Nirmal Keshava and

nessed the collection of measurements with significantly greater spectral breadth and resolution. It has been motivated by a desire to extract increasingly detailed information about the material properties of pixels in a scene for both civilian and military applications. While multispectral sensing has largely succeeded at classifying whole pixels, further analysis of the constituent substances that comprise a pixel is limited by a relatively low number of spectral measurements. The recognition that pixels of interest are frequently a combination of numerous disparate components has introduced a need to quantitatively decompose, or “unmix,” these mixtures. Collecting data in hundreds of spectral bands, hyperspectral sensors have demonstrated the capability of performing spectral unmixing. In hyperspectral imagery, mixed pixels are a mixture of more than one distinct substance, and they exist for one of two reasons. First, if the spatial resolution of a sensor is low enough that disparate materials can jointly occupy a single pixel, the resulting spectral measurement will be some composite of the individual spectra. This is the case for remote sensing platforms flying at a high altitude or performing wide-area surveillance, where low spatial resolution is common. Second, mixed pixels can result when distinct materials are combined into a homogeneous mixture. This circumstance can occur independent of the spatial resolution of the sensor. Spectral unmixing is the procedure by which the measured spectrum of a mixed pixel is decomposed into a collection of constituent spectra, or endmembers, and a set of corresponding fractions, or abundances, that indicate the proportion of each endmember present in the pixel. Endmembers normally correspond to familiar macroscopic objects in the scene, such as water, soil, metal, vegetation, etc. Broadly speaking, unmixing is a special case of the generalized inverse problem that estimates parameters describing an object using an observation(s) of a signal that has interacted with the object before arriving at the sensor [1]. In the case of hyperspectral sensing in the reflective regime, and ignoring atmospheric effects, the incident signal is electromagnetic radiation that originates from the sun and is measured by a sensor after it has been reflected upwards by natural and man-made materials on the surface of the Earth.