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Image Transformations

Each pixel in a multispectral image is a vector of values, one for each band. A transformation applies weights to those values to create one or more new layers. In a linear transformation, the same coefficient matrix is applied to every pixel: each output layer is a weighted combination of selected input bands. A rotation changes the coordinate axes used to describe the same spectral information; it can make patterns easier to compare or concentrate variation into fewer layers.

Both approaches combine band values, but their goals differ: an index applies a compact recipe to measure a known contrast, while a multivariate transformation often produces several axes that reorganize relationships across a set of bands. For the measurements that go into either operation, see Spectral Bands.

Principal component analysis

Principal component analysis (PCA) derives its axes from the data being analyzed. It estimates how selected bands vary together, then uses the eigenvectors of that covariance matrix as coefficients for a new set of uncorrelated components. The first component follows the direction of greatest variance in the fitted data; later components describe remaining directions of variation. An early NASA/JPL account of multispectral image processing describes this covariance-based rotation (Madura et al., 1978).

Because PCA is fitted from a particular sample, its coefficients depend on the chosen scene or region, bands, and their scaling. A component called “PC1” is therefore not a standard physical quantity: it may emphasize brightness, vegetation contrast, atmosphere, or another pattern in that input. Its name alone does not establish what caused the variation, and a component value is not automatically an index of a physical property. Interpret the loadings and the input data, and be careful when comparing separately fitted transforms.

Tasseled Cap

Tasseled Cap is a designed linear transform. Kauth and Thomas’s 1976 paper introduced axes that summarize patterns in Landsat crop observations, including soil brightness and green and yellow crop development. Later coefficient sets commonly define brightness, greenness, and wetness axes. A USGS comparison found that Landsat Thematic Mapper and Advanced Land Imager coefficient matrices were not interchangeable, even though the sensors have similar resolutions (Yamamoto and Finn, 2012). The coefficients are fixed for a specified transform rather than estimated anew from each image. Research deriving coefficients for Landsat 8 OLI, for example, defines them for that sensor and at-satellite reflectance (Baig et al., 2014).

Use coefficients that match the sensor, band order, and input product for which they were derived. The component names describe spectral patterns the axes are intended to emphasize; a “wetness” score is not a direct measurement of water content, nor is “greenness” a vegetation percentage. The scores are weighted combinations whose interpretation depends on the chosen coefficients and input data.

PCA is useful when the goal is to summarize covariance in a particular sample. Tasseled Cap is useful when a defined, repeatable set of interpretable axes is preferred for compatible data. For the matrix operations and examples in Earth Engine, continue to Spectral Transformations in Earth Engine.

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