On the Properties of the Unifying Theory and the Derived Sub-Pixel Efficacy Region

Concepts of efficiency, approximation, and efficacy are discussed in this chapter while referencing the existing literature. Frameworks grouping classes of interpolation functions are acknowledged. Also, properties of the unifying theory are addressed along with the characteristics of the methodology employed to derive the SRE-based interpolation functions. Properties of the unifying theory are also discussed and the reasoning pertaining to the methodology is provided. As it can be seen in this book, both properties and characteristics featuring the methodological development of the SRE-based interpolation function are quite consistent across interpolators and it is true that the empirical observations provided through this works have set the grounds for the legitimate assertion that there exist a unifying theory for the improvement of the interpolation error. Therefore, such consistency across the spectrum of interpolation functions embraced through the unifying theory shows that the concepts embedded in both properties and characteristics of the methodology employed to derive the SRE-based interpolation functions remain the same across the diversity of model interpolators. It is also due to acknowledge that Section V of the book is devoted to Lagrange and Sinc functions and will further expand on the consistency just mentioned. The last section of this chapter addresses the Fourier properties of the Sub-pixel Efficacy Region.

On the Literature of B-Spline Interpolation Functions

This chapter reviews the extensive and comprehensive literature on B-Splines. In the forthcoming text emphasis is given to hierarchy and formal definition of polynomial interpolation with specific focus to the subclass of functions that are called B-Splines. Also, the literature is reviewed with emphasis on methodologies and applications of B-Splines within a wide array of scientific disciplines. The review is conducted with the intent to inform the reader and also to acknowledge the merit of the scientific community for the great effort devoted to B-Splines. The chapter concludes emphasizing on the proposition that the unifying theory presented throughout this book has for what concerns two specific cases of B-Spline functions: univariate quadratic and cubic models.

The Conception of the Sub-Pixel Efficacy Region

This chapter presents the mathematical deduction of the Sub-pixel Efficacy Region (SRE) for the case of the trivariate liner interpolation function. The questions that the reader may have at this point are: (i) what is the need of defining the Sub-pixel efficacy Region once again? And (ii) why the Sub-pixel Efficacy Region is to be calculated on the basis of the Intensity-Curvature Functional?

The Main Innovation Determined By the 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 literature that the magnitude of the interpolation error is strictly related to the magnitude of the sampling resolution. While reference to the literature on the Lagrange interpolation function is reported elsewhere (Ciulla & Deek, 2006), the chapter devotes attention to the literature and the applications related to the Sinc function. The core of the chapter reports a section that condenses the message to the reader of this book about the main innovation determined through the Sub-pixel Efficacy Region. It is delivered to the reader the realization that combining signal intensity with the curvature of the interpolation function, the approximation properties of the model function can be improved. This message is linked to the bridging concept between classic and SRE-based interpolation which is that of the curvature of the interpolation function.

The Results of the Sub-Pixel Efficacy Region Based B-Spline Interpolation Functions

The results obtained processing the MRI database with classic and SRE-based one dimensional quadratic and cubic B-Splines are presented in this chapter. The chapter opens up with information relevant to the image resolution of the MRI database employed for validation. The assessment of the performance of the two classes of interpolators (classic and SRE-based) is conducted both quantitatively and qualitatively. The RSME Ratio is plotted to ascertain which ones of the classic or the SRE-based models deliver the smaller interpolation error. Also, the analysis of error images obtained after processing with either of the two model interpolators and the display of the maps of novel re-sampling locations along with spectral power evolutions corroborates the presentation of the characteristic features of the performances of the interpolation functions treated in this chapter.

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

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).

The Theoretical Approach to the Improvement of the Interpolation Error

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.

The Conception of the Intensity-Curvature Functional

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 Intuition

This chapter reports on the initial idea which gave birth to the investigation which subsequently became the unifying theory. The intuition herein illustrated consists of the realization that the following concept might have been supported by reasonable ground of truth after extensive study. The concept is that there exists a region of spatial extent within two nodes (1D), a pixel (2D), or a voxel (3D) where the interpolation function has best approximation properties. Naturally, the adjective best is to be interpreted with its relativity to the potentiality of the specific model interpolation function to determine approximation properties. Such potentiality according to the intuition resides in: (i) the sequel of discrete samples (e.g. the pixel intensities for the two-dimensional case), and (ii) the curvature of the model interpolator as expressed by its second order derivatives. The study in this chapter is initiated for the trivariate linear interpolation function and formalized through a set of definitions, an observation and a theorem.

The Unifying Theory Embraces Lagrange and Sinc Interpolation Functions

This chapter is devoted to the mathematics of the Lagrange and Sinc SRE-based interpolation functions. The organization of the text of this chapter is consistent with that of chapters VII, X, and XIV. The basic aim of this chapter is to employ the methodology outlined in the book such to develop a mathematical formulation that allows interpolation error improvement also for Lagrange and Sinc interpolation functions. This is achieved through two instruments that bridge classic interpolation with the present innovative theory. The instruments are the Intensity-Curvature Functional (?E) and the Sub-pixel Efficacy Region (SRE). Math processes are thus presented that start from the calculation of the intensitycurvature terms and the corresponding Intensity-Curvature Functional, determine the SRE and employ the formula of the unifying theory (see equations [16] and [38] for Lagrange and Sinc respectively) to calculate the novel re-sampling locations for the two model interpolation functions. A section of this chapter is delegated to recall to the reader the characterization of upper and lower bounds of interpolation error improvement and interpolation error respectively. Details of this section are reported elsewhere (Ciulla & Deek, 2006). Finally the theoretical presentation of resilient interpolation is extended also to Lagrange and Sinc as it was already presented for the two linear functions and the two B-Splines that were object of treatise in Parts II, III and IV of this book. Although the logic behind the math of resilient interpolation is explained and characterized, resilient interpolation remains in this book a theoretical conceptualization which looks forward to empirical confirmation.