Abstract Geometric graphs are a special kind of graph with geometric features, which are vital to model many scientific problems. Unlike generic graphs, geometric graphs often exhibit physical symmetries of translations, rotations, and reflections, making them ineffectively processed by current Graph Neural Networks (GNNs). To address this issue, researchers proposed a variety of geometric GNNs equipped with invariant/equivariant properties to better characterize the geometry and topology of geometric graphs. Given the current progress in this field, it is imperative to conduct a comprehensive survey of data structures, models, and applications related to geometric GNNs. In this paper, based on the necessary but concise mathematical preliminaries, we formalize geometric graph as the data structure, on top of which we provide a unified view of existing models from the geometric message passing perspective. Additionally, we summarize the applications as well as the related datasets to facilitate later research for methodology development and experimental evaluation. We also discuss the challenges and future potential directions of geometric GNNs at the end of this survey.

Nuclei segmentation is an important step for quantitative analysis of fluorescence microscopy images. A large volume generally has many different regions containing nuclei with varying spatial characteristics. Automatically identifying nuclei that are challenging to segment can speed up the analysis of biological tissues. Here we show a segmentation technique that provides a metric of segmentation “confidence” for each segmented object in an image volume. This confidence metric can be used either to generate a “confidence map” for visual distinction of reliable from unreliable regions, or in the data space to identify questionable measurements that can be analyzed separately or eliminated from analysis. In an analysis of nuclei in a 3-dimensional image volume, we show that the confidence map correlates well with visual evaluations of segmentation quality, and that the confidence metric correlates well with F1 scores within subregions of the image volume. In addition, we also describe three visualization methods that can visualize the segmentation differences between a segmented volume and a reference volume.

Learning to represent and simulate the dynamics of physical systems is a crucial yet challenging task. Existing equivariant Graph Neural Network (GNN) based methods have encapsulated the symmetry of physics, \emph{e.g.}, translations, rotations, etc, leading to better generalization ability. Nevertheless, their frame-to-frame formulation of the task overlooks the non-Markov property mainly incurred by unobserved dynamics in the environment. In this paper, we reformulate dynamics simulation as a spatio-temporal prediction task, by employing the trajectory in the past period to recover the Non-Markovian interactions. We propose Equivariant Spatio-Temporal Attentive Graph Networks (ESTAG), an equivariant version of spatio-temporal GNNs, to fulfil our purpose. At its core, we design a novel Equivariant Discrete Fourier Transform (EDFT) to extract periodic patterns from the history frames, and then construct an Equivariant Spatial Module (ESM) to accomplish spatial message passing, and an Equivariant Temporal Module (ETM) with the forward attention and equivariant pooling mechanisms to aggregate temporal message. We evaluate our model on three real datasets corresponding to the molecular-, protein- and macro-level. Experimental results verify the effectiveness of ESTAG compared to typical spatio-temporal GNNs and equivariant GNNs.

Automated microscope systems are increasingly used to collect large-scale 3D image volumes of biological tissues. Since cell boundaries are seldom delineated in these images, detection of nuclei is a critical step for identifying and analyzing individual cells. Due to the large intra-class variability in nuclei morphology and the difficulty of generating ground truth annotations, accurate nuclei detection remains a challenging task. We propose a 3D nuclei centroid detection method by estimating the "vector flow" volume where each voxel represents a 3D vector pointing to its nearest nuclei centroid in the corresponding microscopy volume. We then use a voting mechanism to estimate the 3D nuclei centroids from the "vector flow" volume. Our system is trained on synthetic microscopy volumes and tested on real microscopy volumes. The evaluation results indicate our method outperforms other methods both visually and quantitatively.

Piezoelectrets are polymer-foam based space-charge electrets with strong piezoelectric effect. The piezoelectricity in piezoelectrets occurs due to the elastic heterogeneous cellular structure and the regularly arranged dipolar space charges stored therein. Some polymers have been experimented for piezoelectret preparation, where polypropylene (PP) is the mostly applied material at present. PP piezoelectrets have several promising features, such as large piezoelectric d33coefficient, small thickness, light weight, low cost, large area scale, as well as flexibility and even stretchability, which would enable them very suitable for applications in signal sensing and energy harvesting. In this work, the electromechanical properties of flexible and stretchable PP piezoelectrets are introduced and some of their possible applications as wearable physiological-signal sensors and micro-energy harvesters are demonstrated by experiments.