Joan Glaunès

Pierre-Louis Antonsanti, Thomas Benseghir, Vincent Jugnon et al.

Partial shapes correspondences is a problem that often occurs in computer vision (occlusion, evolution in time...). In medical imaging, data may come from different modalities and be acquired under different conditions which leads to variations in shapes and topologies. In this paper we use an asymmetric data dissimilarity term applicable to various geometric shapes like sets of curves or surfaces, assessing the embedding of a shape into another one without relying on correspondences. It is designed as a data attachment for the Large Deformation Diffeomorphic Metric Mapping (LDDMM) framework, allowing to compute a meaningful deformation of one shape onto a subset of the other. We refine it in order to control the resulting non-rigid deformations and provide consistent deformations of the shapes along with their ambient space. We show that partial matching can be used for robust multi-modal liver registration between a Computed Tomography (CT) volume and a Cone Beam Computed Tomography (CBCT) volume. The 3D imaging of the patient CBCT at point of care that we call live is truncated while the CT pre-intervention provides a full visualization of the liver. The proposed method allows the truncated surfaces from CBCT to be aligned non-rigidly, yet realistically, with surfaces from CT with an average distance of 2.6mm(+/- 2.2). The generated deformations extend consistently to the liver volume, and are evaluated on points of interest for the physicians, with an average distance of 5.8mm (+/- 2.7) for vessels bifurcations and 5.13mm (+/- 2.5) for tumors landmarks. Such multi-modality volumes registrations would help the physicians in the perspective of navigating their tools in the patient's anatomy to locate structures that are hardly visible in the CBCT used during their procedures. Our code is available at https://github.com/plantonsanti/PartialMatchingVarifolds.

Anton François, Pietro Gori, Joan Glaunès

In this paper, we propose an implementation of both Large Deformation Diffeomorphic Metric Mapping (LDDMM) and Metamorphosis image registration using a semi-Lagrangian scheme for geodesic shooting. We propose to solve both problems as an inexact matching providing a single and unifying cost function. We demonstrate that for image registration the use of a semi-Lagrangian scheme is more stable than a standard Eulerian scheme. Our GPU implementation is based on PyTorch, which greatly simplifies and accelerates the computations thanks to its powerful automatic differentiation engine. It will be freely available at https://github.com/antonfrancois/Demeter_metamorphosis.

Robert Hu, Siu Lun Chau, Dino Sejdinovic et al.

Kernel matrix-vector multiplication (KMVM) is a foundational operation in machine learning and scientific computing. However, as KMVM tends to scale quadratically in both memory and time, applications are often limited by these computational constraints. In this paper, we propose a novel approximation procedure coined \textit{Faster-Fast and Free Memory Method} ($\fthreem$) to address these scaling issues of KMVM for tall~($10^8\sim 10^9$) and skinny~($D\leq7$) data. Extensive experiments demonstrate that $\fthreem$ has empirical \emph{linear time and memory} complexity with a relative error of order $10^{-3}$ and can compute a full KMVM for a billion points \emph{in under a minute} on a high-end GPU, leading to a significant speed-up in comparison to existing CPU methods. We demonstrate the utility of our procedure by applying it as a drop-in for the state-of-the-art GPU-based linear solver FALKON, \emph{improving speed 1.5-5.5 times} at the cost of $<1\%$ drop in accuracy. We further demonstrate competitive results on \emph{Gaussian Process regression} coupled with significant speedups on a variety of real-world datasets.

Matthis Maillard, Anton François, Joan Glaunès et al.

In medical imaging, most of the image registration methods implicitly assume a one-to-one correspondence between the source and target images (i.e., diffeomorphism). However, this is not necessarily the case when dealing with pathological medical images (e.g., presence of a tumor, lesion, etc.). To cope with this issue, the Metamorphosis model has been proposed. It modifies both the shape and the appearance of an image to deal with the geometrical and topological differences. However, the high computational time and load have hampered its applications so far. Here, we propose a deep residual learning implementation of Metamorphosis that drastically reduces the computational time at inference. Furthermore, we also show that the proposed framework can easily integrate prior knowledge of the localization of topological changes (e.g., segmentation masks) that can act as spatial regularization to correctly disentangle appearance and shape changes. We test our method on the BraTS 2021 dataset, showing that it outperforms current state-of-the-art methods in the alignment of images with brain tumors.

Pierre-Louis Antonsanti, Joan Glaunès, Thomas Benseghir et al.

In computer vision and medical imaging, the problem of matching structures finds numerous applications from automatic annotation to data reconstruction. The data however, while corresponding to the same anatomy, are often very different in topology or shape and might only partially match each other. We introduce a new asymmetric data dissimilarity term for various geometric shapes like sets of curves or surfaces. This term is based on the Varifold shape representation and assesses the embedding of a shape into another one without relying on correspondences between points. It is designed as data attachment for the Large Deformation Diffeomorphic Metric Mapping (LDDMM) framework, allowing to compute meaningful deformation of one shape onto a subset of the other. Registrations are illustrated on sets of synthetic 3D curves, real vascular trees and livers' surfaces from two different modalities: Computed Tomography (CT) and Cone Beam Computed Tomography (CBCT). All experiments show that this data dissimilarity term leads to coherent partial matching despite the topological differences.

Pierre-Louis Antonsanti, Thomas Benseghir, Vincent Jugnon et al.

Automatic annotation of anatomical structures can help simplify workflow during interventions in numerous clinical applications but usually involves a large amount of annotated data. The complexity of the labeling task, together with the lack of representative data, slows down the development of robust solutions. In this paper, we propose a solution requiring very few annotated cases to label 3D pelvic arterial trees of patients with benign prostatic hyperplasia. We take advantage of Large Deformation Diffeomorphic Metric Mapping (LDDMM) to perform registration based on meaningful deformations from which we build an atlas. Branch pairing is then computed from the atlas to new cases using optimal transport to ensure one-to-one correspondence during the labeling process. To tackle topological variations in the tree, which usually degrades the performance of atlas-based techniques, we propose a simple bottom-up label assignment adapted to the pelvic anatomy. The proposed method achieves 97.6\% labeling precision with only 5 cases for training, while in comparison learning-based methods only reach 82.2\% on such small training sets.

Benjamin Charlier, Jean Feydy, Joan Alexis Glaunès et al.

The KeOps library provides a fast and memory-efficient GPU support for tensors whose entries are given by a mathematical formula, such as kernel and distance matrices. KeOps alleviates the major bottleneck of tensor-centric libraries for kernel and geometric applications: memory consumption. It also supports automatic differentiation and outperforms standard GPU baselines, including PyTorch CUDA tensors or the Halide and TVM libraries. KeOps combines optimized C++/CUDA schemes with binders for high-level languages: Python (Numpy and PyTorch), Matlab and GNU R. As a result, high-level "quadratic" codes can now scale up to large data sets with millions of samples processed in seconds. KeOps brings graphics-like performances for kernel methods and is freely available on standard repositories (PyPi, CRAN). To showcase its versatility, we provide tutorials in a wide range of settings online at \url{www.kernel-operations.io}.

Maxime Berar, Françoise Tilotta, Joan Alexis Glaunès et al.

In this paper, we present a computer-assisted method for facial reconstruction. This method provides an estimation of the facial shape associated with unidentified skeletal remains. Current computer-assisted methods using a statistical framework rely on a common set of extracted points located on the bone and soft-tissue surfaces. Most of the facial reconstruction methods then consist of predicting the position of the soft-tissue surface points, when the positions of the bone surface points are known. We propose to use Latent Root Regression for prediction. The results obtained are then compared to those given by Principal Components Analysis linear models. In conjunction, we have evaluated the influence of the number of skull landmarks used. Anatomical skull landmarks are completed iteratively by points located upon geodesics which link these anatomical landmarks, thus enabling us to artificially increase the number of skull points. Facial points are obtained using a mesh-matching algorithm between a common reference mesh and individual soft-tissue surface meshes. The proposed method is validated in term of accuracy, based on a leave-one-out cross-validation test applied to a homogeneous database. Accuracy measures are obtained by computing the distance between the original face surface and its reconstruction. Finally, these results are discussed referring to current computer-assisted reconstruction facial techniques.