Interpolation Procedures

Interpolation Procedures

Carlo Ciulla (Lane College, USA)
DOI: 10.4018/978-1-60566-202-2.ch009
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Abstract

The intent of the present chapter is to expand the treatise that concerns with the presentation of the current literature in signal-image interpolation with specific focus on the classification of the procedures and the relevant applications of interpolation, in relationship to the various scientific disciplines. Also, the present chapter expands on the importance of the error bounds existing in literature for the characterization and quantification of the interpolation error. The discussion is interleaved with the characteristics features of the unifying theory that the book presents such to place the SRE-based interpolation functions within the context of the larger framework that the scientific literature has conceived so far.
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Classification And Applications Of The Interpolation Procedures

Signal-image interpolation is normally used for several procedures, and they are hereto listed.

When pixel-by-pixel correspondence has to be established for the purpose of matching images collected at different times using the same imaging modality (registration) or different imaging modalities (co-registration), interpolation finds immediate application in motion correction techniques.

When signals are to be reconstructed with increased resolution (up-sampling) or decreased resolution (down-sampling).

For the estimation of signals at space-time locations where the signal is unknown because of the limitation imposed by the sampling frequency of the collecting equipment.

Other relevant procedures are: (i) estimation of signal parameters corresponding to particular frequencies of the spectrum through the use of linear interpolation (Borkowski, 2000), and (ii) enhancement of grayscale images and reconstruction of color images through combined use of covariance coefficients and linear interpolation (Li & Orchard, 2001).

Since interpolation descends from approximation theory, it can also be grouped with extrapolation and regression as shown in an early example of a unifying methodology of linear unbiased estimations (Chow & Lin, 1971).

Interpolation techniques currently in use are classified in two main categories: scene-based interpolation and object-based interpolation (Penney et al., 2004). Scene-based interpolation uses the value of the intensity at the pixels (nodes) at the aim to reconstruct the signal. In order to achieve such purpose, linear, quadratic, cubic, Lagrange and Sinc interpolation paradigms are those among the most commonly employed. Object-based interpolation uses intrinsic features captured from the signal to determine the reconstructed signal. This one category has recently extended to morphology-based methods (Lee & Wang, 2000) and feature guidance based methods (Lee & Lin, 2002).

Object-based interpolation was initiated (Raya & Udupa, 1990) evaluated and extended (Herman et al., 1992), and then reinforced (Grevera & Udupa, 1996, 1998). These authors showed that linear interpolation of a surface representing the distance map of the pixels from the boundary of the object in the scene proves superior to simple linear interpolation of the pixel intensities. The aforementioned authors have also reported a statistical comparison of their shape-based method, and this was developed within the three-dimensional context against a series of other methods including the classic trivariate linear interpolation.

Recent applications of interpolation procedures devote attention to a large variety of applications and paradigms for the estimation of missing data. Hereto is given a list of works that have been brought just recently to the attention of the scientific community pointing to the diversity of the approaches and the applications in science.

Li and Orchard (2001) devised an improved paradigm of the bivariate linear function through the application of the adaptive covariance based approach with applications to grey scale image resolution enhancement. Varadarajan and Krolik (2003) have employed interpolation for the improvement of radar detection. Angelini et al. (2003) employed interpolation to estimate the factors of the multivariate analysis in econometric projections and in the work of these authors, the interpolation paradigm is linked indirectly to the signal through the factors of the multivariate analysis at the aim to estimate missing data in very large datasets.

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