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

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

Carlo Ciulla (Lane College, USA)
DOI: 10.4018/978-1-60566-202-2.ch011
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Abstract

This chapter presents results relevant to the evaluation of the performance of classic and SRE-based trivariate interpolation functions. The forthcoming text reports on the validation procedure employed to quantify the interpolation error of the two model functions along with the information on the resolution of the MRI database that was processed. The presentation of the results is both quantitative through the plots of the RSME Ratio and quantitative through the presentation of the spectral power evolutions of functional MRI data. As anticipated in Chapter I the spectral power evolutions were obtained processing the fMRI volume with the misplacement X = 0.49, Y = 0.49, Z = 0.49 employing the two interpolation paradigms (classic and SRE-based). Particularly, the RSME Ratio quantified the relativity of performance between the two types of trivariate linear interpolation functions while processing T1-weigthed, T2-weighted and functional MRI. The spectral power evolutions quantified for the functional MRI the differences in frequency spectral content between the two classes of interpolators. Interesting to note that the spectral power evolutions clearly show in this chapter their capability to reveal differences, between the images obtained through the two different interpolators, which otherwise would not be observed in the image space. Such differences are somehow hidden in the k-space (Fourier domain).
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Validation Procedure

At the aim to show the error improvement obtainable through the methodological approach of the unifying theory the same type of motion correction paradigm employed for the bivariate linear is presented in this chapter for the trivariate linear functions. An original three dimensional image is shifted. Original means prior to any processing. It is then motion corrected of the initial shift and interpolated with the classic trivariate linear function. This procedure is called processing with classic interpolation as the reader may recall from Chapter VIII of the book. The root-mean-square-error (RMSE) is calculated between original and processed images. Concurrently, the original image is processed with the SRE-based trivariate linear interpolation function (that means that the image is calculated at the novel re-sampling locations).

By doing so, the RMSE Ratio calculates a measure of the relativity of performance between classic and SRE-based trivariate linear interpolation functions. The performance is obviously measured on the basis of the interpolation error and so the RMSE Ratio allows elucidating which one of the two interpolation functions delivers best approximation.

Real Magnetic Resonance Imaging data were employed at the aim to study the interpolation error improvement determined through the SRE-based trivariate linear interpolation function and also to study the spectral characteristics of the re-sampled images as described in the following of this chapter. The data set is the same used for the validation of the SRE-based bivariate linear interpolation function and for convenience of the reader it is hereto recalled.

  • T1 MRI (T1-MRI-230 and T1-MRI-450). Matrix resolution: 176 x 208 with 1.00 x 1.00 mm of pixel size, with 75 slices and an inter-slice resolution of 1.00 mm.

  • T2 MRI. Matrix resolution: 177 x 182 with 1.00 x 1.00 mm of pixel size, with 19 slices and an inter-slice resolution of 1.00 mm.

  • Functional MRI (fMRI-TS1). Matrix resolution: 128 x 128 with 1.72 x 1.72 mm of pixel size, with 20 slices and an inter-slice resolution of 1.72 mm.

  • Functional MRI (fMRI-snffM00587). Matrix resolution: 53 x 63 with 3.00 x 3.00 mm of pixel size, with 30 slices and an inter-slice resolution of 3.00 mm.

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Analysis Of The Rmse Ratio Obtained With Mri Data

The plot of the RMSE Ratio versus the concurrent misplacement along X, Y and Z directions is shown for T1-MRI-230, T1-MRI-450, T2-MRI, fMRI-TS1 and fMRI-snffM00587 in figures 1, 2, 3, 4, and 5 respectively. Each figure plots the RMSE ratio that is relevant to the entire MRI volume. The portion of the curves below zero shows that the SRE-based trivariate interpolation function outperforms the classic trivariate interpolation function. The portion of the curves above zero shows the opposite. For the four MRI data sets the range of misplacement was: [(0, 0, 0), (0.99, 0.99, 0.99)] at steps of (0.01, 0.01, 0.01).

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