The Normalized Difference Layer: A Differentiable Spectral Index Formulation for Deep Learning
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| Main Authors: | , , , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866911366330187776 |
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| author | Lotfi, Ali Carter, Adam Meysami, Mohammad Ha, Thuan Nketia, Kwabena Shirtliffe, Steve |
| author_facet | Lotfi, Ali Carter, Adam Meysami, Mohammad Ha, Thuan Nketia, Kwabena Shirtliffe, Steve |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_06777 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Normalized Difference Layer: A Differentiable Spectral Index Formulation for Deep Learning Lotfi, Ali Carter, Adam Meysami, Mohammad Ha, Thuan Nketia, Kwabena Shirtliffe, Steve Computer Vision and Pattern Recognition 68T07, 68U10 I.2.6; I.5.1 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. |
| title | The Normalized Difference Layer: A Differentiable Spectral Index Formulation for Deep Learning |
| topic | Computer Vision and Pattern Recognition 68T07, 68U10 I.2.6; I.5.1 |
| url | https://arxiv.org/abs/2601.06777 |