Fractal models and wavelet analysis for the characterization of engineering surfaces – FRACLETTES

Our approach is centered on a deterministic (non-random) model, based on fractal geometry. It offers the opportunity to generate an exhaustive and rich variety of rough geometries. A corpus (digital database of more than 40,000 curves and surfaces) was thus quickly created, enhanced with an interface for querying it. The geometric characterization (through the joint use of multi-fractal and wavelet analyses) and the study of the differential properties (geometric variations) of the rough shapes generated by this model are at the heart of our approach. Through the study of the differential nature of curves and surfaces, we have defined different roughness types that can be controlled using constraints imposed on the model. Its deterministic aspect is fundamental in this study, to guarantee the continuous dependence between its parameters and the differential properties. This appears essential to be able to formalize and rigorously study these properties, to subsequently propose algorithms adapted to the optimization of rough surfaces (in particular for the case studies cited previously).

A prototype web application has been initiated to store, visualize and browse the corpus of generated rough shapes. The integration of our theoretical results in the form of usable functionalities is currently being finalized, with a view to making the corpus available to the scientific community. This would offer ways to study the impact of the aforementioned phenomena on different types of roughness. User tracking, in the form of profiling (based on professional and query criteria), has been set up, to consider a recommendation system.
Our approach, mainly theoretical, has given rise to two papers published at international conferences. Another is currently being finalized for an international journal. The project and its results have also been presented to our community during at least one of the two annual national events (7 presentations since 2020). To promote its possibilities of exploitation and to make our model and some of the considered methods known to a wider audience, we participated in several science festivals, and submitted an article in the journal “The Conversation”.

In this project, we focused primarily on the theoretical aspects of geometric roughness control and characterization. The formalism we developed based on DCFs opens up both theoretical and practical perspectives.
From a theoretical perspective, DCFs can be considered a generalization of limited expansions with non-integer exponents, reflecting the local behavior of rough curves and surfaces. They are, therefore, central elements for characterizing differential properties. We still need to establish the relationships between our approach and classical methods for characterizing irregularity (Hölder coefficient and singularity spectrum) to better understand and exploit our results. Since DCFs are directly defined from the transformation parameters (eigenvalues ??and vectors) associated with our model, it will then be possible to propose tools for controlling these irregularity characteristics. Through these DCFs, the study of roughness can be approached from two complementary perspectives: the geometric and spectral aspects. Exploring the latter allows us to envisage possible relationships with classic statistical quantifiers of roughness (Sa, Sq, distribution of slopes/normals, etc.), opening up more control possibilities.
This theoretical work will continue as part of the FractOs regional project (launched in September 2024 and focusing on the design of fractal implants for osteogenesis, in collaboration with the CERAMATHS laboratory in Valenciennes). Fractal structures with multi-scale porosity and whose surface roughness should promote adhesion and therefore colonization of the implant by bone cells are already under study. Their fabrication, using 3D printers with resolutions in the order of hundreds of nanometers, is planned. Prospects for extensive collaborations are also being considered (ANR PRC project submitted and currently in phase 2).
Thanks to the communication established around our work, another collaboration (with the universities of Strasbourg, Sherbrooke, and Poitiers) was also initiated. This resulted in a submission to the latest ANR PRC call for proposals (not selected in phase 2). It focuses on exploiting the new possibilities offered by our model for simulating realistic light interactions in virtual scenes. A resubmission is planned for next fall.
Finally, from a practical perspective, making the corpus freely available to researchers, engineers, and industrialists from various disciplines and application areas is a key element in the promotion of this project.

- L. Druoton, C. Roudet, C. Gentil. “A general method for roughness characterization of non-regularly sampled data using wavelets on graphs.” Submission in progress to an international journal.
- C. Poull, C. Gentil, C. Roudet, L. Druoton, M. Roy. Second Order Differential Properties of Tensor Product Fractal Surfaces. 20th International Conference on Computer Graphics Theory and Applications, Feb. 2025, Porto, Portugal.
- C. Poull, C. Gentil, C. Roudet, L. Druoton, M Roy. Differential properties of rough fractal surfaces. JFIG (Journées Françaises de l'Informatique Graphique), Oct. 2024, Strasbourg, France.
- L. Druoton, C. Roudet, C. Gentil. “Détection et caractérisation de singularités de signaux par analyse en ondelettes sur graphe.” In Journées du Groupe de Travail en Modélisation Géométrique (R-GTMG), Marseille, March 2024.
- M. Janbein, C. Gentil, C. Roudet, C. Poull. Pseudo-Curvature of Fractal Curves for Geometric Control of Roughness. 19th International Conference on Computer Graphics Theory and Applications, Feb. 2024, Rome, Italy. pp.177-188.
- C. Poull, C. Gentil, C. Roudet, M. Roy. Méthodes de contrôle de la rugosité à partir des propriétés différentiels de courbes autosimilaires.. Journées du Groupe de Travail en Modélisation Géométrique (GTMG), March 2023, Strasbourg, France.
- C. Poull, M. Janbein, L. Druoton, C. Roudet, S. Lanquetin, M. Roy, C. Gentil. La rugosité des surfaces et ses applications. Journées Françaises d'Informatique Graphique (JFIG), Bordeaux, Nov. 2022
- C. Poull, M. Janbein, C. Gentil. Designing bio-inspired shapes for energy savings. Dijon Science Village, 17-18 Oct. 2022.
- C. Roudet et M. Roy. Images de science : Ces « ondelettes » qui se cachent derrière la 3D, Online review « The Conversation », 14 Dec. 2021.

The area in which this project fits is geometric modeling for Computer Aided Geometric Design (CAGD) and digital simulation. We propose to tackle the problem of digital representation, analysis and characterization of rough surfaces for digital simulation.
Roughness is a complex concept, that is multi-scale, and based on the study of the local behavior of a surface in a given neighborhood. The evaluation of roughness on surfaces is essential for many experimental problems. It explains the numerous studies carried out in the application fields related to physics and mechanics, where the control and management of the surface topographies is a major need for manufacturers.
A large number of standardized conventional parameters are currently available to attempt to appreciate this concept in the different application areas that make use of it. But it is often difficult, for a given application domain or a special need, to know precisely which parameter(s) connect(s) the topography of a surface to the physical phenomena that it undergoes. This is explained because, to a given parameter value can correspond roughness associated with varied geometries and physical properties. This is mainly due to the fact that conventional roughness measurements are, for the most part, based on a global statistical quantification. The geometric characterization that we propose aims to overcome this major drawback. It seems essential to us to more easily establish relations with the physical properties of surfaces. However, we will not focus on the impact of roughness on physical properties, which is very specific to each area.
The objective of this project is twofold:
1) Theoretical: model the roughness and define tools for manipulation, composition, analysis and geometric characterization. To do this, we propose to rely on a generic approach using wavelet analysis and roughness synthesis from deterministic fractal models. We believe that the BC-IFS fractal model (developed in our team) will allow us to define this geometric characterization from the differential properties that can be defined on these surfaces.
2) Practical: generate a corpus (digital database) of rough geometric models in different forms, so that each user can find the models they are used to handling. It will first serve to master the notion of roughness. In a second step, we plan to make it available to researchers, engineers or industrialists (from different disciplines and fields of applications). In particular, they will be able to use it to perform numerical simulations or assess the impact of different types of roughness on the physical properties of an object.
Despite the richness of the roughness generated by deterministic fractal models, this will certainly not be enough to represent all the varieties that can be encountered in reality. However, these rough models could serve as a reference base for generating new families of roughness, from combination operators (addition, multiplication, dilation, reduction, offset, ...).