The Normalized Difference Layer: A Differentiable Spectral Index Formulation for Deep Learning

Normalized difference indices have been a staple in remote sensing for decades. They stay reliable under lighting changes produce bounded values and connect well to biophysical signals. Even so, they are usually treated as a fixed pre processing step with coefficients set to one, which limits how well they can adapt to a specific learning task. In this study, we introduce the Normalized Difference Layer that is a differentiable neural network module. The proposed method keeps the classical idea but learns the band coefficients from data. We present a complete mathematical framework for integrating this layer into deep learning architectures that uses softplus reparameterization to ensure positive coefficients and bounded denominators. We describe forward and backward pass algorithms enabling end to end training through backpropagation. This approach preserves the key benefits of normalized differences, namely illumination invariance and outputs bounded to $[-1,1]$ while allowing gradient descent to discover task specific band weightings. We extend the method to work with signed inputs, so the layer can be stacked inside larger architectures. Experiments show that models using this layer reach similar classification accuracy to standard multilayer perceptrons while using about 75\% fewer parameters. They also handle multiplicative noise well, at 10\% noise accuracy drops only 0.17\% versus 3.03\% for baseline MLPs. The learned coefficient patterns stay consistent across different depths.

Automated Discovery of Parsimonious Spectral Indices via Normalized Difference Polynomials

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.

Probabilistic Analysis of Least Squares, Orthogonal Projection, and QR Factorization Algorithms Subject to Gaussian Noise

In this paper, we extend the work of Liesen et al. (2002), which analyzes how the condition number of an orthonormal matrix Q changes when a column is added ([Q, c]), particularly focusing on the perpendicularity of c to the span of Q. Their result, presented in Theorem 2.3 of Liesen et al. (2002), assumes exact arithmetic and orthonormality of Q, which is a strong assumption when applying these results to numerical methods such as QR factorization algorithms. In our work, we address this gap by deriving bounds on the condition number increase for a matrix B without assuming perfect orthonormality, even when a column is not perfectly orthogonal to the span of B. This framework allows us to analyze QR factorization methods where orthogonalization is imperfect and subject to Gaussian noise. We also provide results on the performance of orthogonal projection and least squares under Gaussian noise, further supporting the development of this theory.