Implicit Curves Research Articles

An effective algorithm for computing the asymptotes of an implicit curve

In this paper, we first summarize the algorithm presented in Blasco and Pérez-Díaz (2014) for computing the generalized asymptotes of algebraic curves implicitly defined. This algorithm is based on the computation of Puiseux series. The main and very important contribution of this paper is a new and efficient method that allows to easily compute all the generalized asymptotes of an algebraic plane curve implicitly defined by just solving a triangular system of equations. The method can be easily generalized to the case of algebraic curves implicitly defined in the n-dimensional space.

Computing symmetries of implicit algebraic surfaces

We present a complete algorithm to compute the rotational, axial, reflectional and central symmetries of an algebraic surface defined by means of its implicit equation. The algorithms rely on the fact that the symmetries of the surface satisfy an algebraic condition verified by the polynomial defining the surface, and use a variety of ideas, some of them extensions to space of ideas already used to compute the symmetries of planar implicit curves, and some of them new. We flesh out our algorithms with several examples.

Deep learning of nonlinear flame fronts development due to Darrieus–Landau instability

The Darrieus–Landau instability is studied using a data-driven, deep neural network approach. The task is set up to learn a time-advancement operator mapping any given flame front to a future time. A recurrent application of such an operator rolls out a long sequence of predicted flame fronts, and a learned operator is required to not only make accurate short-term predictions but also reproduce characteristic nonlinear behavior, such as fractal front structures and detached flame pockets. Using two datasets of flame front solutions obtained from a heavy-duty direct numerical simulation and a light-duty modeling equation, we compare the performance of three state-of-art operator-regression network methods: convolutional neural networks, Fourier neural operator (FNO), and deep operator network. We show that, for learning complicated front evolution, FNO gives the best recurrent predictions in both the short and long term. A consistent extension allowing the operator-regression networks to handle complicated flame front shape is achieved by representing the latter as an implicit curve.

Hidden Degrees of Freedom in Implicit Vortex Filaments

This paper presents a new representation of curve dynamics, with applications to vortex filaments in fluid dynamics. Instead of representing these filaments with explicit curve geometry and Lagrangian equations of motion, we represent curves implicitly with a new co-dimensional 2 level set description. Our implicit representation admits several redundant mathematical degrees of freedom in both the configuration and the dynamics of the curves, which can be tailored specifically to improve numerical robustness, in contrast to naive approaches for implicit curve dynamics that suffer from overwhelming numerical stability problems. Furthermore, we note how these hidden degrees of freedom perfectly map to a Clebsch representation in fluid dynamics. Motivated by these observations, we introduce untwisted level set functions and non-swirling dynamics which successfully regularize sources of numerical instability, particularly in the twisting modes around curve filaments. A consequence is a novel simulation method which produces stable dynamics for large numbers of interacting vortex filaments and effortlessly handles topological changes and re-connection events.

Determining the asymptotic family of an implicit curve

In this paper we deal with the following problem: given an algebraic plane curve C, implicitly defined, we determine its “asymptotic family”, that is, the set of algebraic curves that have the same asymptotic behavior as C.

A Hybrid Method for 3D Reconstruction of MR Images.

Three-dimensional surface reconstruction is a well-known task in medical imaging. In procedures for intervention or radiation treatment planning, the generated models should be accurate and reflect the natural appearance. Traditional methods for this task, such as Marching Cubes, use smoothing post processing to reduce staircase artifacts from mesh generation and exhibit the natural look. However, smoothing algorithms often reduce the quality and degrade the accuracy. Other methods, such as MPU implicits, based on adaptive implicit functions, inherently produce smooth 3D models. However, the integration in the implicit functions of both smoothness and accuracy of the shape approximation may impact the precision of the reconstruction. Having these limitations in mind, we propose a hybrid method for 3D reconstruction of MR images. This method is based on a parallel Marching Cubes algorithm called Flying Edges (FE) and Multi-level Partition of Unity (MPU) implicits. We aim to combine the robustness of the Marching Cubes algorithm with the smooth implicit curve tracking enabled by the use of implicit models in order to provide higher geometry precision. Towards this end, the regions that closely fit to the segmentation data, and thus regions that are not impacted by reconstruction issues, are first extracted from both methods. These regions are then merged and used to reconstruct the final model. Experimental studies were performed on a number of MRI datasets, providing images and error statistics generated from our results. The results obtained show that our method reduces the geometric errors of the reconstructed surfaces when compared to the MPU and FE approaches, producing a more accurate 3D reconstruction.

Hough Transform for Detecting Space Curves in Digital 3D Models

We present and analyse the Hough transform (HT) to recognise and approximate space curves in digital models, a problem that is not currently addressed by the standard HT. Our method works on meshes and point clouds and applies to models even incomplete or affected by noise, thus being suitable for the analysis of digital models deriving from 3D scans. In our approach we take advantage of a recent HT formulation for algebraic curves to define both parametric and implicit space curve representations. We also provide a comparative analysis of the HT-based method when dealing with both curve representations, discussing the computational performance and the approximation accuracy of both strategies.

The relaxed implicit randomized algebraic reconstruction technique for curve and surface reconstruction

Implicit curve and surface reconstruction has wide applications in computer graphics, which attracts great attention of many researchers. In this paper, we propose the relaxed implicit randomized algebraic reconstruction technique (RIR-ART) for curve and surface reconstruction. The RIR-ART method generates a sequence of curves and surfaces by adjusting the control coefficients with the relaxation parameters iteratively. It is proved that the sequence of curves and surfaces generated by the RIR-ART method with the suitable relaxation parameters converge to the least-norm results in terms of the mean square error. The RIR-ART method can eliminate the extra zero-level sets effectively during its iterative process, and converge faster than the implicit progressive-iterative approximation method (Hamza et al. 2020). Numerical examples present the efficiency and effectiveness of the proposed method.

Linear and Quadratic Leap-and-Land Trajectory Tracking Algorithms

Considered in this note are numerical tracking algorithms for the accurate following of implicit curves. We start with a fixed point on the curve, and then systematically place on it additional points, one after the other. In this note we first go over the basic procedure of moving forward tangentially from an already placed point then orthogonally returning to the curve. Next, we further consider higher order forward stepping procedures for greater accuracy. We note, however, that higher order methods, desirable for greater accuracy, may harbor latent instabilities. This note suggests ways of holding such instabilities in check, to have stable and highly accurate tracing methods. The note has several supporting numerical examples, including the rounding of a dynamical “snap-through” point.

Duffing-type Equations: Singular Points of Amplitude Profiles and Bifurcations

We study the Duffing equation and its generalizations with polynomial nonlinearities. Recently, we have demonstrated that metamorphoses of the amplitude response curves, computed by asymptotic methods in implicit form as $F\left( \Omega ,\ A\right) =0$, permit prediction of qualitative changes of dynamics occurring at singular points of the implicit curve $F\left(\Omega ,\ A\right) =0$. In the present work we determine a global structure of singular points of the amplitude profiles computing bifurcation sets, i.e. sets containing all points in the parameter space for which the amplitude profile has a singular point. We connect our work with independent research on tangential points on amplitude profiles, associated with jump phenomena, characteristic for the Duffing equation. We also show that our techniques can be applied to solutions of form $\Omega _{\pm }=f_{\pm }\left( A\right) $, obtained within other asymptotic approaches.

Curvature computations for the intersection curves of hypersurfaces in Euclidean n-space

The aim of this paper is to derive the curvatures and Frenet vectors of the intersection curve of (n−1) transversally intersecting hypersurfaces represented either in implicit or in parametric form in Euclidean n-space. Since the intersection of (n−1) implicit hypersurfaces defines an implicit curve in n-dimensions, the method derived for the implicit case can be considered as a partial answer to the open problem posed by Ron Goldman in 2005. Also, an application is provided for the implicit case where the results are presented by using computer codes currently produced in MATLAB.

High-quality vascular modeling and modification with implicit extrusion surfaces for blood flow computations

Background and ObjectiveHigh-quality vascular modeling is crucial for blood flow simulations, i.e., computational fluid dynamics (CFD). As without an accurate geometric representation of the smooth vascular surface, it is impossible to make meaningful blood flow simulations. The purpose of this work is to develop high-quality vascular modeling and modification method for blood flow computations. MethodsWe develop a new technique for the accurate geometric modeling and modification of vasculatures using implicit extrusion surfaces (IES). In the proposed method, the skeleton of the vascular structure is subdivided into short curve segments, each of which is then represented implicitly locally as the intersection of two mutually orthogonal implicit surfaces defined by distance functions. A set of contour points is extracted and fitted with an implicit curve for accurately specifying the vessel cross-section profile, which is then extruded locally along the skeleton to fill the gaps between two vascular tube cross sections. We also present a new implicit geometric editing technique to modify the constructed vascular model with pathology for virtual stenting. ResultsExperimental results and validations show that accurate vascular models with highly smooth surfaces can be generated by the proposed method. In addition, we conduct some blood flow simulations to indicate the effectiveness of proposed method for hemodynamic simulations. ConclusionsThe proposed technique can achieve precise geometric models of vasculatures with any required degree of smoothness for reliable blood flow simulations.

Effective equation for two coupled oscillators: Towards a global view of metamorphoses of the amplitude profiles

Dynamics of nonlinear coupled driven oscillators is investigated. Recently, we have demonstrated that the amplitude profiles – dependence of the amplitude A on frequency Ω of the driving force, computed by asymptotic methods in implicit form as FA,Ω=0, permit prediction of metamorphoses of dynamics which occur at singular points of the implicit curve FA,Ω=0. In the present study we strive at a global view of singular points of the amplitude profiles computing bifurcation sets, i.e. sets containing all points in the parameter space for which the amplitude profile has a singular point.

Implicit progressive-iterative approximation for curve and surface reconstruction

Implicit curve and surface reconstruction attracts the attention of many researchers and gains a wide range of applications, due to its ability to describe objects with complicated geometry and topology. However, extra zero-level sets or spurious sheets arising in the reconstruction process make the reconstruction result challenging to be interpreted and damage the final result. In this paper, we propose an implicit curve and surface reconstruction method based on the progressive-iterative approximation, named implicit progressive-iterative approximation (I-PIA). The proposed method elegantly and naturally eliminates the spurious sheets without requiring any explicit minimization procedure, thus significantly reducing the computing cost and providing high-quality reconstruction results. Numerical examples are presented to demonstrate the efficiency and effectiveness of the proposed method.

An adaptive bubble method for structural shape and topology optimization

An adaptive bubble method is presented in this paper for shape and topology optimization design of stiffest structures and compliant mechanisms. With the aim of overcoming the shortcomings of the traditional bubble method and retaining all its advantages, the proposed method achieves a comprehensive integration of topological derivative, implicit model and fixed grid. The topological derivative is devoted to the iterative positioning of new holes inside the design domain. Each of the holes is described by smoothly deformable implicit curve (SDIC) characterized by very few parameters and high deformation capacity. Since the structural model could be implicitly constructed using the SDICs, the finite cell method (FCM) is applied to facilitate the mechanical analysis of the implicit model with fixed Eulerian mesh. Two new concepts, i.e., the topological derivative threshold and the total influence region of inserted holes, are further introduced to adaptively control the numbers and positions of new holes so that the numerical stability of topology optimization could be well guaranteed. It is shown that the adaptive bubble method no longer needs the mesh updating process and can perform the merging and separating operations of holes straightforwardly. Numerical examples concerning the stiffest structure design and the compliant mechanism synthesis are tested to demonstrate the universality of the present method, especially its capability of being integrated within CAD systems directly without post-processing.

The complexity of subdivision for diameter-distance tests

We present a general framework for analyzing the complexity of subdivision-based algorithms whose tests are based on the sizes of regions and their distance to certain sets (often varieties) intrinsic to the problem under study. We call such tests diameter-distance tests. We illustrate that diameter-distance tests are common in the literature by proving that many interval arithmetic-based tests are, in fact, diameter-distance tests. For this class of algorithms, we provide both non-adaptive bounds for the complexity, based on separation bounds, as well as adaptive bounds, by applying the framework of continuous amortization.Using this structure, we provide the first complexity analysis for the algorithm by Plantinga and Vegeter for approximating real implicit curves and surfaces. We present both adaptive and non-adaptive a priori worst-case bounds on the complexity of this algorithm both in terms of the number of subregions constructed and in terms of the bit complexity for the construction. Finally, we construct families of hypersurfaces to prove that our bounds are tight.

Fixed-grid hole-shape optimization for opening structures using smoothly deformable implicit curve

Hole-shape optimization for opening structures can effectively alleviate the hole-edge stress concentration; however, it requires high precision of stress analysis and high geometrical deformation capacity of the hole-edge curves. Therefore, the finite cell method is adopted in this work for high-precision numerical analysis within the fixed mesh, and the smoothly deformable implicit curve is proposed to describe the boundary to be optimized. After deriving analytical sensitivity analysis formulas, we built a hole-shape optimization design framework having high precision and high efficiency. Through hole-shape optimization in opening structures under different load boundary conditions, the proposed framework exhibited several advantages such as no mesh updating, simple sensitivity derivation, high analytical precision, and large design space.

Integrated Hybrid Second Order Algorithm for Orthogonal Projection onto a Planar Implicit Curve

The computation of the minimum distance between a point and a planar implicit curve is a very important problem in geometric modeling and graphics. An integrated hybrid second order algorithm to facilitate the computation is presented. The proofs indicate that the convergence of the algorithm is independent of the initial value and demonstrate that its convergence order is up to two. Some numerical examples further confirm that the algorithm is more robust and efficient than the existing methods.

Accurate geometry modeling of vasculatures using implicit fitting with 2D radial basis functions

Accurate vascular geometry modeling is an essential task in computer assisted vascular surgery and therapy. This paper presents a vessel cross-section based implicit vascular modeling technique, which represents a vascular surface as a set of locally fitted implicit surfaces. In the proposed method, a cross-section based technique is employed to extract from each cross-section of the vascular surface a set of points, which are then fitted with an implicit curve represented as 2D radial basis functions. All these implicitly represented cross-section curves are then being considered as 3D cylindrical objects and combined together using a certain partial shape-preserving spline to build a complete vessel branch; different vessel branches are then blended using a extended smooth maximum function to construct the complete vascular tree. Experimental results show that the proposed method can correctly represent the morphology and topology of vascular structures with high level of smoothness. Both qualitative comparison with other methods and quantitative validations to the proposed method have been performed to verify the accuracy and smoothness of the generated vascular geometric models.

An Integrated Approach of Model Reconstruction, Stress Analysis and Optimization Design via Smoothly Deformable Implicit Curves

An Integrated Approach of Model Reconstruction, Stress Analysis and Optimization Design via Smoothly Deformable Implicit Curves