Carlo Ciulla (Lane College, USA)

Source Title: Improved Signal and Image Interpolation in Biomedical Applications: The Case of Magnetic Resonance Imaging (MRI)

Copyright: © 2009
|Pages: 5
DOI: 10.4018/978-1-60566-202-2.ch006

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

∂
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Dedication

Table of Contents

Foreword

Reda Abraham

Preface

Acknowledgment

On the Philosophical Basis Underlying Unifying Theory

Chapter 1

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Chapter 7

The Theoretical Approach to the Improvement of the Interpolation Error: Bivariate Linear Interpolation Function
(pages 58-71)

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Chapter 8

The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function
(pages 72-170)

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Chapter 10

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Chapter 11

The Results of the Sub-Pixel Efficacy Region Based Trivariate Linear Interpolation Function
(pages 188-205)

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Chapter 15

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Chapter 16

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Chapter 17

The Main Innovation Determined By the Sub-Pixel Efficacy Region
(pages 348-352)

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Chapter 18

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Chapter 19

The Results of the Sub-Pixel Efficacy Region Based Lagrange and Sinc Interpolation Functions
(pages 371-470)

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Chapter 20

On the Implications of the Sub-Pixel Efficacy Region and the Bridging Concept of the Unifying Theory
(pages 471-511)

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