Abstract
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.
Abstract (translated)
几十年来，归一化差异指数（Normalized Difference Indices）一直是遥感中的一个重要工具。它们在光照变化下依然可靠，并产生有界值和与生物物理信号的良好连接性。尽管如此，这些指标通常被视为固定的预处理步骤，其中系数被固定为1，这限制了它们适应特定学习任务的能力。在这项研究中，我们引入了一个可微的神经网络模块——归一化差异层（Normalized Difference Layer）。该方法保留了经典思想，但允许从数据中学习波段权重。 提出了一种完整的数学框架，用于将这个层整合到深度学习架构中，并使用softplus重新参数化来确保正系数和有界的分母。描述了前向和反向传递算法，使通过反向传播进行端到端训练成为可能。这种方法保留了归一化差异的关键优势，即光照不变性和输出限制在[-1,1]范围内，同时允许梯度下降发现任务特定的波段权重。还扩展了该方法以适应有符号输入的情况，使得这个层可以嵌入更大的架构中。 实验表明，使用此层的模型达到了与标准多层感知器（MLPs）相似的分类准确率，但使用的参数减少了约75%。它们在处理乘法噪声时表现良好，在10%的噪声情况下，准确性仅下降了0.17%，而基线MLPs则下降了3.03%。学习到的系数模式在不同深度下保持一致。 总结来说，归一化差异层结合了经典方法的优势和现代机器学习技术的能力，提供了一种新颖且有效的遥感图像处理方式。
URL
https://arxiv.org/abs/2601.06777
