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Variational Regularization of 3D Data

Part of the SpringerBriefs in Computer Science Book Series

DOI: 10.1007/978-1-4939-0533-1

ISBN: 978-1-4939-0532-4

Publisher: Springer New York - Springer Journals

Mar 15, 2014

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Chapter 13D Data in Computer Vision and Technology

Montegranario, Hebert; Espinosa, Jairo

Pages: 1 –4

[Three dimensional data appears in a wide range of modern technology applications, all of them are strongly influenced by the problems of computational vision. It is not strange because through vision, we obtain an understanding of what is in the world, where objects are located, and how they are...
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Chapter 2Function Spaces and Reconstruction

Montegranario, Hebert; Espinosa, Jairo

Pages: 5 –17

[A reconstruction problem is solved when we are able to find a function that models or describes the behavior of data. Although in each problem there will be a particular method to obtain or define this function, many of them are included in the theory of function spaces. In this chapter we will...
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Chapter 3Variational Methods

Montegranario, Hebert; Espinosa, Jairo

Pages: 19 –29

[Variatinal calculus is a theoretical and practical tool for dealing with regularization problems. In this chapter we define the differential calculus of functionals, obtaining the fundamental Euler Lagrange equation. We then use these results in the context of the variational properties of...
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Chapter 4Interpolation. From One to Several Variables

Montegranario, Hebert; Espinosa, Jairo

Pages: 31 –43

[We can use variational properties as a thread that conducts from univariate to multivariate approximations. We begin the chapter illustrating some limitations of the classical Lagrange interpolation and the way splines may help to a better formulation of the problem and the concrete algorithm to...
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Chapter 5Functionals and Their Physical Interpretations

Montegranario, Hebert; Espinosa, Jairo

Pages: 45 –49

[As we saw in the former chapter, the variational interpretation of splines, suggests to apply physical or geometrical references to other problems of reconstruction from data. It is possible to go deeper into this method and obtain other models in the form of functionals that could be added or...
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Chapter 6Regularization and Inverse Theory

Montegranario, Hebert; Espinosa, Jairo

Pages: 51 –57

[Now we are going to apply inverse theory to formalize the techniques developed in former chapters, especially the regularization method used for cubic splines. Inverse theory is one of the great achievements of applied mathematics and its applications are at the core of modern technology. Thus...
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Chapter 73D Interpolation and Approximation

Montegranario, Hebert; Espinosa, Jairo

Pages: 59 –67

[In this chapter we extend to n dimensions the spline theory developed in Chap. 4 on cubic splines for interpolation and approximation of functions; in particular we deal with 3D data. As a consequence of interpolation and smoothness conditions we obtain by a constructive proof, the Thin Plate...
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Chapter 8Radial Basis Functions

Montegranario, Hebert; Espinosa, Jairo

Pages: 69 –81

[This last chapter may be seen as a general introduction to meshless methods, not only for surface reconstruction but in other problems of scattered data. The general interpolation and smoothing problems are described in terms of radial basis functions; we then illustrate some characterizations...

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