特征值分析

主成分 (PC) 转换（也称为 Karhunen-Loeve 转换）是一种光谱旋转，可获取光谱相关的图像数据并输出不相关的数据。PC 转换通过特征值分析对输入频段相关性矩阵进行对角化来实现这一点。如需在 Earth Engine 中执行此操作，请对数组图像使用协方差减小器，并对生成的协方差数组使用 eigen() 命令。 请考虑使用以下函数来实现此目的（应用中的示例可作为 Code Editor 脚本和 Colab 笔记本使用）。

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 设置

如需了解 Python API 以及如何使用 geemap 进行交互式开发，请参阅 Python 环境页面。

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)
  )

该函数的输入是均值为零的图片、缩放比例和要执行分析的区域。请注意，输入图像需要先转换为 1 维数组图像，然后使用 ee.Reducer.centeredCovariance() 进行缩减。此求和运算返回的数组是输入的对称方差-协方差矩阵。 使用 eigen() 命令获取协方差矩阵的特征值和特征向量。eigen() 返回的矩阵包含 1 轴的第 0 个位置上的特征值。如前面的函数所示，使用 slice() 来分隔特征值和特征向量。特征向量矩阵沿 0 轴的每个元素都是特征向量。与流苏帽 (TC) 示例中一样，通过将 arrayImage 与特征向量相乘的矩阵乘法来执行转换。 在此示例中，每次特征向量乘法都会产生一个 PC。