Exponential Parameterized Cubic B-Spline Curves And Surfaces

The use of B-spline interpolation function for curves and surfaces has been developed for many reasons. One reason is the higher degree of continuity and smoothness. A general B-Spline is a polynomial curve and its shape is determined by the control points. To interpolate data points, various works have been done by previous researchers who studies B-Spline parameterization. In this thesis, we develop a new way for interpolating cubic B-Spline curve by taking the first and the second derivative at endpoints and only the first derivative at inner points. The proposed method is the extension in the B-spline interpolation technique of using arbitrary derivatives at end points. In developing B-spline curve interpolation method, an algorithm is presented for interpolating data points. The algorithm computes knot values for parameterization methods. These knot values are used in constructing a matrix of B-Spline basis function and derivative of the basis function. Then, we solve it for control points by using the LU decomposition method, such that the curve will pass through the given data points. Selection of proper parametrization technique is critical for curve and surface reconstruction process. The parametrization method used in this study is an exponential parameterization method with a = 0:8. The main advantage of developing B-spline curve interpolation method is that we can generate different shapes of curves by setting different direction at all data points. As an application, we applied the proposed method in curve reconstruction on a road map from given data points and driving directions, and also for path planning in autonomous vehicle with given starting and goal position

Total Least Squares Fitting of Bezier and B-Spline Curves to Ordered Data. Computer Aided Geometric Design

We begin by considering the problem of fitting a single Bézier curve segment to a set of ordered data so that the error is minimized in the total least squares sense. We develop an algorithm for applying the Gauss–Newton method to this problem with a direct method for evaluating the Jacobian based on implicitly differentiating a pseudo-inverse. We then demonstrate the simple extension of this algorithm to B-spline curves. We present some experimental results for both cases

Methods for constraint-based conceptual free-form surface design

Zusammenfassung Der constraint-basierte Entwurf von Freiformfl„chen ist eine m„chtige Methode im Computer gest�tzten Entwurf. Bekannte Realisierungen beschr„nken sich jedoch meist auf Interpolation von Rand- und isoparametrischen Kurven. In diesem Zusammenhang sind die sog. "Multi-patch" Methoden die am weitesten verbreitete Vorgehensweise. Hier versucht man Fl„chenverb„nde aus einem Netz von dreidimensionalen Kurven (oft gemischt mit unstrukturierten Punktewolken) derart zu generieren, dass die Kurven und Punkte von den Fl„chen interpoliert werden. Die Kurven werden als R„nder von rechteckigen oder dreieckigen bi-polynomialen oder polynomialen Fl„chen betrachtet. Unter dieser Einschr„nkung leidet die Flexibilit„t des Verfahrens. In dieser Dissertation schlagen wir vor, beliebige, d.h. auch nicht iso-parametrische, Kurven zu verwenden. Dadurch ergeben sich folgende Vorteile: Erstens kann so beispielsweise eine B-spline Fl„che entlang einer benutzerdefinierten Kurve verformt werden w„hrend andere Kurven oder Punkte fixiert sind. Zweitens, kann eine B-spline Fl„che Kurven interpolieren, die sich nicht auf iso-parametrische Linien der Fl„che abbilden lassen. Wir behandeln drei Arten von Constraints: Inzidenz einer beliebigen Kurve auf einer B-spline Fl„che, Fixieren von Fl„chennormalen entlang einer beliebigen Kurve (dieser Constraint dient zur Herstellung von tangentialen šberg„ngen zwischen zwei Fl„chen) und die sog. Variational Constrains. Letztere dienen unter anderem zur Optimierung der physikalischen und optischen Eigenschaften der Fl„chen. Es handelt sich hierbei um die Gausschen Normalgleichungen der Fl„chenfunktionale zweiter Ordnung, wie sie in der Literatur bekannt sind. Die Dissertation gliedert sich in zwei Teile. Der erste Teil befasst sich mit der Aufstellung der linearen Gleichungssysteme, welche die oben erw„hnten Constraints repr„sentieren. Der zweite Teil behandelt Methoden zum L”sen dieser Gleichungssysteme. Der Kern des ersten Teiles ist die Erweiterung und Generalisierung des auf Polarformen (Blossoms) basierenden Algorithmus f�r Verkettung von Polynomen auf Bezier und B-spline Basis: Gegeben sei eine B-spline Fl„che und eine B-spline Kurve im Parameterraum der Fl„che. Wir zeigen, dass die Kontrollpunkte der dreidimensionalen Fl„chenkurve, welche als polynomiale Verkettung der beiden definiert ist, durch eine im Voraus berechenbare lineare Tranformation (eine Matrix) der Fl„chenkontrollpunkte ausgedr�ckt werden k”nnen. Dadurch k”nnen Inzidenzbeziehungen zwischen Kurven und Fl„chen exakt und auf eine sehr elegante und kompakte Art definiert werden. Im Vergleich zu den bekannten Methoden ist diese Vorgehensweise effizienter, numerisch stabiler und erh”ht nicht die Konditionszahl der zu l”senden linearen Gleichungen. Die Effizienz wird erreicht durch Verwendung von eigens daf�r entwickelten Datenstrukturen und sorgf„ltige Analyse von kombinatorischen Eigenschaften von Polarformen. Die Gleichungen zur Definition von Tangentialit„ts- und Variational Constraints werden als Anwendung und Erweiterung dieses Algorithmus implementiert. Beschrieben werden auch symbolische und numerische Operationen auf B-spline Polynomen (Multiplikation, Differenzierung, Integration). Dabei wird konsistent die Matrixdarstellung von B-spline Polynomen verwendet. Das L”sen dieser Art von Constraintproblemen bedeutet das Finden der Kontrollpunkte einer B-spline Fl„che derart, dass die definierten Bedingungen erf�llt werden. Dies wird durch L”sen von, im Allgemeinen, unterbestimmten und schlecht konditionierten linearen Gleichungssystemen bewerkstelligt. Da in solchen F„llen keine eindeutige, numerisch stabile L”sung existiert, f�hren die �blichen Methoden zum L”sen von linearen Gleichungssystemen nicht zum Erfolg. Wir greifen auf die Anwendung von sog. Regularisierungsmethoden zur�ck, die auf der Singul„rwertzerlegung (SVD) der Systemmatrix beruhen. Insbesondere wird die L-curve eingesetzt, ein "numerischer Hochfrequenzfilter", der uns in die Lage versetzt eine stabile L”sung zu berechnen. Allerdings reichen auch diese Methoden im Allgemeinen nicht aus, eine Fl„che zu generieren, welche die erw�nschten „sthetischen und physikalischen Eigenschaften besitzt. Verformt man eine Tensorproduktfl„che entlang einer nicht isoparametrischen Kurve, entstehen unerw�nschte Oszillationen und Verformungen. Dieser Effekt wird "Surface-Aliasing" genannt. Wir stellen zwei Methoden vor um diese Aliasing-Effekte zu beseitigen: Die erste Methode wird vorzugsweise f�r Deformationen einer existierenden B-spline Fl„che entlang einer nicht isoparametrischen Kurve angewendet. Es erfogt eine Umparametrisierung der zu verformenden Fl„che derart, dass die Kurve in der neuen Fl„che auf eine isoparametrische Linie abgebildet wird. Die Umparametrisierung einer B- spline Fl„che ist keine abgeschlossene Operation; die resultierende Fl„che besitzt i.A. keine B-spline Darstellung. Wir berechnen eine beliebig genaue Approximation der resultierenden Fl„che mittels Interpolation von Kurvennetzen, die von der umzuparametrisierenden Fl„che gewonnen werden. Die zweite Methode ist rein algebraisch: Es werden zus„tzliche Bedingungen an die L”sung des Gleichungssystems gestellt, die die Aliasing-Effekte unterdr�cken oder ganz beseitigen. Es wird ein restriktionsgebundenes Minimum einer Zielfunktion gesucht, deren globales Minimum bei "optimaler" Form der Fl„che eingenommen wird. Als Zielfunktionen werden Gl„ttungsfunktionale zweiter Ordnung eingesetzt. Die stabile L”sung eines solchen Optimierungsproblems kann aufgrund der nahezu linearen Abh„ngigkeit des Gleichungen nur mit Hilfe von Regularisierungsmethoden gewonnen werden, welche die vorgegebene Zielfunktion ber�cksichtigen. Wir wenden die sog. Modifizierte Singul„rwertzerlegung in Verbindung mit dem L-curve Filter an. Dieser Algorithmus minimiert den Fehler f�r die geometrischen Constraints so, dass die L”sung gleichzeitig m”glichst nah dem Optimum der Zielfunktion ist.The constrained-based design of free-form surfaces is currently limited to tensor-product interpolation of orthogonal curve networks or equally spaced grids of points. The, so- called, multi-patch methods applied mainly in the context of scattered data interpolation construct surfaces from given boundary curves and derivatives along them. The limitation to boundary curves or iso-parametric curves considerably lowers the flexibility of this approach. In this thesis, we propose to compute surfaces from arbitrary (that is, not only iso-parametric) curves. This allows us to deform a B-spline surface along an arbitrary user-defined curve, or, to interpolate a B-spline surface through a set of curves which cannot be mapped to iso-parametric lines of the surface. We consider three kinds of constraints: the incidence of a curve on a B-spline surface, prescribed surface normals along an arbitrary curve incident on a surface and the, so-called, variational constraints which enforce a physically and optically advantageous shape of the computed surfaces. The thesis is divided into two parts: in the first part, we describe efficient methods to set up the equations for above mentioned linear constraints between curves and surfaces. In the second part, we discuss methods for solving such constraints. The core of the first part is the extension and generalization of the blossom-based polynomial composition algorithm for B-splines: let be given a B-spline surface and a B-spline curve in the domain of that surface. We compute a matrix which represents a linear transformation of the surface control points such that after the transformation we obtain the control points of the curve representing the polynomial composition of the domain curve and the surface. The result is a 3D B-spline curve always exactly incident on the surface. This, so-called, composition matrix represents a set of linear curve-surface incidence constraints. Compared to methods used previously our approach is more efficient, numerically more stable and does not unnecessarily increase the condition number of the matrix. The thesis includes a careful analysis of the complexity and combinatorial properties of the algorithm. We also discuss topics regarding algebraic operations on B-spline polynomials (multiplication, differentiation, integration). The matrix representation of B-spline polynomials is used throughout the thesis. We show that the equations for tangency and variational constraints are easily obtained re-using the methods elaborated for incidence constraints. The solving of generalized curve-surface constraints means to find the control points of the unknown surface given one or several curves incident on that surface. This is accomplished by solving of large and, generally, under-determined and badly conditioned linear systems of equations. In such cases, no unique and numerically stable solution exists. Hence, the usual methods such as Gaussian elimination or QR-decomposition cannot be applied in straightforward manner. We propose to use regularization methods based on Singular Value Decomposition (SVD). We apply the so-called L-curve, which can be seen as an numerical high-frequency filter. The filter automatically singles out a stable solution such that best possible satisfaction of defined constraints is achieved. However, even the SVD along with the L-curve filter cannot be applied blindly: it turns out that it is not sufficient to require only algebraic stability of the solution. Tensor-product surfaces deformed along arbitrary incident curves exhibit unwanted deformations due to the rectangular structure of the model space. We discuss a geometric and an algebraic method to remove this, so-called, Surface aliasing effect. The first method reparametrizes the surface such that a general curve constraint is converted to iso-parametric curve constraint which can be easily solved by standard linear algebra methods without aliasing. The reparametrized surface is computed by means of the approximated surface-surface composition algorithm, which is also introduced in this thesis. While this is not possible symbolically, an arbitrary accurate approximation of the resulting surface is obtained using constrained curve network interpolation. The second method states additional constraints which suppress or completely remove the aliasing. Formally we solve a constrained least square approximation problem: we minimize an surface objective function subject to defined curve constraints. The objective function is chosen such that it takes in the minimal value if the surface has optimal shape; we use a linear combination of second order surface smoothing functionals. When solving such problems we have to deal with nearly linearly dependent equations. Problems of this type are called ill-posed. Therefore sophisticated numerical methods have to be applied in order to obtain a set of degrees of freedom (control points of the surface) which are sufficient to satisfy given constraints. The remaining unused degrees of freedom are used to enforce an optically pleasing shape of the surface. We apply the Modified Truncated SVD (MTSVD) algorithm in connection with the L-curve filter which determines a compromise between an optically pleasant shape of the surface and constraint satisfaction in a particularly efficient manner

Smooth quasi-developable surfaces bounded by smooth curves

Computing a quasi-developable strip surface bounded by design curves finds wide industrial applications. Existing methods compute discrete surfaces composed of developable lines connecting sampling points on input curves which are not adequate for generating smooth quasi-developable surfaces. We propose the first method which is capable of exploring the full solution space of continuous input curves to compute a smooth quasi-developable ruled surface with as large developability as possible. The resulting surface is exactly bounded by the input smooth curves and is guaranteed to have no self-intersections. The main contribution is a variational approach to compute a continuous mapping of parameters of input curves by minimizing a function evaluating surface developability. Moreover, we also present an algorithm to represent a resulting surface as a B-spline surface when input curves are B-spline curves.Comment: 18 page

L1 Control Theoretic Smoothing Splines

In this paper, we propose control theoretic smoothing splines with L1 optimality for reducing the number of parameters that describes the fitted curve as well as removing outlier data. A control theoretic spline is a smoothing spline that is generated as an output of a given linear dynamical system. Conventional design requires exactly the same number of base functions as given data, and the result is not robust against outliers. To solve these problems, we propose to use L1 optimality, that is, we use the L1 norm for the regularization term and/or the empirical risk term. The optimization is described by a convex optimization, which can be efficiently solved via a numerical optimization software. A numerical example shows the effectiveness of the proposed method.Comment: Accepted for publication in IEEE Signal Processing Letters. 4 pages (twocolumn), 5 figure

Extracting 3D parametric curves from 2D images of Helical objects

Helical objects occur in medicine, biology, cosmetics, nanotechnology, and engineering. Extracting a 3D parametric curve from a 2D image of a helical object has many practical applications, in particular being able to extract metrics such as tortuosity, frequency, and pitch. We present a method that is able to straighten the image object and derive a robust 3D helical curve from peaks in the object boundary. The algorithm has a small number of stable parameters that require little tuning, and the curve is validated against both synthetic and real-world data. The results show that the extracted 3D curve comes within close Hausdorff distance to the ground truth, and has near identical tortuosity for helical objects with a circular profile. Parameter insensitivity and robustness against high levels of image noise are demonstrated thoroughly and quantitatively