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<p>This dissertation presents methods and associated performance bounds for spectral
image processing tasks such as reconstruction and target detection, which are useful
in a variety of applications such as astronomical imaging, biomedical imaging and
remote sensing. The key idea behind our spectral image processing methods is the fact
that important information in a spectral image can often be captured by low-dimensional
manifolds embedded in high-dimensional spectral data. Based on this key idea, our
work focuses on the reconstruction of spectral images from <italic>photon-limited</italic>,
and distorted observations. </p><p>This dissertation presents a partition-based, maximum
penalized likelihood method that recovers spectral images from noisy observations
and enjoys several useful properties; namely, it (a) adapts to spatial and spectral
smoothness of the underlying spectral image, (b) is computationally efficient, (c)
is near-minimax optimal over an <italic>anisotropic</italic> Holder-Besov function
class, and (d) can be extended to inverse problem frameworks.</p><p>There are many
applications where accurate localization of desired targets in a spectral image is
more crucial than a complete reconstruction. Our work draws its inspiration from classical
detection theory and compressed sensing to develop computationally efficient methods
to detect targets from few projection measurements of each spectrum in the spectral
image. Assuming the availability of a spectral dictionary of possible targets, the
methods discussed in this work detect targets that either come from the spectral dictionary
or otherwise. The theoretical performance bounds offer insight on the performance
of our detectors as a function of the number of measurements, signal-to-noise ratio,
background contamination and properties of the spectral dictionary. </p><p>A related
problem is that of level set estimation where the goal is to detect the regions in
an image where the underlying intensity function exceeds a threshold. This dissertation
studies the problem of accurately extracting the level set of a function from indirect
projection measurements without reconstructing the underlying function. Our partition-based
set estimation method extracts the level set of proxy observations constructed from
such projection measurements. The theoretical analysis presented in this work illustrates
how the projection matrix, proxy construction and signal strength of the underlying
function affect the estimation performance.</p>
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