# The Extension of Theory and Methodology to B-Splines

Carlo Ciulla (Lane College, USA)
DOI: 10.4018/978-1-60566-202-2.ch014

## Abstract

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 dimensional quadratic and cubic B-Splines is herein reported along with the most relevant mathematical details. This chapter should be read along with Appendix VI where proofs are given to the assertions herein presented. In either of the two cases: quadratic and cubic B-Spline the math process starts from the calculation of the Intensity-Curvature Functional and continues with the calculation of the Sub-pixel Efficacy Region. Finally, the math process arrives to the calculation of the novel re-sampling locations through the formulas of the unifying theory seen in equations (23) and (33) for the quadratic and the cubic models respectively. The chapter concludes with a section that addresses specifically the theoretical proposition of resilient interpolation for the two classes of B-Splines. This is conducted consistently with Chapters VII and XII of the book choosing to equate the two intensity-curvature terms (before and after interpolation) as the starting point of the math deduction.
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Calculations were developed based on the B-Spline forms: quadratic (h3(x)), and cubic (h4(x)) as per equations (1) and (2), with 1 x 6 and 1 x 6 neighbourhoods’ nodes (see equations (3) and (4)).

- 2a | x |2 + 1/2 (a+1) 0 ≤ | x | ≤ 1/2h3(x) = (1) a | x |2 - (2a + 1/2) | x | + 3/4 (a+1) 1/2 ≤ | x | ≤ 3/21/2 | x |3 - | x |2 + 2/3 0 ≤ | x | ≤ 1 h4(x) = (2) -1/6 | x |3 + | x |2 - 2 | x | + 4/3 1 ≤ | x | ≤ 2

The absolute value of x assumes the meaning of representing intra-nodal misplacement within the pixel or within two nodes. It is assumed that the origin of the coordinate system of the pixel or the intra-nodal distance (sampling step) is at the corner and not at the center. Therefore the value of the misplacement is never negative. For simplicity let us consider the two functions of equations (1) and (2) written as:

h3(x) = f (0) + [ f (1/2) +f (-1/2) ] * [- 2a x2 + 1/2 (a+1) ] + [ f (1/2) + f (-1/2) + f (-1) + f (1) +f (3/2) + f (-3/2) ] * [ a x2 - (2a + 1/2) x + 3/4 (a+1) ] (3) h4(x) = f (0) + [ f (1/2) + f (-1/2) + f (-1) + f (1) ] * [ 1/2 x3 - x2 + 2/3 ] + [f (3/2) + f (- 3/2) +f (1) +f (-1) +f (2) + f (-2)] * [ -1/6 x3 + x2 - 2 x + 4/3 ] (4)

## 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