We consider analysis of noisy and incomplete hyperspectral imagery, with the objective of removing the noise and inferring the missing data. The noise statistics may be wavelength-dependent, and the fraction of data missing (at random) may be substantial, including potentially entire bands, offering the potential to significantly reduce the quantity of data that need be measured. To achieve this objective, the imagery is divided into contiguous three-dimensional (3D) spatio-spectral blocks, of spatial dimension much less than the image dimension. It is assumed that each such 3D block may be represented as a linear combination of dictionary elements of the same dimension, plus noise, and the dictionary elements are learned in situ based on the observed data (no a priori training). The number of dictionary elements needed for representation of any particular block is typically small relative to the block dimensions, and all the image blocks are processed jointly ("collaboratively") to infer the underlying dictionary. We address dictionary learning from a Bayesian perspective, considering two distinct means of imposing sparse dictionary usage. These models allow inference of the number of dictionary elements needed as well as the underlying wavelength-dependent noise statistics. It is demonstrated that drawing the dictionary elements from a Gaussian process prior, imposing structure on the wavelength dependence of the dictionary elements, yields significant advantages, relative to the more-conventional approach of using an i.i.d. Gaussian prior for the dictionary elements; this advantage is particularly evident in the presence of noise. The framework is demonstrated by processing hyperspectral imagery with a significant number of voxels missing uniformly at random, with imagery at specific wavelengths missing entirely, and in the presence of substantial additive noise.

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Nonparametric Bayesian methods are considered for recovery of imagery based upon compressive, incomplete, and/or noisy measurements. A truncated beta-Bernoulli process is employed to infer an appropriate dictionary for the data under test and also for image recovery. In the context of compressive sensing, significant improvements in image recovery are manifested using learned dictionaries, relative to using standard orthonormal image expansions. The compressive-measurement projections are also optimized for the learned dictionary. Additionally, we consider simpler (incomplete) measurements, defined by measuring a subset of image pixels, uniformly selected at random. Spatial interrelationships within imagery are exploited through use of the Dirichlet and probit stick-breaking processes. Several example results are presented, with comparisons to other methods in the literature.

Recent advances in sparse modeling and dictionary learning for discriminative applications show high potential for numerous classification tasks. In this paper, we show that highly accurate material classification from hyperspectral imagery (HSI) can be obtained with these models, even when the data is reconstructed from a very small percentage of the original image samples. The proposed supervised HSI classification is performed using a measure that accounts for both reconstruction errors and sparsity levels for sparse representations based on class-dependent learned dictionaries. Combining the dictionaries learned for the different materials, a linear mixing model is derived for sub-pixel classification. Results with real hyperspectral data cubes are shown both for urban and non-urban terrain.

Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number. A method is presented for sub-pixel mapping and classification in hyperspectral imagery, using learned blockstructured discriminative dictionaries, where each block is adapted and optimized to represent a material in a compact and sparse manner. The spectral pixels are modeled by linear combinations of subspaces defined by the learned dictionary atoms, allowing for linear mixture analysis. This model provides flexibility in the sources representation and selection, thus accounting for spectral variability, small-magnitude errors, and noise. A spatial-spectral coherence regularizer in the optimization allows for pixels classification to be influenced by similar neighbors. We extend the proposed approach for cases for which there is no knowledge of the materials in the scene, unsupervised classification and provide experiments and comparisons with simulated and real data. We also present results when the data have been significantly under-sampled and then reconstructed, still retaining high-performance classification, showing the potential role of compressive sensing and sparse modeling techniques in efficient acquisition/transmission missions for hyperspectral imagery. Alexey Castrodad, Zhengming Xing, John Greer, Edward Bosch, Lawrence Carin, and Guillermo Sapiro Abstract A method is presented for sub-pixel mapping and classification in hyperspectral imagery, using learned blockstructured discriminative dictionaries, where each block is adapted and optimized to represent a material in a compact and sparse manner. The spectral pixels are modeled by linear combinations of subspaces defined by the learned dictionary atoms, allowing for linear mixture analysis. This model provides flexibility in the sources representation and selection, thus accounting for spectral variability, small-magnitude errors, and noise. A spatial-spectral coherence regularizer in the optimization allows for pixels classification to be influenced by similar neighbors. We extend the proposed approach for cases for which there is no knowledge of the materials in the scene, unsupervised classification, and provide experiments and comparisons with simulated and real data. We also present results when the data have been significantly under...