Abstract
In medical image analysis, constructing an atlas, i.e. a mean representative of an ensemble of images, is a critical task for practitioners to estimate variability of shapes inside a population, and to characterise and understand how structural shape changes have an impact on health. This involves identifying significant shape constituents of a set of images, a process called segmentation, and mapping this group of images to an unknown mean image, a task called registration, making a statistical analysis of the image population possible. To achieve this goal, we propose treating these operations jointly to leverage their positive mutual influence, in a hyperelasticity setting, by viewing the shapes to be matched as Ogden materials. The approach is complemented by novel hard constraints on the L ∞ norm of both the Jacobian and its inverse, ensuring that the deformation is a bi-Lipschitz homeomorphism. Segmentation is based on the Potts model, which allows for a partition into more than two regions, i.e. more than one shape. The connection to the registration problem is ensured by the dissimilarity measure that aims to align the segmented shapes. A representation of the deformation field in a linear space equipped with a scalar product is then computed in order to perform a geometry-driven Principal Component Analysis (PCA) and to extract the main modes of variations inside the image population. Theoretical results emphasizing the mathematical soundness of the model are provided, among which existence of minimisers, analysis of a numerical method, asymptotic results and a PCA analysis, as well as numerical simulations demonstrating the ability of the model to produce an atlas exhibiting sharp edges, high contrast and a consistent shape.
Highlights
The registration operation can be viewed as the inclusion of priors to guide the segmentation process, in particular, for the questions of topology-preservation and geometric priors
This paper addressed the twofold question of finding an average representative of a dataset of different subjects and deriving some statistics by identifying the main modes of variation
The problem is envisioned as a joint registration/segmentation one, based on nonlinear elasticity concepts
Summary
Joint image processing models have experienced increasing attention, including combined segmentation/registration models [33, 37] (joint phase field approximation and registration), [52] (model based on metric structure comparison), [29, 68] (level set formulation that merges the piecewise constant Mumford--Shah model with registration principles), [36] (grounded in the expectation maximization algorithm), [28] (based on a nonlocal characterization of weighted-total variation and nonlocal shape descriptors), or [2, 48, 59, 62, 70, 76]; joint image reconstruction and motion estimation [11, 16, 22, 58, 64, 69, 15, 53, 8]; joint reconstruction and registration for postacquisition motion correction [25] with the goal to reconstruct a single motion-free image and retrieve the physiological dynamics through the deformation maps, joint optical flow estimation with phase field segmentation of the flow field [14], or joint segmentation/optimal transport models [12] (to determine the velocity of blood flow in vascular structures). The proposed work adopts this joint model philosophy It aims at addressing the issue of designing a unified variational model for joint segmentation, registration, and atlas generation by exploiting the strong correlation between the two former tasks reducing error propagation, in the medical imaging setting. The latter one requires the mapping of a group of images to a mean representative, which is an additional unknown of the problem, the subsequent goal being to extract a relevant hidden structure from this ensemble of images. This way of looking at shapes entails substantial modifications in the design of the functional to be minimized and in the search for an appropriate representative of the deformation in a vector space
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