Smooth Shapes with Spherical Topology: Beyond Traditional Modeling, Efficient Deformation, and Interaction

Existing shape models with spherical topology are typically designed either in the discrete domain using interpolating polygon meshes or in the continuous domain using smooth but non-interpolating schemes such as subdivision or NURBS. Both polygon models and subdivision methods require a large number of parameters to model smooth surfaces. NURBS need fewer parameters but have a complicated rational expression and non-uniform shifts in their formulation. We present a new method to construct deformable closed surfaces, which includes exact spheres, by combining the best of two worlds: a smooth, interpolating model with a continuously varying tangent plane and well-defined curvature at every point on the surface. Our formulation is considerably simpler than NURBS and requires fewer parameters than polygon meshes. We demonstrate the generality of our method with applications including intuitive user-interactive shape modeling, continuous surface deformation, shape morphing, reconstruction of shapes from parameterized point clouds, and fast iterative shape optimization for image segmentation. Comparisons with discrete methods and non-interpolating approaches highlight the advantages of our framework.

Smoothly Deformable Spheres: Modeling, Deformation, and Interaction

Pablo Garcia-Amorena Garcia, Daniel Andreas Schmitter, Michaël Unser

Existing shape models with spherical topology are typically designed either in the discrete domain using interpolating polygon meshes or in the continuous domain using smooth but non-interpolating schemes such as NURBS. Polygon models and subdivision methods require a large number of parameters to model smooth surfaces. NURBS need fewer parameters but have a complicated rational expression and non-uniform shifts in their formulation. We present a new method to construct deformable closed surfaces, which includes the exact sphere, by combining the best of two worlds: a smooth and interpolating model with a continuously varying tangent plane and well-defined curvature at every point on the surface. Our formulation is simpler than NURBS while it requires fewer parameters than polygon meshes. We demonstrate the generality of our method with applications ranging from intuitive user-interactive shape modeling, continuous surface deformation, reconstruction of shapes from parameterized point clouds, to fast iterative shape optimization for image segmentation.

ACM2016

Global radii of curvature, and the biarc approximation of space curves

Jana Smutny

The distance from self-intersection of a (smooth and either closed or infinite) curve q in three dimensions can be characterised via the global radius of curvature at q(s), which is defined as the smallest possible radius amongst all circles passing through the given point and any two other points on the curve. The minimum value of the global radius of curvature along the curve gives a convenient measure of curve thickness or normal injectivity radius. Given the utility of the construction inherent to global curvature, it is natural to consider variants defined in related ways. The first part of the thesis considers all possible circular and spherical distance functions and the associated, single argument, global radius of curvature functions that are constructed by minimisation over all but one argument. It is shown that among all possible global radius of curvature functions there are only five independent ones. And amongst these five there are two particularly useful ones for characterising thickness of a curve. We investigate the geometry of how these two functions, ρpt and ρtp, can be achieved. Properties and interrelations of the divers global radius of curvature functions are illustrated with the simple examples of ellipses and helices. It is known that any Lipschitz continuous curve with positive thickness actually has C1,1-regularity. Accordingly, C1,1 is the natural space in which to carry out computations involving self-avoiding curves. The second part of the thesis develops the mathematical theory of biarcs, which are a geometrically elegant way of discretizing C1,1 space curves. A biarc is a pair of circular arcs joined in a C1 fashion according to certain matching rules. We establish a self-contained theory of the geometry of biarc interpolation of point-tangent data sampled from an underlying base curve, and demonstrate that such biarc curves have attractive convergence properties in both a pointwise and function-space sense, e.g. the two arcs of the biarc interpolating a coalescent point-tangent data pair on a C2-curve approach the osculating circle of the curve at the limit of the data points, and for a C1,1-base curve and a sequence of (possibly non-uniform) meshes, the interpolating biarc curves approach the base curve in the C1-norm. For smoother base curves, stronger convergence can be obtained, e.g. interpolating biarc curves approach a C2 base curve in the C1,1-norm. The third part of the thesis concerns the practical utility of biarcs in computation. It is shown that both the global radius of curvature function ρpt and thickness can be evaluated efficiently (and to an arbitrarily small, prescribed precision) on biarc curves. Moreover, both the notion of a contact set, i.e. the set of points realising thickness, and an approximate contact set can be defined rigorously. The theory is then illustrated with an application to the computation of ideal shapes of knots. Informally ideal knot shapes can be described as the configuration allowing a given knot to be tied with the shortest possible piece of rope of prescribed thickness. The biarc discretization is combined with a simulated annealing code to obtain approximate ideal shapes. These shapes provide rigorous upper bounds for rope length of ideal knots. The approximate contact set and the function ρpt evaluated on the computed shapes allow us to assess closeness of the computations to ideality. The high accuracy of the computations reveal various, previously unrecognized, features of ideal knot shapes.

EPFL2004

Implicit meshes

Slobodan Ilic

This thesis proposes novel ways both to represent the static surfaces, and to parameterize their deformations. This can be used both by automated algorithms for efficient 3–D shape reconstruction, and by graphics designers for editing and animation. Deformable 3–D models can be represented either as traditional explicit surfaces, such as triangulated meshes, or as implicit surfaces. Explicit surfaces are widely accepted because they are simple to deform and render, however fitting them involves minimizing a non-differentiable distance function. By contrast, implicit surfaces allow fitting by minimizing a differentiable algebraic distance, but they are harder to meaningfully deform and render. Here we propose a method that combines the strength of both representations to avoid their drawbacks, and in this way build robust surface representation, called implicit mesh, suitable for automated shape recovery from video sequences. This surface representation lets us automatically detect and exploit silhouette constraints in uncontrolled environments that may involve occlusions and changing or cluttered backgrounds, which limit the applicability of most silhouette based methods. We advocate the use of Dirichlet Free Form Deformation (DFFD) as generic surface deformation technique that can be used to parameterize objects of arbitrary geometry defined as explicit meshes. It is based on the small set of control points and the generalized interpolant. Control points become model parameters and their change causes model's shape modification. Using such parameterization the problem dimensionality can be dramatically reduced, which is desirable property for most optimization algorithms, thus makes DFFD good tool for automated fitting. Combining DFFD as a generic parameterization method for explicit surfaces and implicit meshes as a generic surface representation we obtained a powerfull tool for automated shape recovery from images. However, we also argue that any other avaliable surface parameterization can be used. We demonstrate the applicability of our technique to 3–D reconstruction of the human upper-body including – face, neck and shoulders, and the human ear, from noisy stereo and silhouette data. We also reconstruct the shape of a high resolution human faces parametrized in terms of a Principal Component Analysis model from interest points and automatically detected silhouettes. Tracking of deformable objects using implicit meshes from silhouettes and interest points in monocular sequences is shown in following two examples: Modeling the deformations of a piece of paper represented by an ordinary triangulated mesh; tracking a person's shoulders whose deformations are expressed in terms of Dirichlet Free Form Deformations.