Jiri Kosinka

Volumes with piecewise quadratic medial surface transforms: Computation of boundaries and trimmed offsets

MOS surfaces (i.e., medial surface transforms obeying a sum of squares condition) are rational surfaces in R^3^,^1 which possess rational envelopes of the associated two-parameter families of spheres. Moreover, all offsets of the envelopes admit rational parameterizations as well. Recently, it has been proved that quadratic triangular Bezier patches in R^3^,^1 are MOS surfaces. Following this result, we describe an algorithm for computing an exact rational envelope of a two-parameter family of spheres given by a quadratic patch in R^3^,^1. The paper focuses mainly on the geometric aspects of the algorithm. Since these patches are capable of producing C^1 smooth approximations of medial surface transforms of spatial domains, we use this algorithm to generate rational approximations of envelopes of general medial surface transforms. One of the main advantages of this approach to offsetting is the fact that the trimming procedure becomes considerably simpler.

Cluster-Based Point Set Saliency

We propose a cluster-based approach to point set saliency detection, a challenge since point sets lack topological information. A point set is first decomposed into small clusters, using fuzzy clustering. We evaluate cluster uniqueness and spatial distribution of each cluster and combine these values into a cluster saliency function. Finally, the probabilities of points belonging to each cluster are used to assign a saliency to each point. Our approach detects fine-scale salient features and uninteresting regions consistently have lower saliency values. We evaluate the proposed saliency model by testing our saliency-based keypoint detection against a 3D interest point detection benchmark. The evaluation shows that our method achieves a good balance between false positive and false negative error rates, without using any topological information.

Conversion of trimmed NURBS surfaces to Catmull-Clark subdivision surfaces

This paper introduces a novel method to convert trimmed NURBS surfaces to untrimmed subdivision surfaces with Bezier edge conditions. We take a NURBS surface and its trimming curves as input, from this we automatically compute a base mesh, the limit surface of which fits the trimmed NURBS surface to a specified tolerance. We first construct the topology of the base mesh by performing a cross-field based decomposition in parameter space. The number and positions of extraordinary vertices required to represent the trimmed shape can be automatically identified by smoothing a cross field bounded by the parametric trimming curves. After the topology construction, the control point positions in the base mesh are calculated based on the limit stencils of the subdivision scheme and constraints to achieve tangential continuity across the boundary. Our method provides the user with either an editable base mesh or a fine mesh whose limit surface approximates the input within a certain tolerance. By integrating the trimming curve as part of the desired limit surface boundary, our conversion can produce gap-free models. Moreover, since we use tangential continuity across the boundary between adjacent surfaces as constraints, the converted surfaces join with G 1 continuity.

Control vectors for splines

Traditionally, modelling using spline curves and surfaces is facilitated by control points. We propose to enhance the modelling process by the use of control vectors. This improves upon existing spline representations by providing such facilities as modelling with local (semi-sharp) creases, vanishing and diagonal features, and hierarchical editing. While our prime interest is in surfaces, most of the ideas are more simply described in the curve context. We demonstrate the advantages provided by control vectors on several curve and surface examples and explore avenues for future research on control vectors in the contexts of geometric modelling and finite element analysis based on splines, and B-splines and subdivision in particular. We extend traditional splines based on control points by incorporating control vectors.Our paradigm allows combining several spline constructions into one formulation.We can model curves and surfaces that are not possible with existing techniques.

Cubic subdivision schemes with double knots

We investigate univariate and bivariate binary subdivision schemes based on cubic B-splines with double knots. It turns out that double knots change the behaviour of a uniform cubic scheme from primal to dual. We focus on the analysis of new bivariate cubic schemes with double knots at extraordinary points. These cubic schemes produce C^1 surfaces with the original Doo-Sabin weights.

Semi-sharp Creases on Subdivision Curves and Surfaces

We explore a method for generalising Pixar semi‐sharp creases from the univariate cubic case to arbitrary degree subdivision curves. Our approach is based on solving simple matrix equations. The resulting schemes allow for greater flexibility over existing methods, via control vectors. We demonstrate our results on several high‐degree univariate examples and explore analogous methods for subdivision surfaces.

Creases and boundary conditions for subdivision curves

Our goal is to find subdivision rules at creases in arbitrary degree subdivision for piece-wise polynomial curves, but without introducing new control points e.g. by knot insertion. Crease rules are well understood for low degree (cubic and lower) curves. We compare three main approaches: knot insertion, ghost points, and modifying subdivision rules. While knot insertion and ghost points work for arbitrary degrees for B-splines, these methods introduce unnecessary (ghost) control points. The situation is not so simple in modifying subdivision rules. Based on subdivision and subspace selection matrices, a novel approach to finding boundary and sharp subdivision rules that generalises to any degree is presented. Our approach leads to new higher-degree polynomial subdivision schemes with crease control without introducing new control points.

Subdivision Surfaces with Creases and Truncated Multiple Knot Lines

We deal with subdivision schemes based on arbitrary degree B‐splines. We focus on extraordinary knots which exhibit various levels of complexity in terms of both valency and multiplicity of knot lines emanating from such knots. The purpose of truncated multiple knot lines is to model creases which fair out. Our construction supports any degree and any knot line multiplicity and provides a modelling framework familiar to users used to B‐splines and NURBS systems.

Medial axis transforms yielding rational envelopes

Minkowski Pythagorean hodograph (MPH) curves provide a means for representing domains with rational boundaries via the medial axis transform. Based on the observation that MPH curves are not the only curves that yield rational envelopes, we define and study rational envelope (RE) curves that generalise MPH curves while maintaining the rationality of their associated envelopes.To demonstrate the utility of RE curves, we design a simple interpolation algorithm using RE curves, which is in turn used to produce rational surface blends between canal surfaces. Additionally, we initiate the study of rational envelope surfaces as a surface analogy to RE curves. We define rational envelope (RE) curves as a generalisation of MPH curves.RE curves are used in a simple canal surface blending scheme.We initiate the study of RE surfaces.

Converting a CAD model into a non-uniform subdivision surface

CAD models generally consist of multiple NURBS patches, both trimmed and untrimmed. There is a long-standing challenge that trimmed NURBS patches cause unavoidable gaps in the model. We address this by converting multiple NURBS patches to a single untrimmed NURBS-compatible subdivision surface in a three stage process. First, for each patch, we generate in domain space a quadrangulation that follows boundary edges of the patch and respects the knot spacings along edges. Second, the control points of the corresponding subdivision patch are computed in model space. Third, we merge the subdivision patches across their common boundaries to create a single subdivision surface. The converted model is gap-free and can maintain inter-patch continuity up to C 2 . We address the unavoidable gaps in trimmed NURBS by converting them to subdivision.A CAD model of multiple NURBS patches is converted to a subdivision control mesh.The result is a gap-free subdivision surface that approximates the input shape.The chosen non-uniform quadrilateral subdivision is NURBS-compatible.It can match exactly any untrimmed NURBS patch and patch edges curves.