Eigen Analysis

  • The Principal Components (PC) transform is a spectral rotation that converts spectrally correlated image data into uncorrelated data.

  • The PC transform diagonalizes the input band correlation matrix using Eigen-analysis.

  • In Earth Engine, this transformation is achieved by using a covariance reducer on an array image and the eigen() command on the resulting covariance array.

  • The input to the function is a mean zero image, a scale, and a region for analysis, and the transformation involves obtaining eigenvalues and eigenvectors of the covariance matrix.

The principal components (PC) transform (also known as the Karhunen-Loeve transform) is a spectral rotation that takes spectrally correlated image data and outputs uncorrelated data. The PC transform accomplishes this by diagonalizing the input band correlation matrix through Eigen-analysis. To do this in Earth Engine, use a covariance reducer on an array image and the eigen() command on the resultant covariance array. Consider the following function for that purpose (an example of it in application is available as a Code Editor script and a Colab notebook).

Code Editor (JavaScript)

var getPrincipalComponents = function(centered, scale, region) {
  // Collapse the bands of the image into a 1D array per pixel.
  var arrays = centered.toArray();

  // Compute the covariance of the bands within the region.
  var covar = arrays.reduceRegion({
    reducer: ee.Reducer.centeredCovariance(),
    geometry: region,
    scale: scale,
    maxPixels: 1e9
  });

  // Get the 'array' covariance result and cast to an array.
  // This represents the band-to-band covariance within the region.
  var covarArray = ee.Array(covar.get('array'));

  // Perform an eigen analysis and slice apart the values and vectors.
  var eigens = covarArray.eigen();

  // This is a P-length vector of Eigenvalues.
  var eigenValues = eigens.slice(1, 0, 1);
  // This is a PxP matrix with eigenvectors in rows.
  var eigenVectors = eigens.slice(1, 1);

  // Convert the array image to 2D arrays for matrix computations.
  var arrayImage = arrays.toArray(1);

  // Left multiply the image array by the matrix of eigenvectors.
  var principalComponents = ee.Image(eigenVectors).matrixMultiply(arrayImage);

  // Turn the square roots of the Eigenvalues into a P-band image.
  var sdImage = ee.Image(eigenValues.sqrt())
      .arrayProject([0]).arrayFlatten([getNewBandNames('sd')]);

  // Turn the PCs into a P-band image, normalized by SD.
  return principalComponents
      // Throw out an an unneeded dimension, [[]] -> [].
      .arrayProject([0])
      // Make the one band array image a multi-band image, [] -> image.
      .arrayFlatten([getNewBandNames('pc')])
      // Normalize the PCs by their SDs.
      .divide(sdImage);
};

Python setup

See the Python Environment page for information on the Python API and using geemap for interactive development.

import ee
import geemap.core as geemap

Colab (Python)

def get_principal_components(centered, scale, region):
  # Collapse bands into 1D array
  arrays = centered.toArray()

  # Compute the covariance of the bands within the region.
  covar = arrays.reduceRegion(
      reducer=ee.Reducer.centeredCovariance(),
      geometry=region,
      scale=scale,
      maxPixels=1e9,
  )

  # Get the 'array' covariance result and cast to an array.
  # This represents the band-to-band covariance within the region.
  covar_array = ee.Array(covar.get('array'))

  # Perform an eigen analysis and slice apart the values and vectors.
  eigens = covar_array.eigen()

  # This is a P-length vector of Eigenvalues.
  eigen_values = eigens.slice(1, 0, 1)
  # This is a PxP matrix with eigenvectors in rows.
  eigen_vectors = eigens.slice(1, 1)

  # Convert the array image to 2D arrays for matrix computations.
  array_image = arrays.toArray(1)

  # Left multiply the image array by the matrix of eigenvectors.
  principal_components = ee.Image(eigen_vectors).matrixMultiply(array_image)

  # Turn the square roots of the Eigenvalues into a P-band image.
  sd_image = (
      ee.Image(eigen_values.sqrt())
      .arrayProject([0])
      .arrayFlatten([get_new_band_names('sd')])
  )

  # Turn the PCs into a P-band image, normalized by SD.
  return (
      # Throw out an an unneeded dimension, [[]] -> [].
      principal_components.arrayProject([0])
      # Make the one band array image a multi-band image, [] -> image.
      .arrayFlatten([get_new_band_names('pc')])
      # Normalize the PCs by their SDs.
      .divide(sd_image)
  )

The input to the function is a mean zero image, a scale and a region over which to perform the analysis. Note that the input imagery first needs to be converted to a 1-D array image and then reduced using ee.Reducer.centeredCovariance(). The array returned by this reduction is the symmetric variance-covariance matrix of the input. Use the eigen() command to get the eigenvalues and eigenvectors of the covariance matrix. The matrix returned by eigen() contains the eigenvalues in the 0-th position of the 1-axis. As shown in the earlier function, use slice() to separate the eigenvalues and the eigenvectors. Each element along the 0-axis of the eigenVectors matrix is an eigenvector. As in the tasseled cap (TC) example, perform the transformation by matrix multiplying the arrayImage by the eigenvectors. In this example, each eigenvector multiplication results in a PC.