Position emission tomography (PET) is a powerful functional imaging modality with wide uses in fields such as oncology, cardiology, and neurology. Motivated by imaging datasets from a psoriasis clinical trial and a cohort of Alzheimer's disease (AD) patients, several interesting methodological challenges were identified in various steps of quantitative analysis of PET data. In Chapter 1, we consider a classification scenario of bivariate thresholding of a predictor using an upper and lower cutpoints, as motivated by an image segmentation problem of the skin. We introduce a generalization of ROC analysis and the concept of the parameter path in ROC space of a classifier. Using this framework, we define the optimal ROC (OROC) to identify and assess performance of optimal classifiers, and describe a novel nonparametric estimation of OROC which simultaneous estimates the parameter path of the optimal classifier. In simulations, we compare its performance to alternative methods of OROC estimation. In Chapter 2, we develop a novel method to normalize PET images as an essential preprocessing step for quantitative analysis. We propose a method based on application of functional data analysis to image intensity distribution functions, assuming that that individual image density functions are variations from a template density. By modeling the warping functions using a modified function-on-scalar regression, the variations in density functions due to nuisance parameters are estimated and subsequently removed for normalization. Application to our motivating data indicate persistence of residual variations in standardized image densities. In Chapter 3, we propose a nonlinear mixed effects framework to model amyloid-beta (Aβ), an important biomarker in AD. We incorporate the hypothesized functional form of Aβ trajectory by assuming a common trajectory model for all subjects with variations in the location parameter, and a mixture distribution for the random effects of the location parameter address our empirical findings that some subjects may not accumulate Aβ. Using a Bayesian hierarchical model, group differences are specified into the trajectory parameters. We show in simulation studies that the model closely estimates the true parameters under various scenarios, and accurately estimates group differences in the age of onset.