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Predicting low-energy molecular conformations given a molecular graph is an important but challenging task in computational drug discovery. Existing state-of-the-art approaches either resort to large scale transformer-based models that diffuse over conformer fields, or use computationally expensive methods to generate initial structures and diffuse over torsion angles. In this work, we introduce Equivariant Transformer Flow (ET-Flow). We showcase that a well-designed flow matching approach with equivariance and harmonic prior alleviates the need for complex internal geometry calculations and large architectures, contrary to the prevailing methods in the field. Our approach results in a straightforward and scalable method that directly operates on all-atom coordinates with minimal assumptions. With the advantages of equivariance and flow matching, ET-Flow significantly increases the precision and physical validity of the generated conformers, while being a lighter model and faster at inference. Code is available https://github.com/shenoynikhil/ETFlow., Comment: NeurIPS 2024

Predicting molecular impact on cellular function is a core challenge in therapeutic design. Phenomic experiments, designed to capture cellular morphology, utilize microscopy based techniques and demonstrate a high throughput solution for uncovering molecular impact on the cell. In this work, we learn a joint latent space between molecular structures and microscopy phenomic experiments, aligning paired samples with contrastive learning. Specifically, we study the problem ofContrastive PhenoMolecular Retrieval, which consists of zero-shot molecular structure identification conditioned on phenomic experiments. We assess challenges in multi-modal learning of phenomics and molecular modalities such as experimental batch effect, inactive molecule perturbations, and encoding perturbation concentration. We demonstrate improved multi-modal learner retrieval through (1) a uni-modal pre-trained phenomics model, (2) a novel inter sample similarity aware loss, and (3) models conditioned on a representation of molecular concentration. Following this recipe, we propose MolPhenix, a molecular phenomics model. MolPhenix leverages a pre-trained phenomics model to demonstrate significant performance gains across perturbation concentrations, molecular scaffolds, and activity thresholds. In particular, we demonstrate an 8.1x improvement in zero shot molecular retrieval of active molecules over the previous state-of-the-art, reaching 77.33% in top-1% accuracy. These results open the door for machine learning to be applied in virtual phenomics screening, which can significantly benefit drug discovery applications.

Scaling deep learning models has been at the heart of recent revolutions in language modelling and image generation. Practitioners have observed a strong relationship between model size, dataset size, and performance. However, structure-based architectures such as Graph Neural Networks (GNNs) are yet to show the benefits of scale mainly due to the lower efficiency of sparse operations, large data requirements, and lack of clarity about the effectiveness of various architectures. We address this drawback of GNNs by studying their scaling behavior. Specifically, we analyze message-passing networks, graph Transformers, and hybrid architectures on the largest public collection of 2D molecular graphs. For the first time, we observe that GNNs benefit tremendously from the increasing scale of depth, width, number of molecules, number of labels, and the diversity in the pretraining datasets. We further demonstrate strong finetuning scaling behavior on 38 highly competitive downstream tasks, outclassing previous large models. This gives rise to MolGPS, a new graph foundation model that allows to navigate the chemical space, outperforming the previous state-of-the-arts on 26 out the 38 downstream tasks. We hope that our work paves the way for an era where foundational GNNs drive pharmaceutical drug discovery.

Featurizing microscopy images for use in biological research remains a significant challenge, especially for large-scale experiments spanning millions of images. This work explores the scaling properties of weakly supervised classifiers and self-supervised masked autoencoders (MAEs) when training with increasingly larger model backbones and microscopy datasets. Our results show that ViT-based MAEs outperform weakly supervised classifiers on a variety of tasks, achieving as much as a 11.5% relative improvement when recalling known biological relationships curated from public databases. Additionally, we develop a new channel-agnostic MAE architecture (CA-MAE) that allows for inputting images of different numbers and orders of channels at inference time. We demonstrate that CA-MAEs effectively generalize by inferring and evaluating on a microscopy image dataset (JUMP-CP) generated under different experimental conditions with a different channel structure than our pretraining data (RPI-93M). Our findings motivate continued research into scaling self-supervised learning on microscopy data in order to create powerful foundation models of cellular biology that have the potential to catalyze advancements in drug discovery and beyond., Comment: CVPR 2024 Highlight. arXiv admin note: text overlap with arXiv:2309.16064

The emergence of foundation models in Computer Vision and Natural Language Processing have resulted in immense progress on downstream tasks. This progress was enabled by datasets with billions of training examples. Similar benefits are yet to be unlocked for quantum chemistry, where the potential of deep learning is constrained by comparatively small datasets with 100k to 20M training examples. These datasets are limited in size because the labels are computed using the accurate (but computationally demanding) predictions of Density Functional Theory (DFT). Notably, prior DFT datasets were created using CPU supercomputers without leveraging hardware acceleration. In this paper, we take a first step towards utilising hardware accelerators by introducing the data generator PySCF$_{\text{IPU}}$ using Intelligence Processing Units (IPUs). This allowed us to create the dataset QM1B with one billion training examples containing 9-11 heavy atoms. We demonstrate that a simple baseline neural network (SchNet 9M) improves its performance by simply increasing the amount of training data without additional inductive biases. To encourage future researchers to use QM1B responsibly, we highlight several limitations of QM1B and emphasise the low-resolution of our DFT options, which also serves as motivation for even larger, more accurate datasets. Code and dataset are available on Github: http://github.com/graphcore-research/pyscf-ipu, Comment: 15 pages, 7 figures. NeurIPS 2023 Track Datasets and Benchmarks

In the field of Machine Learning Interatomic Potentials (MLIPs), understanding the intricate relationship between data biases, specifically conformational and structural diversity, and model generalization is critical in improving the quality of Quantum Mechanics (QM) data generation efforts. We investigate these dynamics through two distinct experiments: a fixed budget one, where the dataset size remains constant, and a fixed molecular set one, which focuses on fixed structural diversity while varying conformational diversity. Our results reveal nuanced patterns in generalization metrics. Notably, for optimal structural and conformational generalization, a careful balance between structural and conformational diversity is required, but existing QM datasets do not meet that trade-off. Additionally, our results highlight the limitation of the MLIP models at generalizing beyond their training distribution, emphasizing the importance of defining applicability domain during model deployment. These findings provide valuable insights and guidelines for QM data generation efforts., Comment: Accepted at NeurIPS 2023 AI4D3 and AI4S workshops

Recently, pre-trained foundation models have enabled significant advancements in multiple fields. In molecular machine learning, however, where datasets are often hand-curated, and hence typically small, the lack of datasets with labeled features, and codebases to manage those datasets, has hindered the development of foundation models. In this work, we present seven novel datasets categorized by size into three distinct categories: ToyMix, LargeMix and UltraLarge. These datasets push the boundaries in both the scale and the diversity of supervised labels for molecular learning. They cover nearly 100 million molecules and over 3000 sparsely defined tasks, totaling more than 13 billion individual labels of both quantum and biological nature. In comparison, our datasets contain 300 times more data points than the widely used OGB-LSC PCQM4Mv2 dataset, and 13 times more than the quantum-only QM1B dataset. In addition, to support the development of foundational models based on our proposed datasets, we present the Graphium graph machine learning library which simplifies the process of building and training molecular machine learning models for multi-task and multi-level molecular datasets. Finally, we present a range of baseline results as a starting point of multi-task and multi-level training on these datasets. Empirically, we observe that performance on low-resource biological datasets show improvement by also training on large amounts of quantum data. This indicates that there may be potential in multi-task and multi-level training of a foundation model and fine-tuning it to resource-constrained downstream tasks.

Positional and structural encodings (PSE) enable better identifiability of nodes within a graph, rendering them essential tools for empowering modern GNNs, and in particular graph Transformers. However, designing PSEs that work optimally for all graph prediction tasks is a challenging and unsolved problem. Here, we present the Graph Positional and Structural Encoder (GPSE), the first-ever graph encoder designed to capture rich PSE representations for augmenting any GNN. GPSE learns an efficient common latent representation for multiple PSEs, and is highly transferable: The encoder trained on a particular graph dataset can be used effectively on datasets drawn from markedly different distributions and modalities. We show that across a wide range of benchmarks, GPSE-enhanced models can significantly outperform those that employ explicitly computed PSEs, and at least match their performance in others. Our results pave the way for the development of foundational pre-trained graph encoders for extracting positional and structural information, and highlight their potential as a more powerful and efficient alternative to explicitly computed PSEs and existing self-supervised pre-training approaches. Our framework and pre-trained models are publicly available at https://github.com/G-Taxonomy-Workgroup/GPSE. For convenience, GPSE has also been integrated into the PyG library to facilitate downstream applications., Comment: Accepted at ICML 2024; 34 pages, 6 figures

We present GPS++, a hybrid Message Passing Neural Network / Graph Transformer model for molecular property prediction. Our model integrates a well-tuned local message passing component and biased global attention with other key ideas from prior literature to achieve state-of-the-art results on large-scale molecular dataset PCQM4Mv2. Through a thorough ablation study we highlight the impact of individual components and find that nearly all of the model's performance can be maintained without any use of global self-attention, showing that message passing is still a competitive approach for 3D molecular property prediction despite the recent dominance of graph transformers. We also find that our approach is significantly more accurate than prior art when 3D positional information is not available., Comment: arXiv admin note: text overlap with arXiv:2212.02229

It has been increasingly demanding to develop reliable methods to evaluate the progress of Graph Neural Network (GNN) research for molecular representation learning. Existing GNN benchmarking methods for molecular representation learning focus on comparing the GNNs' performances on some node/graph classification/regression tasks on certain datasets. However, there lacks a principled, task-agnostic method to directly compare two GNNs. Additionally, most of the existing self-supervised learning works incorporate handcrafted augmentations to the data, which has several severe difficulties to be applied on graphs due to their unique characteristics. To address the aforementioned issues, we propose GraphAC (Graph Adversarial Collaboration) -- a conceptually novel, principled, task-agnostic, and stable framework for evaluating GNNs through contrastive self-supervision. We introduce a novel objective function: the Competitive Barlow Twins, that allow two GNNs to jointly update themselves from direct competitions against each other. GraphAC succeeds in distinguishing GNNs of different expressiveness across various aspects, and has demonstrated to be a principled and reliable GNN evaluation method, without necessitating any augmentations., Comment: 11th International Conference on Learning Representations (ICLR 2023) Machine Learning for Drug Discovery (MLDD) Workshop. 17 pages, 6 figures, 4 tables

This technical report presents GPS++, the first-place solution to the Open Graph Benchmark Large-Scale Challenge (OGB-LSC 2022) for the PCQM4Mv2 molecular property prediction task. Our approach implements several key principles from the prior literature. At its core our GPS++ method is a hybrid MPNN/Transformer model that incorporates 3D atom positions and an auxiliary denoising task. The effectiveness of GPS++ is demonstrated by achieving 0.0719 mean absolute error on the independent test-challenge PCQM4Mv2 split. Thanks to Graphcore IPU acceleration, GPS++ scales to deep architectures (16 layers), training at 3 minutes per epoch, and large ensemble (112 models), completing the final predictions in 1 hour 32 minutes, well under the 4 hour inference budget allocated. Our implementation is publicly available at: https://github.com/graphcore/ogb-lsc-pcqm4mv2.

Graph Neural Networks (GNNs) that are based on the message passing (MP) paradigm generally exchange information between 1-hop neighbors to build node representations at each layer. In principle, such networks are not able to capture long-range interactions (LRI) that may be desired or necessary for learning a given task on graphs. Recently, there has been an increasing interest in development of Transformer-based methods for graphs that can consider full node connectivity beyond the original sparse structure, thus enabling the modeling of LRI. However, MP-GNNs that simply rely on 1-hop message passing often fare better in several existing graph benchmarks when combined with positional feature representations, among other innovations, hence limiting the perceived utility and ranking of Transformer-like architectures. Here, we present the Long Range Graph Benchmark (LRGB) with 5 graph learning datasets: PascalVOC-SP, COCO-SP, PCQM-Contact, Peptides-func and Peptides-struct that arguably require LRI reasoning to achieve strong performance in a given task. We benchmark both baseline GNNs and Graph Transformer networks to verify that the models which capture long-range dependencies perform significantly better on these tasks. Therefore, these datasets are suitable for benchmarking and exploration of MP-GNNs and Graph Transformer architectures that are intended to capture LRI., Comment: NeurIPS 2022 Track on Datasets and Benchmarks. Open-sourced at: https://github.com/vijaydwivedi75/lrgb

We propose a recipe on how to build a general, powerful, scalable (GPS) graph Transformer with linear complexity and state-of-the-art results on a diverse set of benchmarks. Graph Transformers (GTs) have gained popularity in the field of graph representation learning with a variety of recent publications but they lack a common foundation about what constitutes a good positional or structural encoding, and what differentiates them. In this paper, we summarize the different types of encodings with a clearer definition and categorize them as being $\textit{local}$, $\textit{global}$ or $\textit{relative}$. The prior GTs are constrained to small graphs with a few hundred nodes, here we propose the first architecture with a complexity linear in the number of nodes and edges $O(N+E)$ by decoupling the local real-edge aggregation from the fully-connected Transformer. We argue that this decoupling does not negatively affect the expressivity, with our architecture being a universal function approximator on graphs. Our GPS recipe consists of choosing 3 main ingredients: (i) positional/structural encoding, (ii) local message-passing mechanism, and (iii) global attention mechanism. We provide a modular framework $\textit{GraphGPS}$ that supports multiple types of encodings and that provides efficiency and scalability both in small and large graphs. We test our architecture on 16 benchmarks and show highly competitive results in all of them, show-casing the empirical benefits gained by the modularity and the combination of different strategies., Comment: In Proceedings of NeurIPS 2022

Molecular property prediction is one of the fastest-growing applications of deep learning with critical real-world impacts. Including 3D molecular structure as input to learned models improves their performance for many molecular tasks. However, this information is infeasible to compute at the scale required by several real-world applications. We propose pre-training a model to reason about the geometry of molecules given only their 2D molecular graphs. Using methods from self-supervised learning, we maximize the mutual information between 3D summary vectors and the representations of a Graph Neural Network (GNN) such that they contain latent 3D information. During fine-tuning on molecules with unknown geometry, the GNN still generates implicit 3D information and can use it to improve downstream tasks. We show that 3D pre-training provides significant improvements for a wide range of properties, such as a 22% average MAE reduction on eight quantum mechanical properties. Moreover, the learned representations can be effectively transferred between datasets in different molecular spaces., Comment: 39th International Conference on Machine Learning (ICML 2022). Also accepted at NeurIPS 2021 ML4PH, AI4S, and SSL workshops and as oral at ELLIS ML4Molecules. 24 pages, 7 figures, 18 tables

In recent years, the Transformer architecture has proven to be very successful in sequence processing, but its application to other data structures, such as graphs, has remained limited due to the difficulty of properly defining positions. Here, we present the $\textit{Spectral Attention Network}$ (SAN), which uses a learned positional encoding (LPE) that can take advantage of the full Laplacian spectrum to learn the position of each node in a given graph. This LPE is then added to the node features of the graph and passed to a fully-connected Transformer. By leveraging the full spectrum of the Laplacian, our model is theoretically powerful in distinguishing graphs, and can better detect similar sub-structures from their resonance. Further, by fully connecting the graph, the Transformer does not suffer from over-squashing, an information bottleneck of most GNNs, and enables better modeling of physical phenomenons such as heat transfer and electric interaction. When tested empirically on a set of 4 standard datasets, our model performs on par or better than state-of-the-art GNNs, and outperforms any attention-based model by a wide margin, becoming the first fully-connected architecture to perform well on graph benchmarks., Comment: Accepted in Proceedings of NeurIPS 2021

The lack of anisotropic kernels in graph neural networks (GNNs) strongly limits their expressiveness, contributing to well-known issues such as over-smoothing. To overcome this limitation, we propose the first globally consistent anisotropic kernels for GNNs, allowing for graph convolutions that are defined according to topologicaly-derived directional flows. First, by defining a vector field in the graph, we develop a method of applying directional derivatives and smoothing by projecting node-specific messages into the field. Then, we propose the use of the Laplacian eigenvectors as such vector field. We show that the method generalizes CNNs on an $n$-dimensional grid and is provably more discriminative than standard GNNs regarding the Weisfeiler-Lehman 1-WL test. We evaluate our method on different standard benchmarks and see a relative error reduction of 8% on the CIFAR10 graph dataset and 11% to 32% on the molecular ZINC dataset, and a relative increase in precision of 1.6% on the MolPCBA dataset. An important outcome of this work is that it enables graph networks to embed directions in an unsupervised way, thus allowing a better representation of the anisotropic features in different physical or biological problems., Comment: 11 pages, 10 pages appendix, 6 figures, subtitle: Anisotropic aggregation in graph neural networks via directional vector fields

Graph Neural Networks (GNNs) have been shown to be effective models for different predictive tasks on graph-structured data. Recent work on their expressive power has focused on isomorphism tasks and countable feature spaces. We extend this theoretical framework to include continuous features - which occur regularly in real-world input domains and within the hidden layers of GNNs - and we demonstrate the requirement for multiple aggregation functions in this context. Accordingly, we propose Principal Neighbourhood Aggregation (PNA), a novel architecture combining multiple aggregators with degree-scalers (which generalize the sum aggregator). Finally, we compare the capacity of different models to capture and exploit the graph structure via a novel benchmark containing multiple tasks taken from classical graph theory, alongside existing benchmarks from real-world domains, all of which demonstrate the strength of our model. With this work, we hope to steer some of the GNN research towards new aggregation methods which we believe are essential in the search for powerful and robust models., Comment: 34th Conference on Neural Information Processing Systems (NeurIPS 2020)

Current research in convolutional neural networks (CNN) focuses mainly on changing the architecture of the networks, optimizing the hyper-parameters and improving the gradient descent. However, most work use only 3 standard families of operations inside the CNN, the convolution, the activation function, and the pooling. In this work, we propose a new family of operations based on the Green's function of the Laplacian, which allows the network to solve the Laplacian, to integrate any vector field and to regularize the field by forcing it to be conservative. Hence, the Green's function (GF) is the first operation that regularizes the 2D or 3D feature space by forcing it to be conservative and physically interpretable, instead of regularizing the norm of the weights. Our results show that such regularization allows the network to learn faster, to have smoother training curves and to better generalize, without any additional parameter. The current manuscript presents early results, more work is required to benchmark the proposed method., Comment: 11 pages, 3 figures

In recent years, there has been a rapid progress in solving the binary problems in computer vision, such as edge detection which finds the boundaries of an image and salient object detection which finds the important object in an image. This progress happened thanks to the rise of deep-learning and convolutional neural networks (CNN) which allow to extract complex and abstract features. However, edge detection and saliency are still two different fields and do not interact together, although it is intuitive for a human to detect salient objects based on its boundaries. Those features are not well merged in a CNN because edges and surfaces do not intersect since one feature represents a region while the other represents boundaries between different regions. In the current work, the main objective is to develop a method to merge the edges with the saliency maps to improve the performance of the saliency. Hence, we developed the gradient-domain merging (GDM) which can be used to quickly combine the image-domain information of salient object detection with the gradient-domain information of the edge detection. This leads to our proposed saliency enhancement using edges (SEE) with an average improvement of the F-measure of at least 3.4 times higher on the DUT-OMRON dataset and 6.6 times higher on the ECSSD dataset, when compared to competing algorithm such as denseCRF and BGOF. The SEE algorithm is split into 2 parts, SEE-Pre for preprocessing and SEE-Post pour postprocessing.

Current saliency methods require to learn large scale regional features using small convolutional kernels, which is not possible with a simple feed-forward network. Some methods solve this problem by using segmentation into superpixels while others downscale the image through the network and rescale it back to its original size. The objective of this paper is to show that saliency convolutional neural networks (CNN) can be improved by using a Green's function convolution (GFC) to extrapolate edges features into salient regions. The GFC acts as a gradient integrator, allowing to produce saliency features by filling thin edges directly inside the CNN. Hence, we propose the gradient integration and sum (GIS) layer that combines the edges features with the saliency features. Using the HED and DSS architecture, we demonstrated that adding a GIS layer near the network's output allows to reduce the sensitivity to the parameter initialization, to reduce the overfitting and to improve the repeatability of the training. By simply adding a GIS layer to the state-of-the-art DSS model, there is an absolute increase of 1.6% for the F-measure on the DUT-OMRON dataset, with only 10ms of additional computation time. The GIS layer further allows the network to perform significantly better in the case of highly noisy images or low-brightness images. In fact, we observed an F-measure improvement of 5.2% when noise was added to the dataset and 2.8% when the brightness was reduced. Since the GIS layer is model agnostic, it can be implemented into different fully convolutional networks. A major contribution of the current work is the first implementation of Green's function convolution inside a neural network, which allows the network to operate in the feature domain and in the gradient domain at the same time, thus improving the regional representation via edge filling., Comment: 15 pages, 11 figures

Recent work in graph neural networks (GNNs) has led to improvements in molecular activity and property prediction tasks. Unfortunately, GNNs often fail to capture the relative importance of interactions between molecular substructures, in part due to the absence of efficient intermediate pooling steps. To address these issues, we propose LaPool (Laplacian Pooling), a novel, data-driven, and interpretable hierarchical graph pooling method that takes into account both node features and graph structure to improve molecular representation. We benchmark LaPool on molecular graph prediction and understanding tasks and show that it outperforms recent GNNs. Interestingly, LaPool also remains competitive on non-molecular tasks. Both quantitative and qualitative assessments are done to demonstrate LaPool's improved interpretability and highlight its potential benefits in drug design. Finally, we demonstrate LaPool's utility for the generation of valid and novel molecules by incorporating it into an adversarial autoencoder., Comment: 11 pages, with Appendices

In computer vision, the gradient and Laplacian of an image are used in different applications, such as edge detection, feature extraction, and seamless image cloning. Computing the gradient of an image is straightforward since numerical derivatives are available in most computer vision toolboxes. However, the reverse problem is more difficult, since computing an image from its gradient requires to solve the Laplacian equation, also called Poisson equation. Current discrete methods are either slow or require heavy parallel computing. The objective of this paper is to present a novel fast and robust method of solving the image gradient or Laplacian with minimal error, which can be used for gradient domain editing. By using a single convolution based on a numerical Green's function, the whole process is faster and straightforward to implement with different computer vision libraries. It can also be optimized on a GPU using fast Fourier transforms and can easily be generalized for an n dimension image. The tests show that, for images of resolution 801x1200, the proposed GFC can solve 100 Laplacian in parallel in around 1.0 milliseconds ms. This is orders of magnitude faster than our nearest competitor which requires 294ms for a single image. Furthermore, we prove mathematically and demonstrate empirically that the proposed method is the least error solver for gradient domain editing. The developed method is also validated with examples of Poisson blending, gradient removal, and the proposed gradient domain merging GDM. Finally, we present how the GDM can be leveraged in future works for convolutional neural networks CNN., Comment: 17 pages, single column scientific paper. Patent submitted

Computer vision is a growing field with a lot of new applications in automation and robotics, since it allows the analysis of images and shapes for the generation of numerical or analytical information. One of the most used method of information extraction is image filtering through convolution kernels, with each kernel specialized for specific applications. The objective of this paper is to present a novel convolution kernels, based on principles of electromagnetic potentials and fields, for a general use in computer vision and to demonstrate its usage for shape and stroke analysis. Such filtering possesses unique geometrical properties that can be interpreted using well understood physics theorems. Therefore, this paper focuses on the development of the electromagnetic kernels and on their application on images for shape and stroke analysis. It also presents several interesting features of electromagnetic kernels, such as resolution, size and orientation independence, robustness to noise and deformation, long distance stroke interaction and ability to work with 3D images, Comment: Keywords: Shape analysis; Stroke analysis; Computer vision; Electromagnetic potential field; Feature extraction; Image filtering; Image convolution Published in PolyPublie: https://publications.polymtl.ca/3162/

In Computer Vision, edge detection is one of the favored approaches for feature and object detection in images since it provides information about their objects boundaries. Other region-based approaches use probabilistic analysis such as clustering and Markov random fields, but those methods cannot be used to analyze edges and their interaction. In fact, only image segmentation can produce regions based on edges, but it requires thresholding by simply separating the regions into binary in-out information. Hence, there is currently a gap between edge-based and region-based algorithms, since edges cannot be used to study the properties of a region and vice versa. The objective of this paper is to present a novel spatial probability analysis that allows determining the probability of inclusion inside a set of partial contours (strokes). To answer this objective, we developed a new approach that uses electromagnetic convolutions and repulsion optimization to compute the required probabilities. Hence, it becomes possible to generate a continuous space of probability based only on the edge information, thus bridging the gap between the edge-based methods and the region-based methods. The developed method is consistent with the fundamental properties of inclusion probabilities and its results are validated by comparing an image with the probability-based estimation given by our algorithm. The method can also be generalized to take into consideration the intensity of the edges or to be used for 3D shapes. This is the first documented method that allows computing a space of probability based on interacting edges, which opens the path to broader applications such as image segmentation and contour completion., Comment: Keywords: Computer vision; Stroke analysis; Partial contour; Probability of inclusion; Edge interaction; Image convolution; Electromagnetic potential field