The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Carlo Ciulla (Lane College, USA)
DOI: 10.4018/978-1-60566-202-2.ch008
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Abstract

This is the first of the chapters of the book that present results obtained processing the MRI database with both classic and SRE-based interpolation paradigms. The focus of this chapter is on the bivariate linear interpolation function. An overview on the validation paradigm is given along with an explanation of the simulations that were conducted in order to validate the SRE-based bivariate liner interpolation function. Subsequently, results with real MRI are shown quantitatively through plots of the metric called RMSE Ratio which was employed to assess which one between classic and SRE-based interpolation furnishes lower root-mean-square-error (RMSE). Qualitative demonstration of the results with MRI is also given. The chapter also discusses and evaluates the case of interpolation error improvement when the novel re-sampling location within the pixel is placed on the x axis of the coordinate system (i.e. yr0 = 0) and consistently, both quantitative and qualitative results are presented through plots of the RSME Ratio and figures respectively.
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Validation Paradigm

Ciulla and Deek (2005) showed quantification of the advantages of the Sub-pixel Efficacy Region (SRE) and also presented the validation paradigm adopted in order to demonstrate the improvement of the bivariate liner interpolation function. For convenience to the reader the metric (RMSE Ratio) employed to assess as to which one of the two model interpolation functions delivers the smallest error is herein recalled along with the procedure employed for its calculation.

The image was misplaced of (x0 y0), then motion correction and interpolation was performed in two different ways: (i) through the classic form of bivariate linear function, and (ii) through the SRE-based bivariate linear function. The latter consists of the classic bivariate interpolation function calculated at the novel re-sampling location. After processing the image with a given misplacement or rotation, the ratio between the two resulting RMSE values was calculated with: (i) the classic bivariate interpolation function (RMSEbefore), and (ii) the SRE-based bivariate interpolation function (RMSEafter) respectively. The ratio was then subtracted from the numerical value of one such to obtain the:

RMSE Ratio = (1 - RMSEbefore / RMSEafter)

A negative value of the RMSE Ratio shows that the SRE-based function outperforms the classic in terms of interpolation error and a positive value shows the opposite.

For the reminder of the book the reader is referred to above definition of RMSE Ratio. This metric is consistently employed in these works to assess the relativity of performance of the pair of interpolation functions: classic and SRE-based.

Ciulla and Deek (2005) reported empirical confirmation of what was foreseen through the two conceptions that set the ground upon which the theory was developed. This was done through the employment of diverse image data among which T1-weighted MRI. The same validation paradigm is herein adopted to further illustrate the results of the application of the Sub-pixel Efficacy Region (SRE) to the bivariate linear interpolation function. This presentation starts with the results obtained on signals of known nature and continues extensively while employing T1-weighted and T2-weighted MRI along with functional MRI data (T2*).

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Simulations

Ciulla and Deek (2005) presented results obtained through simulations performed employing the sin (x, y) function along with the plots of the Intensity-Curvature Functional ΔE after re-sampling and also after motion correction and interpolation through the bivariate linear interpolation function in its classic and the SRE-based forms. The results were corroborated with RMSE plots showing the improvement obtained through the SRE. Also presentation of the behavior of the x and y coordinates of the novel re-sampling locations was given for the cos(x, y) function.

As shown in Figure 1, this section extends the former study and thus presents two more functions: cos (x, y) and ln (x, y). Figures 2 and 3 show the plots of the Intensity-Curvature Functional for re-sampled images (Figure 2) and for re-sampled and motion corrected images (Figure 3).

Figure 1.

2D images in their original form. (a), cos (x, y), (b) ln (x, y)

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