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Automated Discovery of Parsimonious Spectral Indices via Normalized Difference Polynomials

Ali Lotfi, Adam Carter, Thuan Ha, Mohammad Meysami, Kwabena Nketia, Steve Shirtliffe

TL;DR

This work addresses the automatic discovery of compact, interpretable spectral indices for vegetation classification by constructing a bounded embedding from pairwise normalized differences $ND_{ij}$ and forming degree-2 polynomials. The feature space includes linear, squared, and cross terms, yielding 1080 candidates for Sentinel-2 with 10 bands, which are reduced via three feature-selection strategies to small index sets. In Kochia detection, a single degree-2 index $ND_{b4,b5} \cdot ND_{b7,b8}$ delivers 96.26% test accuracy, and eight indices push accuracy to 97.70%, with all selected indices involving red-edge to NIR bands, indicating discriminative spectral interactions. The approach offers high interpretability, transferability across sensors, and practical deployability in platforms like Google Earth Engine, and is released as open-source software (ndindex).

Abstract

We introduce an automated way to find compact spectral indices for vegetation classification. The idea is to take all pairwise normalized differences from the spectral bands and then build polynomial combinations up to a fixed degree, which gives a structured search space that still keeps the illumination invariance needed in remote sensing. For a sensor with $n$ bands this produces $\binom{n}{2}$ base normalized differences, and the degree-2 polynomial expansion gives 1,080 candidate features for the 10-band Sentinel-2 configuration we use here. Feature selection methods (ANOVA filtering, recursive elimination, and $L_1$-regularized SVM) then pick out small sets of indices that reach the desired accuracy, so the final models stay simple and easy to interpret. We test the framework on Kochia (\textit{Bassia scoparia}) detection using Sentinel-2 imagery from Saskatchewan, Canada ($N = 2{,}318$ samples, 2022--2024). A single degree-2 index, the product of two normalized differences from the red-edge bands, already reaches 96.26\% accuracy, and using eight indices only raises this to 97.70\%. In every case the chosen features are degree-2 products built from bands $b_4$ through $b_8$, which suggests that the discriminative signal comes from spectral \emph{interactions} rather than individual band ratios. Because the indices involve only simple arithmetic, they can be deployed directly in platforms like Google Earth Engine. The same approach works for other sensors and classification tasks, and an open-source implementation (\texttt{ndindex}) is available.

Automated Discovery of Parsimonious Spectral Indices via Normalized Difference Polynomials

TL;DR

This work addresses the automatic discovery of compact, interpretable spectral indices for vegetation classification by constructing a bounded embedding from pairwise normalized differences and forming degree-2 polynomials. The feature space includes linear, squared, and cross terms, yielding 1080 candidates for Sentinel-2 with 10 bands, which are reduced via three feature-selection strategies to small index sets. In Kochia detection, a single degree-2 index delivers 96.26% test accuracy, and eight indices push accuracy to 97.70%, with all selected indices involving red-edge to NIR bands, indicating discriminative spectral interactions. The approach offers high interpretability, transferability across sensors, and practical deployability in platforms like Google Earth Engine, and is released as open-source software (ndindex).

Abstract

We introduce an automated way to find compact spectral indices for vegetation classification. The idea is to take all pairwise normalized differences from the spectral bands and then build polynomial combinations up to a fixed degree, which gives a structured search space that still keeps the illumination invariance needed in remote sensing. For a sensor with bands this produces base normalized differences, and the degree-2 polynomial expansion gives 1,080 candidate features for the 10-band Sentinel-2 configuration we use here. Feature selection methods (ANOVA filtering, recursive elimination, and -regularized SVM) then pick out small sets of indices that reach the desired accuracy, so the final models stay simple and easy to interpret. We test the framework on Kochia (\textit{Bassia scoparia}) detection using Sentinel-2 imagery from Saskatchewan, Canada ( samples, 2022--2024). A single degree-2 index, the product of two normalized differences from the red-edge bands, already reaches 96.26\% accuracy, and using eight indices only raises this to 97.70\%. In every case the chosen features are degree-2 products built from bands through , which suggests that the discriminative signal comes from spectral \emph{interactions} rather than individual band ratios. Because the indices involve only simple arithmetic, they can be deployed directly in platforms like Google Earth Engine. The same approach works for other sensors and classification tasks, and an open-source implementation (\texttt{ndindex}) is available.
Paper Structure (17 sections, 1 theorem, 22 equations, 5 figures, 6 tables, 3 algorithms)

This paper contains 17 sections, 1 theorem, 22 equations, 5 figures, 6 tables, 3 algorithms.

Table of Contents

  1. Normalized Difference Polynomial Framework
  2. Degree 1 (Linear terms):
  3. Degree 2 (Quadratic terms):
  4. Method 1: SelectKBest with ANOVA F-Statistic
  5. Method 2: Recursive Feature Elimination (RFE)
  6. Method 3: L1-Regularized SVM
  7. Application: Kochia Detection with Sentinel-2 Imagery
  8. Numerical Results
  9. Classification Performance
  10. The Single-Index Kochia Classifier
  11. Spectral Separability
  12. Confusion Matrix Analysis
  13. Generalization Analysis
  14. Diminishing Returns Analysis
  15. Discussion
  16. ...and 2 more sections

Key Result

Proposition 1.3

Let $p(b_1, \ldots, b_n)$ be a normalized difference polynomial of degree $d$ with finitely many nonzero coefficients. Then:

Figures (5)

  • Figure 1: Classification accuracy as a function of the number of spectral indices. The single-index model ($k=1$) achieves 96.26% test accuracy, with diminishing returns for additional indices. The dashed red line indicates the 85% threshold; all models substantially exceed this baseline.
  • Figure 2: Spectral separability of Kochia and Crop samples in the feature space defined by the two most discriminative polynomial indices. The near-linear arrangement of points and clear class separation explain the effectiveness of linear SVM classification.
  • Figure 3: Decision boundaries for Kochia classification with $k = 1, 2, 3, 4$ spectral indices, projected onto the two most discriminative features. The consistency of boundaries across model complexities demonstrates that additional indices refine rather than fundamentally alter the classification geometry.
  • Figure 4: Train-test accuracy gap versus number of spectral indices. All models exhibit gaps below 2% (green zone), indicating robust generalization without overfitting. The minimum gap occurs at $k = 5$ and $k = 6$ (0.80%).
  • Figure 5: Marginal improvement in test accuracy per additional spectral index. The dashed line indicates the 0.5% diminishing returns threshold. Most improvements fall below this threshold, supporting the selection of parsimonious models.

Theorems & Definitions (8)

  • Definition 1.1: Normalized Difference
  • Definition 1.2: Normalized Difference Polynomial of Degree $d$
  • Remark
  • Proposition 1.3: Boundedness of ND Polynomials
  • proof
  • Definition 2.1: Pixel-Level Embedding
  • Remark
  • Definition 2.2: Class-Level Embedding