On the Implications of the Sub-Pixel Efficacy Region and the Bridging Concept of the Unifying Theory

On the Implications of the Sub-Pixel Efficacy Region and the Bridging Concept of the Unifying Theory

Carlo Ciulla (Lane College, USA)
DOI: 10.4018/978-1-60566-202-2.ch020
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Abstract

The last chapter of the book reports on concluding remarks recalling to the reader the message given through these works and also recalling the proposed novelty. The novelty is discussed within the context of the current literature with specific attention to other works devoted to the improvement of the interpolation error. The reader is acknowledged that the methodological approach outlined through the theory can be seen as a viable pathway to follow in order to conceptualize interpolation in an innovative and alternative manner. This descends from the adoption of the mathematical formulation that is dependent on the joint information content of node intensity and curvature of the interpolation function and has brought to the determination of a viable option to adopt when re-sampling the signal (image). Re-sampling inherent to interpolation can be performed at sub-pixel locations that are not necessarily the same as the given misplacement neither are they necessarily the same pixel-by-pixel. This is because of the variability of pixel intensity and curvature of the interpolation function at the neighborhood and between neighborhoods and also because such variability corresponds to various signal shape characteristics. The reader is informed that within the context of a paradigm to be used for the improvement of the interpolation error it is of relevance to include the curvature in the methodology that is chosen to improve the approximation capability of a given interpolation function. The focus is also towards the evidence that local re-sampling is capable of changing the band-pass filtering property of the interpolation functions. Also, a study is undertaken to determine how beneficial is the application of the Sub-pixel Efficacy Region in the estimation of signals at unknown time-space locations and this is done for the one-dimensional interpolation functions presented in the book. It is also shown that the SRE-based interpolation functions are capable to determine error improvement and to be more accurate with respect to the classic interpolation functions in the estimation of signals at locations that are not captured by the sampling frequency because of the Nyquist’s theorem constraint. Also, based on the same data, the effect of the sampling resolution is studied on the interpolation error and the interpolation error improvement. Consequentially, it is outlined the licit conclusion that under the umbrella of the unifying framework proposed within the theory, the sampling resolution influences both interpolation error and interpolation error improvement obtained from the SRE-based functions. Finally the chapter reports on the investigation of the influence of the SCALE parameter and on the performance comparison across classic and SRE-based interpolation functions. The SCALE parameter is employed to scale the convolution of the pixel intensities determined through the polynomials forms: quadratic and cubic B-Splines, Lagrange, and also to scale the numerical values of the sums of cosine and sine functions of the Sinc interpolation function.
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Novel Conceptions And Limitations Of The Unifying Theory

Ciulla and Deek (2006) report a review of Lagrange polynomials for image processing applications (Francesconi et al., 1993; Fuchs & Delyon, 2002; Lehmann et al., 1999; Xie & Zhou, 2001; Ye, 2003). Similarities between Lagrange and Sinc interpolation functions were investigated in literature (Lehmann et al., 1999; Wittaker, 1915) along with an interesting study that presents work on the improvement of the stop-band characteristics of the Lagrange function (Dumpster & Murphy, 1999).

This book presents an improved version of the Lagrange interpolation and this is done through the extension of the theoretical framework herein outlined. As far as the more general description of benefits and limitations of the application of the SRE to the interpolation functions, the knowledge reported in the book shall be connected to the knowledge reported earlier (Ciulla & Deek, 2006).

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