PhD positions

Efficient geometric processing for Design and Manufacturing relies on a number of fundamental primitives or predicates that must be executed extremely fast and very accurately. Such predicates are critical for several algorithms developed within GRAPES, regardless of the underlying representation. We develop components of a toolbox for ray-shooting, surface-surface intersection, computing distances and tangents, and detecting self-intersection. Our methods handle objects given by powerful and novel representations: point clouds, simplicial/curved meshes, and matrix representation, using advanced algebraic techniques like syzygies, fitting and interpolation. We target problems from our industrial partners e.g. swept volume computation, computation with offsets, and self-intersection. Further details shall be specified in relation to the student’s profile.

Expected Results: We exploit various algebraic formulations to optimise our approach, with respect to the degree of involved polynomials and the complexity of the geometric object. We develop a prototype implementation of the toolbox to use for validation and experimentation, leading up to a high-performance implementation, based on the generic programming paradigm that exploits multi-core architectures.

The project will take place at ATHENA RC at the ErGA Lab. The PhD will be awarded by the tadalafil tabs of the National Kapodistrian University of Athens.

Secondments are planned at INRIA (Sophia-Antipolis, France) and at industrial partner RISC-SW (Linz, Austria).

3D data are gaining increasing a en on due to the recent availability of 3D sensors and their use in innovative applications such as self-driving cars. Their complexity makes selection of a suitable representation difficult, especially if one wishes to maintain translation and rotation invariance under varying sampling density. Deep learning has made impressive progress on representing audio and 2D visual data, but its development on 3D data is not as mature. Operating on multiple 2D views suffers from information loss due to projection. Methods operating on 3D point clouds directly usually limit local interaction between points. We remedy this issue by generalising convolution to 3D while maintaining a sparse representation, not necessarily on the same point set. Further, existing architectures typically focus on tasks like classification or semantic part segmentation, and are fully supervised. This PhD investigates and designs deep learning models for meaningful embeddings of 3D shapes in vector spaces, using novel, powerful architectures including autoencoders and generative adversarial networks and targeting new tasks like metric learning for similarity retrieval. Weaker supervision settings like self-supervised, semi-supervised, weakly supervised and few-shot learning will be investigated, allowing the use of unlabeled 3D data.

Expected Results: Adapt fully supervised architectures to shape classification, segmentation. Explore metric learning for retrieval and generative modelling. Introduce weaker supervision (semi, few-shot learning) so as to use unlabelled 3D data. Validation and training relies on the creation and curation of a 3D shape dataset.

The project will take place at ATHENA RC at the ErGA Lab. The PhD will be awarded by the Dept. of Informatics and Telecommunications of the National Kapodistrian University of Athens.

Secondments are planned at Imperial College London (London, UK) and at industrial partner 3DI (London, UK).

This project has a very strong foot in Algebraic Geometry with applications to Machine Learning as a motivation. The extraction of geometric primitives from 3D point clouds is an important problem in reverse engineering. These 3D point clouds are typically obtained by means of accurate 3D scanners and there exist several methods for performing the geometric primitives extraction. Key ingredients in this approach are geometric routines that are capable of producing an instance of a given type of shape from a small number of points. The most used types of shapes are planes, spheres, cylinders, cones and tori. While devising such routines is relatively straightforward for planes and spheres, the cases of cylinders, cones and tori are much more difficult but have been worked out recently. The goal of this PhD is to investigate to what extent one can deepen these results, and also to extend them to other kind of low degree surfaces.

Expected Results: We expect to deliver algorithmic criteria for reconstructing specific surfaces given a cloud of points in 3D, as well as topological, geometric, and algebraic criteria for distinguishing them (degree, genus, Betti numbers, etc.). Generalizations to higher dimensions and non necessarily smooth algebraic objects and implementations are also part of the research plan.

The project will take place at the Mathematics Dept of the University of Barcelona.

Secondments will take place at INRIA (Sophia-Antipolis, France) and at Vilnius University (Vilnius, Lithuania).

Learning models of temporal point clouds is at the core of geometric and geographical studies. Morphological changes are typically derived by calculating distances between points from different acquisitions. This thesis is at the crossroad of ML and 3D visualisation, aiming at predicting temporal point clouds. We define a model for temporal point clouds based on shapes, topology, and sampled properties, and we will study different approaches to predict morphological changes. Usually, the interpretation of the properties and surface changes relies on a human, while gathering visual information contained in the data requires expertise by the user. To assist users we analyse and study interactive visualisation to improve the expert understanding and the performance of ML.

Expected Results: ML models for predicting temporal point clouds and tools for assisting humans in visualising surface changes when predicting rock falls. The software should help humans identify characteristics of geomorphological models.

The project will take place at the Mathematics Dept of the University of Barcelona.

Secondments will take place at ATHENA RC (Athens, Greece) and at industrial partner RISC-Software (Linz, Austria).

Curvilinear meshes offer more flexibility and better accuracy compared to (more common) simplicial meshes. Their generation is an active area of research: it requires to conform curved elements to a given boundary geometry, enforce a smooth and non-degenerate Jacobian everywhere, and keep the mesh as coarse as possible. Several attempts exist to accommodate these three properties. Building on a new variational approach, called Curved Optimal Delaunay Triangulation (CODT), that has been recently introduced to generate such meshes, the main objective of this project will be to develop new methods in order to generate a valid high-order curved meshes made of Bézier triangles. For that purpose, algebraic and subdivision techniques will be investigated for the detection of self-intersection and surface/surface intersection loci, for the preservation of sharp features and for controlling regularity.

Expected Results: A reliable and robust algorithm to detect intersection and self-intersection of low degree Bézier triangles, and an improved CODT method preserving the topology and sharp features of the input data.

The project will take place at Inria Sophia Antipolis, the PhD will be awarded by the University of Nice.

Secondments are planned at U. Barcelona (Barcelona, Spain) and at the industrial partner RISC-SW (Linz, Austria).

Subdivision surfaces offer great flexibility in capturing irregular topologies combined with higher order smoothness. For instance, Loop and Catmull-Clark subdivision schemes provide C^2 smoothness everywhere except at extraordinary vertices, where the generated surfaces are C^1 smooth. The combination of flexibility and smoothness leads to their frequent use in geometric modelling and makes them an ideal candidate for performing isogeometric analysis of higher order problems with irregular topologies, e.g. thin shell problems. In the neighbourhood of extraordinary vertices, however, the resulting surfaces (and the functions defined on them) are non-polynomial. This has several implications on algorithmic aspects, including stable evaluation, numerical integration, computation of derivatives, etc. The project goal is to study existing, and develop new constructions of modelling and analysis-suitable subdivision surfaces and solids.

Expected Results: New constructions of smooth, or approximately smooth, polynomial-based surfaces and solids that approximate well subdivision surfaces are envisaged. Their use in modelling, e.g., in reconstruction problems with irregular topology, will be demonstrated. Moreover, the constructions will be utilised in isogeometric thin shells.

The project will take place at Inria Sophia Antipolis, the PhD will be awarded by the University of Nice.

Secondments are planned at UTV (Rome, Italy), U. Strathclyde (Glasgow, UK) and at the industrial partner ITI (Cambridge, UK).

The structure-and-motion problem is the problem of obtaining information about a three-dimensional configuration – consisting of points, lines, or more complicated geometric figures – from its two-dimensional projections. The “motion”-aspect arises when we want to compute the position of the camera in relation to the spatial configuration. The question can be reduced to systems of polynomial equations. The goal of the proposed PhD project is to study instances of such problems from a theoretical and practical point of view. Relevant questions are: how much information is needed so that the solution is determined uniquely or at least such that there are only finitely many solutions? In the second case, how many solutions do exists, and how can we compute them efficiently? As a new aspect, we plan to exploit special structure of the arising systems, especially symmetries, using tools from algebra and kinematics.

Expected Results: Algorithms for computing algebraic representations of 3D-objects from known algebraic representations of their 2D-pictures. Proof-of-concept examples showing the efficiency of these algorithms. Understanding of the limitations of the method. Formulae for the dimension and degree of the solution set of classes of structure-and-motion problems.

Secondments are planned at Vilnius University (Vilnius, Lithuania) and at the industrial partner ModuleWorks (Aachen, Germany).

For geometric data, ML methods so far have mostly been applied to analytic tasks. Only recently, first synthesis tasks have been addressed leading to generation of new 3D models in a data driven manner. The design process for new models with these techniques, however, is mostly restricted to a ”modelling-by-example” paradigm where (partial) models are taken as input and analogous shapes are created. This refers to the traditional criticism that internal (hidden) representations within a NN often lack any intuitive interpretability. Our goal is to explore NN architectures and training algorithms that promote meaningful internal representations such that they can be used as control handles for interactive design.

Expected Results: Algorithms to train and apply data-driven methods (e.g.~deep NN) for generative geometric design tasks such as shape deformation and style editing. Proof-of-Concept implementation for demonstration and evaluation.

Secondments are planned at the industrial member GeometryFactory (Sophia-Antipolis, France) and at the industrial partner ModuleWorks (Aachen, Germany).

Our goal is to explore the evolution of neural network architectures, by replacing core components with refinable geometric representations that can be gradually adapted over training time. In the context of curriculum learning, investigate how, once a plateau is reached during training, to punctuate this equilibrium by representing the trained model exactly in a new model with larger capacity. Investigate various candidate representations for adding such degrees of freedom, in particular Locally Refinable splines due to their versatility and adaptability. Investigate which components (convolutional masks, activation functions, etc.) in the architecture can benefit the most, and investigate the use of reinforcement learning for developing refinement strategies. Apply the results to various generative modelling problems in computer aided design (e.g. reverse engineering) and big spatial data modelling (e.g. void filling in digital elevation models).

Expected Results: Techniques and software library for evolutionary neural network design for geometry, addressing applications in CAD and big spatial data modelling.

The project will take place at SINTEF, the PhD will be awarded by the Dept. of Mathematics of the University of Oslo.

Secondments are planned at RWTH Aachen University (Aachen, Germany) and at the industrial partner 3DI (London, UK).

Shape optimisation in complex engineering systems depends on the robustness and efficiency of the involved simulation and geometric-modelling tools as well as their seamless integration. This project aims to combine novel concepts and methodologies in computational mechanics and parametric modelling with ML for robust and efficient exploration of the associated design space. Our plan is underpinned by three key strategic pillars: (i) IGA for seamlessly integrating the adopted geometric representations with boundary-element marine-hydrodynamic solvers (ii) locally refinable spline spaces for developing parametric modellers that can deliver the same accuracy but at a considerable lower cost in comparison with the industrial standard (NURBS) and (iii) ML for revealing the intrinsic nature of the associated design spaces.

Expected Results: Novel shape-optimisation platform for marine-hydrodynamic applications based on: (i) seamless CAD-CAx integration, (ii) parametric modelling via novel geometric representation enabling local-refinability, (iii) efficient exploration of design space through dimensionality reduction using shape-quality and hydrodynamic-performance criteria.

Secondments are planned at INRIA (Sophia-Antipolis, France) and at the industrial partner ITI (Cambridge, UK).

Geometric deep learning is a new paradigm applying deep NNs beyond Euclidean-structured data. In extending the success of deep learning on image analysis to geometric data the difficulty is that, when representing geometry in a way acceptable by NNs, structure is lost, so one cannot deal with deformable (nor rigid) transforms. We redefine basic ingredients of deep NNs in a geometrically meaningful manner. We developed the first (Geodesic) CNN architecture for shapes modelled as manifolds, discretised as triangular meshes, with convolution and pooling operations that are automatically invariant to deformations. GCNNs achieve correspondence results in line with the extrinsic method or with significantly less parameters and very small training. They can learn significant deviations from the isometric deformation model and cope with geometric/topological noise, partiality, clutter.

Expected Results: Construction of intrinsic generative models (variational autoencoders, generative adversarial networks) for shapes. There is no canonical ordering of vertices, so we compute the correspondence between input and generated shape. We study spectral and spatial formulations of intrinsic generative models, and topology generations. Applications to 3D scan completion and dynamic fusion, pose and style transfer, surface generation from pointsets and 2D images.

Secondments are planned at Imperial College London (London, UK), RWTH (Aachen, Germany), and at the industrial partner 3DI (London, UK).

Barycentric rational interpolation has recently been shown to be a promising alternative to polynomial, spline, and other interpolation methods, regarding approximation order, stability, and efficient evaluation. While this approach is well understood by now in the functional setting, the main goal of this project is to consider the parametric setting. We will first extend the idea of barycentric rational interpolation to open curves and parametric freeform surfaces and then investigate the generalisation to closed curves and surfaces. To achieve this goal, we will develop the mathematical background as well as efficient algorithms for handling this kind of curves and surfaces, so that they can be used for geometric design and engineering analysis.

Expected Results: We will develop a new kind of rational curve and surface representation with favourable properties. We will derive theoretical results regarding approximation order and stability, as well as practical results in terms of efficient evaluation and manipulation algorithms.

Secondments are planned at SINTEF (Oslo, Norway) and at the industrial member GeometryFactory (Sophia-Antipolis, France).

Multi-degree splines are of interest in several contexts, including IGA. But they constitute an almost unexplored area when multivariate unstructured meshes are needed. We shall extend univariate multi-degree splines to several dimensions beyond tensor-product structure. Theoretical properties shall be investigated and manipulation algorithms shall be provided. The new spline technologies should offer optimal approximation and be compatible with local refinement (T-, LR-, hierarchical meshes). Our methods should meet the requirements of both geometric design and engineering analysis, faithful to IGA. Research follows a constructive approach and we shall further investigate the potential of the Bézier extraction operator in the multivariate multi-degree setting.

Expected Results: A multivariate spline framework for local k-refinement compatible with local mesh refinement in IGA.

Secondments are planned at SINTEF (Oslo, Norway), U. Straclyde (Glasgow, UK), and at the industrial member ITI (Cambridge, UK).

It was known for decades that circular quad meshes produce smooth cyclidic splines, composed of principal patches of Dupin cyclides. Due to topological and shape constraints they appeared too rigid and nonflexible for modelling. At the same time, certain attempts to extended regular cyclidic splines to surfaces of arbitrary topology find empirical justification in architecture. We shall use systematic M\”obius invariants based on quaternions to generalise circular quad meshes to quasi-circular allowing arbitrary even-sided faces. Cyclidic splines of arbitrary topology will use these and a subdivision-like procedure for filling multisided holes. Approximation, free-form modelling and shape optimisation of such splines shall be considered. We also explore 3D circular hexahedron meshes generating 3D cyclidic splines composed of principal Dupin volumes. Locally they represent 3-orthogonal coordinate systems for IGA.

Expected Results: We will generalise cyclidic splines based on circular quad meshes to surfaces of arbitrary topology based on quasi-circular meshes and explore their extensions to 3D cyclidic spline frameworks. Practical applications such as surface offsetting or the computation of surface intersections will also be investigated.

Secondments are planned at JKU (Linz, Austria) and at the industrial member ModuleWorks (Aachen, Germany).

We explore the 3D reconstruction of large-scale outdoor scenes, from raw measurement data. Our focus is on 3D vector maps: semantic-aware representations that exhibit effective complexity-distortion tradeoffs. Our motivation stems from domain-specific applications e.g. urban planning and safer transportation. Departing from current approaches that determine a single prior for regularising the inherent ill-posed nature of reconstruction, we plan to find by supervised ML a series of priors that locally adapt to the semantic class of objects. Resilience to missing data will be tackled via data-driven completion, and data consolidation will be achieved via joint learning and regularisation based on curved geometric primitives. Reconstruction is performed via curved meshes. We finally learn the classes and meta-parameters of error metrics, best suited to trade data fidelity for resilience to defect-laden measurement data.

Expected Results: Learning both priors and error metrics for reconstruction should yield a pliant approach, able to reconstruct large-scale scene with heterogeneous types of objects, ranging from free-form surfaces to trees with fractal dimension, through highly-structured buildings or outdoor furniture. We deliver a C++ prototype implementation for CGAL.

The project will take place at the industrial member GeometryFactory, the PhD will be awarded by the University of Nice.

Secondments are planned at RWTH (Aachen, Germany) and at USI (Lugano, Switzerland).

Synthetic tasks in 3d geometry often rely on effective and accurate editing of the underlying representation. However, editing 3D shape representations, and in particular 3D meshes, is a very tedious task. Designers spend a significant amount of time repurposing existing 3D models, manually repositioning one by one the vertices of the mesh to produce a new shape. A tool that enables easier and faster editing would greatly increase the capacity at which new shapes could be generated, enabling us to move towards interactive design. Our main tool shall be Neural Networks and the corresponding internal representations.

Recent works utilize Neural Networks and learn meaningful intermediate representations of 3D shapes by decomposing them into sets of primitive objects, such as 3D Gaussians, or by enclosing them inside coarse meshes with semantic control points. The goal of this project is to exploit and improve such meaningful internal representations, and use them as conditioning inputs for part-specific editing and generation operations.

General collaborative framework with the PhD thesis undertaken by K. Tertikas (ESR2) on Geometric Learning. Applications may include industrial CAD parts (e.g. in the aerospace industry), and structural bioinformatics, namely geometric modeling of proteins, and molecular binding sites.

Expected Results: Algorithms for shape editing and shape generation. Proof-of-Concept UI to showcase editing and generative capabilities.

The project will take place at ATHENA RC at the ErGA Lab.

Duration: 8.5 months

Our goal is to explore NN architectures and training that allow for fast and realistic predictions of key engineering data during interactive shape design. In particular, we are interested in revealing the physical interaction of contacting shapes during the CAD design phase.

Existing methods for contact simulation based on standard discretization methods are time intensive therefore cannot be used in an interactive design phase. To advance and fuse interactive design and ML-based analysis we shall employ and combine recent methodologies, namely, Physics Informed Neural Networks1 (PINNS) and Isogeometric Analysis2 (IGA).

The use of PINNs has the potential to outperform classical simulations in terms of evaluation time, while preserving sufficient accuracy in early design stages. Coupled with IGA, they can be applied directly on the CAD model under construction. This data-driven exploration of the design space shall facilitate the early optimization of the model’s parameters especially for highly standardized parts like Machine Elements.

Expected Results: Provide a fast, adequately accurate approximation of the shape deformation in real-time while interactively designing mechanical assemblies.

The project will take place at Inria Sophia Antipolis.

Duration: 5.8 months.

Develop generative ML models and train them in a mixed-initiative way while keeping Geometry at its kernel to develop meaningful internal representations, that can be used as control handles for interactive design to generate optimal and diverse designs that go beyond the ”modelling-by-example” paradigm, where (partial or parent) models are taken as input and shapes strongly similar to the parent ones are created. In the presence of annotated data, learn embeddings that group shapes according to their attributes. Analogously to word embeddings in NLP (Natural Language Processing), our embeddings induce a semantic shape similarity metric and enable meaningful (local) interpolation in latent space. In the absence of annotated data, learn abstract representations on the basis of the co-occurrence of features across different models. Apply functional maps to establish, train and validate correspondences between objects. Learn embeddings of shape editing operators instead of static shapes and investigate more sophisticated representations of such operations based on handles and control structures.

Expected Results:Algorithms to train and apply data-driven methods, e.g., variational auto-encoders and Generative Adversarial Networks (GANs), for generative geometric design tasks in the context of industrial engineering applications with focus in ship and blade design.

Duration: 6.9 months.
Short Visit: 1 month, Co-Design Lab, Dept Mech Eng, University California, Berkeley (US)

Project Collaborators:
• Dr Kosa Goucher-Lambert, Cognition and Computation in Design (Co-Design) Lab, Department of Mechanical Engineering, the University of California, Berkeley, US.
• Dr Faez Ahmed, Design Computation & Digital Engineering (DeCoDE) Lab, Massachusetts Institute of Technology (MIT), Cambridge, US.

Trivariate subdivision algorithms offer an interesting approach to the approximation of volumetric data, as needed for instance for interactive 3D model design, 3D printing or isogeometric analysis. We want to optimize an existing Catmull-Clark-type algorithm with respect to its spectral properties and develop a new Doo-Sabin-type scheme. Unlike in the bivariate case, the variety of structurally different mesh layouts that need to be addressed is not finite. Moreover, an elegant simplification of the setting in the spirit of the discrete Fourier transform is not available so that an individual treatment of each case seems to be unavoidable. To cope with this situation, we want to determine suitable weights for a series of specific cases using standard tools from linear algebra and optimization and then apply learning methods to generalize the findings to new scenarios without having to repeat the tedious analysis repeatedly.

Expected Results: Algorithms to train and apply data-driven methods (e.g., deep NN) for finding suitable 3D subdivision weights. Analysis of the geometric and analytic properties of these weights. Proof-of-Concept implementation for demonstration and evaluation.

Duration: 4 months.