PCA (Principal Component Analysis)

Overview

The pca module implements a Principal Component Analysis (PCA) workflow designed for dimensionality reduction, spectral-spatial information compression, and uncorrelated feature extraction from multi-spectral satellite imagery. Satellite sensors capture surface reflectance across multiple overlapping spectral bands, which often results in high data redundancy and strong correlation between adjacent channels.

This module unifies a six-band multi-spectral stack—typically comprising the visible spectrum (Red, Green, Blue), Near-Infrared (NIR), and Short-Wave Infrared (SWIR1, SWIR2)—and projects it into a new coordinate space. The resulting orthogonal axes, or Principal Components (PCs), are ordered by the amount of total variance they explain, isolating dominant spatial patterns and suppressing high-frequency sensor noise.

This mathematical inversion identifies the dominant spatial patterns that account for the most variability across the band dimension, generating coherent eigenimages that capture unique landscape characteristics.

       [6 Native Spatial Bands] (Red, Green, Blue, NIR, SWIR1, SWIR2)
                  │
                  ▼
   ┌─────────────────────────────┐
   │ 2D Array Flattening Block   │ ──► Each band reshaped to (1, N) vector
   └──────────────┬──────────────┘
                  │
                  ▼
   ┌─────────────────────────────┐
   │ Matrix Construction ($X$)    │ ──► Dimensions: $\mathbb{R}^{6 \times N}$ ($m=6$ bands, $n=N$ pixels)
   └──────────────┬──────────────┘
                  │
                  ▼
   ┌─────────────────────────────┐
   │ Broadcast Variable Centering│ ──► Subtract mean pixel brightness: $X_c = X - \bar{x}$
   └──────────────┬──────────────┘
                  │
                  ▼
   ┌─────────────────────────────┐
   │  Singular Value Decomposition│ ──► $X_c = U \Sigma V^T$ (Avoids $N \times N$ Covariance Matrix)
   └──────────────┬──────────────┘
                  │
        ┌─────────┴─────────┐
        ▼                   ▼
   [Left Singular Vectors] [Right Singular Vectors ($V^T$)]
    Spectral Loadings       6 Spatial Eigenimages (PC1 - PC6)
   ($U \in \mathbb{R}^{6 \times 6}$)       Shape: $(6, N) \to$ Reshaped to $(H, W)$

Comprehensive Mathematical Foundations

Spatial Data Matrix Construction

Let each input band $b$ (where $b \in \{1, 2, \dots, 6\}$) represent a discrete 2D image matrix of height $H$ and width $W$. Each band matrix is flattened into a single row vector of length $N$, where $N = H \times W$. The combined multi-spectral data matrix $X$ is constructed by stacking these row vectors vertically:

$$X = \begin{bmatrix} \text{band}_1 \\ \text{band}_2 \\ \vdots \\ \text{band}_6 \end{bmatrix} \in \mathbb{R}^{6 \times N}$$

In this framework, the matrix contains $m = 6$ samples (the spectral bands) and $n = N$ features (the individual pixel coordinates).

Broadcast Variable Centering

To eliminate global illumination offsets, the data must be centered before decomposition. The algorithm calculates the mean value $\bar{x}_j$ for each pixel variable $j$ across the 6 spectral samples:

$$\bar{x}_j = \frac{1}{m} \sum_{i=1}^{m} X_{ij}$$

The centered data matrix $X_c$ is formed by subtracting this mean vector from each row of $X$:

$$X_c = X - \begin{bmatrix} \bar{x}_1 & \bar{x}_2 & \dots & \bar{x}_N \end{bmatrix}_{1 \times N}$$

This centering operation is broadcast across all rows, centering each individual pixel variable around zero.

Resolving the Dimensionality Boundary via SVD

The classical covariance matrix $C$ for the pixel features is defined as:

$$C = \frac{1}{m - 1} X_c^T X_c \in \mathbb{R}^{N \times N}$$

Because a typical satellite scene contains millions of pixels ($N > 10^6$), building and decomposing this $N \times N$ matrix directly is computationally prohibitive. To bypass this bottleneck, PCACalculator uses Singular Value Decomposition (SVD) directly on the centered data matrix $X_c$:

$$X_c = U \Sigma V^T$$

Where:

• $U \in \mathbb{R}^{6 \times 6}$ is an orthogonal matrix containing the left singular vectors, which represent the spectral loadings for each band.

• $\Sigma \in \mathbb{R}^{6 \times 6}$ is a diagonal matrix containing the sorted singular values ($\sigma_1 \ge \sigma_2 \ge \dots \ge \sigma_6 \ge 0$).

• $V \in \mathbb{R}^{N \times 6}$ contains the right singular vectors, which correspond to the spatial eigenvectors of the massive $C$ matrix.

The eigenvalues $\lambda_k$ of the underlying covariance matrix are directly related to the singular values $\sigma_k$ by:

$$\lambda_k = \frac{\sigma_k^2}{m - 1}$$

Quantifying Explained Variance

The $k$-th row of the transposed right singular matrix $V^T$ represents the $k$-th principal component vector, which has a length of $N$. The proportion of total spatial variance captured by this component is calculated directly from its singular value:

$$VE_k = \frac{\lambda_k}{\sum_{j=1}^{6} \lambda_j} = \frac{\sigma_k^2}{\sum_{j=1}^{6} \sigma_j^2}$$

Class Specification: PCACalculator

Scientific and Interpretation Profiles

When a principal component vector of length $N$ is reshaped back to its original dimensions $(H, W)$, it forms a spatial map known as an eigenimage. These components separate different types of information based on how variation is distributed across the scene:

Principal Component 1 (PC1) — Albedo and Illumination Map

PC1 captures the most dominant spatial variation common to all six input bands, typically accounting for more than 90% of the total scene variance. Because land cover features generally reflect light with similar broad trends under uniform lighting, the PC1 image behaves like a panchromatic brightness map. It highlights overall surface albedo, topographic shading, and solar illumination while minimizing compositional differences.

Principal Component 2 (PC2) — Compositional and Vegetation Contrast

PC2 highlights the second most dominant axis of variation, focusing on strong contrasts between different wavelengths. In landscapes with active vegetation, PC2 typically captures the sharp divergence between high near-infrared ($NIR$) reflectance and the strong visible light absorption of chlorophyll. This makes it an effective index for mapping biomass distribution and separating vegetative cover from urban surfaces or open water.

Principal Components 3 to 6 (PC3–PC6) — Residuals and Sensor Noise

These higher-order components capture progressively smaller variations in the data. While PC3 often highlights subtle moisture or mineral variations across the short-wave infrared spectrum ($SWIR$), components PC4 through PC6 are typically dominated by high-frequency sensor noise, atmospheric striping, and random background variations. Consequently, these late-stage components can generally be discarded during data compression workflows without losing meaningful information.

Interface Architecture

Constructor Method Signature (__init__)

• Input Arguments:

  • red_path (str | Path): File path to the visible Red band raster layer.

  • green_path (str | Path): File path to the visible Green band raster layer.

  • blue_path (str | Path): File path to the visible Blue band raster layer.

  • nir_path (str | Path): File path to the Near-Infrared band raster layer.

  • swir1_path (str | Path): File path to the Short-Wave Infrared 1 band raster layer.

  • swir2_path (str | Path): File path to the Short-Wave Infrared 2 band raster layer.

  • selectBand (Literal["red","green","blue","nir","swir1","swir2", None]): Optional parameter. Selects a specific input band to map against the component outputs during specialized diagnostic profiling.

Processing Pipeline Lifecycle (process())

1. Ingests the six target spectral bands and flattens each 2D matrix into a continuous 1D array.

2. Combines the flattened arrays into a single unified matrix of shape (6, N_pixels).

3. Passes this data matrix to sklearn.decomposition.PCA(n_components=6). The model centers the variables and executes an optimized SVD.

4. Extracts the six spatial eigenvectors from pca.components_ and stores this (6, N_pixels) array in self._output.

Visualization Lifecycle (_export_file)

Generates a structured $6 \times 2$ grid layout to support visual data analysis:

• Left Column Panels: Displays the six individual principal component vectors, reshaped back to the original image dimensions (Height, Width).

• Right Column Panels: Plots the corresponding frequency histograms for each component, helping analysts inspect the distribution of variance across the different axes.

Return Value

Returns a floating-point numpy.ndarray of shape (6, N_pixels). Each row in this matrix contains the continuous spatial weights for one of the six principal components.

Operational Implementation

from pathlib import Path
from fezrs.tools.pca import PCACalculator

# Initialize the spatial PCA transformation engine
pca_transformer = PCACalculator(
    red_path=Path("./landsat/LC08_B04_Red.tif"),
    green_path=Path("./landsat/LC08_B03_Green.tif"),
    blue_path=Path("./landsat/LC08_B02_Blue.tif"),
    nir_path=Path("./landsat/LC08_B05_NIR.tif"),
    swir1_path=Path("./landsat/LC08_B06_SWIR1.tif"),
    swir2_path=Path("./landsat/LC08_B07_SWIR2.tif")
)

# Run the SVD engine, construct diagnostic histograms, and save the 6-panel visualization layout
pca_transformer.execute(
    output_path="./exports/pca_analytics/",
    title="Spatial PCA Matrix Decomposition",
    dpi=500
)

Analytical Performance Reference

The table below outlines the general characteristics and typical interpretation profiles of the resulting principal components:

Component	Target Variance Share	Spatial Contrast Profile	Primary Analytical Applications
PC1	Typical $\ge 90\%$	High structural detail; behaves like a panchromatic brightness map.	Topographic mapping, shadow analysis, and baseline albedo feature extraction.
PC2	Typical $5\% - 8\%$	High contrast between visible light absorption and near-infrared ($NIR$) plateau.	Biomass delineation, vegetation health mapping, and land-cover classification.
PC3	Typical $1\% - 3\%$	Captures variations across infrared bands ($SWIR1 / SWIR2$).	Soil moisture profiling, surface water mapping, and mineral identification.
PC4 – PC6	Distributive $\le 1\%$	Low structural coherence; dominated by random high-frequency sensor patterns.	Noise filtering, data compression filtering, and system calibration diagnostics.