Capturing the Multiscale Anatomical Shape Variability with Polyaffine Transformation Trees

International audienceMandible fractures are classified depending on their location. In clinical practice, locations are grouped into regions at different scales according to anatomical, functional and esthetic considerations. Implant design aims at defining the optimal implant for each patient. Emerging population-based techniques analyze the anatomical variability across a population and perform statistical analysis to identify an optimal set of implants. Current efforts are focused on finding clusters of patients with similar characteristics and designing one implant for each cluster. Ideally, the description of anatomical variability is directly connected to the clinical regions. This connection is what we present here, by introducing a new registration method that builds upon a tree of locally affine transformations that describes variability at different scales. We assess the accuracy of our method on 146 CT images of femurs. Two medical experts provide the ground truth by manually measuring six landmarks. We illustrate the clinical importance of our method by clustering 43 CT images of mandibles for implant design. The presented method does not require any application-specific input, which makes it attractive for the analysis of other multiscale anatomical structures. At the core of our new method lays the introduction of a new basis for stationary velocity fields. This basis has very close links to anatomical substructures. In the future, this method has the potential to discover the hidden and possibly sparse structure of the anatomy

Higher-Order Momentum Distributions and Locally Affine LDDMM Registration

To achieve sparse parametrizations that allows intuitive analysis, we aim to represent deformation with a basis containing interpretable elements, and we wish to use elements that have the description capacity to represent the deformation compactly. To accomplish this, we introduce in this paper higher-order momentum distributions in the LDDMM registration framework. While the zeroth order moments previously used in LDDMM only describe local displacement, the first-order momenta that are proposed here represent a basis that allows local description of affine transformations and subsequent compact description of non-translational movement in a globally non-rigid deformation. The resulting representation contains directly interpretable information from both mathematical and modeling perspectives. We develop the mathematical construction of the registration framework with higher-order momenta, we show the implications for sparse image registration and deformation description, and we provide examples of how the parametrization enables registration with a very low number of parameters. The capacity and interpretability of the parametrization using higher-order momenta lead to natural modeling of articulated movement, and the method promises to be useful for quantifying ventricle expansion and progressing atrophy during Alzheimer's disease

Statistical analysis of organs' shapes and deformations: the Riemannian and the affine settings in computational anatomy

International audienceComputational anatomy is an emerging discipline at the interface of geometry, statistics and medicine that aims at analyzing and modeling the biological variability of organs' shapes at the population level. Shapes are equivalence classes of images, surfaces or deformations of a template under rigid body (or more general) transformations. Thus, they belong to non-linear manifolds. In order to deal with multiple samples in non-linear spaces, a consistent statistical framework on Riemannian manifolds has been designed over the last decade. We detail in this chapter the extension of this framework to Lie groups endowed with the affine symmetric connection, a more invariant (and thus more consistent) but non-metric structure on transformation groups. This theory provides strong theoretical bases for the use of one-parameter subgroups and diffeomorphisms parametrized by stationary velocity fields (SVF), for which efficient image registration methods like log-Demons have been developed with a great success from the practical point of view. One can further reduce the complexity with locally affine transformations , leading to parametric diffeomorphisms of low dimension encoding the major shape variability. We illustrate the methodology with the modeling of the evolution of the brain with Alzheimer's disease and the analysis of the cardiac motion from MRI sequences of images

Geodesics, Parallel Transport & One-parameter Subgroups for Diffeomorphic Image Registration

International audienceThe aim of computational anatomy is to develop models for understanding the physiology of organs and tissues. The diffeomorphic non-rigid registration is a validated instrument for the detection of anatomical changes on medical images and is based on a rich mathematical background. For instance, the ''large deformation diffeomoprhic metric mapping'' framework defines a Riemannian setting by providing an opportune right invariant metric on the tangent space, and solves the registration problem by computing geodesics parametrized by time-varying velocity fields. In alternative, stationary velocity fields have been proposed for the diffeomorphic registration based on the one-parameter subgroups from Lie groups theory. In spite of the higher computational efficiency, the geometrical setting of the latter method is more vague, especially regarding the relationship between one-parameter subgroups and geodesics. In this study, we present the relevant properties of the Lie groups for the definition of geometrical properties within the one-parameter subgroups parametrization, and we define the geometrical structure for computing geodesics and for parallel transporting. The theorethical results are applied to the image registration context, and discussed in light of the practical computational problems

HNSF Log-Demons: Diffeomorphic demons registration using hierarchical neighbourhood spectral features

© 2021 The Authors. Many biomedical applications require accurate non-rigid image registration that can cope with complex deformations. However, popular diffeomorphic Demons registration algorithms suffer from difficulties for complex and serious distortions since they only use image greyscale and gradient information. To address these difficulties, a new diffeomorphic Demons registration algorithm is proposed using hierarchical neighbourhood spectral features namely HNSF Log-Demons in this paper. In view of three important properties of hierarchical neighbourhood spectral features based on line graph such as rotation invariance, invariance of linear changes of brightness, and robustness to noise, the hierarchical neighbourhood spectral features of a reference image and a moving image is first extracted and these novel spectral features are incorporated into the energy function of the diffeomorphic registration framework to improve the capability of capturing complex distortions. Secondly, the Nystr ö o ̈ m approximation based on random singular value decomposition is employed to effectively enhance the computational efficiency of HNSF Log-Demons. Finally, the hybrid multi-resolution strategy based on wavelet decomposition in the registration process is utilised to further improve the registration accuracy and efficiency. Experimental results show that the proposed HNSF Log-Demons not only effectively ensures the generation of smooth and reversible deformation field, but also achieves better performance than state-of-the-art algorithms.National Natural Science Foundation of China. Grant Numbers: 61762058, 61861024, 61871259; Natural Science Foundation of Gansu Province of China. Grant Number: 20JR5RA404; Natural Science Basic Research Program of Shaanxi. Grant Number: 2021JC-47

LCC-Demons: a robust and accurate symmetric diffeomorphic registration algorithm

International audienceNon-linear registration is a key instrument for computational anatomy to study the morphology of organs and tissues. However, in order to be an effective instrument for the clinical practice, registration algorithms must be computationally efficient, accurate and most importantly robust to the multiple biases affecting medical images. In this work we propose a fast and robust registration framework based on the log-Demons diffeomorphic registration algorithm. The transformation is parameterized by stationary velocity fields (SVFs), and the similarity metric implements a symmetric local correlation coefficient (LCC). Moreover, we show how the SVF setting provides a stable and consistent numerical scheme for the computation of the Jacobian determinant and the flux of the deformation across the boundaries of a given region. Thus, it provides a robust evaluation of spatial changes. We tested the LCC-Demons in the inter-subject registration setting, by comparing with state-of-the-art registration algorithms on public available datasets, and in the intra-subject longitudinal registration problem, for the statistically powered measurements of the longitudinal atrophy in Alzheimer's disease. Experimental results show that LCC-Demons is a generic, flexible, efficient and robust algorithm for the accurate non-linear registration of images, which can find several applications in the field of medical imaging. Without any additional optimization, it solves equally well intra & inter-subject registration problems, and compares favorably to state-of-the-art methods

Regional flux analysis for discovering and quantifying anatomical changes: An application to the brain morphometry in Alzheimer's disease

International audienceIn this study we introduce the regional flux analysis, a novel approach to deformation based morphometry based on the Helmholtz decomposition of deformations parameterized by stationary velocity fields. We use the scalar pressure map associated to the irrotational component of the deformation to discover the critical regions of volume change. These regions are used to consistently quantify the associated measure of volume change by the probabilistic integration of the flux of the longitudinal deformations across the boundaries. The presented framework unifies voxel-based and regional approaches, and robustly describes the volume changes at both group-wise and subject-specific level as a spatial process governed by consistently defined regions. Our experiments on the large cohorts of the ADNI dataset show that the regional flux analysis is a powerful and flexible instrument for the study of Alzheimer's disease in a wide range of scenarios: cross-sectional deformation based morphometry, longitudinal discovery and quantification of group-wise volume changes, and statistically powered and robust quantification of hippocampal and ventricular atrophy

Geometry-Aware Multiscale Image Registration via OBBTree-Based Polyaffine Log-Demons

Non-linear image registration is an important tool in many areas of image analysis. For instance, in morphometric studies of a population of brains, free-form deformations between images are analyzed to describe the structural anatomical variability. Such a simple deformation model is justified by the absence of an easy expressible prior about the shape changes. Applying the same algorithms used in brain imaging to orthopedic images might not be optimal due to the difference in the underlying prior on the inter-subject deformations. In particular, using an un-informed deformation prior often leads to local minima far from the expected solution. To improve robustness and promote anatomically meaningful deformations, we propose a locally affine and geometry-aware registration algorithm that automatically adapts to the data. We build upon the log-domain demons algorithm and introduce a new type of OBBTree-based regularization in the registration with a natural multiscale structure. The regularization model is composed of a hierarchy of locally affine transformations via their logarithms. Experiments on mandibles show improved accuracy and robustness when used to initialize the demons, and even similar performance by direct comparison to the demons, with a significantly lower degree of freedom. This closes the gap between polyaffine and non-rigid registration and opens new ways to statistically analyze the registration results