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Unformatted text preview: ral Imaging Plotting Spectra in Spectral Space
The spectral plots on the previous pages provide a convenient way to visualize the differences in spectral properties between different materials, especially when we are comparing only a few spectra. Spectral plots are an important tool to use when you explore a hyperspectral image. But to understand how a computer compares and discriminates among a large number of spectra, it is useful to consider other conceptual ways of representing spectra. A reflectance spectrum consists of a set of reflectance values, one for each spectral channel (band). Each of these channels can be considered as one dimension in a hypothetical ndimensional spectral space, where n is the number of spectral channels. If we plot the measured reflectance value for each spectral channel on its respective coordinate axis, we can use these coordinates to specify the location of a point in spectral space that Point representing mathematically represents that particular spectrum (0.8, 0.7) spectrum. A simple twoband example is shown in the illustration. The designated point 0.7 g can also be treated mathematically as the end tin en point of a vector that begins at the origin of es pr re t r u m the coordinate system. Spectra with the same c or ct shape but differing overall reflectance (alpe s Ve bedo) plot as vectors with the same orientation 0.8 but with endpoints at different distances from the origin. Shorter spectral vectors represent Reflectance in Band 1 darker spectra and longer vectors represent Ndimensional plot of a reflectance brighter spectra. It may be difficult to visualize such a plot for an image involving more than three wavelength bands, but it is mathematically possible to construct a hyperdimensional spectral space defined by dozens or hundreds of mutuallyperpendicular coordinate axes. Each spectrum being considered occupies a position in this ndimensional spectral space. Similarity between spectra can be judged by the relative closeness of these positions (spectral distance) or by how small the angle is between the spectral vectors. The spectral reflectance curves shown on the previous pages for various materials represent “averages” or “typical examples”. All natural materials exhibit some variability in composition and structure that results in variability in their reflectance spectra. If we obtain spectra from a number of examples of a material, the resulting spectral points will define a small cloud in ndimensional spectral space, rather than plotting at one single location.
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spectrum for a hypothetical 2band case (n = 2). Reflectance in Band 2 Introduction to Hyperspectral Imaging Spatial Resolution and Mixed Spectra
An imaging spectrometer makes spectral measurements of many small patches of the Earth’s surface, each of which is represented as a pixel (raster cell) in the hyperspectral image. The size of the ground area represented by a single set of spectral measurements defines the spatial resolution of the image and depends on the sensor design and the height of the sensor above the surface. NASA’s Airborne Visible/Infrared Imaging Spectrometer (AVIRIS), for example, has a spatial resolution of 20 meters when flown at its typical altitude of 20 kilometers, but a 4meter resolution when flown at an altitude of 4 kilometers. When the size of the ground resolution cell is large, it is more likely that more than one material contributes to an individual spectrum measured by the sensor. The result is a c omposite o r mixed s pectrum, and the “pure” spectra that contribute to the mixture are called endmember spectra.
60 Reflectance (%) B
40 20 C = 60% A + 40% B C A 0 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 Wavelength (micrometers)
Example of a composite spectrum (C) that is a linear mixture of two spectra: A (dry soil) and B (green vegetation). Spectral mixtures can be macroscopic or intimate. In a macroscopic mixture each reflected photon interacts with only one surface material. The energy reflected from the materials combines additively, so that each material’s contribution to the composite spectrum is directly proportional to its area within the pixel. An example of such a linear mixture is shown in the illustration Spectrum A above, which could represent a patchwork of vegetation and bare soil. In spectral space each endmember spectrum defines the end Mixing of a mixing line (for two endmembers) or space the corner of a mixing space (for greater numbers of endmembers). Later we will discuss Spectrum C how the endmember fractions can be calcuSpectrum B lated for each pixel. In an intimate mixture, such as the microscopic mixture of mineral Reflectance in Band 1 particles found in soils, a single photon inNdimensional plot of three endteracts with more than one material. Such member spectra for a hypothetical 2band case. All spectra that are mixtures are nonlinear in character and theremixtures of A, B, and C alone must fore more difficult to unravel.
lie within the mix...
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This note was uploaded on 12/16/2010 for the course ENV 148 taught by Professor Chang during the Spring '10 term at APU Japan.
 Spring '10
 Chang

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