Abstract:The bioluminescence tomography is a novel molecular imaging technology for small animal studies. Known reconstruction methods require the completely measured data on the external surface, although only partially measured data is available in practice. In this work, we formulate a mathematical model for BLT from partial data and generalize our previous results on the solution uniqueness to the partial data case. Then we extend two of our reconstruction methods for BLT to this case. The first method is a variant of the well-known EM algorithm. The second one is based on the Landweber scheme. Both methods allow the incorporation of knowledgebased constraints. Two practical constraints, the source non-negativity and support constraints, are introduced to regularize the BLT problem and produce stability. The initial choice of both methods and its influence on the regularization and stability are also discussed. The proposed algorithms are evaluated and validated with intensive numerical simulation and a physical phantom experiment. Quantitative results including the location and source power accuracy are reported. Various algorithmic issues are investigated, especially how to avoid the inverse crime in numerical simulations.

This paper considers a distributionally robust version of a quadratic knapsack problem. In this model, a subsets of items is selected to maximizes the total profit while requiring that a set of knapsack constraints be satisfied with high probability. In contrast to the stochastic programming version of this problem, we assume that only part of the information on random data is known, i.e., the first and second moment of the random variables, their joint support, and possibly an independence assumption. As for the binary constraints, special interest is given to the corresponding semidefinite programming (SDP) relaxation. While in the case that the model only has a single knapsack constraint we present an SDP reformulation for this relaxation, the case of multiple knapsack constraints is more challenging. Instead, two tractable methods are presented for providing upper and lower bounds (with its associated conservative solution) on the SDP relaxation. An extensive computational study is given to illustrate the tightness of these bounds and the value of the proposed distributionally robust approach.

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