Subdivision Curves Research Articles

Matrix-valued 4-point spline and 3-point non-spline interpolatory curve subdivision schemes

The objective of this paper is to study and construct matrix-valued templates for interpolatory curve subdivision. Since our investigation of this problem was motivated by the need of such subdivision stencils as boundary templates for interpolatory surface subdivision, we provide both spline and non-spline templates that are necessarily symmetric, due to the lack of direction-orientation in carrying out surface subdivision in general. For example, the minimum-supported Hermite interpolatory C 1 cubic spline curve subdivision scheme, with the skew-symmetric basis function for interpolating first derivatives, does not meet the symmetry specification. Non-spline C 2 interpolatory templates constructed in this paper are particularly important, due to their smaller support needed to minimize undesirable surface oscillations, when adopted as boundary templates for interpolatory C 2 surface subdivision. The curve subdivision templates introduced in this paper are adopted as boundary stencils for interpolatory surface subdivision with matrix-valued templates.

Ringing: Frugal subdivision of curves and surfaces

Ringing reduces the working memory needed during the rendering of subdivision curves, surfaces, and animations. For example, it only stores 4D vertices when rendering the dth level of subdivision of a polygon using four-point, cubic, or quintic B-spline subdivision masks. Ringing can be used on CPUs and GPUs.

Formula omitted]-splines

Both the 4-point and the uniform cubic B -spline subdivisions double the number of vertices of a closed-loop polygon k P and produce sequences of vertices f j and b j respectively. We study the J -spline subdivision scheme J s , introduced by Maillot and Stam, which blends these two methods to produce vertices of the form v j = ( 1 − s ) f j + s b j . Iterative applications of J s yield a family of limit curves, the shape of which is parameterized by s. They include four-point subdivision curves ( J 0 ), uniform cubic B -spline curves ( J 1 ), and uniform quintic B -spline curves ( J 1.5 ). We show that the limit curve is at least C 1 when − 1.7 ≤ s ≤ 5.8 , C 2 when 0 < s < 4 , C 3 when 1 < s ≤ 2.8 , and C 4 when s = 3 / 2 , even though a 4-point yields only C 1 curves and a cubic B -spline yields only C 2 curves. We generalize the J s scheme to a two-parameter family J a , b and propose data-dependent and data-independent solutions for computing values of parameters a and b that minimize various objective functions (distance to the control vertices, deviation from the control polygon, change in surface area, and popping when switching levels of subdivision in multi-resolution rendering). We extend the J -spline subdivision to open curves and to a smooth surface subdivision scheme for quad-meshes with arbitrary connectivity.

Four-point curve subdivision based on iterated chordal and centripetal parameterizations

Dubuc's interpolatory four-point scheme inserts a new point by fitting a cubic polynomial to neighbouring points over uniformly spaced parameter values. In this paper we replace uniform parameter values by chordal and centripetal ones. Since we update the parameterization at each refinement level, both schemes are non-linear. Because of this data-dependent parameterization, the schemes are only invariant under solid body and isotropic scaling transformations, but not under general affine transformations. We prove convergence of the two schemes and bound the distance between the limit curve and the initial control polygon. Numerical examples indicate that the limit curves are smooth and that the centripetal one is tighter, as suggested by the distance bounds. Similar to cubic spline interpolation, the use of centripetal parameter values for highly non-uniform initial data yields better results than the use of uniform or chordal ones.

Adaptive geometry compression based on four-point interpolatory subdivision schemes with labels

We propose an adaptive geometry compression method with labels based on four-point interpolatory subdivision schemes. It can work on digital curves of arbitrary dimensions. With the geometry compression method, a digital curve is adaptively compressed into several segments with different compression levels. Each segment is a four-point subdivision curve with a subdivision step. Labels are recorded in data compression to facilitate merging the segments in data decompression. We provide high-speed four-point interpolatory subdivision curve generation methods for efficiently decompressing the compressed data. For an arbitrary positive integer k, formulae for the number of resultant control points of a four-point subdivision curve after k subdivision steps are provided. Some formulae for calculating points at the kth subdivision step are also presented. The time complexity of the new approaches is O(n), where n is the number of points in the given digital curve. Examples are provided to illustrate the efficiency of the proposed approaches.

Subdivision of curves and surfaces: An overview

Subdivision schemes are widely used in various applications such as data-fitting, computer graphics and solid modeling. In this paper we present the basic ideas of subdivision schemes for curves; both interpolatory and corner-cutting schemes, as well as their adaptation to finite sequences. We conclude with examples of specific applications for these subdivision schemes and provide an example of surface subdivision.

A framework for quad/triangle subdivision surface fitting: Application to mechanical objects

In this paper we present a new framework for subdivision surface approximation of three-dimensional models represented by polygonal meshes. Our approach, particularly suited for mechanical or Computer Aided Design (CAD) parts, produces a mixed quadrangle-triangle control mesh, optimized in terms of face and vertex numbers while remaining independent of the connectivity of the input mesh. Our algorithm begins with a decomposition of the object into surface patches. The main idea is to approximate the region boundaries first and then the interior data. Thus, for each patch, a first step approximates the boundaries with subdivision curves (associated with control polygons) and creates an initial subdivision surface by linking the boundary control points with respect to the lines of curvature of the target surface. Then, a second step optimizes the initial subdivision surface by iteratively moving control points and enriching regions according to the error distribution. The final control mesh defining the whole model is then created assembling every local subdivision control meshes. This control polyhedron is much more compact than the original mesh and visually represents the same shape after several subdivision steps, hence it is particularly suitable for compression and visualization tasks. Experiments conducted on several mechanical models have proven the coherency and the efficiency of our algorithm, compared with existing methods.

Two-order Hermite vector-interpolating subdivision schemes

A family of two-order Hermite vector-interpolating subdivision schemes is proposed and its convergence and continuity are analyzed. The iterative level can be estimated for given error. The sufficient conditions of C2 continuity are proved. Geometric features of subdivision curves, such as line segments, cusps and inflection points, are obtained by appending some conditions to initial vectorial Hermite sequence. An algorithm is presented for generating geometric features. For an initial sequence of two-order Hermite elements from unit circle, the numerical error of the 4th subdivided level is O(10−4).

Automated 2-D Cephalometric Analysis on X-ray Images by a Model-Based Approach

Craniofacial landmark localization and anatomical structure tracing on cephalograms are two important ways to obtain the cephalometric analysis. In order to computerize them in parallel, a model-based approach is proposed to locate 262 craniofacial feature points, including 90 landmarks and 172 auxiliary points. In model training, 12 landmarks are selected as reference points and used to divide every training shape to 10 regions according to the anatomical knowledge; principle components analysis is employed to characterize the region shape variations and the statistical grey profile of every feature point. Locating feature points on an input image is a two-stage procedure. First, we identify the reference landmarks by image processing and pattern matching techniques, so that the shape partition is performed on the input image. Then, for each region, its feature points are located by a modified active shape model. All craniofacial anatomical structures can be traced out by connecting the located points with subdivision curves according to the prior knowledge. Users are permitted to modify the results interactively in many different ways. Experimental results show the advantage and reliability of the proposed method.

Smoothness Analysis of Subdivision Schemes by Proximity

Linear curve subdivision schemes may be perturbed in various ways, for example, by modifying them such as to work in a manifold, surface, or group. The analysis of such perturbed and often nonlinear schemes "T" is based on their proximity to the linear schemes "S" which they are derived from. This paper considers two aspects of this problem: One is to find proximity inequalities which together with Ck smoothness of S imply Ck smoothness of T. The other is to verify these proximity inequalities for several ways to construct the nonlinear scheme T analogous to the linear scheme S. The first question is treated for general k, whereas the second one is treated only in the case k = 2. The main result of the paper is that convergent geodesic/projection/Lie group analogues of a certain class of factorizable linear schemes have C2 limit curves.

Intrinsic subdivision with smooth limits for graphics and animation

This article demonstrates the definition of subdivision processes in nonlinear geometries such that smoothness of limits can be proved. We deal with curve subdivision in the presence of obstacles, in surfaces, in Riemannian manifolds, and in the Euclidean motion group. We show how to model kinematic surfaces and motions in the presence of obstacles via subdivision. As to numerics, we consider the sensitivity of the limit's smoothness to sloppy computing.

Effect of the two Hassan-Dodgson Parameters on the Shape of 3-point Ternary Interpolatory Subdivision Curve

Article Effect of the two Hassan-Dodgson Parameters on the Shape of 3-point Ternary Interpolatory Subdivision Curve was published on September 1, 2006 in the journal International Journal of Nonlinear Sciences and Numerical Simulation (volume 7, issue 3).

Convergence and [formula omitted] analysis of subdivision schemes on manifolds by proximity

Curve subdivision schemes on manifolds and in Lie groups are constructed from linear subdivision schemes by first representing the rules of affinely invariant linear schemes in terms of repeated affine averages, and then replacing the operation of affine average either by a geodesic average (in the Riemannian sense or in a certain Lie group sense), or by projection of the affine averages onto a surface. The analysis of these schemes is based on their proximity to the linear schemes which they are derived from. We verify that a linear scheme S and its analogous nonlinear scheme T satisfy a proximity condition. We further show that the proximity condition implies the convergence of T and continuity of its limit curves, if S has the same property, and if the distances of consecutive points of the initial control polygon are small enough. Moreover, if S satisfies a smoothness condition which is sufficient for its limit curves to be C 1 , and if T is convergent, then the curves generated by T are also C 1 . Similar analysis of C 2 smoothness is postponed to a forthcoming paper.

A new subdivision based approach for piecewise smooth approximation of 3D polygonal curves

This paper presents an algorithm dealing with the data reduction and the approximation of 3D polygonal curves. Our method is able to approximate efficiently a set of straight 3D segments or points with a piecewise smooth subdivision curve, in a near optimal way in terms of control point number. Our algorithm is a generalization for subdivision rules, including sharp vertex processing, of the Active B-Spline Curve developed by Pottmann et al. We have also developed a theoretically demonstrated approach, analysing curvature properties of B-Splines, which computes a near optimal evaluation of the initial number and positions of control points. Moreover, our original Active Footpoint Parameterization method prevents wrong matching problems occurring particularly for self-intersecting curves. Thus, the stability of the algorithm is highly increased. Our method was tested on different sets of curves and gives satisfying results regarding to approximation error, convergence speed and compression rate. This method is in line with a larger 3D CAD object compression scheme by piecewise subdivision surface approximation. The objective is to fit a subdivision surface on a target patch by first fitting its boundary with a subdivision curve whose control polygon will represent the boundary of the surface control polyhedron.

C1 Interpolatory Subdivision with Shape Constraints for Curves

We derive two reformulations of the C1 Hermite subdivision scheme introduced by Merrien. One where we separate computation of values and derivatives and one based of refinement of a control polygon. We show that the latter leads to a subdivision matrix which is totally positive. Based on this we give algorithms for constructing subdivision curves that preserve positivity, monotonicity, and convexity.

Education-driven research in CAD

We argue for a new research category, named education-driven research (EDR), which fills the gap between traditional field-specific research that is not concerned with educational objectives and research in education that focuses on fundamental teaching and learning principles and possibly on their customization to broad areas (such as mathematics or physics), but not to specific disciplines (such as CAD). The objective of EDR is to simplify the formulation of the underlying theoretical foundations and of specific tools and solutions in a specialized domain, so as to make them easy to understand and internalize. As such, EDR is a difficult and genuine research activity, which requires a deep understanding of the specific field and can rarely be carried out by generalists with primary expertise in broad education principles. We illustrate the concept of EDR with three examples in CAD: (1) the Split and Tweak subdivisions of a polygon and its use for generating curves, surfaces, and animations; (2) the construction of a topological partition of a plane induced by an arbitrary arrangement of edges; and (3) a romantic definition of the minimal and Hausdorff distances. These examples demonstrate the value of using analogies, of introducing evocative terminology, and of synthesizing the simplest fundamental building blocks. The intuitive understanding provided by EDR enables the students (and even the instructor) to better appreciate the limitations of a particular solution and to explore alternatives. In particular, in these examples, EDR has allowed the author to: (1) reduce the cost of evaluating a cubic B-spline curve; (2) develop a new subdivision curve that is better approximated by its control polygon than either a cubic B-spline or an interpolating 4-point subdivision curve; (3) discover how a circuit inclusion tree may be used for identifying the faces in an arrangement; and (4) rectify a common misconception about the computation of the Hausdorff error between triangle meshes. We invite the scientific community to encourage the development of EDR by publishing its results as genuine research contributions in peer-reviewed professional journals.

L-SYSTEM DESCRIPTION OF SUBDIVISION CURVES

In recent years, subdivision has emerged as a major geometric modeling technique. Algorithms for generating subdivision curves are often specified in terms of iterated matrix multiplication. Each multiplication maps a globally indexed sequence of points that represents a coarser approximation of the curve onto a longer sequence that represents a finer approximation. Unfortunately, this use of matrices and indices obscure the local and stationary character of typical subdivision rules. We introduce parametric context-sensitive L-systems with affine geometry interpretation as an alternative technique for specifying and generating subdivision curves. This technique is illustrated using Chaikin, cubic B-spline, and Dyn-Levin-Gregory (4-point) subdivision schemes as examples. L-systems formalize subdivision algorithms in an intuitive, concise, index-free manner, reflecting the parallel and local character of these algorithms. Furthermore, L-system specification of subdivision algorithms directly leads to their computer implementation.

Philosophy in the Flesh: The Embodied Mind and its Challenge to Western Thought

Splines and subdivision curves are flexible tools in the design and manipulation of curves in Euclidean space. In this paper we study generalizations of interpolating splines and subdivision schemes to the Riemannian manifold of shell surfaces in which the associated metric measures both bending and membrane distortion. The shells under consideration are assumed to be represented by Loop subdivision surfaces. This enables the animation of shells via the smooth interpolation of a given set of key frame control meshes. Using a variational time discretization of geodesics efficient numerical implementations can be derived. These are based on discrete geodesic interpolation, a discrete geometric logarithm, a discrete exponential map, and discrete parallel transport. With these building blocks at hand discrete Riemannian cardinal splines and three different types of discrete, interpolatory subdivision schemes are defined. Numerical results for two different subdivision shell models underline the potential of this approach in key frame animation.

Linear combination of transformations

Geometric transformations are most commonly represented as square matrices in computer graphics. Following simple geometric arguments we derive a natural and geometrically meaningful definition of scalar multiples and a commutative addition of transformations based on the matrix representation, given that the matrices have no negative real eigenvalues. Together, these operations allow the linear combination of transformations. This provides the ability to create weighted combination of transformations, interpolate between transformations, and to construct or use arbitrary transformations in a structure similar to a basis of a vector space. These basic techniques are useful for synthesis and analysis of motions or animations. Animations through a set of key transformations are generated using standard techniques such as subdivision curves. For analysis and progressive compression a PCA can be applied to sequences of transformations. We describe an implementation of the techniques that enables an easy-to-use and transparent way of dealing with geometric transformations in graphics software. We compare and relate our approach to other techniques such as matrix decomposition and quaternion interpolation.

Taxonomy of interpolation constraints on recursive subdivision curves

This paper is the first of two, which together describe and classify the various situations that any complete study of interpolation constraints for a recursive subdivision surface needs to consider. They do so in the form of a systematic taxonomy of situations. Presented here are curve cases, which provide good illustrations of principles which will be used in both contexts; surfaces will be addressed in the second paper. Known results are classified and open questions identified.