Magnetic Resonance Imaging and the Signal-Image Processing Techniques Developed Under the Umbrella of the Unifying Theory

Magnetic Resonance Imaging and the Signal-Image Processing Techniques Developed Under the Umbrella of the Unifying Theory

Carlo Ciulla (Lane College, USA)
DOI: 10.4018/978-1-60566-202-2.ch001
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Abstract

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 instance a section that summarizes on how the literature is reviewed throughout the book and also the MRI database employed to validate the unifying theory. Following this presentation, the chapter introduces to the reader the concept behind the basic issue relating to signal-image interpolation, which is the preservation of the signal (image) energy after processing with interpolation. This concept is strictly related to the approximation properties of the interpolation functions and consequently to the interpolation error. This concept is widely accepted in literature and is viewed in this works with a different and innovative perspective. The discussion undertaken in this chapter then introduces the signal-image processing techniques that were developed under the umbrella of the unifying theory. These are: (i) the SRE-based interpolation functions, (ii) the spectral power evolutions, (iii) a three layered artificial neural network with the added capability to generate its internal architecture during the learning process such to adjust to the given pattern classification task.
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Introduction

Since its inception (Lauterbur, 1973; Mansfield & Grannell, 1973) Magnetic Resonance Imaging (MRI) has produced a beneficial and revolutionary trend in several biomedical imaging related applications. Diagnostic imaging, among others, is one of the most relevant. Following the discovery of MRI and parallel to the widespread use of computer technology, an immense research literature has been produced on signal-image processing techniques devoted to Magnetic Resonance Imaging. Several text books exist in literature that explain the concepts behind the MRI physics and the concepts that drive the need and the types of signal-image processing techniques that were and are being developed (Cho et al., 1993; Haacke et al., 1999; Jezzard et al., 2001; Liang & Lauterbur, 2000).

Interpolation is indeed one of the most relevant signal processing applications related to MRI and also to any scientific discipline that requires the estimation of the signal at time-space locations where the signal is not known because of the limitations to sampling as imposed by the recording (imaging) equipment. This fact should provide with an immediate answer as to what is the relevance of the contents of this manuscript. The main objective of this book is to present a unifying theory, supported through empirical validation, for the improvement of the approximation properties of mathematical functions employed in signal-image interpolation to estimate the value of the signal at those space-time locations that are not sampled. Given the relevance of MRI, this book presents the validation of the unifying theory employing Magnetic Resonance Imaging signals (images).

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 instance a section that summarizes on how the literature is reviewed throughout the book and also the MRI database employed to validate the unifying theory. Following this presentation, the chapter introduces to the reader the concept behind the basic issue relating to signal-image interpolation, which is the preservation of the signal (image) energy after processing with interpolation. This concept is strictly related to the approximation properties of the interpolation functions and consequently to the interpolation error. This concept is widely accepted in literature and is viewed in this works with a different and innovative perspective. The discussion undertaken in this chapter then introduces the signal-image processing techniques that were developed under the umbrella of the unifying theory. These are: (i) the SRE-based interpolation functions, (ii) the spectral power evolutions, (iii) a three layered artificial neural network with the added capability to generate its internal architecture during the learning process such to adjust to the given pattern classification task.

The chapter also introduces the very basic concept of the unifying theory which is the distinctive feature of the SRE-based interpolation functions: the local curvature of the function. The main implication of the theory is introduced as it relates to the possibility to calculate the re-sampling locations through a computational approach based on the combined information content of signal intensities and curvature (second order derivative) of the interpolation function. Finally the chapter gives a general overview on the implications of the unifying theory in signal-image processing and particularly in motion correction in functional MRI. The discussion highlights that the creation of signal-image artifacts, which are detrimental to signal reconstruction, can derive directly from the approximate nature of interpolation, but can be reduced through SRE-based interpolation.

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