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

Author(s):  
Carlo Ciulla

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.

Author(s):  
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 proposed novelty. The novelty is discussed within the context of the current literature with specific attention to other works devoted to the improvement of the interpolation error. The reader is acknowledged that the methodological approach outlined through the theory can be seen as a viable pathway to follow in order to conceptualize interpolation in an innovative and alternative manner. This descends from the adoption of the mathematical formulation that is dependent on the joint information content of node intensity and curvature of the interpolation function and has brought to the determination of a viable option to adopt when re-sampling the signal (image). Re-sampling inherent to interpolation can be performed at sub-pixel locations that are not necessarily the same as the given misplacement neither are they necessarily the same pixel-by-pixel. This is because of the variability of pixel intensity and curvature of the interpolation function at the neighborhood and between neighborhoods and also because such variability corresponds to various signal shape characteristics. The reader is informed that within the context of a paradigm to be used for the improvement of the interpolation error it is of relevance to include the curvature in the methodology that is chosen to improve the approximation capability of a given interpolation function. The focus is also towards the evidence that local re-sampling is capable of changing the band-pass filtering property of the interpolation functions. Also, a study is undertaken to determine how beneficial is the application of the Sub-pixel Efficacy Region in the estimation of signals at unknown time-space locations and this is done for the one-dimensional interpolation functions presented in the book. It is also shown that the SRE-based interpolation functions are capable to determine error improvement and to be more accurate with respect to the classic interpolation functions in the estimation of signals at locations that are not captured by the sampling frequency because of the Nyquist’s theorem constraint. Also, based on the same data, the effect of the sampling resolution is studied on the interpolation error and the interpolation error improvement. Consequentially, it is outlined the licit conclusion that under the umbrella of the unifying framework proposed within the theory, the sampling resolution influences both interpolation error and interpolation error improvement obtained from the SRE-based functions. Finally the chapter reports on the investigation of the influence of the SCALE parameter and on the performance comparison across classic and SRE-based interpolation functions. The SCALE parameter is employed to scale the convolution of the pixel intensities determined through the polynomials forms: quadratic and cubic B-Splines, Lagrange, and also to scale the numerical values of the sums of cosine and sine functions of the Sinc interpolation function.


Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 767
Author(s):  
Alexandra Băicoianu ◽  
Cristina Maria Păcurar ◽  
Marius Păun

The present paper concretizes the models proposed by S. Ri and N. Secelean. S. Ri proposed the construction of the fractal interpolation function(FIF) considering finite systems consisting of Rakotch contractions, but produced no concretization of the model. N. Secelean considered countable systems of Banach contractions to produce the fractal interpolation function. Based on the abovementioned results, in this paper, we propose two different algorithms to produce the fractal interpolation functions both in the affine and non-affine cases. The theoretical context we were working in suppose a countable set of starting points and a countable system of Rakotch contractions. Due to the computational restrictions, the algorithms constructed in the applications have the weakness that they use a finite set of starting points and a finite system of Rakotch contractions. In this respect, the attractor obtained is a two-step approximation. The large number of points used in the computations and the graphical results lead us to the conclusion that the attractor obtained is a good approximation of the fractal interpolation function in both cases, affine and non-affine FIFs. In this way, we also provide a concretization of the scheme presented by C.M. Păcurar .


Fractals ◽  
2005 ◽  
Vol 13 (01) ◽  
pp. 33-41 ◽  
Author(s):  
ZHIGANG FENG ◽  
LIXIN TIAN ◽  
JIANLI JIAO

Fractal interpolation function (FIF) is continuous on its interval of definition. As a special kind of continuous function, FIFs' integrations on various scales and Fourier transform are studied in this paper. All of them can be expressed by the parameters of the corresponding iterative function systems.


2011 ◽  
Vol 07 (03) ◽  
pp. 771-792
Author(s):  
ALMASA ODŽAK ◽  
LEJLA SMAJLOVIĆ

We prove that there exists an entire complex function of order one and finite exponential type that interpolates the Li coefficients λF(n) attached to a function F in the class [Formula: see text] that contains both the Selberg class of functions and (unconditionally) the class of all automorphic L-functions attached to irreducible, cuspidal, unitary representations of GL n(ℚ). We also prove that the interpolation function is (essentially) unique, under generalized Riemann hypothesis. Furthermore, we obtain entire functions of order one and finite exponential type that interpolate both archimedean and non-archimedean contribution to λF(n) and show that those functions can be interpreted as zeta functions built, respectively, over trivial zeros and all zeros of a function [Formula: see text].


Author(s):  
Linxia Gu ◽  
Ashok V. Kumar

A method is presented for the solution of Poisson’s Equations using a Lagrangian formulation. The interpolation functions are the Lagrangian operation of those used in the classical finite element method, which automatically satisfy boundary conditions exactly even though there are no nodes on the boundaries of the domain. The integration is introduced in an implicit way by using approximated step functions. Classical surface integration terms used in the weak form are unnecessary due to the interpolation function in the Lagrangian formulation. Furthermore, the Lagrangian formulation simplified the connection between the mesh and the solid structures, thus providing a very easy way to solve the problems without a conforming mesh.


Author(s):  
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 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.


Author(s):  
Carlo Ciulla

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).


Author(s):  
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 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.


Fractals ◽  
2003 ◽  
Vol 11 (01) ◽  
pp. 1-7 ◽  
Author(s):  
M. ANTONIA NAVASCUÉS ◽  
M. VICTORIA SEBASTIÁN

Fractal interpolation functions (FIFs) provide new methods of approximation of experimental data. In the present paper, a fractal technique generalizing cubic spline functions is proposed. A FIF f is defined as the fixed point of a map between spaces of functions. The properties of this correspondence allow to deduce some inequalities that express the sensitivity of these functions and their derivatives to those changes in the parameters defining them. Under some hypotheses on the original function, bounds of the interpolation error for f, f′ and f′′ are obtained. As a consequence, the uniform convergence to the original function and its derivative as the interpolation step tends to zero is proved. According to these results, it is possible to approximate, with arbitrary accuracy, a smooth function and its derivatives by using a cubic spline fractal interpolation function (SFIF).


2019 ◽  
Vol 4 (10) ◽  
pp. 42-51
Author(s):  
Abdarrhim Mohammed Ahmed ◽  
Saleh Y. Barony

As Classical and Spline finite strip method based on stiffness and mixed variational formulation principle become important tool for continuum structural analysis , especially in the field of plate bending problems , a lot of researches has been focused on interpolation functions in order to improve the efficiency and increase the reliability of the method. The main objective of this paper is to introduce and propose a new spline interpolation function in the light of combination techniques of basic splines through introduction and brief review of previous studies in this field. This work which uses abbreviated form of augmented matrix proposed by authors published in past time , reveals a very good accordance results compared with the analytical and published solutions of different plate bending problems.


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