In many medical, security or genomics applications, decisions must be made about the state of an object. An example is determination of a tumor's physiological state from CT perfusion imaging data, which is a dynamic imaging modality used for monitoring physiological changes in the body over a period of time. Such decision making problems are challenging since the observations are high-dimensional and training data is very limited. Furthermore, the true object of interest (e.g. a physiological parameter in CT perfusion imaging) is not directly observable but it is indirectly related to the observations through a sensing process. When this sensing mechanism is known, the conventional approach is to invert it and design a classifier in the reconstructed domain. The first goal of this thesis is to develop novel inversion technics for CT perfusion imaging which will improve subsequent decision making in the reconstructed domain. Motivated by CT perfusion imaging and other problems that share the same structure, the second goal is to develop tools to exploit knowledge of the latent sensing structure and to contribute to the understanding of fundamental performance limits in the setting of high-dimensional decision making with limited training data.

The contributions of this thesis towards these two main goals are presented in two parts. In the first part, a detailed account of tracer kinetic methods for modeling the CT perfusion (CTp) imaging data is provided and it is shown that this type of modeling yields a linear sensing structure. Then, novel algorithms that incorporate spatial and temporal correlations into the reconstruction process are developed to solve the high-dimensional inversion problem. It is also shown that the new reconstruction method results in improved estimation of important physiological parameters which are used in a rectal cancer study to decide whether a voxel is in the cancerous region.

In the second part of the thesis an abstract classification problem with an underlying sensing structure is considered to explore fundamental limits to classification performance when data dimensionality is much larger than the number of training examples. To this end, the asymptotic performance of various classification strategies that incorporate different levels of prior knowledge on the sensing mechanism is analyzed. In particular it is first proven that strategies based on ignoring the sensing structure and naively estimating all model parameters will result in a classification performance asymptotically no better than guessing. Then it is shown that projection-based classification rules that properly utilize knowledge of the sensing structure can attain Bayes-optimal risk. Finally, the theoretical findings are validated through simulations and also the advantage of the sensing-aware classification approach over a well-known learning-based method (support vector machines), which ignores the underlying structure in the data, is demonstrated.