PCA-Based Band Selection (Acosta Variant)

The PCA-Based Band Selection, inspired by the Acosta variant, applies Principal Component Analysis (PCA) to identify the most spectrally informative and contrasting structures in a satellite image. This method does not directly use the raw PCA projections but instead selects a single principal component that maximizes the spectral contrast between two user-defined bands—one representing a known reflectance feature and the other an absorption feature.

How it works #

1. Principal Component Decomposition The input image is first reduced to its selected bands and centered (mean-subtracted). PCA is then computed using the covariance matrix of the centered image, resulting in:
   • A set of orthogonal eigenvectors (defining the direction of each PC in feature space)Corresponding eigenvalues, which measure the variance explained by each component $\text{PCs}=X·$ where $X$ is the centered image matrix and $E$ is the eigenvector matrix.
   • User-defined Band Roles Unlike generic PCA, this method relies on the user explicitly defining:
     • One band as the reflectance reference
     • One band as the absorption reference
     These are used to guide the selection of the most relevant PC.
   • Contrast Maximization Across PCs For each principal component (i.e., each eigenvector), the absolute difference in loading between the two user-selected bands is computed: $D_{\text{PC}}=|{\text{loading}}_{\text{reflectance}}-{\text{loading}}_{\text{absorption}}|$ The PC with the maximum contrast $D_{\text{PC}}$ is selected, as it best captures the divergence between absorption and reflectance behaviors.
   • Weight Bands by Importance Optionally, a scoring system can be used to rank bands by their significance: ${\text{Score}}_j=\left(\frac{\left|{\text{loading}}_j\right|}{\sum _k\left|{\text{loading}}_k\right|}\right)\times \left(\frac{{\lambda }_{\text{PC}}}{\sum {\lambda }_{\text{PC}}}\right)$ Where:
     • $\left|{\text{loading}}_j\right|$: absolute contribution of band j to the component
     • ${\lambda }_{\text{PC}}$: eigenvalue (variance explained) of that component
     This helps identify not just contrastive bands, but also statistically stable and relevant ones.
   • Band Inversion and Addition The selected PC is scaled (inverted if necessary) so that the direction of variation aligns with physical expectations—e.g., higher values indicating stronger absorption. The result is added back as a new band to the image for analysis or visualization.

Technical advantages #

• Captures complex spectral interactions using PCA, yet retains physical interpretability.
• Allows users to target specific spectral features by defining reflectance and absorption anchors.
• Produces a single, optimized spectral feature for further analysis (e.g., classification, thresholding).