Mixed Pixels Endmembers & Spectral Unmixing Mixed Pixel Analysis 1 Mixed Pixels and Spectral Unmixing Spectral Mixtures Areal Aggregate Intimate TYPES of MIXTURES Areal Aggregate Intimate Pixel 1 Pixel 2 Pixel 3 Sand Clay 50/50 mixtures of sand and clay Linear mixing models (areal and aggregate) Mixed Pixel Analysis 2 Page 1 1
Mixed Pixels and Spectral Unmixing Areal mixture where the various materials are physically separated into discrete areas within the pixel. In this case the reflectance properties of the materials are independent of one another and will mix additively. Example: a pixel spanning the boundary between a wooded area and a plowed field. TYPES of MIXTURES Areal Aggregate Intimate Pixel 1 Pixel 2 Pixel 3 50/50 mixtures of sand and clay Sand Clay Single pixel Mixed Pixel Analysis 3 Mixed Pixels and Spectral Unmixing Aggregate mixture materials are intermixed, but are locally aggregated. Example: a pixel covering a portion of a tennis court. The bound lines and the colored asphalt are physically separate and reflect nearly independently of one another even though the features are much smaller than the size of the pixel. Single pixel TYPES of MIXTURES Areal Aggregate Intimate Pixel 1 Pixel 2 Pixel 3 50/50 mixtures of sand and clay Sand Clay Mixed Pixel Analysis 4 Page 2 2
Mixed Pixels and Spectral Unmixing Intimate mixture materials are intermixed, on a scale small enough that the reflectance of the combined materials cannot be assumed to be a linear combination of the individual reflectances Example 1: a pixel covering an area where sediments from two distinct mineral sources are mixing. TYPES of MIXTURES Areal Aggregate Intimate Pixel 1 Pixel 2 Pixel 3 50/50 mixtures of sand and clay Sand Clay Single pixel Mixed Pixel Analysis 5 Mixed Pixels and Spectral Unmixing Intimate mixture Example 2: a pixel in an orchard where the pixel is a mixture of the reflectance of the individual trees and the background grass & soil. Light reflecting from the understory is very likely to interact with the tree canopy before being detected. Similarly, light that transmits through the canopy is likely to interact with the understory before being detected. Single pixel TYPES of MIXTURES Areal Aggregate Intimate Pixel 1 Pixel 2 Pixel 3 50/50 mixtures of sand and clay Sand Clay Mixed Pixel Analysis 6 Page 3 3
Mixed Pixels and Spectral Unmixing (cont d) N R fr i i i 1 Where: R is the effective reflectance of the mixed pixel, R i is the reflectance of the i th material (end member), f i is the spatial fraction covered by the i th material, and N is the number of materials in the pixel. Mixed Pixel Analysis 7 Mixed Pixels and Spectral Unmixing (cont d) Since L = mr + b (i.e., radiance is a linear function of Reflectance N N N L f L f ( mr b) m f R b f mr b i i i i i i i i 1 i 1 i 1 where L i is the radiance from a pure pixel of material/end member i. Mixed Pixel Analysis 8 Page 4 4
Mixed Pixels and Spectral Unmixing (cont d) If we can claim to know the spectral reflectance or radiance for the (pure) materials potentially in each pixel (i.e., the end members) And if The materials can be considered to mix linearly Then we can write M simultaneous linear equations in N unknowns (i.e., the fractions are the only unknowns). Mixed Pixel Analysis 9 Mixed pixel analysis Mixed Pixel Analysis 10 Page 5 5
Spectral Mixture Analysis Consider linear mixing of 3 end members in 2 bands. All combinations of these lie along lines connecting the end members in spectral space (no matter how many bands). Band 2 2 3 All combinations lie within the area defined by the outermost pair-wise combinations of end members. 1 Band 1 Mixed Pixel Analysis 11 Spectral Mixture Analysis (cont d) A ternary diagram maps space to a linear combination of end member fractions in a geometric representation with 100% of end members at the extremes. Mixtures of 2 end members lie along the solid lines Mixtures of more than 2 in the interior. A geometric mixture model with 4 end members would be a pyramid with 3 sides and a base having end members at each apex. 2 2 @ 50%; 1 @ 25%; 3 @ 25% 1 0.5 3 Mixed Pixel Analysis 12 Page 6 6
Mixed Pixels and Spectral Unmixing (cont d) SHADOW In order to account for brightness variations due to solar illumination effects and mixed pixels containing shadows, a shadow end member may be introduced. The shadow end member typically is assigned the spectral reflectance expected from a dark shadow element. Ideally this would be a zero reflectance point (a dark object). In cases where shade fractions are needed, but not of interest, the shade fraction can be redistributed to the other fractions. i.e., Each fraction is increased according to + Mixed Pixel Analysis 13 Mixed Pixels and Spectral Unmixing (cont d) In general, to avoid over-fitting the end member model, a smaller number of end members is preferred. Adding more end members to the model will always reduce the residual error, but often amounts to trying to fit information to the noise. Because it is difficult to isolate pure end members (i.e., true extremes), it is often reasonable to allow fractions slightly less than zero or greater than one. For this reason, the partially constrained model is often most appropriate. Mixed Pixel Analysis 14 Page 7 7
Mixed Pixels and Spectral Unmixing (cont d) End member concepts Band 2 End members are assumed to be spectral extrema representing spectrally idealized examples of a land cover type. E 1 E 2 E 3 requires more than 100% of E 2 Band 1 requires less than 0% of E 2 All combinations of end members must lie inside of a convex hull made up of the end members. End member fractions may have values greater than 1 and less than zero. Mixed Pixel Analysis 15 Linked Models Rather than try to unmix a complex image simultaneously with many possible end members, it is often desirable to unmix on a smaller number of end members in a localized region and then link the solutions together. We can use masks to avoid unmixing the same area more than once. Mixed Pixel Analysis 16 Page 8 8
Residual Error Use error vectors as a means of analyzing fraction maps The error vector for each pixel is the vector comprised of the difference between the image radiance (reflectance) vector and the vector predicted by the fraction model. e L EF Make a map of the magnitude of the errors as an indication of locations where the model is inadequate Compare the spectral shape of the error to the spectral features of interest. Mixed Pixel Analysis 17 Residual Error (cont d) The Forested Wetland class is modeled as a combination of green leaf reflectance and 56% shade. The difference between the measured and the modeled spectra is the residual error. Mixed Pixel Analysis 18 Page 9 9
Residual Error (cont d) If a target material is only represented by a small number of pixels, it may be more effective to leave it out of the end member analysis and just look closely at the error vectors. An error may be spectrally localized to characteristic features in the target material. What is an end member and what use are end member maps. Fraction maps Associations as end members Combining fractions to form associations Class maps with transition classes Mixed Pixel Analysis 19 Truth Fraction Maps Fractions Labels Mixed Pixel Analysis 20 Page 10 10