Multi-directional Geodesic Neural Networks via Equivariant Convolution

We propose a novel approach for performing convolution of signals on curved surfaces and show its utility in a variety of geometric deep learning applications. Key to our construction is the notion of directional functions defined on the surface, which extend the classic real-valued signals and which can be naturally convolved with with real-valued template functions. As a result, rather than trying to fix a canonical orientation or only keeping the maximal response across all alignments of a 2D template at every point of the surface, as done in previous works, we show how information across all rotations can be kept across different layers of the neural network. Our construction, which we call multi-directional geodesic convolution, or directional convolution for short, allows, in particular, to propagate and relate directional information across layers and thus different regions on the shape. We first define directional convolution in the continuous setting, prove its key properties and then show how it can be implemented in practice, for shapes represented as triangle meshes. We evaluate directional convolution in a wide variety of learning scenarios ranging from classification of signals on surfaces, to shape segmentation and shape matching, where we show a significant improvement over several baselines

Surface-guided computing to analyze subcellular morphology and membrane-associated signals in 3D

Signal transduction and cell function are governed by the spatiotemporal organization of membrane-associated molecules. Despite significant advances in visualizing molecular distributions by 3D light microscopy, cell biologists still have limited quantitative understanding of the processes implicated in the regulation of molecular signals at the whole cell scale. In particular, complex and transient cell surface morphologies challenge the complete sampling of cell geometry, membrane-associated molecular concentration and activity and the computing of meaningful parameters such as the cofluctuation between morphology and signals. Here, we introduce u-Unwrap3D, a framework to remap arbitrarily complex 3D cell surfaces and membrane-associated signals into equivalent lower dimensional representations. The mappings are bidirectional, allowing the application of image processing operations in the data representation best suited for the task and to subsequently present the results in any of the other representations, including the original 3D cell surface. Leveraging this surface-guided computing paradigm, we track segmented surface motifs in 2D to quantify the recruitment of Septin polymers by blebbing events; we quantify actin enrichment in peripheral ruffles; and we measure the speed of ruffle movement along topographically complex cell surfaces. Thus, u-Unwrap3D provides access to spatiotemporal analyses of cell biological parameters on unconstrained 3D surface geometries and signals.Comment: 49 pages, 10 figure

Evaluation of different statistical shape models for segmentation of the left ventricular endocardium from magnetic resonance images

International audienceStatistical shape models (SSMs) represent a powerful tool used in patient-specific modeling to segment medical images because they incorporate a-priori knowledge that guide the model during deformation. Our aim was to evaluate segmentation accuracy in terms of left ventricular (LV) volumes obtained using four different SSMs versus manual gold standard tracing on cardiac magnetic resonance (CMR) images. A database of 3D echocardiographic (3DE) LV surfaces obtained in 435 patients was used to generate four different SSMs, based on cardiac phase selection. Each model was scaled and deformed to detect LV endocardial contours in the enddiastolic (ED) and end-systolic (ES) frames of a CMR short-axis (SAX) stack for 15 patients with normal LV function. Linear correlation and Bland–Altman analyses versus gold-standard showed in all cases high correlation (r²>0.95), non-significant biases and narrow limits of agreement

QUANTIFICATION OF MYOCARDIAL MECHANICS IN LEFT VENTRICLES UNDER INOTROPIC STIMULATION AND IN HEALTHY RIGHT VENTRICLES USING 3D DENSE CMR

Statistical data from clinical studies indicate that the death rate caused by heart disease has decreased due to an increased use of evidence-based medical therapies. This includes the use of magnetic resonance imaging (MRI), which is one of the most common non-invasive approaches in evidence-based health care research. In the current work, I present 3D Lagrangian strains and torsion in the left ventricle of healthy and isoproterenol-stimulated rats, which were investigated using Displacement ENcoding with Stimulated Echoes (DENSE) cardiac magnetic resonance (CMR) imaging. With the implementation of the 12-segment model, a detailed profile of regional cardiac mechanics was reconstructed for each subject. Statistical analysis revealed that isoproterenol induced a significant change in the strains and torsion in certain regions at the mid-ventricle level. In addition, I investigated right ventricular cardiac mechanics with the methodologies developed for the left ventricle. This included a comparison of different regions within the basal and mid-ventricular regions. Despite no regional variation found in the peak circumferential strain, the peak longitudinal strain exhibited regional variation at the anterior side of the RV due to the differences in biventricular torsion, mechanism of RV free wall contraction, and fiber architecture at RV insertions. Future applications of the experimental work presented here include the construction and validation of biventricular finite element models. Specifically, the strains predicted by the models will be statistically compared with experimental strains. In addition, the results of the present study provide an essential reference of RV baseline evaluated with DENSE MRI, a highly objective technique