The Extension of Theory and Methodology to B-Splines

The Extension of Theory and Methodology to B-Splines

Carlo Ciulla (Lane College, USA)
DOI: 10.4018/978-1-60566-202-2.ch014
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Abstract

The organization of the chapter is similar to that of Chapters VII and X. The methodological approach to extend the unifying theory to the one dimensional quadratic and cubic B-Splines is herein reported along with the most relevant mathematical details. This chapter should be read along with Appendix VI where proofs are given to the assertions herein presented. In either of the two cases: quadratic and cubic B-Spline the math process starts from the calculation of the Intensity-Curvature Functional and continues with the calculation of the Sub-pixel Efficacy Region. Finally, the math process arrives to the calculation of the novel re-sampling locations through the formulas of the unifying theory seen in equations (23) and (33) for the quadratic and the cubic models respectively. The chapter concludes with a section that addresses specifically the theoretical proposition of resilient interpolation for the two classes of B-Splines. This is conducted consistently with Chapters VII and XII of the book choosing to equate the two intensity-curvature terms (before and after interpolation) as the starting point of the math deduction.
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Quadratic And Cubic B-Splines

Calculations were developed based on the B-Spline forms: quadratic (h3(x)), and cubic (h4(x)) as per equations (1) and (2), with 1 x 6 and 1 x 6 neighbourhoods’ nodes (see equations (3) and (4)).

- 2a | x |2 + 1/2 (a+1) 0 ≤ | x | ≤ 1/2h3(x) = (1) a | x |2 - (2a + 1/2) | x | + 3/4 (a+1) 1/2 ≤ | x | ≤ 3/21/2 | x |3 - | x |2 + 2/3 0 ≤ | x | ≤ 1 h4(x) = (2) -1/6 | x |3 + | x |2 - 2 | x | + 4/3 1 ≤ | x | ≤ 2

The absolute value of x assumes the meaning of representing intra-nodal misplacement within the pixel or within two nodes. It is assumed that the origin of the coordinate system of the pixel or the intra-nodal distance (sampling step) is at the corner and not at the center. Therefore the value of the misplacement is never negative. For simplicity let us consider the two functions of equations (1) and (2) written as:

h3(x) = f (0) + [ f (1/2) +f (-1/2) ] * [- 2a x2 + 1/2 (a+1) ] + [ f (1/2) + f (-1/2) + f (-1) + f (1) +f (3/2) + f (-3/2) ] * [ a x2 - (2a + 1/2) x + 3/4 (a+1) ] (3) h4(x) = f (0) + [ f (1/2) + f (-1/2) + f (-1) + f (1) ] * [ 1/2 x3 - x2 + 2/3 ] + [f (3/2) + f (- 3/2) +f (1) +f (-1) +f (2) + f (-2)] * [ -1/6 x3 + x2 - 2 x + 4/3 ] (4)

Complete Chapter List

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Dedication
Table of Contents
Foreword
Reda Abraham
Acknowledgment
On the Philosophical Basis Underlying Unifying Theory
Chapter 1
Carlo Ciulla
The forthcoming text of this chapter is intended to raise the debate about the content of the book. To fulfill this purpose, is presented in first... Sample PDF
Magnetic Resonance Imaging and the Signal-Image Processing Techniques Developed Under the Umbrella of the Unifying Theory
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Chapter 2
The Intuition  (pages 23-30)
Carlo Ciulla
This chapter reports on the initial idea which gave birth to the investigation which subsequently became the unifying theory. The intuition herein... Sample PDF
The Intuition
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Chapter 3
Carlo Ciulla
The preceding chapter is to be viewed as a purely theoretical math intuition and the claim consists in that of the existence of a region within the... Sample PDF
The Conception of the Intensity-Curvature Functional
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Chapter 4
Carlo Ciulla
This chapter presents the mathematical deduction of the Sub-pixel Efficacy Region (SRE) for the case of the trivariate liner interpolation function.... Sample PDF
The Conception of the Sub-Pixel Efficacy Region
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Chapter 5
Carlo Ciulla
This chapter continues the investigation conducted through deduction from the previous three chapters. Chapter II of the book has brought to light... Sample PDF
Assessment of the Truth Foreseen in the Intuition
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Chapter 6
The Notion  (pages 52-56)
Carlo Ciulla
The goal of this chapter is to discuss the practical implications of the notion. This chapter concludes the logical reasoning outlined through the... Sample PDF
The Notion
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Chapter 7
Carlo Ciulla
In the sections of this chapter the reader will be introduced to the sequence of mathematical processes which, starting from a model interpolation... Sample PDF
The Theoretical Approach to the Improvement of the Interpolation Error: Bivariate Linear Interpolation Function
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Chapter 8
Carlo Ciulla
This is the first of the chapters of the book that present results obtained processing the MRI database with both classic and SRE-based... Sample PDF
The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function
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Chapter 9
Interpolation Procedures  (pages 172-179)
Carlo Ciulla
The intent of the present chapter is to expand the treatise that concerns with the presentation of the current literature in signal-image... Sample PDF
Interpolation Procedures
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Chapter 10
Carlo Ciulla
The organization of this chapter is similar to that of Chapter VII where it was outlined the theoretical approach to the improvement of the... Sample PDF
The Extension of the Theory to the Trivariate Linear Interpolation Function
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Chapter 11
Carlo Ciulla
This chapter presents results relevant to the evaluation of the performance of classic and SRE-based trivariate interpolation functions. The... Sample PDF
The Results of the Sub-Pixel Efficacy Region Based Trivariate Linear Interpolation Function
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Chapter 12
Carlo Ciulla
As well accepted in literature it should be almost impossible to reconstruct the signal intensity value at time-space location where the sampling... Sample PDF
Equating the Two Intensity-Curvature Terms, Before and After Interpolation, Attempting to Obtain Resilient Interpolation: Trivariate Linear Interpolation Function
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Chapter 13
Carlo Ciulla
This chapter reviews the extensive and comprehensive literature on B-Splines. In the forthcoming text emphasis is given to hierarchy and formal... Sample PDF
On the Literature of B-Spline Interpolation Functions
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Chapter 14
Carlo Ciulla
The organization of the chapter is similar to that of Chapters VII and X. The methodological approach to extend the unifying theory to the one... Sample PDF
The Extension of Theory and Methodology to B-Splines
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Chapter 15
Carlo Ciulla
The results obtained processing the MRI database with classic and SRE-based one dimensional quadratic and cubic B-Splines are presented in this... Sample PDF
The Results of the Sub-Pixel Efficacy Region Based B-Spline Interpolation Functions
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Chapter 16
Carlo Ciulla
Concepts of efficiency, approximation, and efficacy are discussed in this chapter while referencing the existing literature. Frameworks grouping... Sample PDF
On the Properties of the Unifying Theory and the Derived Sub-Pixel Efficacy Region
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Chapter 17
Carlo Ciulla
This chapter introduces the reader to Section V of the book. The chapter opens up with a discussion on the undeniable evidence reported in... Sample PDF
The Main Innovation Determined By the Sub-Pixel Efficacy Region
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Chapter 18
Carlo Ciulla
This chapter is devoted to the mathematics of the Lagrange and Sinc SRE-based interpolation functions. The organization of the text of this chapter... Sample PDF
The Unifying Theory Embraces Lagrange and Sinc Interpolation Functions
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Chapter 19
Carlo Ciulla
This chapter presents results obtained processing the MRI dataset with classic and SRE-based models of one dimensional Lagrange and Sinc... Sample PDF
The Results of the Sub-Pixel Efficacy Region Based Lagrange and Sinc Interpolation Functions
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Chapter 20
Carlo Ciulla
The last chapter of the book reports on concluding remarks recalling to the reader the message given through these works and also recalling the... Sample PDF
On the Implications of the Sub-Pixel Efficacy Region and the Bridging Concept of the Unifying Theory
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Appendix Section
About the Author