On the Properties of the Unifying Theory and the Derived Sub-Pixel Efficacy Region

On the Properties of the Unifying Theory and the Derived Sub-Pixel Efficacy Region

Carlo Ciulla (Lane College, USA)
DOI: 10.4018/978-1-60566-202-2.ch016
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Abstract

Concepts of efficiency, approximation, and efficacy are discussed in this chapter while referencing the existing literature. Frameworks grouping classes of interpolation functions are acknowledged. Also, properties of the unifying theory are addressed along with the characteristics of the methodology employed to derive the SRE-based interpolation functions. Properties of the unifying theory are also discussed and the reasoning pertaining to the methodology is provided. As it can be seen in this book, both properties and characteristics featuring the methodological development of the SRE-based interpolation function are quite consistent across interpolators and it is true that the empirical observations provided through this works have set the grounds for the legitimate assertion that there exist a unifying theory for the improvement of the interpolation error. Therefore, such consistency across the spectrum of interpolation functions embraced through the unifying theory shows that the concepts embedded in both properties and characteristics of the methodology employed to derive the SRE-based interpolation functions remain the same across the diversity of model interpolators. It is also due to acknowledge that Section V of the book is devoted to Lagrange and Sinc functions and will further expand on the consistency just mentioned. The last section of this chapter addresses the Fourier properties of the Sub-pixel Efficacy Region.
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Efficiency, Approximation And Efficacy

When the efficiency of algorithms for alignment of brain images was studied, a comparative evaluation of the interpolation function computational demand was reported (Joyeux & Penczeck, 2002; Meijering et al., 2001; Ostuni et al., 1997). Several interpolation methods have been analyzed at the aim of finding the trade off between accuracy and computational cost, which is defined as the function having time and number of multiplications and additions as dependent and independent variables respectively (Thevenaz et al., 2000). Extensive evaluation of accuracy and efficiency of interpolation paradigms was reported (Lehmann et al., 1999) through the analysis of: (i) the Fourier properties, (ii) the visual quality, (iii) the quantitative error estimation, and (iv) the computational demand. An approach was designed to equate the effects of an ideal interpolator for the linear paradigm (Wang, 2001), while focusing on error estimations and minimizations over intervals rather than over the entire interpolation curve.

This book introduces the concept of interpolation function efficacy as the capability to generate minimal approximation. For the interpolation functions seen in the preceding chapters (bivariate and trivariate linear, quadratic and cubic B-Spline) it was shown that the capability of the interpolation function to determine minimal approximation is dependent on: (i) the magnitude of the misplacement, (ii) the relationships between intensity values at the target pixel to re-sample and the intensity values at the neighboring pixels, and (iii) the local curvature of the function.

This constitutes of a confirmation and also of an extension of the findings reported while studying the Newton-based description of the interpolation error given within the context of mathematical analysis (Agarwal & Wong, 1993; Thevenaz et al., 2000). According to these studies the interpolation error depends: (i) the on step size (resolution), (ii) the location of the re-sampled point relative to the initial coordinate system (misplacement) and (iii) the value of the interpolation function at the node points. It was also reported a specific expression of the Hermite polynomial interpolation error formulated as dependent on optimized constants (Agarwal & Wong, 1993). These constants minimize the approximation of the kernel and are dependent on both resolution and misplacement.

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