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.
Novel Conceptions And Limitations Of The Unifying Theory
Ciulla and Deek (2006) report a review of Lagrange polynomials for image processing applications (Francesconi et al., 1993; Fuchs & Delyon, 2002; Lehmann et al., 1999; Xie & Zhou, 2001; Ye, 2003). Similarities between Lagrange and Sinc interpolation functions were investigated in literature (Lehmann et al., 1999; Wittaker, 1915) along with an interesting study that presents work on the improvement of the stop-band characteristics of the Lagrange function (Dumpster & Murphy, 1999).
This book presents an improved version of the Lagrange interpolation and this is done through the extension of the theoretical framework herein outlined. As far as the more general description of benefits and limitations of the application of the SRE to the interpolation functions, the knowledge reported in the book shall be connected to the knowledge reported earlier (Ciulla & Deek, 2006).