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The movement of a garment has infinite degrees of freedom. This originates the main difficulties for its robotic handling. Automated manipulation of clothing requires to forecast and understand its position while the clothing is grabbed. We will address these questions and we will show some solutions obtained within the Clothilde project at IRI (CSIC-UPC).

In the era of GPT models, one gets notoriously confronted with the question of how much inductive (geometric) bias can help when scaling up deep neural architectures. In this lecture, we first contrast modern geometric deep learning approaches, such as group equivariant learning, with large-scale Transformer success stories. Secondly, we introduce Clifford (geometric) algebras as an emerging paradigm within the deep learning community. To delve deeper, we commence with a comprehensive overview of modern (plane-based) geometric algebra, which is founded on the representation of isometries as elements within the Pin(p, q, r) group. Subsequently, we propose the concept of group actions and Clifford group equivariant layers. Lastly, we discuss the potential application of these layers when scaling up neural networks, which, in turn, could potentially pave the way for the development of geometric GPTs in the future.

Modern products are increasingly designed, optimised, and manufactured using geometric and physical models held on computer, so-called digital twins. As these continue to grow in scale, complexity and ambition, gaps appear in the digital thread connecting these models, where engineers must manually intervene, and transform their geometry to suit each tool or analysis model. This talk will describe some of the geometry reasoning technology ITI has developed to address these gaps, how hybrid geometry representations have become key to our approach, and where the future opportunities might lie.

In this talk, we explore AI methods for predicting stresses when the parameters describing the shape of the component change. We focus on two techniques, one using mesh deformation and the other using image-based approaches using pretrained convolutional neural networks. We provide numerical results to predict the Van Mises stresses for a parameterized two-dimensional component.

In this work, we introduce ShipHullGAN, a generic parametric modeller built using deep convolutional generative adversarial networks (GANs) for the versatile representation and generation of ship hulls. We trained ShipHullGAN on a large dataset of 52,591 physically validated designs from a wide range of existing ship types, including container ships, tankers, bulk carriers, tugboats, and crew supply vessels. We developed a new shape extraction and representation strategy to convert all training designs into a common geometric representation of the same resolution, as typically GANs can only accept vectors of fixed dimension as input. A space-filling layer is placed right after the generator component to ensure that the trained generator can cover all design classes. During training, designs are provided in the form of a shape-signature tensor (SST) which harnesses the compact geometric representation using geometric moments that further enable the inexpensive incorporation of physics-informed elements in ship design. We have shown through extensive comparative studies and optimisation cases that ShipHullGAN can generate designs with augmented features resulting in versatile design spaces that produce traditional and novel designs with geometrically valid and practically feasible shapes.

Food recognition presents a formidable challenge in computer vision and machine learning, stemming from the intrinsic complexity of food images. The complexity is characterized by high inter-class similarity and intra-class variability, the fine-grained nature of food classes, visual differences arising from different cooking styles, lack of distinctive spatial layouts and rigid structures, and randomly distributed ingredients across the food platter. As is the case with any deep learning problem, the training data plays a pivotal role in food recognition. In this talk, we address two data-centric questions that deepen our comprehension of food data. First, we investigate the issue of data labelling by harnessing self-supervised learning algorithms. We present “All4One”, a novel neighbour-based contrastive SSL approach that employs “centroids” generated through a self-attention mechanism to reduce the distance between different neighbour representations. Second, we tackle the challenge posed by the fine-grained nature of classes through subset learning-based methods. We introduce “Dining on Details”, an expert learning framework for food classification that incorporates lexical information extracted using LLMs. Both proposed solutions shed light on critical aspects of data-centred food recognition and pave the way for more accurate and robust food recognition models.

The ability to generate realistic and diverse 3D shapes has the potential to significantly accommodate the workflow of artists and content creators and potentially enable new levels of creativity through “generative art”. However, despite the impressive progress in generative models that can generate plausible 3D shapes of high fidelity, an important aspect of the generation process is to enable control on what parts of the generated object should change. Therefore, having a basic understanding of the decomposition of an object into parts facilitates controlling what to edit. In the first part of my talk, I will present PartNeRF, which is a part-aware generative model for editable 3D shape synthesis that is trained using posed images. The key idea is to generate objects as a set of locally defined NeRFs, that emerge automatically during training. Our formulation enables part-level control, hence permitting several editing operations such as rigid and non rigid transformations on parts, mixing parts from different objects etc. Although PartNeRF, can reliably edit the appearance and the shape of the generated instance, it is an unconditional generative model. However, being able to generate realistic 3D geometries from versatile user inputs is a crucial task that can unlock numerous editing capabilities. To address this, we introduced PASTA, which is an autoregressive transformer architecture for controllable shape generations that focuses on yielding detailed geometries. PASTA comprises two main components: An autoregressive transformer that generates objects as a sequence of cuboidal primitives and a blending network, implemented with a transformer decoder that composes the sequences of cuboids and synthesizes high quality meshes for each object. Our model is trained in two stages: First we train our autoregressive generative model using only annotated cuboidal parts as supervision and next, we train our blending network using explicit 3D supervision, in the form of watertight meshes. Our model, trained with various conditionings, can be used to perform shape generation from diverse inputs e.g. from scratch, from a partial object, from text and images, as well size-guided generation. Moreover, as PASTA considers the underlying part-based structure of a 3D object, we are able to select a specific part and produce shapes with meaningful variations of this part. Finally, I will present two works for controllable scene synthesis. First, I will talk about ATISS, which is a novel autoregressive transformer architecture for creating diverse and plausible synthetic indoor environments that poses scene synthesis as sequence generation and generates rooms as unordered sets of objects. Next, I will also talk about our recent work, CC3D, which is a conditional generative model for scene synthesis that does not require 3D labelled bounding boxes as supervision and instead can be trained using only posed images.

Within the first year of our life, we develop a common-sense understanding of the physical behavior of the world, which relies heavily on our ability to properly reason about the arrangement of objects in a scene. While this seems to be a fairly easy task for the human brain, computer vision algorithms struggle to form such high-level reasoning. Therefore, the research community shifted their attention to the development of primitive-based methods that seek to represent objects as semantically consistent part arrangements. However, due to the simplicity of existing primitive representations, these methods fail to accurately reconstruct 3D shapes using a small number of primitives/parts. In the first part of my talk, I will introduce several compositional representations that have been explored in the literature. I will address the trade-off between reconstruction quality and number of parts and present Neural Parts, a novel 3D primitive representation that defines primitives using an Invertible Neural Network (INN) which implements homeomorphic mappings between a sphere and the target object. Since a homeomorphism does not impose any constraints on the primitive shape, our model effectively decouples geometric accuracy from parsimony and as a result captures complex geometries with an order of magnitude fewer primitives. Next, I will present our recent work ALTO, that can produce high-fidelity reconstruction of implicit 3D surfaces from noisy point clouds. Our key idea is to alternate back and forth between point and grid latents before converging to a final feature representation that is easy to decode. ALTO is able to yield high-quality 3D meshes while being up to 10 times faster than prior works. Finally, I will present COPILOT, a transformer-based architecture capable of performing collision prediction and localization simultaneously given either multi-view or single-view egocentric video observations. COPILOT accumulates information across multi-view inputs through a novel 4D space-time-viewpoint attention mechanism and provides rich information regarding imminent collisions.

Nowadays, production of goods and industrial components relies on a mixture of digital and classical fabrication steps. AI and ML seem to be right tools to automate the existing workflows to reduce involvement of human beings and optimize the outcome of the manufacturing industry. In this presentation, the audience will be immersed in the existing challenges and opportunities that can be reached with the AI. Generic industrial examples and AI try-outs at ModuleWorks will be presented.

In my talk, I will try to motivate the relevance of the Castelnuovo-Mumford regularity as an invariant that governs the computations with polynomial systems. Namely, I will discuss the relation of this invariant with elimination matrices, Groebner bases, resultants and incremental algorithms for solving polynomial systems. I will also deal with the possible extensions of this invariant to multigraded and sparse systems, and motivate it with many examples coming from applications.

Impressive progress in generative models and implicit representations gave rise to methods that can generate 3D shapes of high quality. However, being able to locally control and edit shapes is another essential property that can unlock several content creation applications. Local control can be achieved with part-aware models, but existing methods require 3D supervision and cannot produce textures.
In the first part of the talk, we present PartNeRF, a novel part-aware generative model for editable 3D shape synthesis that does not require any explicit 3D supervision. Our model generates objects as a set of locally defined NeRFs, augmented with an affine transformation. This enables several editing operations such as applying transformations on parts, mixing parts from different objects etc. Evaluations on various ShapeNet categories demonstrate the ability of our model to generate editable 3D objects of improved fidelity, compared to previous part-based generative approaches that require 3D supervision or models relying on NeRFs.
Existing 3D-aware generative methods – including PartNeRF – are trained using synthetic data, and require multiple views per object instance and known camera poses. However, from a user’s perspective, it would be extremely beneficial to have the ability to generate 3D shapes using only in-the-wild images. In the second part of the talk, we focus on the task of generating 3D shapes from in-the-wild images and present our preliminary results.

Characterizes degenerate cases of cubic plane curves by practicable formulas in terms of the coefficients of the polynomials describing the curve. We will use minors of the well-known matrices associated with the cubic curves such as Hessian matrix.

In the domain of computer vision, decomposing 3D scenes into object instances is a fundamental challenge. In addition, achieving accurate segmentation of 3D scenes without the laborious task of annotating data is paramount for applications spanning robotics to augmented reality. This presentation introduces a paradigm-shifting approach to unsupervised 3D scene segmentation, driven by object-level generative modeling that transcends the necessity for scene-level supervision. By parsing the point cloud of a 3D scene and utilizing the object likelihood of a variational autoencoder model, we discover and segment the objects inside the scene. The empirical validation highlights the effectiveness of our unsupervised 3D scene segmentation approach, paving the way for profound advancements in 3D scene understanding, all without being tethered to the traditional way of exhaustive annotations.

We present new results about the partitioning of measures with polynomials. Namely, for any Borel probability measure $\mu$ on $\R^n$ there exists a set $X\subset \R^n$ of $n+1$ points such that any $n$-variate quadratic polynomial $P$ that is nonnegative on $X$ (i.e. $P(x)\geq 0$, for every $x \in X$) satisfies $\mu\{P\geq 0\}\geq \frac{2}{(n+1)(n+2)}$.
More generally, given an absolutely continuous probability measure $\mu$ on $\R^n$ and $D\leq 2k$, for every $\delta>0$ there exists a set $X\subset \R^n$ with $|X|\leq \binom{n+2k}{n}-n-1$ such that any $n$-variate polynomial $P$ of degree $D$ that is nonnegative on $X$ satisfies $\mu\{P\geq 0\}\geq \frac{1}{\binom{n+2k}{n}+1} – \delta$. These statements are analogues of the celebrated \emph{centerpoint theorem}, which corresponds to the case of linear polynomials. Our results follow from new estimates on the Carath\’eodory numbers of real Veronese varieties, or alternatively, from bounds on the convex symmetric rank of real symmetric tensors.This is a joint work with Alfredo Hubard and Roman Karasev.

Geometrically smooth spline functions are piecewice polynomial functions defined on a mesh, that satisfy properties of differentiability across shared edges; their unstructured nature provides them a wide range of application. We start introducing a G1 spline construction defined by means of smoothing masks on quad meshes approximating the well-known Catmull-Clark surface. Furthermore, we provide a set of G1 basis functions, describe their properties and analyze their space, to be applied in point data fitting problems and Isogeometric Analysis simulations.

The task of identifying an algebraic surface from a single branching curve (apparent contour) can be simplified by recovering a homogeneous equation in four variables from its branching curve (which is defined by the discriminant).
Currently, successful reconstructions have been achieved for smooth surfaces, surfaces with ordinary singularities, ruled surfaces, and the latest work focuses on reconstructing Darboux cyclides.
There is used the fact that the Darboux Cyclides have a singularity along the absolute conic in order to recognize them up to Euclidean similarity transformations.
By applying the same concept, it becomes feasible to reconstruct a planar map from a given branching curve. The stepping stone is the ramification curve, which is obtained as the linear normalization of the branching curve, which lies on a parametric surface. The composition of the parametrization with linear normalization yields a planar map that shares the same branching curve as the parametric surface.

A neural network consisting of piecewise affine building blocks, such as fully-connected layers and ReLU activations, is itself a piecewise affine function supported on a polyhedral complex. This complex has been previously studied to characterize theoretical properties of neural networks, but, in practice, extracting it remains a challenge due to its high combinatorial complexity. A natural idea described in previous works is to subdivide the regions via intersections with hyperplanes induced by each neuron. However, we argue that this view leads to computational redundancy. Instead of regions, we propose to subdivide edges, leading to a novel method for polyhedral complex extraction. A key to this are sign-vectors, which encode the combinatorial structure of the complex. Our approach allows to use standard tensor operations on a GPU, taking seconds for millions of cells on a consumer grade machine. Motivated by the growing interest in neural shape representation, we use the speed and differentiability of our method to optimize geometric properties of the complex.

In this work, we introduce ShipHullGAN, a generic parametric modeller built using deep convolutional generative adversarial networks (GANs) for the versatile representation and generation of ship hulls. At a high level, the new model intends to address the current conservatism in the parametric ship design paradigm, where parametric modellers can only handle a particular ship type. We trained ShipHullGAN on a large dataset of 52,591 physically validated designs from a wide range of existing ship types, including container ships, tankers, bulk carriers, tugboats, and crew supply vessels. We developed a new shape extraction and representation strategy to convert all training designs into a common geometric representation of the same resolution, as typically GANs can only accept vectors of fixed dimension as input. A space-filling layer is placed right after the generator component to ensure that the trained generator can cover all design classes. During training, designs are provided in the form of a shape-signature tensor (SST) which harnesses the compact geometric representation using geometric moments that further enable the inexpensive incorporation of physics-informed elements in ship design. We have shown through extensive comparative studies and optimisation cases that ShipHullGAN can generate designs with augmented features resulting in versatile design spaces that produce traditional and novel designs with geometrically valid and practically feasible shapes.
Video abstract: https://youtu.be/LT9Z52vBgzI

Universal approximation theorems are results that demonstrate the density of a certain class of functions within a space of interest. We will begin our discussion with the early works of Cybenko, Hornik, and others, on feed-forward neural networks, up to more recent results that go beyond the Euclidean setting. Subsequently, we will illustrate how we can use existing universal approximation theorems to prove new ones, even in the light of a certain kind of symmetry.

We discuss the classical methods in curve design such as Bezier and B-Spline. We also discuss some extension to the periodic case and other recent and notable alternatives. The common properties of these tools are that even though they are very intuitive tools, they lack direct control over a curve. On the other hand, the Barycentric rational form can be used to complement the previous methods, since due to the interpolatory property, we can force the curve to pass through specific points. We describe the correspondence of the classical method and the interpolation method to offer new editing possibilities of a curve.

A computational result about recognition of any implicit form of Dupin cyclides will be presented. Applications will include: singularity investigation, illustration of properties of Dupin cyclides suggested in a classical paper of G. Darboux, and investigation of Dupin orthogonal coordinate systems.

Locally Refined B-splines (LR B-splines) [Dokken et al.,Polynomial splines over locally refined box-partitions, CAGD, 2013] have proven to be a very flexible and powerful framework in several applications, such as, geometric modeling, data analysis, visualization and numerical simulation. In this work we extend LR B-splines to allow for non-uniform polynomial degree, where we combine degree elevation with local h-refinement, resulting in local k-refinement, that is, increasing the polynomial degree as well as the smoothness at newly introduced knot lines. The non-uniform degree LR B-splines are scaled by weights to maintain the nestedness of the resulting space and the non-negative partition of unity giving the convex hull property, which can be computed by means of certain two-scale relationship. In this talk, we discuss the theoretical properties of the new spline spaces ensuring the fundamental properties like local support, (scaled) partition of unity, etc., and their application in several 2D model problems, illustrating the efficacy of the proposed adaptive refinement methodology in the context of isogeometric analysis.

Shape reconstruction from raw point sets is a pivotal task in the realms of computer graphics and computer vision. This presentation presents the intricate journey of transforming unstructured point data into coherent 3D representations of objects. We explored both explicit and implicit methods to reconstruct shapes. For explicit reconstruction, we present Bézier primitive segmentation on 3D point clouds. Different from previous primitive segmentations that treat different primitives separately, we take inspiration from Bézier decomposition on NURBS models and transfer it to guide point cloud segmentation casting off primitive types. Experiments show superior performance over previous work in terms of segmentation, with a substantially faster inference speed. For implicit reconstruction, we present a method for reconstructing an isotropic surface triangle mesh from an unoriented point cloud whose density adapts to an estimate of the local feature size (LFS). Instead of performing dense reconstruction followed by remeshing, our approach reconstructs both an implicit function and an LFS-aware mesh sizing function, which are then used for producing the final LFS-aware reconstructed mesh. Our approach’s added value is generating isotropic meshes directly from 3D point clouds with an LFS-aware density. Our experiments demonstrate robustness to noise, outliers, and missing data.

André Lieutier (Geometry Engineering) – From topological inference to meshing algorithms

In this tutorial we consider the general problem of retrieving a topologically correct model from partial geometric information. Applications areas span from automated 3D scene faithful reconstruction from scanned points clouds to abstract submanifolds inference from high dimensional data in the context of machine learning.
In a first part, the tutorial will recall several topological notions and consider their possible inference from data. In a second part, we focus on the computation of simplicial complexes homeomorphic to manifolds known through point samples, in other words, manifolds triangulation algorithms with topological guarantees.

Leonidas Guibas (Stanford university) – Learning to Vary and Varying to Learn

Variations in how a 3D object or scene is presented to a learning algorithm, for example the coordinate system chosen, can significantly impact the learned outcome. At the same time, the space of intrinsic variations of the scene itself is intimately connected with the semantics of the scene and the continuous or discrete attributes we are trying to learn. Certain variations may be just nuisance while others crucial to the task at hand. The talk highlight a few efforts that address this issue of disentangling how variations of a scene or of its presentation affect learning tasks, aiming to show the value of low dimensional parametrizations of the semantic tangent space of object and scene representations, towards both analysis and synthesis learning tasks.

A first vignette looks at how to design neural architectures that are invariant or equivariant to spatial transforms like rigid motions that affect scene presentation and introduces the vector neuron machinery for neural components that operate on vectors and not scalars. In the second vignette we look at how to shape neural scene representations such as NeRFs to better reflect the compositional structure of the scene by doing contrastive learning based on co-variation patterns between scene points or pixels (in scene views) with high mutual information. These learned “resonances” allow efficient propagation of sparse annotations or semantically coherent scene edits. Finally, in the third vignette we look at how language that differentiates related object geometries can be used for segmentation as well as for the generation of object variants based on written/spoken instructions.

Christoph Hofer (RISC-Software) – Numerical and automatic differentiation

In this tutorial session, we will investigate different ways to program the computer to calculate derivatives of functions. To be precise we consider difference-quotients, the complex step, and automatic/algorithmic differentiation methods. We discuss their advantages, disadvantages, and limitations. Besides the algorithmic realization, we will also look at their mathematical properties. The different algorithms and examples will be implemented in small programming exercises.

The preferred language for the exercises is python, but in principle the participants are free to use any language as long as it provides operator overloading, graphs of functions/line charts can be plotted, and complex numbers are supported. For python various plotting libraries exist, a common one is plotly (https://plotly.com/python/). A very basic understanding of python numpy is advantageous. The participants, who use python should have the packages numpy, plotly and pandas installed. I advise to use PyCharm as an IDE.

XMANAI team (Theodore Dalamagas, Eleni Lavasa, Vasilis Gkolemis – Athena RC) – XAI application in Industrial Metrology — SLIDES: Interpretable Machine Learning, eXplainable Artificial Intelligence Application in Metrology

Nowadays, quality Control of manufactured parts (e.g., automotive, aeronautical and energy sectors) is performed with advanced optical technology (3D laser scanning) in the Industry 4.0 era to achieve dimensional quality requirements and tolerances with high precision and in minimum time. Each part under metrological study is captured in a 3D Point Cloud, which is subsequently analyzed with domain-specific software to obtain the dimensional measurement and estimate the associated accuracy of the measurement (tolerance). In this tutorial we examine the data science applications for industrial metrology scenarios: starting from the manipulation of Point Cloud data, moving on to the prediction of an instrument’s accuracy in point capturing using popular Machine Learning algorithms, and closing with the application of post-hoc explainability techniques to interpret the predictions of the ML models. Our goal is to showcase the benefits explainable AI (XAI) has to offer in complex industrial settings such as Industrial Metrology.

Tutorial outline:

• Part A (1h)
  1. Introduction to Explainable AI and the XMANAI research project (5 min)
  2. Industrial Metrology: the Use Case of UNIMETRIK pilot in XMANAI (15 min)
  3. Lab#1: Point Cloud analysis and feature extraction (30 min)
     Software: Python3 / numpy, pandas, jupyter notebook, open3d
  4. Discussion (10min)
• Part B (1.5h)
  1. Lab#2: Building a predictor for point capturing accuracy (30 min)
     Software: Python3 / numpy, pandas, jupyter, scikit-learn
  2. Discussion: Quantitative & Qualitative model evaluation (15 min)
  3. Lab#3: Explainability study of the predictive models (30 min)
     Software: Python3 / scikit-learn (permutation importance, pdp), SHAP
  4. Discussion: Assessing the quality/impact of explanations (15 min)

Claudio Mancinelli (University of Genoa) – Vector Graphics on Discrete Surfaces, (pdf version)

Software packages for the generation of vector graphics in 2D are available in many forms nowadays. The concept of geometric primitive is at the core of this technology. Essentially, geometric primitives are basic geometric shapes that can be used and combined to generate more complex ones. Such primitives are defined in a plane endowed with the Euclidean metric, together with a Cartesian coordinate system, and the algorithms to generate them are based on this structure.

In this talk, we will start by describing the theoretical and computational challenges that one needs to overcome when trying to bring this technology onto a (discrete) surface. The theoretical ones, are mainly due to the fact that we can no longer rely on the Euclidean metric. From a computational point of view instead, one needs to ensure the correctness, robustness and efficiency of its implementation, in order to reproduce the behaviour of the 2D drawing systems. We will then present some examples of publicly available implementations in which both of these problems have been successfully addressed.

Arturs Berzins (SINTEF) – Neural Implicit Shape Editing using Boundary Sensitivity.

Neural fields are receiving increased attention as a geometric representation due to their ability to compactly store detailed and smooth shapes and easily undergo topological changes. Compared to classic geometry representations, however, neural representations do not allow the user to exert intuitive control over the shape. Motivated by this, we leverage boundary sensitivity to express how perturbations in parameters move the shape boundary. This allows to interpret the effect of each learnable parameter and study achievable deformations. With this, we perform geometric editing: finding a parameter update which best approximates a globally prescribed deformation. Prescribing the deformation only locally allows to deform the rest of the shape according to some prior, such as semantics or deformation rigidity. Different to previous efforts, our method is model-agnostic and can be applied to any pre-trained NN and update it in-place. Furthermore, we show how boundary sensitivity helps to optimize and constrain objectives (such as surface area and volume), which are difficult to compute without first converting to another representation, such as a mesh.

Rao Fu (GF) – Local feature size adaptive surface reconstruction from unoriented point sets

This work seeks to design a surface reconstruction algorithm from unoriented point sets by adapting to the local feature size. The method has three steps: constructing an envelope, signing the envelope, and adaptive meshing. First, a Delaunay refinement process produces the envelope from the iso-level of the unsigned distance function field. Afterward, we use a globally optimized solver to solve the sign of the envelope by a signing guess from the edges. Finally, we use another Delaunay refinement process to mesh the target shape by a sizing function adaptive to the local feature size. This method is easy to implement and can guarantee the output of an isotropic mesh in a single pass. Experiments prove that our algorithm outperforms two-stage algorithms.

Amrutha Balachandran Nair (Inria) – Mu-Basis of Parametrizations of Surfaces 

Mu-basis is the basis of syzygy module of a parametrization from which we can retrieve the parametrization back. We will be looking at certain parametrizations in two variables, that describes surfaces and how to find mu-basis. In particular we will be looking at translational surfaces and generalized surfaces of revolution. Translational surfaces, which are surfaces generated by two rational curves by sliding one along the other. Generalized surfaces of revolution are surfaces are generated by rotating a rational space curve around another curve.

Konstantinos Tertikas (Athena RC) – Generating Part-Aware Editable 3D Shapes without 3D Supervision 

Impressive progress in generative models and implicit representations gave rise to methods that can generate 3D shapes of high quality. However, being able to locally control and edit shapes is another essential property that can unlock several content creation applications. Local control can be achieved with part-aware models, but existing methods require 3D supervision and cannot produce textures. In this work, we devise a novel part-aware generative model for editable 3D shape synthesis that does not require any explicit 3D supervision. Our model generates objects as a set of locally defined NeRFs, augmented with an affine transformation. This enables several editing operations such as applying transformations on parts, mixing parts from different objects etc. To ensure distinct, manipulable parts we enforce a hard assignment of rays to parts that makes sure that the color of each ray is only determined by a single NeRF. As a result, altering one part does not affect the appearance of the others. Evaluations on various ShapeNet categories demonstrate the ability of our model to generate editable 3D objects of improved fidelity, compared to previous part-based generative approaches that require 3D supervision or models relying on NeRFs.

TBA

Thanasis Zoumpekas (UB) – Temporal 3D Point Cloud Processing and Understanding

Point clouds have emerged as a crucial data type in the field of 3D analysis, owing to their ability to provide a comprehensive and accurate representation of real-world environments. They are utilized in various areas such as autonomous driving, geoscience, construction, manufacturing, and heritage preservation. However, analyzing large and dynamic point clouds poses several challenges, such as dealing with noise, missing data, and temporal consistency. This talk presents some of our studies on processing and understanding temporal and static 3D point clouds, with a focus on geoscience and manufacturing applications. The studies include a proposed framework for rockfall detection, a novel application of temporal 3D point clouds in manufacturing for machining tool identification, and an ongoing study on unsupervised 3D detection of complex geometrical shapes in 3D point cloud scenes both indoors and outdoors.

Francesco Patrizi (University of Florence) – An Algorithmic Introduction to LR B-splines

In this tutorial we shall introduce LR B-splines. These are generalization of B-splines over locally refined quadrilateral meshes and are exploited for a wide range of applications in Computer Aided Design (CAD) and Computer Aided Engineering (CAE), such as isogeometric analysis and quasi-interpolation, in particular for bathymetry and terrain datasets.
There are several ways to introduce the LR B-splines. In this tutorial we will focus on an algorithmic approach: provided an initial set of tensor B-splines on a coarse tensor mesh we shall see the procedure to generate the LR B-splines while updating such mesh with local insertions.

Next, we shall see the main refinement algorithms for LR B-splines, i.e., the procedures that automatically generate LR B-splines on the regions where they are needed to reduce the error in some sense. Namely, we will see 6 strategies: the minimum span, the full span, the structured mesh, the non-nested-support-structured mesh, the hierachical mesh and the effective grading refinements.
Finally, if we have time, we should address the linear dependence problem and how to solve it with the peeling algorithm.

Eriola Hoxhaj (JKU) and Jean Michel Menjanahary (VU) – Reconstruction of Darboux cyclides from a single view apparent contour

The task of recognizing an algebraic surface from its apparent contour can be reduced to the recovering of a homogeneous equation in four variables from its discriminant. We present a reconstruction algorithm that applies to Darboux cyclidic surfaces up to Euclidean transformation similarities.

Carles Checa (Athena RC) and Pablo González-Mazón (Inria) – Approximating swept volumes with Bezier patches

The computation of the volume swept by a tool along a trajectory is an important step in the simulation of a machining process. One approach for this computation consists of sampling the surface of the tool with multiple points at different instants of the motion, and fitting the resulting point cloud with an approximate surface using implicit functions. In our work, we describe the pipeline of this computation with several types of volume representations. Namely, we present an approach based on Bezier patches. In this approach, the machining tool is given as a point cloud that is fitted by a curvilinear mesh, i.e., it is described by Bezier patches. The computation of the swept volume is then performed using algebraic methods over the polynomial parametrizations given by these patches. Finally, we use a routine of ray-tracing with elimination matrices that can be used for both visualization and point cloud reconstruction.

Michelangelo Marsala (Inria) – Point Cloud Data Fitting via G1 Smooth Spline Basis Functions

Geometrically smooth spline functions are piecewice polynomial functions defined on a mesh, that satisfy properties of differentiability across shared edges; their unstructured nature provide them a wide range of application like, for example, the point data fitting. Point cloud fitting is a very important topic especially in CAD theory, which permits in fact to translate CAD data into Bézier representation. In the presentation we consider G1splines on quadrangular mesh defined with quadratic gluing data function along shared edges. We will describe shortly their construction, their properties, analyze their space, and provide dimension formula. After having showed how to construct an efficient basis for the considered space, numerical results will be presented to illustrate the quality of the fitting.  

Krunal Raval (UTV) – Adaptive isogeometric analysis based on Tchebycheffian splines

Tchebycheffian splines are piecewise smooth functions with pieces in Tchebycheff spaces, the natural generalization of polynomial spaces. Under suitable assumptions, Tchebycheffian splines can be represented in terms of B-spline like basis functions, called Tchebycheffian B-splines, with structural similarities and all fundamental properties of classical polynomial B-splines (compact support, non-negativity, partition of unity, etc.). Tchebycheffian splines offer a huge flexibility compared to classical polynomial splines: they are equipped with parameters that can be selected according to a problem-oriented strategy, taking into account the geometrical and/or analytical issues of the specific addressed problem.
While the tensor-product approach can easily build multivariate splines, they lack adequate adaptive local refinement, which could be important in both geometric modelling and numerical simulation. This triggered the interest in alternative (polynomial) spline structures supporting local refinement still retaining the local tensor-product structure. Since Tchebycheffian B-splines are plug-to-plug compatible with polynomial B-splines, they are naturally compatible with the local refinement structures known for polynomial B-splines (Hierarchical mesh, T-spline mesh and Locally Refined(LR-) mesh).  
In this talk, we discuss the use of Tchebycheffian splines equipped with a local tensor-product structure as a possible tool in isogeometric analysis with adaptive refinement.

Shahroz Khan (U.Strathclyde) – Shape-Supervised Dimension Reduction

In shape optimisation problems, subspaces generated with conventional feature extraction-based design space dimension reduction (DSDR) approaches often fail to extract the intrinsic geometric features of the shape that would allow the exploration of diverse but valid candidate solutions. More importantly, they also lack incorporation of any notion of physics against which shape is optimised. To simultaneously tackle these deficiencies, the proposed shape-supervised DSDR uses higher-level information about the shape in terms of its geometric integral properties, such as geometric moments and their invariants. Their usage is based on the fact that moments of a shape are intrinsic features of its geometry, providing a unifying medium between geometry and physics. To enrich the subspace with latent features associated with the shape’s geometrical features and physics, we also evaluate a set of composite geometric moments, using the divergence theorem, for appropriate shape decomposition. These moments are combined with the shape modification function to form a decomposed SSV uniquely representing a shape. Afterwards, the generalised Karhunen–Loève expansion is applied to SSV, embedded in a generalised (disjoint) Hilbert space, which results in a basis of the shape-supervised subspace retaining the highest geometric and physical variance.

Alexandros Georgiou (USI) – Diffusion models for graph-structured data

Diffusion (probabilistic) models form a family of generative models that have found great success in various tasks and, most notably, image synthesis. We will discuss recent results of diffusion models on graph-structured data and some ideas on how these can be improved.