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Ciulla, Carlo. "The Conception of the Intensity-Curvature Functional." Improved Signal and Image Interpolation in Biomedical Applications: The Case of Magnetic Resonance Imaging (MRI). IGI Global, 2009. 31-39. Web. 17 Jan. 2017. doi:10.4018/978-1-60566-202-2.ch003

APA

Ciulla, C. (2009). The Conception of the Intensity-Curvature Functional. In C. Ciulla (Ed.), Improved Signal and Image Interpolation in Biomedical Applications: The Case of Magnetic Resonance Imaging (MRI) (pp. 31-39). Hershey, PA: IGI Global. doi:10.4018/978-1-60566-202-2.ch003

Chicago

Ciulla, Carlo. "The Conception of the Intensity-Curvature Functional." In Improved Signal and Image Interpolation in Biomedical Applications: The Case of Magnetic Resonance Imaging (MRI), ed. Carlo Ciulla, 31-39 (2009), accessed January 17, 2017. doi:10.4018/978-1-60566-202-2.ch003

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Abstract

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 voxel (the three dimensional pixel) where interpolation is most beneficial because it is meant to produce the least approximation of the true intensity value and this region has been named: “Sub-pixel Efficacy Region” (SRE). An energy function will be defined here as the ratio between the energy of the original image and the energy of the interpolated image. This ratio which is called the Intensity-Curvature Functional, is symbolized by the expression ?E = Eo / EIN, and it is prone to be studied to reveal its behavior within the voxel, and it is prone to determine the boundary of the Sub-pixel Efficacy Region within the voxel. In this chapter the Intensity-Curvature Functional will be treated for what concerns the trivariate liner interpolation function such to present its original conception. An application of the Intensity-Curvature Functional for the improvement of the trivariate liner interpolation function was however previously reported (Ciulla & Deek, 2005).

The formulation of the interpolation function h(x, y, z) was given in definition IV of Chapter II. The curvature consists of the second order derivatives of h(x, y, z): ¶^{2} (h(x, y, z))/¶x¶y, ¶^{2}(h (x, y, z))/¶y¶z, and ¶^{2}(h (x, y, z))/¶z¶x. For a voxel, let the energy of the interpolated image be defined as:E_{IN} = E_{IN}(x, y, z) = E_{IN} (Ψ_{xy}) + E_{IN} (Ψ_{zx}) + E_{IN} (Ψ_{yz}) (1) where:^{x/2 y/2 z/2}E_{IN} (Ψ_{xy}) = ∫ ∫ ∫ h(x, y, z) (¶^{2} (h(x, y, z)) /¶x¶y) dx dy dz (2)^{-x/2 -y/2 -z/2}^{x/2 y/2 z/2}E_{IN} (Ψ_{zx}) = ∫ ∫ ∫ h(x, y, z) (¶^{2} (h(x, y, z)) /¶z¶x) dx dy dz (3)^{-x/2 -y/2 -z/2}^{x/2 y/2 z/2}E_{IN} (Ψ_{yz}) = ∫ ∫ ∫ h(x, y, z) (¶^{2} (h(x, y, z)) /¶y¶z) dx dy dz (4)^{-x/2 -y/2 -z/2}For the original image, let the energy be defined as:

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...
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Magnetic Resonance Imaging and the Signal-Image Processing Techniques Developed Under the Umbrella of the Unifying Theory

This chapter reports on the initial idea which gave birth to the investigation which subsequently became the unifying theory. The intuition herein...
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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...
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The Conception of the Intensity-Curvature Functional

This chapter presents the mathematical deduction of the Sub-pixel Efficacy Region (SRE) for the case of the trivariate liner interpolation function....
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This chapter continues the investigation conducted through deduction from the previous three chapters. Chapter II of the book has brought to light...
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The goal of this chapter is to discuss the practical implications of the notion. This chapter concludes the logical reasoning outlined through the...
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In the sections of this chapter the reader will be introduced to the sequence of mathematical processes which, starting from a model interpolation...
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The Theoretical Approach to the Improvement of the Interpolation Error: Bivariate Linear Interpolation Function

The intent of the present chapter is to expand the treatise that concerns with the presentation of the current literature in signal-image...
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The organization of this chapter is similar to that of Chapter VII where it was outlined the theoretical approach to the improvement of the...
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The Extension of the Theory to the Trivariate Linear Interpolation Function

This chapter presents results relevant to the evaluation of the performance of classic and SRE-based trivariate interpolation functions. The...
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The Results of the Sub-Pixel Efficacy Region Based Trivariate Linear Interpolation Function

As well accepted in literature it should be almost impossible to reconstruct the signal intensity value at time-space location where the sampling...
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Equating the Two Intensity-Curvature Terms, Before and After Interpolation, Attempting to Obtain Resilient Interpolation: Trivariate Linear Interpolation Function

This chapter reviews the extensive and comprehensive literature on B-Splines. In the forthcoming text emphasis is given to hierarchy and formal...
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On the Literature of B-Spline Interpolation Functions

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...
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The Extension of Theory and Methodology to B-Splines

The results obtained processing the MRI database with classic and SRE-based one dimensional quadratic and cubic B-Splines are presented in this...
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The Results of the Sub-Pixel Efficacy Region Based B-Spline Interpolation Functions

Concepts of efficiency, approximation, and efficacy are discussed in this chapter while referencing the existing literature. Frameworks grouping...
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On the Properties of the Unifying Theory and the Derived Sub-Pixel Efficacy Region

This chapter introduces the reader to Section V of the book. The chapter opens up with a discussion on the undeniable evidence reported in...
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The Main Innovation Determined By the Sub-Pixel Efficacy Region

This chapter is devoted to the mathematics of the Lagrange and Sinc SRE-based interpolation functions. The organization of the text of this chapter...
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The Unifying Theory Embraces Lagrange and Sinc Interpolation Functions

The last chapter of the book reports on concluding remarks recalling to the reader the message given through these works and also recalling the...
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On the Implications of the Sub-Pixel Efficacy Region and the Bridging Concept of the Unifying Theory