Numerical proper reparametrization of parametric plane curves

We present an algorithm for reparametrizing algebraic plane curves from a numerical point of view. More precisely, given a tolerance ϵ>0 and a rational parametrization P of a plane curve C with perturbed float coefficients, we present an algorithm that computes a parametrization Q of a new plane curve D such that Q is an ϵ –proper reparametrization of D. In addition, the error bound is carefully discussed and we present a formula that measures the “closeness” between the input curve C and the output curve D

PDE-Based Parameterisation Techniques for Planar Multipatch Domains

This paper presents a PDE-based parameterisation framework for addressing the planar surface-to-volume (StV) problem of finding a valid description of the domain's interior given no more than a spline-based description of its boundary contours. The framework is geared towards isogeometric analysis (IGA) applications wherein the physical domain is comprised of more than four sides, hence requiring more than one patch. We adopt the concept of harmonic maps and propose several PDE-based problem formulations capable of finding a valid map between a convex parametric multipatch domain and the piecewise-smooth physical domain with an equal number of sides. In line with the isoparametric paradigm of IGA, we treat the StV problem using techniques that are characteristic for the analysis step. As such, this study proposes several IGA-based numerical algorithms for the problem's governing equations that can be effortlessly integrated into a well-developed IGA software suite. We augment the framework with mechanisms that enable controlling the parametric properties of the outcome. Parametric control is accomplished by, among other techniques, the introduction of a curvilinear coordinate system in the convex parametric domain that, depending on the application, builds desired features into the computed harmonic map, such as homogeneous cell sizes or boundary layers

Piecewise Arc-Length Parameterized NURBS Tool Paths Generation for 3-Axis CNC Machining of Accurate, Smooth Sculptured Surfaces

In current industrial applications many engineering parts having complex shapes are designed using sculptured surfaces in CAD system. Due to the lack of smooth motions and accurate machining of these surfaces using standard linear and circular motions in conventional CNC machines, new commercial CNC systems are equipped with parametric curve interpolation function. However, in some applications these surfaces can be very complex that are susceptible to gouging and due to the approximation of; CL-path in CAM system and path parameter in real –time, high machining accuracy, smooth kinematic and feed-rate profiles, are difficult to achieve. This dissertation focuses on developing algorithms that generate tool paths in NURBS form for smooth, high speed and accurate sculptured surface machining. The first part of the research identifies and eliminates gouge cutter location (CL) point from the tool path. The proposed algorithm uses global optimization technique (Particle Swarm Optimization) to check all the CC-points along a tool-path with high accuracy, and only gouging free CC-points are used to generate the set of valid CL-points. Mathematical models have been developed and implemented to cover most of the cutter shapes, used in the industry. In the second phase of the research, all valid CL-points along the tool-path are used to generate CL-path in B-spline form. The main contribution of this part is to formulate an error function of the offset approximation and to represent it in NURBS form to globally bound the approximation errors. Based on this error function, an algorithm is proposed to generate tool-paths in B-spline from with; globally controlled accuracy, fewer control points and low function degree, compared to its contemporaries. The proposed approach thus presents an error-bounded method for B-spline curve approximation to the ideal CL-path within the accuracy. This part of research has two components, one is for 2½- axis (pocket) and the other one is for 3-axis (surface) CNC machining. The third part deals with the problem of CL-path parameter estimation during machining in real time. Once the gouging free CL-path in NURBS form with globally controlled accuracy is produced, it is re-parameterized with approximate arc-length in the off-line stage. The main features of this work are; (1) sampling points and calculating their approximate arc-lengths within error bound by decomposing the input path into Bezier curve segments, (2) fitting the NURBS curve with approximate arc-length parameter to the sample points until the path and parameterization errors are within the tolerance, and (3) segment the curve into pieces with different feed rates if during machining the cutter trajectory errors are beyond the tolerance at highly curved regions in the NURBS tool path

Coherent multi-dimensional segmentation of multiview images using a variational framework and applications to image based rendering

Image Based Rendering (IBR) and in particular light field rendering has attracted a lot of attention for interpolating new viewpoints from a set of multiview images. New images of a scene are interpolated directly from nearby available ones, thus enabling a photorealistic rendering. Sampling theory for light fields has shown that exact geometric information in the scene is often unnecessary for rendering new views. Indeed, the band of the function is approximately limited and new views can be rendered using classical interpolation methods. However, IBR using undersampled light fields suffers from aliasing effects and is difficult particularly when the scene has large depth variations and occlusions. In order to deal with these cases, we study two approaches: New sampling schemes have recently emerged that are able to perfectly reconstruct certain classes of parametric signals that are not bandlimited but characterized by a finite number of parameters. In this context, we derive novel sampling schemes for piecewise sinusoidal and polynomial signals. In particular, we show that a piecewise sinusoidal signal with arbitrarily high frequencies can be exactly recovered given certain conditions. These results are applied to parametric multiview data that are not bandlimited. We also focus on the problem of extracting regions (or layers) in multiview images that can be individually rendered free of aliasing. The problem is posed in a multidimensional variational framework using region competition. In extension to previous methods, layers are considered as multi-dimensional hypervolumes. Therefore the segmentation is done jointly over all the images and coherence is imposed throughout the data. However, instead of propagating active hypersurfaces, we derive a semi-parametric methodology that takes into account the constraints imposed by the camera setup and the occlusion ordering. The resulting framework is a global multi-dimensional region competition that is consistent in all the images and efficiently handles occlusions. We show the validity of the approach with captured light fields. Other special effects such as augmented reality and disocclusion of hidden objects are also demonstrated

Accurate geometry reconstruction of vascular structures using implicit splines

3-D visualization of blood vessel from standard medical datasets (e.g. CT or MRI) play an important role in many clinical situations, including the diagnosis of vessel stenosis, virtual angioscopy, vascular surgery planning and computer aided vascular surgery. However, unlike other human organs, the vasculature system is a very complex network of vessel, which makes it a very challenging task to perform its 3-D visualization. Conventional techniques of medical volume data visualization are in general not well-suited for the above-mentioned tasks. This problem can be solved by reconstructing vascular geometry. Although various methods have been proposed for reconstructing vascular structures, most of these approaches are model-based, and are usually too ideal to correctly represent the actual variation presented by the cross-sections of a vascular structure. In addition, the underlying shape is usually expressed as polygonal meshes or in parametric forms, which is very inconvenient for implementing ramification of branching. As a result, the reconstructed geometries are not suitable for computer aided diagnosis and computer guided minimally invasive vascular surgery. In this research, we develop a set of techniques associated with the geometry reconstruction of vasculatures, including segmentation, modelling, reconstruction, exploration and rendering of vascular structures. The reconstructed geometry can not only help to greatly enhance the visual quality of 3-D vascular structures, but also provide an actual geometric representation of vasculatures, which can provide various benefits. The key findings of this research are as follows: 1. A localized hybrid level-set method of segmentation has been developed to extract the vascular structures from 3-D medical datasets. 2. A skeleton-based implicit modelling technique has been proposed and applied to the reconstruction of vasculatures, which can achieve an accurate geometric reconstruction of the vascular structures as implicit surfaces in an analytical form. 3. An accelerating technique using modern GPU (Graphics Processing Unit) is devised and applied to rendering the implicitly represented vasculatures. 4. The implicitly modelled vasculature is investigated for the application of virtual angioscopy