A variational model for data fitting on manifolds by minimizing the acceleration of a B\'ezier curve

We derive a variational model to fit a composite B\'ezier 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 B\'ezier 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

Interpolation and fitting on Riemannian manifolds

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.(FSA - Sciences de l'ingénieur) -- UCL, 202

Data Fitting on Manifolds with Composite Bézier-Like Curves and Blended Cubic Splines

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 $\lambda \rightarrow \infty$ and reducing to the well-known cubic smoothing spline when the manifold is Euclidean. The methods are illustrated on synthetic and real data

Piecewise-Bézier C1 Interpolation on Riemannian Manifolds with Application to 2D Shape Morphing

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Piecewise-Bézier C¹ interpolation on Riemannian manifolds with application to 2D shape morphing

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)

Cylindrical Surface Reconstruction by Fitting Paths on Shape Space

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

Blended smoothing splines on Riemannian manifolds

We present a method to compute a fitting curve B to a set of data points d0,...,dm lying on a manifold 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 C1 and is the natural cubic smoothing spline when M is the Euclidean space. We show examples on the sphere S2 as a proof of concept

Differentiable piecewise-Bézier surfaces on Riemannian manifolds

We generalize the notion of Bézier surfaces and surface splines to Riemannian manifolds. To this end we put forward and compare three possible alternative definitions of Bézier surfaces. We furthermore investigate how to achieve C0- and C1-continuity of Bézier surface splines. Unlike in Euclidean space and for one-dimensional Bézier splines on manifolds, C1-continuity cannot be ensured by simple conditions on the Bézier control points: it requires an adaptation of the Bézier spline evaluation scheme. Finally, we propose an algorithm to optimize the Bézier control points given a set of points to be interpolated by a Bézier surface spline. We show computational examples on the sphere, the special orthogonal group, and two Riemannian shape space