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

Cylindrical Surface Reconstruction by Fitting Paths on Shape Space

“…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.…”

“…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).…”

“…, 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.…”

“…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].…”

“…However, when the observed data is acquired as 2D image slices, one can often find convenient planar boundary representations of objects of interest. Then one needs to perform surface reconstruction by computing an optimal fitting between the boundaries and taking into account their parametrization and the non-linearity of their spatial evolution [6].…”