Predicting infant cortical surface development using a 4D varifold-based learning framework and local topography-based shape morphing

Longitudinal neuroimaging analysis methods have remarkably advanced our understanding of early postnatal brain development. However, learning predictive models to trace forth the evolution trajectories of both normal and abnormal cortical shapes remains broadly absent. To fill this critical gap, we pioneered the first prediction model for longitudinal developing cortical surfaces in infants using a spatiotemporal current-based learning framework solely from the baseline cortical surface. In this paper, we detail this prediction model and even further improve its performance by introducing two key variants. First, we use the varifold metric to overcome the limitations of the current metric for surface registration that was used in our preliminary study. We also extend the conventional varifold-based surface registration model for pairwise registration to a spatiotemporal surface regression model. Second, we propose a morphing process of the baseline surface using its topographic attributes such as normal direction and principal curvature sign. Specifically, our method learns from longitudinal data both the geometric (vertices positions) and dynamic (temporal evolution trajectories) features of the infant cortical surface, comprising a training stage and a prediction stage. In the training stage, we use the proposed varifold-based shape regression model to estimate geodesic cortical shape evolution trajectories for each training subject. We then build an empirical mean spatiotemporal surface atlas. In the prediction stage, given an infant, we select the best learnt features from training subjects to simultaneously predict the cortical surface shapes at all later timepoints, based on similarity metrics between this baseline surface and the learnt baseline population average surface atlas. We used a leave-one-out cross validation method to predict the inner cortical surface shape at 3, 6, 9 and 12 months of age from the baseline cortical surface shape at birth. Our method attained a higher prediction accuracy and better captured the spatiotemporal dynamic change of the highly folded cortical surface than the previous proposed prediction method

Sparse Adaptive Parameterization of Variability in Image Ensembles

International audienceThis paper introduces a new parameterization of diffeomorphic deformations for the characterization of the variability in image ensembles. Dense diffeomorphic deformations are built by interpolating the motion of a finite set of control points that forms a Hamiltonian flow of self-interacting particles. The proposed approach estimates a template image representative of a given image set, an optimal set of control points that focuses on the most variable parts of the image, and template-to-image registrations that quantify the variability within the image set. The method automatically selects the most relevant control points for the characterization of the image variability and estimates their optimal positions in the template domain. The optimization in position is done during the estimation of the deformations without adding any computational cost at each step of the gradient descent. The selection of the control points is done by adding a L 1 prior to the objective function, which is optimized using the FISTA algorithm

Using longitudinal metamorphosis to examine ischemic stroke lesion dynamics on perfusion-weighted and relation to final outcome on T2-w images

AbstractWe extend the image-to-image metamorphosis into constrained longitudinal metamorphosis. We apply it to estimate an evolution scenario, in patients with acute ischemic stroke, of both scattered and solitary ischemic lesions visible on serial MR perfusion weighted imaging from acute to subacute stages. We then estimate a patient-specific residual map that enables us to capture the most relevant shape and intensity changes, continuously, as the lesion evolves from acute through subacute to chronic timepoints until merging into the final image. We detect areas with high residuals (i.e., high dynamics) and identify areas that became part of the final T2-w lesion obtained at ≥1month after stroke. This allows the investigation of the dynamic influence of perfusion values on the final lesion outcome as seen on T2-w imaging. The model provides detailed insights into stroke lesion dynamic evolution in space and time that will help identify factors that determine final outcome and identify targets for interventions to improve outcome

Existence and Continuity of Minimizers for the Estimation of Growth Mapped Evolutions for Current Data Term and Couterexamples for Varifold Data Term

In the field of computational anatomy, the complexity of changes occurring during the evolution of a living shape while it is growing, aging or reacting to a disease, calls for more and more accurate models to allow subject comparison. Growth mapped evolutions have been introduced to tackle the loss of homology between two ages of an organism following a growth process that involves creation of new material. They model the evolution of longitudinal shape data with partial mappings. One viewpoint consists in a progressive embedding of the shape into an ambient space on which acts a group of diffeomorphisms. In practice, the shape evolves through a time-varying dynamic called the growth dynamic. The concept of shape space has now been widely studied and successfully applied to analyze the variability of a population of related shapes. Time-varying dynamics subsequently enlarge this framework and open the door to new optimal control problems for the assimilation of longitudinal shape data. We address in this paper an interesting problem in the field of the calculus of variations to investigate the existence and continuity of solutions for the registration of growth mapped evolutions with the growth dynamic. This theoretical question highlights the unexpected role of the data term grounded either on current or varifold representations. Indeed, in this new framework, the spatial regularity of a continuous scenario estimated from a temporal sequence of shapes with the growth dynamic depends on the temporal regularity of the deformation. Current metrics have the property to be more robust to this spatial regularity than varifold metrics. We will establish the existence and continuity of global minimizers for current data term and highlight two counterexamples for varifold data term

Do ideas have shape? Plato's theory of forms as the continuous limit of artificial neural networks

We show that ResNets converge, in the infinite depth limit, to a generalization of image registration algorithms. In this generalization, images are replaced by abstractions (ideas) living in high dimensional RKHS spaces, and material points are replaced by data points. Whereas computational anatomy aligns images via deformations of the material space, this generalization aligns ideas by via transformations of their RKHS. This identification of ResNets as idea registration algorithms has several remarkable consequences. The search for good architectures can be reduced to that of good kernels, and we show that the composition of idea registration blocks with reduced equivariant multi-channel kernels (introduced here) recovers and generalizes CNNs to arbitrary spaces and groups of transformations. Minimizers of L2 regularized ResNets satisfy a discrete least action principle implying the near preservation of the norm of weights and biases across layers. The parameters of trained ResNets can be identified as solutions of an autonomous Hamiltonian system defined by the activation function and the architecture of the ANN. Momenta variables provide a sparse representation of the parameters of a ResNet. The registration regularization strategy provides a provably robust alternative to Dropout for ANNs. Pointwise RKHS error estimates lead to deterministic error estimates for ANNs