A New Fast Algorithm for Linearly Unmixing Hyperspectral Images

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Abstract
Linear spectral unmixing is nowadays an essential tool to analyze remotely sensed hyperspectral images. Although many different contributions have been uncovered during the last two decades, the majority of them are based on dividing the whole process of linearly unmixing a given hyperspectral image into three sequential steps: 1) estimation of the number of endmembers that are present in the hyperspectral image under consideration; 2) extraction of these endmembers from the hyperspectral data set; and 3) calculation of the abundances associated with the endmembers induced in the previous step per each mixed pixel of the image. Although this de facto processing chain has proven to be accurate enough for unmixing most of the images collected by hyperspectral remote sensors, it is also true that it is not exempt of drawbacks, such as the fact that all the possible combinations of algorithms in order to fully unmix a hyperspectral image according to the aforementioned processing chain demand a formidable computational effort, which tends to be higher the better the performance of the designed unmixing chain is. This troublesome issue unfortunately prevents the use of hyperspectral imaging technology in applications under real-time constraints, in which hyperspectral images have to be analyzed in a short period of time. Hence, there is a clear need to face the challenge of fully exploiting the unquestionable benefits of the hyperspectral imaging technology for these applications, but concurrently overcoming the limitations imposed by the computationally complex nature of the processes involved. For this purpose, this paper introduces a novel algorithm named fast algorithm for linearly unmixing hyperspectral images (FUN), which is capable of fully unmixing a hyperspectral image with at least the same accuracy than state-of-the-art approaches while demanding a much lower computational effort, independent of the characteristics of the image under analysis. The FUN algorithm is based on the concept of orthogonal projections and allows performing the estimation of the number of endmembers and their extraction simultaneously, using the modified Gram-Schmidt method. The operations performed by the FUN algorithm are simple and can be easily parallelized. Moreover, this algorithm is able to calculate the abundances using very similar operations, also based on orthogonal projections, which makes it easier to achieve a hardware implementation to perform the entire unmixing process. The benefits of our proposal are demonstrated with a diverse set of artificially generated hyperspectral images and with the well-known AVIRIS Cuprite image, for which the proposed FUN algorithm is able to reduce in a factor of more than 31 times the time required for processing it, while providing a better unmixing performance than traditional methods.

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