RECONSTRUCTING AND SEGMENTING HYPERSPECTRAL IMAGES FROM COMPRESSED MEASUREMENTS Qiang Zhang a Robert Plemmons b David Kittle c David Brady c Sudhakar Prasad d a Biostatistical Sciences, Wake Forest University, Winston-Salem, NC 27157 b Computer Science and Mathematics, Wake Forest University, Winston-Salem, NC 27106 c Electrical and Computer Engineering, Duke University, Durham, NC 27708 d Physics and Astronomy, University of New Mexico, Albuquerque, NM 87131 ABSTRACT A joint reconstruction and segmentation model for hy- perspectral data obtained from a compressive measurement system is described. Although hyperspectral imaging (HSI) technology has incredible potential, its utility is currently lim- ited because of the enormous quantity and complexity of the data it gathers. Yet, often the scene to be reconstructed from the HSI data contains far less information, typically consist- ing of spectrally and spatially homogeneous segments that can be represented sparsely in an appropriate basis. Such vast informational redundancy thus implicitly contained in the HSI data warrants a compressed sensing (CS) strategy that acquires appropriately coded spectral-spatial data from which one can reconstruct the original image more efficiently while still enabling target identification procedures. A coded- aperture snapshot spectral imager (CASSI) that collects com- pressed measurements is considered here, and a joint recon- struction and segmentation model for hyperspectral data ob- tained from CASSI compressive measurements is described. Promising test results on simulated and real data are reported. Index Terms— Hyperspectral images, compressive mea- surements, reconstruction, segmentation, classification, target identification. 1. INTRODUCTION Hyperspectral images are digital images, often taken either from an airplane or satellite, in which each pixel records not just the usual three visible bands of light (red at 650nm, green at 550nm, and blue at 450nm), but on the order of hundreds of wavelengths so that spectroscopy can be conducted on the materials in the object or scene. In air to ground remote sens- ing, the user is then able to identify, for instance, the species of trees and other vegetation, crop health, mineral and soil composition, moisture content of soils and vegetation, and pollution quantities. The technology also has clear military Research supported by the U.S. Air Force Office of Scientific Research (AFOSR). Corresponding author: R. Plemmons, plemmons@wfu.edu, http://www.wfu.edu/˜plemmons and homeland-security applications, as it enables identifica- tion of targets such as buildings and vehicles, even with at- tempts to camouflage, as well as objects in space. It is also possible to detect and identify gas plumes such as those aris- ing from leaks even when the gases are invisible to the human eye. In fact, hyperspectral imaging was used following the attack on the twin towers and the hurricane Katrina disaster to identify dangerous gas leaks, providing guidance and pro- tection to rescuers. We are also working with applications to space situational analysis for the U.S. Air Force [1, 2], where the targets are space objects including assets, such as satel- lites, monitored from the ground by hyperspectral imaging systems. More generally, see [3] and the references therein for a comprehensive overview of recent hyperspectral data analysis and target identification trends. 2. JOINT RECONSTRUCTION AND SEGMENTATION OF SPECTRAL IMAGES TAKEN BY COMPRESSIVE MEASUREMENTS In many applications image targets are sparse in the sense that in some basis they typically occupy a small fraction of the overall region of interest (the target domain). This spar- sity assumption suggests approaching the imaging problem by using the framework of compressed sensing. At the core of compressed sensing lies the following problem (here we focus, as is common in the compressed sensing community, on the discrete setting). Assume f ∈ R n is a signal that is sparse measured by the ℓ 0 quasi-norm; i.e., the number of its nonzero components in f is much less than n, i.e., ‖f ‖ 0 ≪ n. Letting g ∈ R m be the measurement data vector, then the compressive sensing linear inverse problem forward model can be expressed as Hf = g, where H is an m × n sys- tem sensing matrix with m ≪ n. The goal is to recover f , given the data vector g and the sensing matrix H. As m ≪ n, the linear system Hf = g is severely underdetermined and a unique reconstruction of f is in general impossible. How- ever, due to the sparsity of f one can compute f by solving the optimization problem min ‖f ‖ 0 subject to Hf = g. (1)