Computing a compact spline representation of the medial axis transform of a 2D shape (original) (raw)

Medial Meshes -- A Compact and Accurate Representation of Medial Axis Transform

IEEE Transactions on Visualization and Computer Graphics, 2015

The medial axis transform has long been known as an intrinsic shape representation supporting a variety of shape analysis and synthesis tasks. However, for a given shape, it is hard to obtain its faithful, concise and stable medial axis, which hinders the application of the medial axis. In this paper, we introduce the medial mesh, a new discrete representation of the medial axis. A medial mesh is a 2D simplicial complex coupled with a radius function that provides a piecewise linear approximation to the medial axis. We further present an effective algorithm for computing a concise and stable medial mesh for a given shape. Our algorithm is quantitatively driven by a shape approximation error metric, and progressively simplifies an initial medial mesh by iteratively contracting edges until the approximation error reaches a predefined threshold. We further demonstrate the superior efficiency and accuracy of our method over existing methods for medial axis simplification.

Medial axis computation for planar free–form shapes

Computer-Aided Design, 2009

We present a simple, efficient, and stable method for computing-with any desired precision-the medial axis of simply connected planar domains. The domain boundaries are assumed to be given as polynomial spline curves. Our approach combines known results from the field of geometric approximation theory with a new algorithm from the field of computational geometry. Challenging steps are (1) the approximation of the boundary spline such that the medial axis is geometrically stable, and (2) the efficient decomposition of the domain into base cases where the medial axis can be computed directly and exactly. We solve these problems via spiral biarc approximation and a randomized divide & conquer algorithm.

Shape simplification based on the medial axis transform

2003

Abstract We present a new algorithm for simplifying the shape of 3D objects by manipulating their medial axis transform (MAT). From an unorganized set of boundary points, our algorithm computes the MAT, decomposes the axis into parts, then selectively removes a subset of these parts in order to reduce the complexity of the overall shape. The result is simplified MAT that can be used for a variety of shape operations.

B-spline normal multi-scale transforms for planar curves

arXiv (Cornell University), 2013

Normal multi-scale transform [4] is a nonlinear multi-scale transform for representing geometric objects that has been recently investigated [1, 7, 10]. The restrictive role of the exact order of polynomial reproduction P e of the approximating subdivision operator S in the analysis of the S normal multi-scale transform, established in [7, Theorem 2.6], significantly disfavors the practical use of these transforms whenever P e ≪ P. We analyze in detail the normal multiscale transform for planar curves based on B-spline subdivision scheme S p of degree p ≥ 3 and derive higher smoothness of the normal re-parameterization than in [7]. We show that further improvements of the smoothness factor are possible, provided the approximate normals are cleverly chosen. Following [10], we introduce a more general framework for those transforms where more than one subdivision operator can be used in the prediction step, which leads to higher detail decay rates.

A Tracing Algorithm for Constructing Medial Axis Transform of 3D Objects Bound by Free-Form Surfaces

2005

This paper presents an algorithm for generating the Medial Axis Transform(MAT) of 3D objects with free-form boundaries. The algorithm proposed uses the exact representation of the part and generates an approximate rational spline description (to within a defined tolerance) of the MAT. The algorithm generates the MAT by a tracing technique that marches along the object boundary. The level of approximation is controlled by the choice of the step size in the tracing procedure. Criteria based on distance and local curvature of boundary entities are used to identify the junction points and the search for these junction points is done in an efficient way. The algorithm works for multiplyconnected objects as well. Results of implementation are provided.

Interior Medial Axis Transform computation of 3D objects bound by free-form surfaces

Computer-aided Design, 2010

This paper presents an algorithm for generating the Interior Medial Axis Transform (iMAT) of 3D objects with free-form boundaries. The algorithm proposed uses the exact representation of the part and generates an approximate rational spline description of the iMAT. The algorithm generates the iMAT by a tracing technique that marches along the object's boundary. The level of approximation is controlled by the choice of the step size in the tracing procedure. Criteria based on distance and local curvature of boundary entities are used to identify the junction points and the search for these junction points is done in an efficient way. The algorithm works for multiply-connected objects as well. Results of the implementation are provided.

Constructing medial axis transform of extruded and revolved 3D objects with free-form boundaries

Computer-aided Design, 2005

This paper presents an algorithm for generating the Medial Axis Transform (MAT) of 3D objects with free form boundaries that are obtained by extrusion along a line or revolution about an axis. The algorithm proposed uses the exact representation of the part and generates an approximate rational spline description (to within a defined tolerance) of the MAT. The algorithm uses the 2D MAT of the profile being extruded or revolved to identify the limiting entities (junction points, seams and points of extremal maximum curvature) of the 3D MAT. It is shown that the MAT points of the profile face are sufficient to determine the topology and geometry of the MAT of this class of solids. The algorithm works for multiply-connected objects as well. Results of implementation are presented and use of the algorithm to handle general solids is discussed. q

Shape Projectors for Landmark-Based Spline Curves

IEEE Signal Processing Letters, 2017

We present a generic method to construct orthogonal projectors for 2D landmark-based parametric spline curves. We construct vector spaces that define a geometric transformation (e.g., affine, similarity, scaling) that is applied to a reference curve. These vector spaces contain all parametric curves up to the chosen transformation. We define the vector spaces implicitly through an orthogonal projection operator and present a theorem that characterizes the projector for landmark-based spline curves which are popular for the user-interactive analysis of biomedical images. Finally, we show how shape priors are constructed with the spline projector and provide an example of application for the segmentation of microscopy images in biology.

Algorithms for computing shape preserving spline approximations to data

Numerische Mathematik, 1985

We treat the problem of approximating data that are sampled with error from a function known to be convex and increasing. The approximating function is a polynomial spline with knots at the data points. This paper presents results (analogous to those in [7] and [9]) that describe some approximation properties of polynomial splines and algorithms for determining the existence of a "shape-preserving" approximant for given data.

A local fitting algorithm for converting planar curves to B-splines

Computer Aided Geometric Design, 2008

In this paper we present a local fitting algorithm for converting smooth planar curves to B-splines. For a smooth planar curve a set of points together with their tangent vectors are first sampled from the curve such that the connected polygon approximates the curve with high accuracy and inflexions are detected by the sampled data efficiently. Then, a G 1 continuous Bézier spline curve is obtained by fitting the sampled data with shape preservation as well as within a prescribed accuracy. Finally, the Bézier spline is merged into a C 2 continuous B-spline curve by subdivision and control points adjustment. The merging is guaranteed to be within another error bound and with no more inflexions than the Bézier spline. In addition to shape preserving and error control, this conversion algorithm also benefits that the knots are selected automatically and adaptively according to local shape and error bound. A few experimental results are included to demonstrate the validity and efficiency of the algorithm.