主要元件 (PC) 轉換 (也稱為 Karhunen-Loeve 轉換) 是一種光譜旋轉，可擷取光譜相關聯的圖像資料，並輸出不相關的資料。PC 轉換會透過特徵值分析，將輸入頻帶相關矩陣對角化，以達成這項目標。如要在 Earth Engine 中執行這項操作，請在陣列圖像上使用共變異數縮減器，並在產生的共變異數陣列上使用 eigen() 指令。請考慮使用下列函式來達成這個目的 (應用程式中的範例可做為 Code Editor 指令碼和 Colab 筆記本)。
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); };
import ee import geemap.core as geemap
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) )
函式輸入內容為平均值為零的圖片、比例和要執行分析的區域。請注意，輸入的圖像必須先轉換為 1D 陣列圖像，然後再使用 ee.Reducer.centeredCovariance() 進行縮減。這項縮減作業傳回的陣列，是輸入內容的對稱差異-共變異矩陣。使用 eigen() 指令取得協方差矩陣的固有值和固有向量。eigen() 傳回的矩陣包含 1 軸第 0 個位置的固有值。如前述函式所示，請使用 slice() 分隔特徵值和特徵向量。沿著特徵向量陣列的 0 軸的每個元素都是特徵向量。如同流蘇帽 (TC) 範例所示，透過矩陣將 arrayImage 乘以特徵向量，執行轉換作業。在本例中，每個特徵向量相乘結果都會產生 PC。