Parametric Surface Diffeomorphometry for Low Dimensional Embeddings of Dense Segmentations and Imagery

Abstract

In the field of Computational Anatomy, biological form (including our focus, neuroanatomy) is studied quantitatively through the action of the diffeomorphism group on example anatomies – a technique called diffeomorphometry. Here we design an algorithm within this framework to pass from dense objects common in neuromaging studies (binary segmentations, structural images) to a sparse representation defined on the surface boundaries of anatomical structures, and embedded into the low dimensional coordinates of a parametric model. Our main new contribution is to introduce an expanded group action to simultaneously deform surfaces through direct mapping of points, as well as images through functional composition with the inverse. This allows us to index the diffeomorphisms with respect to 2-dimensional surface geometries like subcortical gray matter structures, but explicitly map onto cost functions determined by noisy 3-dimensional measurements. We consider models generated from empirical covariance of training data, as well as bandlimited (Laplace-Beltrami eigenfunction) models when no such data is available. We show applications to noisy or anomalous segmentations, and other typical problems in neuroimaging studies. We reproduce statistical results detecting changes in Alzheimer’s disease, despite dimensionality reduction. Lastly we apply our algorithm to the common problem of segmenting subcortical structures from T1 MR images.

Entorhinal and transentorhinal atrophy in mild cognitive impairment using longitudinal diffeomorphometry

“…We choose to parametrize v t by a function p 0 supported on the boundary of an atlas surface as in [26]. Describing this surface parametrically through a function $f:U\subset {\mathrm{boldR}}^2\to {\mathrm{boldR}}^3$, our velocity can be written as$v\left(x\right)=\underset{U}{\int }K\left(x,f\left(u\right)\right)p_0\left(u\right)du$This representation is optimal when images to be matched are piecewise constant functions [27] and is a parsimonious model otherwise.…”

“…We choose to parametrize v t by a function p 0 supported on the boundary of an atlas surface as in [26]. Describing this surface parametrically through a function f : U3R 2 /R 3 , our velocity can be written as…”

On variational solutions for whole brain serial-section histology using a Sobolev prior in the computational anatomy random orbit model

“…Since Christensen’s early work [21], diffeomorphic transformation has become the de-facto standard as diffeomorphisms generate one-to-one and onto correspondences between coordinate systems. Herein we focus on the diffeomorphometry orbit model [22, 23] of computational anatomy [24], where the space of dense volume imagery is modelled as a Riemannian orbit of an atlas under the diffeomorphism group. We use the large deformation diffeomorphic metric mapping (LDDMM) algorithm first derived for dense imagery by Beg [25] to retrieve the unknown high-dimensional reparameterization of the template coordinates.…”

A Fully-Automated Subcortical and Ventricular Shape Generation Pipeline Preserving Smoothness and Anatomical Topology

“…In other words, when the segmentations are noisy (like those from the second dataset that the pipeline failed on), the resulting surfaces will inherit the noise (inaccuracy) of the segmentations from MALF. A potential solution is to utilize a much more robust variant of the LDDMM-surface matching, such as the one proposed by Tward and colleagues (Tward et al, 2016 ). Investigation of more advanced surface matching algorithms that are capable of maintaining a high fidelity to the segmentation while being robust to noisy subregions of the segmentations will be one of our future efforts.…”