73 research outputs found
Reverse engineering of CAD models via clustering and approximate implicitization
In applications like computer aided design, geometric models are often
represented numerically as polynomial splines or NURBS, even when they
originate from primitive geometry. For purposes such as redesign and
isogeometric analysis, it is of interest to extract information about the
underlying geometry through reverse engineering. In this work we develop a
novel method to determine these primitive shapes by combining clustering
analysis with approximate implicitization. The proposed method is automatic and
can recover algebraic hypersurfaces of any degree in any dimension. In exact
arithmetic, the algorithm returns exact results. All the required parameters,
such as the implicit degree of the patches and the number of clusters of the
model, are inferred using numerical approaches in order to obtain an algorithm
that requires as little manual input as possible. The effectiveness, efficiency
and robustness of the method are shown both in a theoretical analysis and in
numerical examples implemented in Python
High-performance geometric vascular modelling
Image-based high-performance geometric vascular modelling and reconstruction is an essential component of computer-assisted surgery on the diagnosis, analysis and treatment of cardiovascular diseases. However, it is an extremely challenging task to efficiently reconstruct the accurate geometric structures of blood vessels out of medical images. For one thing, the shape of an individual section of a blood vessel is highly irregular because of the squeeze of other tissues and the deformation caused by vascular diseases. For another, a vascular system is a very complicated network of blood vessels with different types of branching structures. Although some existing vascular modelling techniques can reconstruct the geometric structure of a vascular system, they are either time-consuming or lacking sufficient accuracy. What is more, these techniques rarely consider the interior tissue of the vascular wall, which consists of complicated layered structures. As a result, it is necessary to develop a better vascular geometric modelling technique, which is not only of high performance and high accuracy in the reconstruction of vascular surfaces, but can also be used to model the interior tissue structures of the vascular walls.This research aims to develop a state-of-the-art patient-specific medical image-based geometric vascular modelling technique to solve the above problems. The main contributions of this research are:- Developed and proposed the Skeleton Marching technique to reconstruct the geometric structures of blood vessels with high performance and high accuracy. With the proposed technique, the highly complicated vascular reconstruction task is reduced to a set of simple localised geometric reconstruction tasks, which can be carried out in a parallel manner. These locally reconstructed vascular geometric segments are then combined together using shape-preserving blending operations to faithfully represent the geometric shape of the whole vascular system.- Developed and proposed the Thin Implicit Patch method to realistically model the interior geometric structures of the vascular tissues. This method allows the multi-layer interior tissue structures to be embedded inside the vascular wall to illustrate the geometric details of the blood vessel in real world
Changing representation of curves and surfaces: exact and approximate methods
Το κύριο αντικείμενο μελέτης στην παρούσα διατριβή είναι η αλλαγή αναπαράστασης
γεωμετρικών αντικειμένων από παραμετρική σε αλγεβρική (ή πεπλεγμένη) μορφή.
Υπολογίζουμε την αλγεβρική εξίσωση παρεμβάλλοντας τους άγνωστους συντελεστές
του πολυωνύμου δεδομένου ενός υπερσυνόλου των μονωνύμων του. Το τελευταίο
υπολογίζεται απο το Newton πολύτοπο της αλγεβρικής εξίσωσης που υπολογίζεται
από μια πρόσφατη μέθοδο πρόβλεψης του συνόλου στήριξης της εξίσωσης. H μέθοδος
πρόβλεψης του συνόλου στήριξης βασίζεται στην αραιή (ή τορική) απαλοιφή: το
πολύτοπο υπολογίζεται από
το Newton πολύτοπο της αραιής απαλοίφουσας αν θεωρίσουμε την παραμετροποίηση ως
πολυωνυμικό σύστημα. Στα μονώνυμα που αντιστοιχούν στα ακέραια σημεία του
Newton πολυτόπου δίνονται τιμές ώστε να σχηματίσουν έναν αριθμητικό πίνακα. Ο
πυρήνα του πίνακα αυτού, διάστασης 1 σε ιδανική περίπτωση, περιέχει τους
συντελεστές των
μονωνύμων στην αλγεβρική εξίσωση. Υπολογίζουμε τον πυρήνα του πίνακα είτε
συμβολικά είτε αριθμητικά εφαρμόζοντας την μέθοδο του singular value
decomposition (SVD). Προτείνουμε τεχνικές για να διαχειριστούμε την περίπτωση
ενός πολυδιάστατου πυρήνα το οποίο εμφανίζεται όταν το προβλεπόμενο σύνολο
στήριξης είναι ένα υπερσύνολο του
πραγματικού. Αυτό δίνει έναν αποτελεσματικό ευαίσθητο-εξόδου αλγόριθμο
υπολογισμού της αλγεβρικής εξίσωσης. Συγκρίνουμε διαφορετικές προσεγγίσεις
κατασκευής του πίνακα μέσω των λογισμικών Maple και SAGE. Στα πειράματά μας
χρησιμοποιήθηκαν ρητές
καμπύλες και επιφάνειες καθώς και NURBS. Η μέθοδός μας μπορεί να εφαρμοστεί σε
πολυώνυμα ή ρητές παραμετροποιήσεις επίπεδων καμπυλών ή (υπερ)επιφανειών
οποιασδήποτε διάστασης συμπεριλαμβανομένων και των περιπτώσεων με
παραμετροποίηση σεσημεία βάσης που εγείρουν σημαντικά ζητήματα για άλλες
μεθόδους αλγεβρικοποίησης.
Η μέθοδος έχει τον εξής περιορισμό: τα γεωμετρικά αντικείμενα πρέπει να
αναπαριστώνται από βάσεις μονωνύμων που στην περίπτωση τριγωνομετρικών
παραμετροποιήσεων θα πρέπει να μπορούν να μετασχηματιστούν σε ρητές
συναρτήσεις. Επιπλέον η τεχνική που
προτείνουμε μπορεί να εφαρμοστεί σε μη γεωμετρικά προβλήματα όπως ο
υπολογισμόςτης διακρίνουσας ενός πολυωνύμου με πολλές μεταβλητές ή της
απαλοίφουσας ενός συστήματος πολυωνύμων με πολλές μεταβλητές.The main object of study in our dissertation is the representation change of
the geometric objects
from the parametric form to implicit. We compute the implicit equation
interpolating the
unknown coefficients of the implicit polynomial given a superset of its
monomials. The latter is
derived from the Newton polytope of the implicit equation obtained by the
recently developed
method for support prediction. The support prediction method we use relies on
sparse (or
toric) elimination: the implicit polytope is obtained from the Newton polytope
of the sparse
resultant of the system in parametrization, represented as polynomials. The
monomials that
correspond to the lattice points of the Newton polytope are suitably evaluated
to build a numeric
matrix, ideally of corank 1. Its kernel contains their coefficients in the
implicit equation.
We compute kernel of the matrix either symbolically, or numerically, applying
singular value
decomposition (SVD). We propose techniques for handling the case of the
multidimensional
kernel space, caused by the predicted support being a superset of the actual.
This yields an
efficient, output-sensitive algorithm for computing the implicit equation. We
compare different
approaches for constructing the matrix in Maple and SAGE software. In our
experiments we
have used classical algebraic curves and surfaces as well as NURBS. Our method
can be
applied to polynomial or rational parametrizations of planar curves or
(hyper)surfaces of any
dimension including cases of parameterizations with base points which raise
important issues
for other implicitization methods. The method has its limits: geometric objects
have to be presented
using monomial basis; in the case of trigonometric parametrizations they have
to be
convertible to rational functions. Moreover, the proposed technique can be
applied for nongeometric
problems such as the computation of the discriminant of a multivariate
polynomial
or the resultant of a system of multivariate polynomials
Superficies de Bézier en el Espacio de Lorentz-Minkowski
El objetivo principal de este trabajo es estudiar qué cambios experimenta la
geometría diferencial de las superficies de Bézier al ser introducidas en el espacio de
Lorentz-Minkowski.
Específicamente se estudiarán resultados relacionados con el área de las superficies
para poder tratar el Problema de Plateau-Bézier en estas condiciones. Para ello
reuniremos una serie de condiciones sobre las redes de puntos que nos permitirán
acotar las áreas de las superficies con un borde fijo.The main objective of this work is to study how changes the differential geometry
of Bézier surfaces when they are studied in Lorentz-Minkowski space.
Results about the area of the surfaces will be studied specifically in order to
deal with the Plateau-Bézier problem in this particular conditions. We will gather a
number of conditions over the control points which will allow us to delimit the area
value of prescribed border surfaces.Universidad de Sevilla. Grado en Matemática
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A Graphical Method of Computing Offset Curves
Computation of offset curves is an operation critical to many computer-aided design and manufacturing (CAD/CAM) applications. Though simple on the surface, differences between the straightforward mathematical definition and the demands of CAD/CAM environment in the formulation and expression of an offset curve create a problem for which only complicated, approximate solutions are presently available. This thesis explores one of the newest methods of offset curve computation, using graphics hardware to directly compute the offset curve for arbitrary input geometry. Linear segments of the input curve are represented as meshes in 3D space, and the rendering process is used to create a field of depth values from which the offset curve is extracted as an isoline. This results in significant performance enhancements over previous, similar methods. Combined with a quantification of the errors involved in a graphical approach, these advances bring the technique closer to industrial readiness. Algorithm performance is shown to be linear with respect to geometric complexity of the input curve
Bernstein Polynomial-Based Method for Solving Optimal Trajectory Generation Problems
The article of record as published may be found at http://dx.doi.org/10.3390/s22051869This paper presents a method for the generation of trajectories for autonomous system operations. The proposed method is based on the use of Bernstein polynomial approximations to transcribe infinite dimensional optimization problems into nonlinear programming problems. These, in turn, can be solved using off-the-shelf optimization solvers. The main motivation for this approach is that Bernstein polynomials possess favorable geometric properties and yield computationally efficient algorithms that enable a trajectory planner to efficiently evaluate and enforce constraints along the vehicles� trajectories, including maximum speed and angular rates as well as minimum distance between trajectories and between the vehicles and obstacles. By virtue of these properties and algorithms, feasibility and safety constraints typically imposed on autonomous vehicle operations can be enforced and guaranteed independently of the order of the polynomials. To support the use of the proposed method we introduce BeBOT (Bernstein/B�zier Optimal Trajectories), an open-source toolbox that implements the operations and algorithms for Bernstein polynomials. We show that BeBOT can be used to efficiently generate feasible and collision-free trajectories for single and multiple vehicles, and can be deployed for real-time safety critical applications in complex environments.This research was supported by the Office of Naval Research, grants N000141912106, N000142112091 and N0001419WX00155. Antonio Pascoal was supported by H2020-EU.1.2.2-FET Proactive RAMONES, under Grant GA 101017808 and LARSyS-FCT under Grant UIDB/50009/2020. Isaac Kaminer was supported by the Office of Naval Research grant N0001421WX01974.This research was supported by the Office of Naval Research, grants N000141912106, N000142112091 and N0001419WX00155. Antonio Pascoal was supported by H2020-EU.1.2.2-FET Proactive RAMONES, under Grant GA 101017808 and LARSyS-FCT under Grant UIDB/50009/2020. Isaac Kaminer was supported by the Office of Naval Research grant N0001421WX01974
A unified Pythagorean hodograph approach to the medial axis transform and offset approximation
AbstractAlgorithms based on Pythagorean hodographs (PH) in the Euclidean plane and in Minkowski space share common goals, the main one being rationality of offsets of planar domains. However, only separate interpolation techniques based on these curves can be found in the literature. It was recently revealed that rational PH curves in the Euclidean plane and in Minkowski space are very closely related. In this paper, we continue the discussion of the interplay between spatial MPH curves and their associated planar PH curves from the point of view of Hermite interpolation. On the basis of this approach we design a new, simple interpolation algorithm. The main advantage of the unifying method presented lies in the fact that it uses, after only some simple additional computations, an arbitrary algorithm for interpolation using planar PH curves also for interpolation using spatial MPH curves. We present the functionality of our method for G1 Hermite data; however, one could also obtain higher order algorithms
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