<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> / Dissertation