Joint registration of a stack of 2D histological sections to recover 3D structure (3D histology reconstruction) finds application in areas such as atlas building and validation of in vivo imaging. Straighforward pairwise registration of neighbouring sections yields smooth reconstructions but has well-known problems such as banana effect (straightening of curved structures) and z-shift (drift). While these problems can be alleviated with an external, linearly aligned reference (e.g., Magnetic Resonance images), registration is often inaccurate due to contrast differences and the strong nonlinear distortion of the tissue, including artefacts such as folds and tears. In this paper, we present a probabilistic model of spatial deformation that yields reconstructions for multiple histological stains that that are jointly smooth, robust to outliers, and follow the reference shape. The model relies on a spanning tree of latent transforms connecting all the sections and slices, and assumes that the registration between any pair of images can be see as a noisy version of the composition of (possibly inverted) latent transforms connecting the two images. Bayesian inference is used to compute the most likely latent transforms given a set of pairwise registrations between image pairs within and across modalities. Results on synthetic deformations on multiple MR modalities, show that our method can accurately and robustly register multiple contrasts even in the presence of outliers. The 3D histology reconstruction of two stains (Nissl and parvalbumin) from the Allen human brain atlas, show its benefits on real data with severe distortions. We also provide the correspondence to MNI space, bridging the gap between two of the most used atlases in histology and MRI. Data is available at https://openneuro.org/datasets/ds003590 and code at https://github.com/acasamitjana/3dhirest.
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