Medical image reconstruction is the process of reproducing an image of an object from the measurements produced from a scanner through some physical process. Deep image reconstruction, which is utilizing deep learning in image reconstruction, has surpassed what is possible compared to classical approaches. However, one requirement of deep learning is a large dataset that covers as many cases as possible and covers different demographics, disease conditions, hardware configurations, and acquisition schemes. Thus, a major limiting factor in deep learning in medical image reconstruction is the training data available. In this dissertation, I address the challenge of imperfect and small data in deep learning. This is addressed through the use of Bayesian deep learning. Bayesian deep learning allows a practitioner to estimate uncertainty to quantify possible sources of error and reduce the tendency of models to overfit. This is used in two application domains: simultaneous positron emission tomography and magnetic resonance imaging (PET/MRI) attenuation correction and in sub-Nyquist accelerated MRI. The use of uncertainty in PET/MRI attenuation correction has allowed the correction of intrinsic dataset errors due to bowel air movement, lack of bone signal, misregistration, and metal implant artifacts. The use of uncertainty in sub-Nyquist accelerated MRI has reduced the overfitting behavior of reconstruction models and allowed for higher possible acceleration factors.

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