# The Theoretical Approach to the Improvement of the Interpolation Error: Bivariate Linear Interpolation Function

Carlo Ciulla (Lane College, USA)
DOI: 10.4018/978-1-60566-202-2.ch007

## Abstract

In the sections of this chapter the reader will be introduced to the sequence of mathematical processes which, starting from a model interpolation function yield to the corresponding SRE-based paradigm. Particularly, this chapter addresses the development of the SRE-based bivariate interpolation function. The mathematical procedure is consistently iterated in the rest of the book for all the other model functions that the unifying theory embraces. The first step of the procedure is that of the calculation of the intensity-curvature terms and through their ratio the Intensity-Curvature Functional is calculated for the model function. The second step is that of calculating the first order partial derivatives of the Intensity-Curvature Functional. Thirdly, the polynomial consisting of the first order partial derivatives is solved to obtain the Sub-pixel Efficacy Region. At this point, the formula of the unifying theory (equation [21]) sets the stage to obtain the novel re-sampling locations. Worth noting that this formula can be adapted to cover cases of one, two, and three dimensional interpolation functions and it is also consistently employed for linear quadratic cubic and trigonometric (Sinc) models. This shall be manifest throughout the remainder of the book. The remainder of this chapter discusses on the nature of the SRE, also makes a connection with Chapter XX of the book within the context of the relationship existing between resolution and interpolation error, and in the last section, the concept of resilient interpolation is introduced and the relevant math is illustrated for the case of the bivariate linear function.
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## Scope

The basic aim of the theory is that of determining the mathematical characterization of the relationship existing between misplacement and pixel intensity at the re-sampling location and those neighboring pixel intensities. This is done in the Section II of the book for the bivariate linear interpolation function. This characterization should be dependent on the local properties of the interpolation function and independent from any constant shifts and also dependent on the local properties of the discrete signal.

## Complete Chapter List

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Dedication
Foreword
Reda Abraham
Acknowledgment
On the Philosophical Basis Underlying Unifying Theory
Chapter 1
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Chapter 2
The Intuition  (pages 23-30)
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Chapter 6
The Notion  (pages 52-56)
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Chapter 7
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Chapter 8
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Chapter 9
Interpolation Procedures  (pages 172-179)
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Chapter 10
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Chapter 11
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Chapter 15
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Chapter 16
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Chapter 17
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Chapter 18
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Chapter 19
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Chapter 20
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Appendix Section