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: 7
DOI: 10.4018/978-1-60566-202-2.ch012

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TopFrom equations (11) and (10) of Chapter IV it can be written that the intensity curvature terms before (E_{o}), and after interpolation (E_{IN}) are:E_{IN} = E_{IN}(x, y, z) = - ω_{f} H_{xyz} (x, y, z) [ θ_{xy} + θ_{xz} + θ_{yz} - x (1 - ζ_{x} ×) - y (1 - ζ_{y} ×) - z (1 - ζ_{z} ×) ]*(1)* E_{o} = E_{o}(x, y, z) = - f (0, 0, 0) x y z ω_{f} [θ_{xy} + θ_{yz} + θ_{zx} ] *(2)* Where through equations (9) through (12) of Chapter III it is posited that:

Let us proceed by posing E_{o}(x, y, z) = E_{IN}(x, y, z), and solving the resulting equation in the unknown f(0, 0, 0). The problem herein addressed can be formulated in the following terms.

*For the given re-sampling location what is the true pixel (signal) intensity value? Can the model interpolation function calculate that signal intensity value that is the one that would be sampled?* It is reasonable to attempt to answer the question by posing in equality the two intensity curvature terms which is like assuming that there exists a pixel (signal) intensity value that remains the same before and after interpolation. This pixel (signal) intensity value is not the one we start from (before interpolation) nor is the value that it would be obtained through re-sampling the signal through the model interpolation function at the given placement.

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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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