The Notion

The Notion

Carlo Ciulla (Lane College, USA)
DOI: 10.4018/978-1-60566-202-2.ch006
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Abstract

The goal of this chapter is to discuss the practical implications of the notion. This chapter concludes the logical reasoning outlined through the process of deduction that begun in Chapter II while reporting on the intuition. The logical reasoning was necessary in order to arrive to a determination as to what is the meaning of the intuition. Along the way it was possible to introduce the novel conception of Intensity-Curvature Functional and to connect such conception to the initial intuition of SRE. At that point the intuition was assessed in its truth and thus it was manifest to reconsider the formulation of the SRE. The implication of doing so rests in that the formulation and the conceptualization of the SRE given in Chapter IV is useful for the remainder of the book in presenting the unifying theory for the improvement of the interpolation error. The process of deduction (logical reasoning) has finally arrived to the notion that the curvature of the model interpolation function is quite relevant to the extent of improving the approximation properties of the function. The realization of the notion was already anticipated in Chapter V and this chapter further extends on the notion through the illustrations of its practical implications. In other words the process of deduction has arrived to a beneficial conclusion and for this reason it proves itself to have been useful in order to derive the notion.
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The Curvature Of The Interpolation Function

This chapter concludes Section I of the book deriving the notion from the evaluation of the truth foreseen in the intuition. The presentation is consequential to what already seen in the previous four chapters and it is particularly focused on the trivariate liner interpolation function.

The conception of the Intensity-Curvature Functional that was given in Chapter III has shown that it possible to give a mathematical formulation that is such to measure the energy level of the signal (image) before and after interpolation. The Intensity-Curvature Functional is thus capable to determine the energy level change of the signal (image) when is subject to re-sampling at any intra-voxel location through the model interpolation function. Based on this conception it was derived the empirical formulation of the Sub-pixel Efficacy Region (SRE), which deduction is subsequent to the study of the Intensity-Curvature Functional. Such formulation asserts that the Sub-pixel Efficacy Region is that spatial set of intra-voxel points where the energy level change of the signal (image) is minimum or maximum. This descends from the fact that the set of points that belong to the SRE are derived through the solution of the polynomial system of first order partial derivatives of the Intensity-Curvature Functional (ΔE) and that is the task that corresponds to the identification of the extremes of ΔE.

In Chapter II however, the definition given to the SRE was purely theoretical and was derived from the intuition. Definition VII seen in Chapter II was given within the context of the hyperbolic paraboloid function h = h(X, Y, Z) that exists for each (X, Y, Z) of the voxel Ψ and that is capable to estimate the signal at any intra-voxel location. The Sub-pixel Efficacy Region in Ψ was thus defined as that intra-voxel set of points:

Ω = { (X, Y, Z): | ∂2h /δxδy |z=Z = | ∂2h /δyδx |z=Z ≤ Z θxy and| ∂2h /δxδz |y=Y = | ∂2h /δzδx |y=Y ≤ Y θxz and| ∂2h /δyδz |x=X = | ∂2h /δzδy |x=X ≤ X θyz }.

Also, for convenience to the reader, it is herein recalled from definition V of Chapter II that:

2h /δxδy = ∂2h /∂y∂x = [f(0,0,0) – f(1,0,0) – f(0,1,0) + f(1,1,0)] + Z ωf2h/∂x∂z = ∂2h/∂z∂x = [f(0,0,0) – f(1,0,0) – f(0,0,1) + f(1,0,1)] + Y ωf2h /∂y∂z = ∂2h /∂z∂y = [f(0,0,0) – f(0,1,0) – f(0,0,1) + f(0,1,1)] + X ωf

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