Fitting Smooth Paths on Riemannian Manifolds: Endometrial Surface Reconstruction and Preoperative MRI-Based Navigation

Arnould, Antoine; Gousenbourger, Pierre-Yves; Samir, Chafik; Absil, Pierre-Antoine; Canis, Michel

doi:10.1007/978-3-319-25040-3_53

Cited by 10 publications

(18 citation statements)

References 14 publications

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“…We present two examples of cylindrical surface reconstruction and compare the fitting paths on shape space obtained with (i) the solution from [2], (ii) the solution from [6] and (iii) a piecewise-geodesics method. The level-set curves were segmented manually by an expert from MRI slices.…”

Section: Resultsmentioning

confidence: 99%

“…Third, thanks to a modified version of [15] to compute a geodesic path between any two points on shape space, we applied the three methods to construct S MRI , for each example. The solution from [2] corresponds to the first two rows, [6] to the rows 3 and 4 and piecewise-geodesics to rows 5 and 6. To give an idea about the quality of the reconstructed surface, we show an example of S MRI (b) reconstructed from a set of curves (a), the norm of the fitting path's velocity η(t) (c), the norm of its acceleration η(t) (d) , the Laplacian map ∆S MRI (e), the norm of the gradient along the radial curves ∇ r S MRI (f), the norm of the gradient along the circular curves ∇ θ S MRI (g), and ∇ r,θ S MRI as a function (2D map) of (r, θ ) (h).…”

Section: Resultsmentioning

confidence: 99%

“…, n. In [2,6], methods to do so were proposed for a general Riemannian manifold M. In both works, the strategy was to formulate a C 1 piecewise-Bézier interpolation problem for the case where M reduces to a Euclidean space, then obtain equations that govern the control points of the sought Bézier curves, and finally generalize those equations to Riemannian manifolds. In this section, we briefly summarize the Euclidean foundations of two of these methods, referring the reader to [6] and [2] for details. Then, in the next section, we work out concretely the extension of the two methods to the specific case where the manifold M is the shape space of curves.…”

Section: The Path Fitting Problem On the Euclidean Spacementioning

confidence: 99%

“…A and C are tridiagonal matrices of coefficients (see [2] for details) with A invertible which makes the solution unique:…”

Section: The Path Fitting Problem On the Euclidean Spacementioning

confidence: 99%

“…It is worth noting that the solutions presented above can be expressed on some general Riemannian manifolds embedded in a Euclidean space E [2,6]. For such a generalization, one needs to define and compute the exponential map, the logarithmic map and an appropriate metric [1,12].…”

Section: The Path Fitting Problem On the Shape Spacementioning

confidence: 99%

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Cylindrical Surface Reconstruction by Fitting Paths on Shape Space

Samir¹,

Gousenbourger²,

Joshi³

2015

Procedings of the Proceedings of the 1st International Workshop on DIFFerential Geometry in Computer Vision for Analysis of Sha

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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.

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Section: Resultsmentioning

confidence: 99%

Section: Resultsmentioning

confidence: 99%

Section: The Path Fitting Problem On the Euclidean Spacementioning

confidence: 99%

“…A and C are tridiagonal matrices of coefficients (see [2] for details) with A invertible which makes the solution unique:…”

Section: The Path Fitting Problem On the Euclidean Spacementioning

confidence: 99%

Section: The Path Fitting Problem On the Shape Spacementioning

confidence: 99%

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Cylindrical Surface Reconstruction by Fitting Paths on Shape Space

Samir¹,

Gousenbourger²,

Joshi³

2015

Procedings of the Proceedings of the 1st International Workshop on DIFFerential Geometry in Computer Vision for Analysis of Sha

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Generalization of the Neville–Aitken interpolation algorithm on Grassmann manifolds: Applications to reduced order model

Hamidi

Hamdouni

Falaize

2021

Numerical Methods in Fluids

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An extension of the well-known Neville-Aitken's algorithm for interpolation on the Grassmann manifold G m (R n ) in the framework of parametric model order reduction is presented. Interpolation points on G m (R n ) are the subspaces spanned by bases obtained by Proper Orthogonal Decomposition of available solutions associated with the chosen parameter sampling. The Neville-Aitken's algorithm is performed recursively via the geodesic barycenter of two points. Three CFD applications are presented: (i) the Von Karman vortex shedding street, (ii) the lid-driven cavity with inflow and (iii) the flow induced by a rotating solid. Numerical results are relevant with respect to accuracy while the asymptotic complexity is comparable to the state of the art.

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Bézier Curves and $$C^{2}$$ Interpolation in Riemannian Symmetric Spaces

Samir¹,

Adouani²

2019

Lecture Notes in Computer Science

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We consider the problem of interpolating a finite set of observations at given time instant. In this paper, we introduce a new method to compute the optimal intermediate control points that define a C 2 interpolating Bézier curve. We prove this concept for interpolating data points belonging to a Riemannian symmetric spaces. The main property of the proposed method is that the control points minimize the mean square acceleration. Moreover, potential applications of fitting smooth paths on Riemannian manifold include applications in robotics, animations, graphics, and medical studies.

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Fitting Smooth Paths on Riemannian Manifolds: Endometrial Surface Reconstruction and Preoperative MRI-Based Navigation

Cited by 10 publications

References 14 publications

Cylindrical Surface Reconstruction by Fitting Paths on Shape Space

Cylindrical Surface Reconstruction by Fitting Paths on Shape Space

Generalization of the Neville–Aitken interpolation algorithm on Grassmann manifolds: Applications to reduced order model

Bézier Curves and $$C^{2}$$ Interpolation in Riemannian Symmetric Spaces

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