We present USLR, a computational framework for longitudinal registration of brain MRI scans to estimate nonlinear image trajectories that are smooth across time, unbiased to any timepoint, and robust to imaging artefacts. It operates on the Lie algebra parameterisation of spatial transforms (which is compatible with rigid transforms and stationary velocity fields for nonlinear deformation) and takes advantage of log-domain properties to solve the problem using Bayesian inference. USRL estimates rigid and nonlinear registrations that: (i) bring all timepoints to an unbiased subject-specific space; and (i) compute a smooth trajectory across the imaging time-series. We capitalise on learning-based registration algorithms and closed-form expressions for fast inference. A use-case Alzheimer's disease study is used to showcase the benefits of the pipeline in multiple fronts, such as time-consistent image segmentation to reduce intra-subject variability, subject-specific prediction or population analysis using tensor-based morphometry. We demonstrate that such approach improves upon cross-sectional methods in identifying group differences, which can be helpful in detecting more subtle atrophy levels or in reducing sample sizes in clinical trials. The code is publicly available in https://github.com/acasamitjana/uslr

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