consists of the bands nearest to the cluster centroids. In this formulation, ABC supplies a global linear-redundancy coordinate, MI supplies class relevance, and VIF reduces multicollinearity before clustering.

These two uses clarify that ABC is not tied to a single selection protocol. It can function as a direct thresholding statistic (Deb et al., 24 Jan 2025) or as one coordinate in a multi-criterion selection space (Deb et al., 26 Sep 2025).

4. Redundancy, multicollinearity, and spectral interpretation

The operational rationale for ABC in hyperspectral band selection is linear redundancy reduction. If band $j$8 is strongly correlated in magnitude with many other bands, then $j$9 is large and the band is regarded as offering limited unique information. Conversely, low $r_{ij}$0 indicates that the band is less linearly dependent on the rest of the spectrum and is therefore more likely to contribute diverse information. In (Deb et al., 24 Jan 2025), this is linked directly to reducing spectral redundancy, reducing computational load, and mitigating the Hughes phenomenon.

This rationale is sharpened by the global correlation-matrix analysis in (Stepanov et al., 2021). For a correlation matrix $r_{ij}$1, with off-diagonal mean correlation $r_{ij}$2 and dispersion $r_{ij}$3, the largest eigenvalue satisfies

$r_{ij}$4

and, more generally,

$r_{ij}$5

When $r_{ij}$6 is large, the first principal component is necessarily dominant. When $r_{ij}$7 is small and $r_{ij}$8, the leading eigenvector $r_{ij}$9 is forced to align closely with the diagonal direction $i$0 through the lower bound

$i$1

up to the paper’s stated minimum with the alternative term. Interpreted for spectral bands, nearly uniform loadings mean that the first principal component behaves as a common-band mode (Stepanov et al., 2021).

This suggests a useful conceptual bridge. A bandwise ABC selector tries to keep bands with small local average dependence, whereas the global mean-correlation analysis explains what happens when the correlation matrix as a whole has a large off-diagonal mean: a distinctly large first eigenvalue and, under modest dispersion, nearly uniform band loadings. Put differently, low per-band ABC seeks local diversity, while high global ABC signals a strong common mode.

The use of absolute correlation is also deliberate. Both (Deb et al., 24 Jan 2025) and (Deb et al., 26 Sep 2025) treat strong positive and strong negative correlations as equally redundant. This is defensible for multicollinearity control, but it also means that sign-based complementarity is not preserved by the statistic itself.

5. Experimental behavior in hyperspectral classification

The thresholded ABC selector in (Deb et al., 24 Jan 2025) was evaluated on two standard benchmark datasets. Pavia University (PA), captured by the ROSIS sensor, has spatial size $i$2, 103 bands after removing broken bands, 9 classes, and wavelength range approximately $i$3–$i$4. Salinas Valley (SA), captured by AVIRIS, has spatial size $i$5, 204 used bands after removing 20 bands for water absorption and low SNR from the original 224, 16 classes, and wavelength range approximately $i$6–$i$7 (Deb et al., 24 Jan 2025).

With $i$8, ABC-based selection retained 54 bands for PA and 88 bands for SA. On SA, the reported OA values were 89.08 for PCA, 90.41 for the Similarity-Based method, and 89.33 for the ABC method; the corresponding Kappa values were 0.88, 0.89, and 0.88. On PA, the OA values were 81.45 for PCA, 91.00 for the Similarity-Based method, and 89.56 for ABC, with Kappa values 0.74, 0.88, and 0.86. Visual classification maps were reported to be more consistent with ground truth for ABC and the Similarity-Based method than for PCA, particularly on PA (Deb et al., 24 Jan 2025). The paper’s conclusion is accordingly limited and comparative: ABC-based selection is competitive, improves over PCA on PA, and attains results close to the Similarity-Based baseline on both datasets.

The later ABC-MI-VIF method in (Deb et al., 26 Sep 2025) extends the empirical scope to WHU-Hi-LongKou, Pavia University, Salinas, and Oil Spill. Using SVM with RBF, with $i$9 and gamma set by grid search, 10% training samples per class, and 10 runs averaged, the paper reports that SA with 20 or 21 selected bands yields OA around 92.2–92.7% for ABC-MI variants, compared with 93.18% using all bands. On PA, 14 or 15 selected bands yield OA around 90.7–93.0%, compared with 94.41% using all bands. The ablation studies are especially informative. For PA with 20 bands selected, ABC-MI(0.00) achieved OA=93.42, ABC-only (no VIF) achieved OA=91.94, MI-only (no VIF) achieved OA=92.35, and ABC-MI without VIF achieved OA=92.29. For SA, ABC-MI(0.05) achieved OA=92.74, ABC-only (no VIF) achieved OA=92.82, MI-only (no VIF) achieved OA=90.65, and ABC-MI without VIF achieved OA=91.24 (Deb et al., 26 Sep 2025).

The same study reports substantial VIF-driven pruning before clustering: PA was reduced from 103 to 34 bands for $$\text{ABC}_i = \frac{1}{N-1}\sum_{j=1,\, j\ne i}^{N} |r_{ij}|.$$0, SA from 204 to 72, OS from 190 to 58, and LongKou from 270 to 50 for $$\text{ABC}_i = \frac{1}{N-1}\sum_{j=1,\, j\ne i}^{N} |r_{ij}|.$$1 (Deb et al., 26 Sep 2025). This supports the paper’s claim that multicollinearity reduction can make the ABC-MI search space more manageable while preserving competitive classification performance.

6. Limitations, edge cases, and extensions

Several limitations recur across the literature. First, ABC is a linear dependence measure because it is built from Pearson correlation. Both (Deb et al., 24 Jan 2025) and (Deb et al., 26 Sep 2025) note that nonlinear redundancy may therefore remain unaddressed. Second, because ABC averages absolute correlations, negatively correlated bands are treated as redundant in exactly the same way as positively correlated bands. This is reasonable for multicollinearity control, but it may discard bands whose signed contrast is useful for classification (Deb et al., 24 Jan 2025, Deb et al., 26 Sep 2025).

Third, low redundancy does not guarantee discriminative value. (Deb et al., 24 Jan 2025) states explicitly that strongly correlated bands may still carry class-discriminative signals, and that removing too many such bands can reduce class separability, especially for subtle materials whose signatures occupy adjacent wavelengths. The empirical role of MI in (Deb et al., 26 Sep 2025) addresses this limitation directly: ABC alone measures redundancy, while MI contributes class relevance.

Fourth, thresholding remains problematic in direct ABC selection. In (Deb et al., 24 Jan 2025), $$\text{ABC}_i = \frac{1}{N-1}\sum_{j=1,\, j\ne i}^{N} |r_{ij}|.$$2 is fixed experimentally; it is neither learned nor cross-validated, no sweep over $$\text{ABC}_i = \frac{1}{N-1}\sum_{j=1,\, j\ne i}^{N} |r_{ij}|.$$3 is reported, and the paper identifies elimination of the threshold as future work. By contrast, (Deb et al., 26 Sep 2025) shifts the parameter burden to the user-chosen number of clusters $$\text{ABC}_i = \frac{1}{N-1}\sum_{j=1,\, j\ne i}^{N} |r_{ij}|.$$4 and an optional VIF tolerance $$\text{ABC}_i = \frac{1}{N-1}\sum_{j=1,\, j\ne i}^{N} |r_{ij}|.$$5, while claiming a parameter-free variant when $$\text{ABC}_i = \frac{1}{N-1}\sum_{j=1,\, j\ne i}^{N} |r_{ij}|.$$6. The papers therefore reflect different senses of “parameter-free”: threshold removal in one case, optional tolerance elimination in another.

Fifth, robustness and estimation issues remain open. (Deb et al., 24 Jan 2025) reports no explicit robustness tests or statistical significance tests. It also notes that correlations computed over all pixels assume stationarity, so spatial heterogeneity or class imbalance can bias estimates; stratified or region-wise ABC is suggested as a possible mitigation. (Deb et al., 26 Sep 2025) similarly remarks that correlation estimation can be sensitive to noise and that practical performance depends on MI estimation details, which are not specified beyond theoretical formulas.

The extensions proposed in the source material follow directly from these limitations. For redundancy modeling, (Deb et al., 24 Jan 2025) suggests Spearman’s $$\text{ABC}_i = \frac{1}{N-1}\sum_{j=1,\, j\ne i}^{N} |r_{ij}|.$$7, Kendall’s $$\text{ABC}_i = \frac{1}{N-1}\sum_{j=1,\, j\ne i}^{N} |r_{ij}|.$$8, partial correlation, MI, and mRMR-style criteria. For threshold selection, it mentions cross-validation on OA/Kappa, elbow or percentile selection on $$\text{ABC}_i = \frac{1}{N-1}\sum_{j=1,\, j\ne i}^{N} |r_{ij}|.$$9, stability selection, and optimization-based correlation minimization. For computational scaling, it suggests random pixel subsampling, streaming or blocked covariance computation, and low-rank approximations. These proposals indicate that ABC is best viewed not as a complete theory of band utility, but as a simple and interpretable linear-redundancy primitive that can be thresholded directly, embedded in broader dependence-measure pipelines, or related to correlation-spectrum structure through the mean-correlation framework of (Stepanov et al., 2021).