I defended my PhD thesis on November 7th. You may download the manuscript (in French) here.

Abstract

Fibrous structures, which consist of an assembly of flexible slender objects, are ubiquitous in our environment, notably in biological systems such as plants or hair. Over the past few years, various techniques have been developed for digitalizing fibers, either through manual synthesis or with the help of automatic capture. Concurrently, advanced physics based models for the dynamics of entangled fibers have been introduced in order to animate these complex objects automatically.

The goal of this thesis is to bridge the gap between those two areas: on the one hand, the geometric representation of fibers; on the other hand, their dynamic simulation. More precisely, given an input fiber geometry assumed to represent a mechanical system in stable equilibrium under external forces (gravity, contact forces), we are interested in the mapping of such a geometry onto the static configuration of a physics-based model for a fiber assembly. Our goal thus amounts to computing the parameters of the fibers that ensure the equilibrium of the given geometry. We propose to solve this inverse problem by modeling a fiber assembly physically as a discrete collection of super-helices subject to frictional contact.

We propose two main contributions. The first one deals with the problem of converting the digitalized geometry of fibers, represented as a space curve, into the geometry of the super-helix model, namely a G^1 piecewise helical curve. For this purpose we introduce the 3d floating tangents algorithm, which relies upon the co-helicity condition recently stated by Ghosh. More precisely, our method consists in interpolating N+1 tangents distributed on the initial curve by N helices, while minimizing points displacement. Furthermore we complete the partial proof of Ghosh for the co-helicity condition to prove the validity of our algorithm in the general case. The efficiency and accuracy of our method are then demonstrated on various data sets, ranging from synthetic data created by an artist to real data captures such as hair, muscle fibers or lines of the magnetic field of a star.

Our second contribution is the computation of the geometry at rest of a super-helix assembly, so that the equilibrium configuration of this system under external forces matches the input geometry. First, we consider a single fiber subject to forces deriving from a potential, and show that the computation is trivial in this case. We propose a simple criterion for stating whether the equilibrium is stable, and if not, we show how to stabilize it. Next, we consider a fiber assembly subject to dry frictional contact (Signorini-Coulomb law). Considering the material as homogeneous, with known mass and stiffness, and relying on an estimate of the geometry at rest, we build a well-posed convex quadratic optimization problem with second order cone constraints. For an input geometry consisting of a few thousands of fibers subject to tens of thousands frictional contacts, we compute within a few seconds a plausible approximation of both the geometry of the fibers at rest and the contact forces at play.

We finally apply the combination of our two contributions to the automatic synthesis of natural hairstyles. Our method is used to initialize a physics hair engine with the hair geometry taken from the latest captures of real hairstyles, which can be subsequently animated physically.

Comittee

• Bruno Lévy (Directeur de recherche, Inria Nancy), Reviewer
• Doug James (Associate Professor, Cornell University), Reviewer
• Roberto Santoprete (Chercheur, L'Oréal Recherche), Examiner
• Jean-Claude Léon (Professeur, Grenoble INP), Examiner
• Joëlle Thollot (Professeur, Grenoble INP), Thesis Director
• Florence Bertails-Descoubes (Chargée de recherche, Inria Rhônes-Alpes), Thesis co-advisor

Stable Inverse Dynamic Curves

Alexandre Derouet-Jourdan, Florence Bertails-Descoubes & Joëlle Thollot. ACM Transactions on Graphics (Proceedings of the ACM SIGGRAPH Asia 2010 Conference).
[Project website] [PDF] [Movie] (supplemental material available here)

Summary: 2d animation is a traditional but fascinating domain that has recently regained popularity both in animated movies and video games. This paper introduces a method for automatically converting a smooth sketched curve into a 2d dynamic curve at stable equilibrium under gravity. The curve can then be physically animated to produce secondary motions in 2d animations or simple video games. Our approach proceeds in two steps. We first present a new technique to fit a smooth piecewise circular arcs curve to a sketched curve. Then we show how to compute the physical parameters of a dynamic rod model (super-circle) so that its stable rest shape under gravity exactly matches the fitted circular arcs curve. We demonstrate the interactivity and controllability of our approach on various examples where a user can intuitively setup efficient and precise 2d animations by specifying the input geometry.

3D Inverse Dynamic Modeling of Strands

Alexandre Derouet-Jourdan, Florence Bertails-Descoubes & Joëlle Thollot. ACM SIGGRAPH 2011 Posters.

Summary: In this poster, we propose a new method to automatically and consistently convert 3D splines into dynamic rods at rest under gravity, bridging the gap between the modeling of 3D strands (such as hair, plants) and their physics-based animation.

Floating tangents for approximating spatial curves with G1 piecewise helices

Alexandre Derouet-Jourdan, Florence Bertails-Descoubes & Joëlle Thollot. Computer-Aided Geometric Design, Volume 30, Issue 5, June 2013.

[Editor link]

Summary: Curves are widely used in computer science to describe real-life objects such as slender deformable structures. Using only 3 parameters per element, piecewise helices offer an interesting and compact way of representing digital curves. In this paper, we present a robust and fast algorithm to approximate Bézier curves with G1 piecewise helices. Our approximation algorithm takes a Bézier spline as input along with an integer N and returns a piecewise helix with N elements that closely approximates the input curve. The key idea of our method is to take N+1 evenly distributed points along the curve, together with their tangents, and interpolate these tangents with helices by slightly relaxing the points. Building on previous work, we generalize the proof for Ghosh's co-helicity condition, which serves us to guarantee the correctness of our algorithm in the general case. Finally, we demonstrate both the efficiency and robustness of our method by successfully applying it on various datasets of increasing complexity, ranging from synthetic curves created by an artist to automatic image-based reconstructions of real data such as hair, heart muscular fibers or magnetic field lines of a star.

Inverse Dynamic Hair Modeling with Frictional Contact

Alexandre Derouet-Jourdan, Florence Bertails-Descoubes, Gilles Daviet & Joëlle Thollot. ACM Transactions on Graphics (Proceedings of the ACM SIGGRAPH Asia 2013 Conference).

[HAL Link]

Summary: In the latest years, considerable progress has been achieved for accurately acquiring the geometry of human hair, thus largely improving the realism of virtual characters. In parallel, rich physics-based simulators have been successfully designed to capture the intricate dynamics of hair due to contact and friction. However, at the moment there exists no consistent pipeline for converting a given hair geometry into a realistic physics-based hair model. Current approaches simply initialize the hair simulator with the input geometry in the absence of external forces. This results in an undesired sagging effect when the dynamic simulation is started, which basically ruins all the efforts put into the accurate design and/or capture of the input hairstyle. In this paper we propose the first method which consistently and robustly accounts for surrounding forces --- gravity and frictional contacts, including hair self-contacts --- when converting a geometric hairstyle into a physics-based hair model. Taking an arbitrary hair geometry as input together with a corresponding body mesh, we interpret the hair shape as a static equilibrium configuration of a hair simulator, in the presence of gravity as well as hair-body and hair-hair frictional contacts. Assuming that hair parameters are homogeneous and lie in a plausible range of physical values, we show that this large underdetermined inverse problem can be formulated as a well-posed constrained optimization problem, which can be solved robustly and efficiently by leveraging the frictional contact solver of the direct hair simulator. Our method was successfully applied to the animation of various hair geometries, ranging from synthetic hairstyles manually designed by an artist to the most recent human hair data automatically reconstructed from capture.