User Manual - pancroma

cover types, a linear mixing model can be formulated. This linear mixing modelcan be used to determine the fractions of each cover type present in the image.For example, consider the six Landsat visible plus NIR bands. A reflectancevalue can be computed for each pixel in the image, a different (usually)reflectance for each band corresponding to that pixel. Let's assume that thesereflectances, measured by the satellite sensors are completely the result ofreflectances from four cover types. As a result, each cover type will contribute afraction F (referred to as endmember Abundance) of its individual reflectance Xat that wavelength to the measured reflectance B for each multispectral Landsatband.Let's further assume that we know the reflectances X in each multispectralwavelength for each of the four end members. These can be obtained fromlibraries as described in previous sections, or from actual ground measurements.F and B can be considered column vectors, F composed of the four unknownAbundances and B composed of the six reflectances (one for each multispectralband) measured by the satellite. We can write the linear relationship among X, Fand B as an equation as follows:X 00 * F 0 + X 01 * F 1 + X 02 * F 2 + X 03 * F 3 = B 0This is just the standard linear mixing equation. We can formulate a set of linearequations that relate the reflectances in each wavelength of the end members tothe measured reflectances Y. The relationship is shown in matrix form below.Endmember X i Type ImageLibrary Spectra Fraction ReflectancesX F Bend1 end2 end3 end4reflectance at wavelength 0 |X00 X01 X02 X03| |F0| |B0|reflectance at wavelength 1 |X10 X11 X12 X13| * |F1| = |B1|reflectance at wavelength 2 |X20 X21 X22 X23| |F2| |B2|reflectance at wavelength 3 |X30 X31 X32 X33| |F3| |B3|reflectance at wavelength 4 |X40 X41 X42 X43| |B4|reflectance at wavelength 5 |X50 X51 X52 X53| |B5|Note that if we had only four equations we could solve for the fractions F exactly.We can get a better solution by including all of the multispectral data we have,and then solving the set of over determined equations using linear regression.Note that the system of equations that we have to solve applies to one pixel only.We will have to repeat the solution process for every pixel in the image. This isobviously a computationally expensive task but it can be accomplished by thePANCROMA TM regression solver.The solution yields four sets of fractions F, one for each pixel in the image.These fractions are usually depicted graphically by scaling them so that therange from 0 to 255 and then rendering them as four grayscale images. The267