عنوان

Fitting splines to a parametric function /

(Number (ISBN

3030125513

(Number (ISBN

3030125521

(Number (ISBN

9783030125516

(Number (ISBN

9783030125523

Erroneous ISBN

3030125505

Erroneous ISBN

9783030125509

Title Proper

Fitting splines to a parametric function /

General Material Designation

[Book]

First Statement of Responsibility

Alvin Penner.

Place of Publication, Distribution, etc.

Cham, Switzerland :

Name of Publisher, Distributor, etc.

Springer,

Date of Publication, Distribution, etc.

2019.

Specific Material Designation and Extent of Item

1 online resource (xii, 79 pages) :

Other Physical Details

illustrations (some color)

Series Title

SpringerBriefs in computer science,

ISSN of Series

2191-5768

Text of Note

Includes bibliographical references.

Text of Note

Intro; Preface; Contents; 1 Introduction; 2 Least Squares Orthogonal Distance Fitting; 2.1 Definition of Error Function; 2.2 Character of Solution; 2.3 Optimization of F(a, u); References; 3 General Properties of Splines; References; 4 ODF Using a Cubic Bézier; 4.1 Fitting a Function with a Double Inflection Point; 4.2 Initializing a Cubic Bézier; 4.3 Optimizing the Fit; 4.4 Continuity of the rms Error; 4.5 Character of Coalescing Solutions; References; 5 Topology of Merges/Crossovers; 5.1 Center of Mass Fit; 5.2 Example of Two Types of Merge/Crossover; 5.3 Response to Change in g(t)

Text of Note

5.4 Distinguishing Between Type 1 and Type 2 EventsReferences; 6 ODF Using a 5-Point B-Spline; 6.1 Initializing a 5-Point B-Spline; 6.2 Basis Functions of a 5-Point B-Spline; 6.3 Decomposition into Two Bézier Segments; 6.4 ODF Results for a 5-Point B-Spline; 7 ODF Using a 6-Point B-Spline; 7.1 Initializing a 6-Point B-Spline; 7.2 Basis Functions of a 6-Point B-Spline; 7.3 Decomposition into Three Bézier Segments; 7.4 ODF Results for a 6-Point B-Spline; 8 ODF Using a Quartic Bézier; 8.1 Initializing a Quartic Bézier; 8.2 ODF Results for a Quartic Bézier; 8.3 Enumeration of Solutions

0

8

Text of Note

This Brief investigates the intersections that occur between three different areas of study that normally would not touch each other: ODF, spline theory, and topology. The Least Squares Orthogonal Distance Fitting (ODF) method has become the standard technique used to develop mathematical models of the physical shapes of objects, due to the fact that it produces a fitted result that is invariant with respect to the size and orientation of the object. It is normally used to produce a single optimum fit to a specific object; this work focuses instead on the issue of whether the fit responds continuously as the shape of the object changes. The theory of splines develops user-friendly ways of manipulating six different splines to fit the shape of a simple family of epiTrochoid curves: two types of Bézier curve, two uniform B-splines, and two Beta-splines. This work will focus on issues that arise when mathematically optimizing the fit. There are typically multiple solutions to the ODF method, and the number of solutions can often change as the object changes shape, so two topological questions immediately arise: are there rules that can be applied concerning the relative number of local minima and saddle points, and are there different mechanisms available by which solutions can either merge and disappear, or cross over each other and interchange roles. The author proposes some simple rules which can be used to determine if a given set of solutions is internally consistent in the sense that it has the appropriate number of each type of solution.

Source for Acquisition/Subscription Address

Springer Nature

Stock Number

com.springer.onix.9783030125516

Title

Fitting splines to a parametric function.

International Standard Book Number

9783030125509

Spline theory.

Computer Graphics.

Image Processing and Computer Vision.

Spline theory.

COM012000

UML

UML

Number

Edition

Class number

Penner, Alvin

Date of Transaction

20200823082954.0

Cataloguing Rules (Descriptive Conventions))

pn

Electronic name

[Book]

Y