Medical imaging — MRI, CT, PET, ultrasound, and optical modalities — involves reconstructing internal anatomy from indirect measurements. The raw data are projections, frequency-domain samples, or photon counts that must be inverted to produce the images clinicians interpret. This inverse problem is typically ill-posed: the data are insufficient to uniquely determine the image, and noise further degrades the solution. Bayesian inference provides a principled framework for regularization by encoding prior beliefs about image properties — smoothness, sparsity, anatomical structure — into the reconstruction process.
Bayesian Image Reconstruction
In the Bayesian formulation, the observed data y are related to the unknown image x through a forward model (likelihood) that captures the physics of the imaging system, plus noise. The prior P(x) encodes expectations about the image — for instance, that natural images have sparse gradients (total variation prior), that anatomical structures follow atlas-based templates, or that tissue properties fall within physiologically plausible ranges.
Full Bayesian Reconstruction P(x | y) = P(y | x) · P(x) / P(y)
MAP estimation reduces to an optimization problem and is widely used in clinical practice. Full Bayesian reconstruction goes further by characterizing the entire posterior distribution, providing voxel-wise uncertainty maps that quantify confidence in the reconstructed image. This uncertainty information is valuable for clinical decision-making but computationally demanding.
MRI: Compressed Sensing and Beyond
MRI acquisition is inherently slow because it samples the spatial frequency domain (k-space) sequentially. Compressed sensing MRI, grounded in Bayesian sparse signal recovery, demonstrated that images can be accurately reconstructed from far fewer k-space samples than the Nyquist criterion requires — dramatically reducing scan times. Bayesian compressed sensing extends this by providing uncertainty estimates on the reconstruction, identifying regions where the image may be unreliable.
Modern approaches combine deep neural networks with Bayesian principles. Learned priors — where a neural network trained on thousands of images replaces hand-crafted priors — achieve superior reconstruction quality. Bayesian neural networks and Monte Carlo dropout provide uncertainty estimates alongside the reconstructed images, flagging potential artifacts that could be mistaken for pathology.
PET and SPECT: Photon-Limited Imaging
Positron emission tomography (PET) and single-photon emission computed tomography (SPECT) detect individual photons emitted by radioactive tracers. The data follow Poisson statistics, making the likelihood naturally non-Gaussian. Bayesian reconstruction with Poisson likelihoods and anatomically-informed priors (using co-registered CT or MRI as structural guides) produces higher-quality images than conventional filtered back-projection, enabling lower radiation doses or shorter scan times.
Segmentation and Analysis
Beyond reconstruction, Bayesian methods drive image segmentation — the delineation of anatomical structures and pathological regions. Markov random field (MRF) priors encourage spatial coherence in label assignments. Bayesian atlas-based segmentation combines probabilistic atlases (priors on anatomy) with image intensity models (likelihoods) to segment brain structures, tumors, or cardiac chambers. FreeSurfer, the most widely used brain MRI analysis suite, relies on Bayesian segmentation at its core.
"Every medical image is an answer to an inverse problem. Bayesian inference is the language in which the question — and its uncertainty — are most naturally expressed." — Karl Friston, on the Bayesian foundations of neuroimaging analysis
Current Frontiers
Bayesian methods are central to quantitative MRI, where physical tissue parameters (T1, T2, diffusion coefficients) are estimated from multi-contrast acquisitions with full uncertainty propagation. Information field theory applies Bayesian inference to continuous image fields. And real-time Bayesian inference enables adaptive imaging protocols that focus acquisition on regions of diagnostic uncertainty.