DEBLURRING AND SPARSE UNMIXING OF HYPERSPECTRAL IMAGES USING MULTIPLE POINT SPREAD FUNCTIONS SEBASTIAN BERISHA * , JAMES G. NAGY † , AND ROBERT J. PLEMMONS ‡ Abstract. This paper is concerned with deblurring and spectral analysis of ground-based as- tronomical images of space objects. A numerical approach is provided for deblurring and sparse unmixing of ground-based hyperspectral images (HSI) of objects taken through atmospheric tur- bulence. Hyperspectral imaging systems capture a 3D datacube (tensor) containing: 2D spatial information, and 1D spectral information at each spatial location. Pixel intensities vary with wave- length bands providing a spectral trace of intensity values, and generating a spatial map of spectral variation (spectral signatures of materials). The deblurring and spectral unmixing problem is quite challenging since the point spread function (PSF) depends on the imaging system as well as the seeing conditions and is wavelength varying. We show how to efficiently construct an optimal Kro- necker product-based preconditioner, and provide numerical methods for estimating the multiple PSFs using spectral data from an isolated (guide) star for joint deblurring and sparse unmixing the HSI datasets in order to spectrally analyze the image objects. The methods are illustrated with numerical experiments on a commonly used test example, a simulated HSI of the Hubble Space Telescope satellite. Key words. image deblurring, hyperspectral imaging, preconditioning, least squares, ADMM, Kronecker product AMS Subject Classifications: 65F20, 65F30 1. Introduction. Information about the material composition of an object is contained most unequivocally in the spectral profiles of the brightness at the different surface pixels of the object. Thus by acquiring the surface brightness distribution in narrow spectral channels, as in hyperspectral image (HSI) data cubes, and by per- forming spectral unmixing on such data cubes, one can infer material identities as functions of position on the object surface [1]. Spectral unmixing involves the com- putation of the fractional contribution of elementary spectra, called endmembers. By assuming the measured spectrum of each mixed pixel in an HSI is a linear combination of spectral signatures of endmembers, the underlying image model can be formulated as a linear mixture of endmembers with nonnegative and sparse coefficients G = XM + N , where M ∈R Nm×Nw represents a spectral library containing N m spectral signatures of endmembers with N w spectral bands or wavelengths, G ∈R Np×Nw is the observed data matrix (each row contains the observed spectrum of a given pixel, and we use N p to denote the number of pixels in each image), and X ∈R Np×Nm contains the frac- tional abundances of endmembers (each column contains the fractional abundances of a given endmember). Here, we assume X is a sparse matrix and N ∈R Np×Nw is a matrix representing errors or noise affecting the measurements at each spectral * Department of Mathematics and Computer Science, Emory University. Email: sber- ish@emory.edu. † Department of Mathematics and Computer Science, Emory University. Email: nagy@mathcs.emory.edu. Research supported in part by the AFOSR under grant FA9550-09-1-0487, and by the US National Science Foundation under grant no. DMS-1115627. ‡ Department of Computer Science, Wake Forest University, Winston-Salem, NC, USA. Email: plemmons@wfu.edu. His work was supported by grant no. FA9550-11-1-0194 from the US Air Force Office of Scientific Research and by contract no. HM1582-10-C-0011 from the US National Geospatial- Intelligence Agency. 1