Compact Spectral Indices for Vegetation Classification

• The paper demonstrates that compact spectral indices, derived via neural networks, polynomial search, and genetic programming, maximize class separability and maintain invariance to illumination.
• It uses controlled search spaces and feature selection to combine 2–4 spectral bands, reducing data dimensionality while preserving near-full multispectral performance.
• Empirical validations show these indices mimic key spectroscopic features, simplify computation, and generalize effectively across sensors and remote sensing applications.

Compact spectral indices for vegetation classification are algebraic combinations of spectral bands, primarily constructed to maximize class separability of vegetation types or states, while maintaining invariance to illumination and simplifying downstream inference. These indices are typically constrained to use only 2–4 input bands (often in ratio or normalized difference form) and are discovered or optimized using neural networks, explainable AI (XAI), polynomial expansions, or soft computing approaches such as genetic programming. Compact indices provide high discriminative power and computational efficiency for remote-sensing tasks including vegetation type discrimination, crop mapping, and materials identification.

1. Principles of Compact Spectral Index Construction

Compact spectral indices are formed by combining reflectance values from multispectral or hyperspectral sensors using algebraic formulas designed for invariance to multiplicative effects from illumination or sensor gain. The fundamental building block is the normalized difference:

$$\mathrm{ND}_{i,j} = \frac{b_i - b_j}{b_i + b_j + \varepsilon}$$

where $b_i$, $b_j$ are reflectance in bands $i$ and $j$, and $\varepsilon$ is a small constant to avoid singularity (Lotfi et al., 26 Dec 2025). This structure ensures

$$(b_i, b_j) \mapsto (\alpha b_i, \alpha b_j) \implies \mathrm{ND}_{i,j} \text{ is unchanged}$$

indicating robust illumination invariance.

Higher-order compact indices can be constructed by polynomially combining normalized differences, e.g., products:

$$\mathrm{P}_{(i,j),(k,\ell)} = \mathrm{ND}_{i,j} \times \mathrm{ND}_{k,\ell}$$

This preserves the invariance and allows capturing spectral interactions relevant for more complex vegetation discrimination.

2. Discovery via Neural Networks and Explainable AI

End-to-end neural network (NN) training on raw reflectance spectra can automatically recover or invent compact indices. In a one-hidden-layer NN on hyperspectral vegetation scenes, the first-layer weights concentrate almost entirely on two bands corresponding to the classic normalized vegetation difference index (NDVI):

$$\mathrm{NDVI} = \frac{R_\mathrm{NIR} - R_\mathrm{Red}}{R_\mathrm{NIR} + R_\mathrm{Red}}$$

where, for AVIRIS, $R_\mathrm{Red} \approx 656$ nm and $R_\mathrm{NIR} \approx 802$ nm. The network weights empirically yield $w_{656\;\mathrm{nm}} = -0.68$, $w_{802\;\mathrm{nm}} = +0.72$, all other weights $\approx 0$ (Basener, 2022). Similarly, when trained on polymer spectra, the NN discovers analogous two-band indices uniquely diagnostic for each polymer, by selecting sharp absorption features and constructing paired differences.

Explainable AI methods such as Shapley Value Sampling (SVS), applied to GRU-based crop classifiers, identify the most influential bands per crop and guide the selection or modification of vegetation indices (Najjar et al., 2024). SVS attribute scores reveal which classic or variant indices (NDVI, NDRE, NDMI, etc.) best exploit the most discriminative spectral regions, enabling construction of highly compact, class-relevant VIs.

3. Automated Index Generation and Selection via Polynomial Search Spaces

Structured enumeration of all pairwise normalized difference indices followed by polynomial expansion to degree $d$ yields a controlled search space for index discovery. For $n$ bands, this generates $m = \binom{n}{2}$ base indices and $2m + \binom{m}{2}$ degree-2 products (e.g., 1080 candidate indices for Sentinel-2):

• Degree-1: $\mathrm{ND}_{i,j}$ (total $m$)
• Degree-2 squares: $\mathrm{ND}_{i,j}^2$ (total $m$)
• Degree-2 cross-products: $\mathrm{ND}_{i,j}\,\mathrm{ND}_{k,\ell}$, $(i,j)<(k,\ell)$ (total $\binom{m}{2}$)

Successive feature selection—ANOVA filtering, recursive elimination (RFE), and $L_1$-regularized SVM—reduces the candidate set to ultra-compact subsets (1–8 indices), with accuracy often matching models using all spectral bands (Lotfi et al., 26 Dec 2025). For Kochia detection, a single product of red-edge normalized differences achieves 96.26% test accuracy:

$$\phi_1(b) = \mathrm{ND}_{4,5}\,\mathrm{ND}_{7,8} = \frac{b_4 - b_5}{b_4 + b_5}\;\frac{b_7 - b_8}{b_7 + b_8}$$

4. Genetic Programming for Index Evolution

Genetic programming (GP) constructs indices as symbolic expressions combining spectral bands through arithmetic and nonlinear operators (protected $+\,-\,\times\,/$, $\mathrm{rlog}$, $\mathrm{srt}$). The fitness function is class-separability:

$$S(I) = \frac{|\mu_a-\mu_b|}{\max(\sigma_a,\,\sigma_b)}$$

where $\mu_{a/b}$ and $\sigma_{a/b}$ are mean and standard deviation of candidate index $I(B)$ over classes $a$ and $b$ (Albarracín et al., 2020).

GPVI indices, typically containing only 2–4 bands and 10–20 operations, outperform classical NDVI/EVI by 4–10 percentage points:

$$I_{\mathrm{FS}}(B) = \mathrm{rlog}\!\Bigl(\frac{B_4}{B_7}\Bigr) - \mathrm{srt}(B_7-B_4)$$

GPVI deployment is comparable to classical indices: one pass per pixel, tractable computational complexity, and operationally robust across different biome comparison tasks.

5. Operational Performance and Practical Reduction

Empirical validation demonstrates that compact indices, discovered either via NN/XAI, polynomial search, or GP, retain nearly all of the discriminative power of the raw multispectral stack. For crop mapping using Sentinel-2, a single index (NDMI) achieves 0.65 overall accuracy—just 2 percentage points below the full-band baseline—while two indices (NDRE + NDMI2) surpass the baseline at 0.70 (Najjar et al., 2024). For vegetation and polymer classification, NN-learned indices yield >99.9% test accuracy when only the top one or two indices are used (Basener, 2022).

Reducing input dimensionality lowers computation, storage, and I/O costs, which is especially advantageous for real-time on-board edge inference in UAV or satellite applications. Parsimonious indices also generalize better with limited training data and enable direct deployment via simple arithmetic formulas in platforms like Google Earth Engine (Lotfi et al., 26 Dec 2025).

6. Biological and Physical Interpretability

Examination of weight profiles or evolved formulas reveals that compact indices invariably reflect known spectroscopic features: chlorophyll absorption at the red edge, C–H overtone, or SWIR transitions relevant to leaf structure or polymer backbone chemistry. NN-derived and GP-evolved indices correspond to positive weight lobes at absorption peaks and negative lobes at troughs, providing direct human-interpretability and bridging data-driven learning and classical physics-based spectroscopy (Basener, 2022, Albarracín et al., 2020).

7. Generalization Across Sensors, Problems, and Modalities

Compact index construction is extensible to arbitrary multispectral sensors—compute all pairwise normalized differences, polynomially combine as needed, and apply feature selection. For most practical setups, degree-2 products represent an optimal trade-off between richness and computational tractability; higher degrees risk overfitting and excessive complexity (Lotfi et al., 26 Dec 2025). The methodologies are directly applicable to other remote sensing tasks, including plastics sorting, litter detection, or any pixelwise classification involving spectral separability. Open-source libraries (e.g., "ndindex" [Editor’s term]) facilitate rapid, reproducible derivation and deployment of these indices.

In summary, compact spectral indices—whether recovered by neural networks, XAI, polynomial search, or genetic programming—capture the essential spectral contrasts necessary for robust vegetation classification, retain physical meaning, and dramatically simplify the operational pipeline for remote sensing practitioners.