In applications that involve interactive curve and surface modeling, the intuitive manipulation of shapes is crucial. For instance, user interaction is facilitated if a geometrical object can be manipulated through control points that interpolate the shape itself. Additionally, models for shape representation often need to provide local shape control and they need to be able to reproduce common shape primitives such as ellipsoids, spheres, cylinders, or tori. We present a general framework to construct families of compactly-supported interpolators that are piecewise-exponential polynomial. They can be designed to satisfy regularity constraints of any order and they enable one to build parametric deformable shape models by suitable linear combinations of interpolators. They allow to change the resolution of shapes based on the refinability of B-splines. We illustrate their use on examples to construct shape models that involve curves and surfaces with applications to interactive modeling and character design.
Recently, a method to build piecewise-polynomial interpolators has been presented in @cite_23 @cite_19 and its bivariate generalization was proposed in @cite_12 . The present work is the continuation of our previous efforts to, first, generalize the popular Catmull-Rom @cite_15 and Keys @cite_24 @cite_33 interpolators for practical applications @cite_37 @cite_2 @cite_4 @cite_36 and, next, to go one step further and construct families of interpolators that allow varying the resolution of a shape @cite_22 @cite_13 . Here, the novelty w.r.t. @cite_36 is that the presented framework allows one to vary the resolution of shapes which facilitates shape design in practice, as illustrated in .
@cite_37 @cite_4 @cite_33 @cite_22 @cite_36 @cite_24 @cite_19 @cite_23 @cite_2 @cite_15 @cite_13 @cite_12
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We propose a new parametric 3D snake with cylindrical topology. Its construction is based on interpolatory basis functions which facilitates user-interaction because the control points of the snake directly lie on the surface of the deformable cylinder. We prove that the basis functions exactly reproduce a cylinder and propose a new parametrization as a tensor-product spline surface. We provide explicit formulas for the energy function based on Green's theorem that speed up the computation of the optimization algorithm. We have implemented the proposed framework as a freely available open-source plugin for the bioimaging platform Icy. Its utility has been tested on phantom data as well as on real 3D data to segment the spinal cord and the descending aorta. We present a new trigonometric basis function that is capable of perfectly reproducing circles, spheres and ellipsoids while at the same time being interpolatory. Such basis functions have the advantage that they allow to construct shapes through a sequence of control points that lie on their contour (2-D) or surface (3-D) which facilitates user-interaction, especially in 3-D. Our piecewise exponential basis function has finite support, which enables local control for shape modification. We derive and prove all the necessary properties of the kernel to represent shapes that can be smoothly deformed and show how idealized shapes such as ellipses and spheres can be constructed. We establish a link between classical osculatory interpolation and modern convolution-based interpolation and use it to show that two well-known cubic convolution schemes are formally equivalent to two osculatory interpolation schemes proposed in the actuarial literature about a century ago. We also discuss computational differences and give examples of other cubic interpolation schemes not previously studied in signal and image processing. Interpolatory basis functions are helpful to specify parametric curves or surfaces that can be modified by simple user-interaction. Their main advantage is a characterization of the object by a set of control points that lie on the shape itself (i.e., curve or surface). In this paper, we characterize a new family of compactly supported piecewise-exponential basis functions that are smooth and satisfy the interpolation property. They can be seen as a generalization and extension of the Keys interpolation kernel using cardinal exponential B-splines. The proposed interpolators can be designed to reproduce trigonometric, hyperbolic, and polynomial functions or combinations of them. We illustrate the construction and give concrete examples on how to use such functions to construct parametric curves and surfaces. Cubic convolution interpolation is a new technique for resampling discrete data. It has a number of desirable features which make it useful for image processing. The technique can be performed efficiently on a digital computer. The cubic convolution interpolation function converges uniformly to the function being interpolated as the sampling increment approaches zero. With the appropriate boundary conditions and constraints on the interpolation kernel, it can be shown that the order of accuracy of the cubic convolution method is between that of linear interpolation and that of cubic splines. A one-dimensional interpolation function is derived in this paper. A separable extension of this algorithm to two dimensions is applied to image data. In this paper we consider the problem of designing piecewise polynomial local interpolants of non-uniformly spaced data. We provide a constructive approach that, for any assigned degree of polynomial reproduction, continuity order, and support width, allows for generating the fundamental spline functions of minimum degree having the desired properties. Finally, the proposed construction is extended to handle open sets of data and to the case of multiple knots. This paper presents a general framework for the construction of piecewise-polynomial local interpolants with given smoothness and approximation order, defined on non-uniform knot partitions. We design such splines through a suitable combination of polynomial interpolants with either polynomial or rational, compactly supported blending functions. In particular, when the blending functions are rational, our approach provides spline interpolants having low, and sometimes minimum degree. Thanks to its generality, the proposed framework also allows us to recover uniform local interpolating splines previously proposed in the literature, to generalize them to the non-uniform case, and to complete families of arbitrary support width. Furthermore it provides new local interpolating polynomial splines with prescribed smoothness and polynomial reproduction properties. 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. In CAGD the design of a surface that interpolates an arbitrary quadrilateral mesh is definitely a challenging task. The basic requirement is to satisfy both criteria concerning the regularity of the surface and aesthetic concepts.With regard to aesthetic quality, it is well known that interpolatory methods often produce shape artifacts when the data points are unevenly spaced. In the univariate setting, this problem can be overcome, or at least mitigated, by exploiting a proper non-uniform parametrization, that accounts for the geometry of the data. Recently the same principle has been generalized and proven to be effective in the context of bivariate interpolatory subdivision schemes.In this paper, we propose a construction for parametric surfaces of good aesthetic quality and high smoothness interpolating quadrilateral meshes of arbitrary topology. In the classical tensor product setting the same parameter interval must be shared by an entire row or column of mesh edges. Conversely, in this paper, we assign a different parameter interval to each mesh edge. This particular structure, which we call an augmented parametrization, allows us to interpolate each section polyline at parameters values that prevent wiggling of the resulting curve or other interpolation artifacts and yields high quality interpolatory surfaces.The proposed method is generalization of the local univariate spline interpolants introduced in (2013a) and (2014), that have arbitrary continuity and arbitrary order of polynomial reproduction. The generated surfaces retain the same smoothness of the underlying class of univariate splines in the regular regions of the mesh (where, locally, all vertices have valence 4). Mesh regions containing vertices of valence other than 4 are covered with suitably defined surface patches joining the neighboring regular ones with G 1 - or G 2 -continuity. Interpolatory methods often produce shape artifacts for unevenly-spaced data points.A proper non-uniform parametrization can help overcome interpolation artifacts.Parametric surfaces of good aesthetic quality and high smoothness are constructed.The method relies on non-uniform parametrization and locally supported splines.The surfaces interpolate quadrilateral meshes with arbitrary topology.