The use of subdivision schemes in applied and real-world contexts requires the development of conceptually simple algorithms that can be converted into fast and efficient implementation procedures. In the domain of interpolatory subdivision schemes, there is a demand for developing an algorithm capable of (i) reproducing all types of conic sections whenever the input data (in our case point-normal pairs) are arbitrarily sampled from them, (ii) generating a visually pleasing limit curve without creating unwanted oscillations, and (iii) having the potential to be naturally and easily extended to the bivariate case. In this paper we focus on the construction of an interpolatory subdivision scheme that meets all these conditions simultaneously. At the center of our construction lies a conic fitting algorithm that requires as few as four point-normal pairs for finding new edge points (and associated normals) in a subdivision step. Several numerical results are included to showcase the validity of our algorithm.

The aim of this work is to provide new characterizations of planar quintic Pythagorean-hodograph curves. The first two are algebraic and consist of two and three equations, respectively, in terms of the edges of the Bézier control polygon as complex numbers. These equations are symmetric with respect to the edge indices and cover curves with generic as well as degenerate control polygons. The last two characterizations are geometric and rely both on just two auxiliary points outside the control polygon. One requires two (possibly degenerate) quadrilaterals to be similar, and the other highlights two families of three similar triangles. All characterizations are a step forward with respect to the state of the art, and they can be linked to the well-established counterparts for planar cubic Pythagorean-hodograph curves. The key ingredient for proving the aforementioned results is a novel general expression for the hodograph of the curve.

We present a new family of active surfaces for the semiautomatic segmentation of volumetric objects in 3D biomedical images. We represent our deformable model by a subdivision surface encoded by a small set of control points and generated through a geometric refinement process. The subdivision operator confers important properties to the surface such as smoothness, reproduction of desirable shapes and interpolation of the control points. We deform the subdivision surface through the minimization of suitable gradient-based and region-based energy terms that we have designed for that purpose. In addition, we provide an easy way to combine these energies with convolutional neural networks. Our active subdivision surface satisfies the property of multiresolution, which allows us to adopt a coarse-to-fine optimization strategy. This speeds up the computations and decreases its dependence on initialization compared to singleresolution active surfaces. Performance evaluations on both synthetic and real biomedical data show that our active subdivision surface is robust in the presence of noise and outperforms current state-of-the-art methods. In addition, we provide a software that gives full control over the active subdivision surface via an intuitive manipulation of the control points.

We present a new exponential B-spline basis that enables the construction of active contours for the analysis of biomedical images. Our functions generalize the well-known polynomial Hermite B-splines and provide us with a direct control over the tangents of the parameterized contour, which is absent in traditional spline-based active contours. Our basis functions have been designed to perfectly reproduce elliptical and circular shapes. Moreover, they can approximate any closed curve up to arbitrary precision by increasing the number of anchor points. They are therefore well-suited to the segmentation of the roundish objects that are commonly encountered in the analysis of bioimages. We illustrate the performance of an active contour built using our functions on some examples of real biological data.

Subdivision schemes are nowadays customary in curve and surface modeling. In this paper the problem of designing interpolatory subdivision schemes is considered. The idea is to modify a given approximating subdivision scheme just enough to satisfy the interpolation requirement. From an algebraic point of view this leads to the solution of a generalized Bezout polynomial equation possibly involving more than two polynomials. By exploiting the matrix counterpart of this equation it is shown that small-degree solutions can be generally found by inverting an associated structured matrix of Toeplitz-like form. If the approximating scheme is defined in terms of a free parameter, then the inversion can be performed by numeric-symbolic methods.

In this paper we propose a segmentation method to locate faults and/or large variations in order to approximate a surface from Lagrange dataset.