The Main Innovation Determined By the Sub-Pixel Efficacy Region

The Main Innovation Determined By the Sub-Pixel Efficacy Region

Carlo Ciulla (Lane College, USA)
DOI: 10.4018/978-1-60566-202-2.ch017
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Abstract

This chapter introduces the reader to Section V of the book. The chapter opens up with a discussion on the undeniable evidence reported in literature that the magnitude of the interpolation error is strictly related to the magnitude of the sampling resolution. While reference to the literature on the Lagrange interpolation function is reported elsewhere (Ciulla & Deek, 2006), the chapter devotes attention to the literature and the applications related to the Sinc function. The core of the chapter reports a section that condenses the message to the reader of this book about the main innovation determined through the Sub-pixel Efficacy Region. It is delivered to the reader the realization that combining signal intensity with the curvature of the interpolation function, the approximation properties of the model function can be improved. This message is linked to the bridging concept between classic and SRE-based interpolation which is that of the curvature of the interpolation function.
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Literature

For a discussion on Lagrange approximation, literature and more generally on interpolation approximation error bounds the reader is referred to Ciulla and Deek (2006). For what concerns the present chapter it is necessary to recall that the literature extensively admits and reports the evidence that the goodness of approximation of any interpolation function is inversely proportional to the sampling step (resolution). Conceptually this is straightforward since the degree of uncertainty of the estimate of the model interpolation increases proportionally to the length of the curve connecting two adjacent nodes. That is to say that the degree of uncertainty about the true local curvature between two adjacent nodes increases with the increase of the sampling step. Such increase is detrimental because of two fundamental reasons. Both reasons descend from the nature of the ill posed problem that the model interpolator attempt to solve. The first and main reason is that the true signal is unknown and from this concept it descends that the true local curvature is consequentially unknown. The second reason is consequential to the first one and it reveals itself by the nature of the task the interpolator has to perform which is that of approximation. To approximate within a large sampling step is thus far more difficult thus prone to a larger error because the length of the curve that the model interpolator has to reconstruct is extended. This concept applies in either image space or k-space (frequency domain).

Nevertheless, as demonstrated by early experimental research (Whittaker, 1915, 1935), the ideal approach to interpolation remains the Sinc interpolation function. The reason why Sinc interpolation proves to be superior stands in its theoretical background as it considers the signal to be defined from -∞ to +∞ through infinite convolution of the sample node intensities. The typical Sinc function is: ∞ ∞ x (t) = Σ -∞ x (nT) sinc(π(t-nT)/T) = Σ -∞ x (nT) sin[ (π(t-nT)/T) ] / (π(t-nT)/T) (1) where T is the sampling period. Two main limitations arise from equation (1) and they are: (i) the number of intensity values (nT) are finite, and (ii) values of the sampling point close or equal to t = nT furnish an undetermined expression of the type 0/0. Methods of approximating the Sinc interpolation function given by equation (1) through a finite summation of samples have been explored (Schanze, 1995). It was also demonstrated that the approximated form of the Sinc function can also be represented by the Lagrange function (Whittaker, 1915, 1935).

Schanze (1995) reported the following expressions of the finite sum approximation of the Sinc function: M –1 x (t) = sin[ πt / N ] * Σn = - L xn (-1n) cot [π (t - n)/ N], N even (2) M –1 x (t) = sin[ πt / N ] * Σn = - L xn (-1n) csc [π (t - n)/ N], N odd (3) Alternative formulations were reported (Candocia & Principe, 1998; Dooley & Nandi, 2000).

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