Pierre-Yves Gousenbourger, Applied Mathematics

PhD thesis

P.-Y. Gousenbourger. Interpolation and fitting on Riemannian manifolds. PhD thesis, UCLouvain., 2020.
The access to constantly increasing computational capacities has revolutionized the way engineering is seen. We are now able to produce a large quantity of data, thanks to cheap sensors. However, processing such data remains costly both in computational time and in energy. One of the reasons is that the structure of the data is often omitted or unknown. The search space becomes so large that finding the solution to simple problems often turns out to be finding a needle in a haystack.
A classical problem in data processing is called the "fitting problem". It consists in fitting a d-dimensional curve to a set of data points associated to d parameters. The curve must pass sufficiently close to the data points while being regular enough. When the underlying structure of the data points (i.e., the manifold) is known, one can impose to the curve to preserve this structure (i.e., to remain on the manifold), such that the search space is drastically reduced.
The goal of this thesis is to develop methods to (approximately) solve this fitting problem; the bet is to require "less" (less computational capabilities, power, storage, time) by leveraging “more” knowledge on the search space. The objective is the following: provide a toolbox that produces a differentiable fitting curve to data points on manifolds, based on very few and simple geometric tools, at low computational cost and storage capacity, all this while maintaining an acceptable quality of the solutions.
The algorithms are applied to different illustrative problems. In 3D shape reconstruction, the data points are organs contours acquired via MRI, and the parameter is the acquisition depth; in wind fields estimation, the data points belong to the manifold of positive semi-definite matrices of given rank, and the parameters are the prevalent wind amplitudes and angles. We also show the performances of our algorithms in applications for parametric model order reduction.
```
@phdthesis{Gousenbourger2020thesis,
  author  = {Gousenbourger, Pierre-Yves},
  title	  = {Interpolation and fitting on Riemannian manifolds},
  school  = {UCLouvain, ICTEAM institute},
  year	  = {2020}
}
```

Journal papers

N. Delpierre, F. Franzini, P.-Y. Gousenbourger, S. Van Emelen, Y. Zech, S. Soares-Frazão. Breaching of sand dikes: Assessment of 1D and 2D numerical models against laboratory experiments. Journal of Hydraulic Research (IAHR), under revision, 2024.
Dike failures often induce highly damaging floods. Improved understanding of the inherent mechanisms is important to predict and prevent dike failures. This research focuses on sand dike failure by overtopping. During the experiments, the evolution of nine sections was monitored using laser-sheet while the water level in the upstream reservoir was captured using ultrasonic gauges. The impact of the grain size was studied by comparing a fine and a coarser sand. Even though the failure mechanisms look similar in both cases, some differences could be observed in the evolution of the geometry of the dike. Finally, the failure was simulated with 1D and 2D models. The 1D model shows good results for the initial stage of the dike evolution and in predicting the evolution of the width of the breach. The 2D model predicts the morphological evolution of the dike more accurately but slightly overestimates the widening of the breach.
```
@article{Bergmann2018a,
  title   = {{Breaching of sand dikes: Assessment of 1D and 2D numerical models against laboratory experiments}},
  author  = {Delpierre, Nathan and Franzini, Fabian and Gousenbourger, Pierre-Yves and Van Emelen, Sylvie and Zech, Yves and Soares-Frazão, Sandra},
  journal = {Journal of Hydraulic Research (IAHR)},
  doi     = {},
  volume  = {under revision},
  number  = {},
  pages   = {},
  year    = {2024}
}
```
P.-Y. Gousenbourger, R. Bergmann. A variational model for data fitting on manifolds by minimizing the acceleration of a Bézier curve. Frontiers in Applied Mathematics and Statistics, 4(59), 2018.
We derive a variational model to fit a composite Bézier curve to a set of data points on a Riemannian manifold. The resulting curve is obtained in such a way that its mean squared acceleration is minimal in addition to remaining close the data points. We approximate the acceleration by discretizing the squared second order derivative along the curve. We derive a closed-form, numerically stable and efficient algorithm to compute the gradient of a Bézier curve on manifolds with respect to its control points, expressed as a concatenation of so-called adjoint Jacobi fields. Several examples illustrate the capabilites and validity of this approach both for interpolation and approximation. The examples also illustrate that the approach outperforms previous works tackling this problem.
```
@article{Bergmann2018a,
  title   = {{A variational model for data fitting on manifolds by minimizing 
             the acceleration of a B\'ezier curve}},
  author  = {Bergmann, Ronny and Gousenbourger, Pierre-Yves},
  journal = {Frontiers in Applied Mathematics and Statistics},
  doi     = {10.3389/fams.2018.00059},
  volume  = {4},
  number  = {59},
  pages   = {1--16},
  year    = {2018}
}
```
P.-Y. Gousenbourger, E. Massart, P.-A. Absil. Data Fitting on Manifolds with Composite Bézier-Like Curves and Blended Cubic Splines. Journal of Mathematical Imaging and Vision, 61(5), pp. 645-671, 2018.
We propose several methods that address the problem of fitting a $C^1$ curve $\gamma$ to time-labeled data points on a manifold. The methods have a parameter, $\lambda$, to adjust the relative importance of the two goals that the curve should meet: being "straight enough" while fitting the data "closely enough". The methods are designed for ease of use: they only require to compute Riemannian exponentials and logarithms, they represent the curve by means of a number of tangent vectors that grows linearly with the number of data points, and, once the representation is computed, evaluating $\gamma(t)$ at any $t$ requires a small number of exponentials and logarithms (independent of the number of data points). Among the proposed methods, the blended cubic spline technique combines the additional properties of interpolating the data when $\lambda \to \infty$ and reducing to the well-known cubic smoothing spline when the manifold is Euclidean. The methods are illustrated on synthetic and real data.
```
@article{Gousenbourger2018,
  author  = {Gousenbourger, Pierre-Yves and Massart, Estelle and Absil, P.-A.},
  title   = {Data fitting on manifolds with composite {B\'e}zier-like curves and blended cubic splines},
  journal = {Journal of Mathematical Imaging and Vision},
  doi     = {10.1007/s10851-018-0865-2},
  volume  = {61},
  number  = {5},
  pages   = {645-671},
  year    = {2018}
}
```
P.-A. Absil, P.-Y. Gousenbourger, P. Striewski, B. Wirth. Differentiable Piecewise-Bézier Surfaces on Riemannian Manifolds. SIAM Journal on Imaging Sciences, 9(4), pp. 1788-1828, 2016.
We propose several methods that address the problem of fitting a $C^1$ curve $\gamma$ to time-labeled data points on a manifold. The methods have a parameter, $\lambda$, to adjust the relative importance of the two goals that the curve should meet: being "straight enough" while fitting the data "closely enough". The methods are designed for ease of use: they only require to compute Riemannian exponentials and logarithms, they represent the curve by means of a number of tangent vectors that grows linearly with the number of data points, and, once the representation is computed, evaluating $\gamma(t)$ at any $t$ requires a small number of exponentials and logarithms (independent of the number of data points). Among the proposed methods, the blended cubic spline technique combines the additional properties of interpolating the data when $\lambda \to \infty$ and reducing to the well-known cubic smoothing spline when the manifold is Euclidean. The methods are illustrated on synthetic and real data.
```
@article{Absil2016,
  author  = {Absil, P.-A. and Gousenbourger, Pierre-Yves and Striewski, Paul and Wirth, Benedikt},
  title   = {{Differentiable Piecewise-B\'ezier Surfaces on Riemannian Manifolds}},
  doi     = {10.1137/16M1057978},
  journal = {SIAM Journal on Imaging Sciences},
  volume  = {9},
  number  = {4},
  pages   = {1788--1828},
  year    = {2016}
}
```

Conference papers

Y. Zech, R. Meurice, N. Delpierre, P.-Y. Gousenbourger, S. Soares-Frazão. Flood interpretation in case of measurement failure. The Wamme River in 2021. RiverFlow, 2024, submitted.
During the extreme flood events that occurred in Belgium in July 2021, several measuring devices failed before the flow peak. As a consequence, it is difficult to interpret the event a posteriori and to reconstruct a reliable time series of water levels and discharges. How-ever, such interpretations are essential to help decision-makers to mitigate similar risks in the future. Most of the time, the measurement failure is clearly visible, and misinterpreta-tion can be easily avoided. However, in some other cases, measurements seem to be rec-orded normally, whereas it was not the case because of clogging of a gauge, or displace-ments of the device. This paper presents an example of such a case and the methodology that was used to cope with this situation.
```
@inproceedings{Zech2024,
  author  = {Zech, Yves and Meurice, Robin and Delpierre, Nathan and Gousenbourger, Pierre-Yves and {S}oares-{F}razão, Sandra},
  title   = {Flood interpretation in case of measurement failure. The {W}amme {R}iver in 2021},
  booktitle = {RiverFlow},
  pages   = {},
  year    = {2024},
  organization = {}
}
```
N. T. Son, P.-Y. Gousenbourger, E. Massart, P.-A. Absil. Online balanced truncation for linear time-varying systems using continuously differentiable interpolation on Grassmann manifold Proceedings of the 6th International Conference on Control, Decision and Information Technologies (CoDIT 2019), 2019, to appear.
We consider model order reduction of linear time-varying systems on a finite time interval using balanced truncation. A standard way to perform MOR is to first numerically integrate the associated pair of differential Lyapunov equations for the two gramians, then compute projection matrices using the square root method, and finally formulate the reduced systems at each time instant of a chosen grid. This approach is well-known for delivering good approximation, but rather costly in computation and storage requirement. Furthermore, if one needs to compute the reduced system for any new time instant that is not included in the chosen grid, the mentioned procedure must be performed again without explicitly making use of the already computed data. For dealing with such a situation, we propose to store the projection matrices corresponding to a simplified sparse time grid and to use them to recover the projection subspaces at any other time instant via curve interpolation on the Grassmann manifold. By doing this, we can avoid the repetition of solving the differential Lyapunov equations which is the most expensive step in the procedure and therefore, as shown in a numerical example, accelerate the online reduction process.
```
@inproceedings{Son2019,
  author  = {Son, Nguyen Thanh and Gousenbourger, Pierre-Yves and Massart, Estelle and Absil, P.-A.},
  title   = {Online balanced truncation for linear time-varying systems using continuously differentiable interpolation on Grassmann manifold},
  booktitle = {2019 6th International Conference on Control, Decision and Information Technologies (CoDIT)},
  pages   = {165--170},
  year    = {2019},
  organization = {IEEE}
}
```
E. Massart, P.-Y. Gousenbourger, N. T. Son, T. Stykel, P.-A. Absil. Interpolation on the manifold of fixed-rank positive-semidefinite matrices for parametric model order reduction: preliminary results. ESANN2019, submitted.
We present several interpolation schemes on the manifold of fixed-rank positive-semidefinite (PSD) matrices. We explain how these techniques can be used for model order reduction of parameterized linear dynamical systems, and obtain preliminary results on an application.
```
@inproceedings{Massart2019a,
  author  = {Massart, Estelle and Gousenbourger, Pierre-Yves and Son, Nguyen Thanh and Stykel, Tatjana and Absil, P.-A.},
  title   = {Interpolation on the manifold of fixed-rank positive-semidefinite matrices for parametric model order reduction: preliminary results},
  booktitle = {ESANN2019},
  pages   = {281--286},
  isbn    = {978-287-587-065-0},
  publisher = {Springer},
  year    = {2019}
}
```
P.-Y. Gousenbourger, L. Jacques, P.-A. Absil. Fast method to fit a $C^1$ piecewise-Bézier function to manifold-valued data points: how suboptimal is the curve obtained on the sphere $\mathbb S^2$? Nielsen, F. and Barbaresco F., editors, GSI2017, pp. 595-603, 2017.
We propose an analysis of the quality of the fitting method proposed in [7]. This method fits smooth paths to manifold-valued data points using $C^1$ piecewise-B ́ezier functions. This method is based on the principle of minimizing an objective function composed of a data-attachment term and a regularization term chosen as the mean squared acceleration of the path. However, the method strikes a tradeoff between speed and accuracy by following a strategy that is guaranteed to yield the optimal curve only when the manifold is linear. In this paper, we focus on the sphere $S^2$. We compare the quality of the path returned by the algorithms from [7] with the path obtained by minimizing, over the same search space of $C^1$ piecewise-B́ezier curves, a finite-difference approximation of the objective function by means of a derivative-free manifold-based optimization method.
```
@inproceedings{Gousenbourger2017a,
  author  = {Gousenbourger, Pierre-Yves and Jacques, Laurent and Absil, P.-A.},
  editor  = {Nielsen, Frank and Barbaresco, Fr{\'e}d{\'e}ric},
  title   = {Fast method to fit a {$C^1$} piecewise-{B\'{e}}zier function to manifold-valued data points: how suboptimal is the curve obtained on the sphere {$\mathbb S^2$}?},
  booktitle = {Geometric Science of Information},
  doi     = {10.1007/978-3-319-68445-1_69},
  publisher = {Springer},
  address = {Berlin, Heidelberg},
  series  = {Lecture Notes in Computer Sciences},
  volume  = {10589},
  pages   = {595--603},
  year    = {2017}
}
```
P.-Y. Gousenbourger, E. Massart, A. Musolas, P.-A. Absil, L. Jacques, J.M. Hendrickx, Y. Marzouk. Piecewise-Bézier $C^1$ smoothing on manifolds with application to wind field estimation. ESANN2017, Springer eds,, pp. 305-301, 2017.
We propose an algorithm for fitting $C^1$ piecewise-Bézier curves to (possibly corrupted) data points on manifolds. The curve is chosen as a compromise between proximity to data points and regularity. We apply our algorithm as an example to fit a curve to a set of low-rank covariance matrices, a task arising in wind field modeling. We show that our algorithm has denoising abilities for this application.
```
@inproceedings{Gousenbourger2017,
  author  = {Gousenbourger, Pierre-Yves and Massart, Estelle and Musolas, Antoni and Absil, P.-A. and Jacques, Laurent and Hendrickx, Julien M and Marzouk, Youssef},
  booktitle = {ESANN2017},
  pages   = {305--310},
  publisher = {Springer},
  title   = {Piecewise-{B}{\'e}zier {$C^1$} smoothing on manifolds with application to wind field estimation},
  year    = {2017}
}
```
P.-A. Absil, P.-Y. Gousenbourger, P. Striewski, B. Wirth. Differentiable piecewise-Bézier interpolation on Riemannian manifolds. ESANN2016, Springer eds,, pp. 95-100, 2016.
We propose a generalization of classical Euclidean piecewise-Bézier surfaces to manifolds, and we use this generalization to compute a $C^1$-surface interpolating a given set of manifold-valued data points associated to a regular 2D grid. We then propose an efficient algorithm to compute the control points defining the surface based on the Euclidean concept of natural $C^2$-splines and show examples on different manifolds.
```
@inproceedings{Absil2016a,
  author    = {Absil, P.-A. and Gousenbourger, Pierre-Yves and Striewski, Paul and Wirth, Benedikt},
  title     = {Differentiable piecewise-{B\'{e}}zier interpolation on {R}iemannian manifolds},
  booktitle = {ESANN2016},
  isbn      = {978--287587027--8},
  publisher = {Springer},
  pages     = {95--100},
  year      = {2016}
}
```
C. Samir, P.-Y. Gousenbourger, S. H. Joshi. Cylindrical Surface Reconstruction by Fitting Paths on Shape Space. EProceedings of the 1st International Workshop on DIFFerential Geometry in Computer Vision for Analysis of Shapes, Images and Trajectories (DIFF-CV 2015), 2015.
We present a differential geometric approach for cylindrical anatomical surface reconstruction from 3D volumetric data that may have missing slices or discontinuities. We extract planar boundaries from the 2D image slices, and parameterize them by an indexed set of curves. Under the SRVF framework, the curves are represented as invariant elements of a nonlinear shape space. Differently from standard approaches, we use tools such as exponential maps and geodesics from Riemannian geometry and solve the problem of surface reconstruction by fitting paths through the given curves. Experimental results show the surface reconstruction of smooth endometrial tissue shapes generated from MRI slices.
```
@inproceedings{Samir2015,
  author = {Samir, Chafik and Gousenbourger, Pierre-Yves and Joshi, Shantanu H.},
  booktitle = {DIFF-CV 2015},
  pages  = {1--10},
  title  = {{Cylindrical Surface Reconstruction by Fitting Paths on Shape Space}},
  year   = {2015}
}
```
A. Arnould, P.-Y. Gousenbourger, C. Samir, P.-A. Absil, M. Canis. Fitting smooth paths on Riemannian manifolds: Endometrial surface reconstruction and preoperative MRI-based navigation. Nielsen, F. and Barbaresco F., editors, GSI2015, pp. 491-498, 2015.
We present a new method to fit smooth paths to a given set of points on Riemannian manifolds using $C^1$ piecewise-Bézier functions. A property of the method is that, when the manifold reduces to a Euclidean space, the control points minimize the mean square acceleration of the path. As an application, we focus on data observations that evolve on certain nonlinear manifolds of importance in medical imaging: the shape manifold for endometrial surface reconstruction; the special orthogonal group $SO(3)$ and the special Euclidean group $SE(3)$ for preoperative MRI-based navigation. Results on real data show that our method succeeds in meeting the clinical goal: combining different modalities to improve the localization of the endometrial lesions.
```
@inproceedings{Arnould2015,
  author  = {Arnould, Antoine and Gousenbourger, Pierre-Yves and Samir, Chafik and Absil, P.-A. and Canis, Michel},
  title   = {{Fitting Smooth Paths on Riemannian Manifolds : Endometrial Surface Reconstruction and Preoperative MRI-Based Navigation}},
  booktitle = {Geometric Science of Information},
  doi     = {10.1007/978-3-319-25040-3_53},
  editor  = {Nielsen, Frank and Barbaresco, Fr{\'e}d{\'e}ric},
  publisher = {Springer},
  address = {Berlin, Heidelberg},
  series  = {Lecture Notes in Computer Sciences},
  volume  = {9389},
  pages   = {491--498},
  year    = {2015}
}
```
P.-Y. Gousenbourger, C. Samir, P.-A. Absil. Piecewise-Bézier $C^1$ interpolation on Riemannian manifolds with application to 2D shape morphing. International Conference on Pattern Recognition (ICPR), pp. 4086-4091, 2014.
We propose a generalization of classical Euclidean piecewise-Bézier surfaces to manifolds, and we use this generalization to compute a $C^1$-surface interpolating a given set of manifold-valued data points associated to a regular 2D grid. We then propose an efficient algorithm to compute the control points defining the surface based on the Euclidean concept of natural $C^2$-splines and show examples on different manifolds.
```
@inproceedings{Gousenbourger2014,
  author = {Gousenbourger, Pierre-Yves and Samir, Chafik and Absil, P.-A.},
  title  = {Piecewise-{B\'e}zier {$C^1$} interpolation on {R}iemannian manifolds with application to {2D} shape morphing},
  booktitle = {International Conference on Pattern Recognition (ICPR)},
  pages  = {4086--4091},
  year   = {2014},
  doi    = {10.1109/ICPR.2014.700},
  isbn   = {9781479952083},
  issn   = {10514651}
}
```

Talks

Data fitting on manifolds: applications, challenges and solutions
ISP Group, Louvain-la-Neuve, Belgium. Dec. 11, 2019.
Storm trajectories prediction, birds migrations follow-up, rigid rotations of 3D objects, wind field estimation, model order reduction of superlarge parameter-dependent dynamical systems, MRI 3D body volumes reconstruction... All these applications have two things in common: first, they have a geometrical data-structure, i.e., the data lives on a (generally) Riemannian manifold; second, they can benefit of parameter(s)-dependent fitting methods somewhere in the process. If data fitting is a basic problem in the Euclidean space (where natural cubic splines and thin plates splines are the superstars in the domain), it become more intricate when data structure constrains the problem.

This talk is an opportunity to present you an efficient, ready-to-use algorithm for data fitting on manifolds based on B\'ezier curves, applied to some of the aforementioned applications.
Data fitting on manifolds by minimizing the mean square acceleration of a Bézier curve
Benelux Meeting 2019, Lommel, Belgium. Mar. 19, 2019.
Curve fitting on manifolds with Bézier and blended curves
Seminar at Research Group "Numerical Mathematics (Partial Differential Equations)", from TU Chemnitz, Chemnitz, Germany. Jul. 18, 2018.
Fast method to fit a $C^1$ piecewise-Bézier function to manifold-valued data points: how suboptimal is the curve obtained on the sphere $\mathbb S^2$?
GSI2015, Paris, France. Nov. 7, 2017.
We propose an analysis of the quality of the fitting method proposed in [7]. This method fits smooth paths to manifold-valued data points using $C^1$ piecewise-B ́ezier functions. This method is based on the principle of minimizing an objective function composed of a data-attachment term and a regularization term chosen as the mean squared acceleration of the path. However, the method strikes a tradeoff between speed and accuracy by following a strategy that is guaranteed to yield the optimal curve only when the manifold is linear. In this paper, we focus on the sphere $S^2$. We compare the quality of the path returned by the algorithms from [7] with the path obtained by minimizing, over the same search space of $C^1$ piecewise-B́ezier curves, a finite-difference approximation of the objective function by means of a derivative-free manifold-based optimization method.
Wind field estimation via $C^1$ Bézier smoothing on manifolds
Workshop on "Regularized Inverse Problem Solving and High-Dimensional Learning Methods" (WIPS), Louvain-la-Neuve, Belgium. Aug. 30, 2017.
Unmanned aerial vehicle control is a hot topic in research and at the crossroad of a lot of disciplines. For instance, safe and reliable navigation of UAVs requires consideration of the surrounding environment, in particular, the external wind conditions. This external wind configuration is usually evaluated by computationnaly expensive efforts and depends on external meteorological parameters. Hopefully, it can be modelled as a Gaussian process characterized by a covariance matrix belonging to the space of PSD matrices of rank r. In this work, we both expoit the manifold structure of this specific space and also propose a method to fit a small set of pre-computed solutions. That way, for a new value of the external meteorological parameters, we are able to recover a sufficiently accurate wind field configuration in a computationnaly tractable effort. Our method is based on manifold-valued Bezier curves. Joint work with E.M. Massart, A. Musolas, P.-A. Absil, J.M. Hendrickx, L. Jacques and Y. Marzouk.
Wind field estimation via $C^1$ Bézier smoothing on manifolds
ESANN2017, Bruges, Belgium. Apr. 27, 2017.
We propose an algorithm for fitting $C^1$ piecewise-Bézier curves to (possibly corrupted) data points on manifolds. The curve is chosen as a compromise between proximity to data points and regularity. We apply our algorithm as an example to fit a curve to a set of low-rank covariance matrices, a task arising in wind field modeling. We show that our algorithm has denoising abilities for this application.
Wind field estimation via $C^1$ Bézier smoothing on manifolds
Benelux Meeting 2017, Spa, Belgium. Mar. 28, 2017.
Interpolation and fitting on manifolds with differentiable piecewise-Bézier functions
ISP Group, Louvain-la-Neuve, Belgium. Mar. 16, 2017.
Fitting and interpolation are well known topics on the Euclidean space. However, when it turns out that the data points are manifold valued (understand: when the data point belongs to a certain manifold), most of the algorithms are no more applicable because even the notion of distance is not as simple as on the Euclidean space.

Manifold-valued data appear in various domains as MRI acquisition (diffusion tensors, segmented shapes,...), fluid mechanics (SDP matrices or values on the Grassmann manifold), matrix completion and many more. A particular example concern the representation of local wind fields in a given domain. These can be expressed as covariance matrices which are SDP and of fixed (small) rank. The wind field changes when the prevalent wind has different orientation or magnitude. The computation of this wind field, however, resort in solving complicated and time-consuming PDEs in fluid mechanics. In order to obtain approached solutions as fast as possible, a solution is to compute several wind field covariance matrices representations for a set of given orientation/magnitude of the prevalent wind and then interpolate (or fit) a manifold-valued curve in the space of fixed rand SDP matrices.

In this talk, I will propose a method to fit manifold-valued data points based on a generalization of Bezier curves. This method is motivated by (but not specific to) the wind field orientation problem. I will try to introduce the concepts of manifolds and Bezier curves as a tutorial to finally show interesting results for that specific application.
Interpolation and fitting on manifolds with differentiable piecewise-Bézier functions
Anuj Srivastava's working group meeting, Skype presentation, Tallahassee, Florida, USA. Mar. 10, 2017.
Differentiable Bézier interpolation on manifolds with B-splines
GAMM 2017, Weimar, Germany. Mar. 6, 2017.
Interpolation and fitting on manifolds with differentiable piecewise-Bézier functions
Workshop on Manifold-Valued Image Processing (MVIP) 2016, Kaiserslautern, Germany. Dec. 2, 2016.
Interpolation on manifolds with differentiable surfaces of Bézier
Benelux Meeting 2016, Soesterberg, The Netherlands. Mar. 24, 2016.
Bézier interpolation on Riemannian manifolds
ASCII Tutorial Seminars, Louvain-la-Neuve, Belgium. Nov. 27, 2015.
Mani-what you say? Manifolds are mathematical sets with a smooth geometry (such as spheres) used in applications like model reduction, optimization and many more. What do manifold-valued objects look like? Can we interpolate them? Is it costly? With Bézier functions, interpolation on manifolds becomes very easy. Bézier? You should remember this from your bachelor, isn't it? From a very few concepts and drawings, a variety of possibilities is open.
Endometriosis: MRI navigation and surface reconstruction on manifolds
GSI2015, Paris, France. Oct. 30, 2015.
We present a new method to fit smooth paths to a given set of points on Riemannian manifolds using $C^1$ piecewise-Bézier functions. A property of the method is that, when the manifold reduces to a Euclidean space, the control points minimize the mean square acceleration of the path. As an application, we focus on data observations that evolve on certain nonlinear manifolds of importance in medical imaging: the shape manifold for endometrial surface reconstruction; the special orthogonal group $SO(3)$ and the special Euclidean group $SE(3)$ for preoperative MRI-based navigation. Results on real data show that our method succeeds in meeting the clinical goal: combining different modalities to improve the localization of the endometrial lesions.
How to design interpolating surfaces on manifolds?
One-Day Workshop on manifolds optimization, Louvain-la-Neuve, Belgium. Sep. 25, 2015.
Interpolation on Riemannian manifolds with a $C^1$ piecewize-Bézier path.
GDR MIA 2014, Paris, France. Nov. 21, 2014.
Interpolation on Riemannian manifolds with a $C^1$ piecewize-Bézier path.
ISP Group, Louvain-la-Neuve, Belgium. Oct. 8, 2014.
Nowadays, more and more problems are solved through specific manifold formulation. This often allows important reduction of computation time and/or memory management compared to classical formulations on the classical Euclidean space (because of non-linear constraints like restricting the solutions to a certain subdomain of a larger ambiant space). Interpolation and optimization tools can be useful for solving some of these problems (like defining the optimal trajectory of a humanitory plane dropping supplies, or fitting two objects orientations). However, current procedures are only defined on the Euclidean space. In this presentation, I focus on interpolation methods and, more precisely, I propose a new general framework to fit a path through a finite set of data points lying on a Riemannian manifold. The path takes the form of a continuously-differentiable concatenation of Riemannian Bézier segments. This framework will be illustrated by results on the Euclidean space, the sphere, the orthogonal group and the shape manifold. The content of this presentation meets also very recent research carried out in this institute for providing novel efficient manifold-based optimization methods.

Posters

Interpolation on the manifold of fixed-rank positive-semidefinite matrices for parametric model order reduction: preliminary results.
Conference ESANN 2019, Bruges, Belgium, 2019.
We present several interpolation schemes on the manifold of fixed-rank positive-semidefinite (PSD) matrices. We explain how these techniques can be used for model order reduction of parameterized linear dynamical systems, and obtain preliminary results on an application.
Blended smoothing splines on Riemannian manifolds.
Conference iTwist 2018, Marseille, France, 2018.
We present a method to compute a fitting curve $B$ to a set of data points $d_0,\dots, d_m$ lying on a manifold $\mathcal M$. That curve is obtained by blending together Euclidean Bézier curves obtained on different tangent spaces. The method guarantees several properties among which $B$ is $C^1$ and is the natural cubic smoothing spline when $\mathcal M$ is the Euclidean space. We show examples on the sphere $S^2$ as a proof of concept.
Data fitting on manifolds with blended cubic splines.
ICML2018, workshop GiMLi, Stockholm, Sweden, 2018.
Interpolation on manifolds using Bézier functions.
Conference iTwist 2018, Aalborg, Denmark, 2016.
Given a set of data points lying on a smooth manifold, we present methods to interpolate those with piecewise Bézier splines. The spline is composed of Bézier curves (resp. surfaces) patched together such that the spline is continuous and differentiable at any point of its domain. The spline is optimized such that its mean square acceleration is minimized when the manifold is the Euclidean space. We show examples on the sphere $S^2$ and on the special orthogonal group $SO(3)$.
Differentiable piecewise-Bézier interpolation on Riemannian manifolds.
Conference ESANN2016, Bruges, Belgium, 2016.
We propose a generalization of classical Euclidean piecewise-Bézier surfaces to manifolds, and we use this generalization to compute a $C^1$-surface interpolating a given set of manifold-valued data points associated to a regular 2D grid. We then propose an efficient algorithm to compute the control points defining the surface based on the Euclidean concept of natural $C^2$-splines and show examples on different manifolds.
Piecewise-Bézier $C^1$ interpolation on Riemannian manifolds with application to 2D shape morphing.
Dysco Day, Gent, Belgium, 2014.
Piecewise-Bézier $C^1$ interpolation on Riemannian manifolds with application to 2D shape morphing.
International Conference on Pattern Recognition (ICPR), pp. 4086-4091, 2014.
We present a new framework to fit a path to a given finite set of data points on a Riemannian manifold. The path takes the form of a continuously-differentiable concatenation of Riemannian Bézier segments. The selection of the control points that define the Bézier segments is partly guided by the differentiability requirement and by a minimal mean squared acceleration objective. We illustrate our approach on specific manifolds: the Euclidean plane (for sanity check), the sphere (as a first nonlinear illustration), the special orthogonal group (with rigid body motion applications), and the shape manifold (with 2D shape morphing applications).