The intent of the present chapter is to expand the treatise that concerns with the presentation of the current literature in signal-image interpolation with specific focus on the classification of the procedures and the relevant applications of interpolation, in relationship to the various scientific disciplines. Also, the present chapter expands on the importance of the error bounds existing in literature for the characterization and quantification of the interpolation error. The discussion is interleaved with the characteristics features of the unifying theory that the book presents such to place the SRE-based interpolation functions within the context of the larger framework that the scientific literature has conceived so far.
Classification And Applications Of The Interpolation Procedures
Signal-image interpolation is normally used for several procedures, and they are hereto listed.
When pixel-by-pixel correspondence has to be established for the purpose of matching images collected at different times using the same imaging modality (registration) or different imaging modalities (co-registration), interpolation finds immediate application in motion correction techniques.
When signals are to be reconstructed with increased resolution (up-sampling) or decreased resolution (down-sampling).
For the estimation of signals at space-time locations where the signal is unknown because of the limitation imposed by the sampling frequency of the collecting equipment.
Other relevant procedures are: (i) estimation of signal parameters corresponding to particular frequencies of the spectrum through the use of linear interpolation (Borkowski, 2000), and (ii) enhancement of grayscale images and reconstruction of color images through combined use of covariance coefficients and linear interpolation (Li & Orchard, 2001).
Since interpolation descends from approximation theory, it can also be grouped with extrapolation and regression as shown in an early example of a unifying methodology of linear unbiased estimations (Chow & Lin, 1971).
Interpolation techniques currently in use are classified in two main categories: scene-based interpolation and object-based interpolation (Penney et al., 2004). Scene-based interpolation uses the value of the intensity at the pixels (nodes) at the aim to reconstruct the signal. In order to achieve such purpose, linear, quadratic, cubic, Lagrange and Sinc interpolation paradigms are those among the most commonly employed. Object-based interpolation uses intrinsic features captured from the signal to determine the reconstructed signal. This one category has recently extended to morphology-based methods (Lee & Wang, 2000) and feature guidance based methods (Lee & Lin, 2002).
Object-based interpolation was initiated (Raya & Udupa, 1990) evaluated and extended (Herman et al., 1992), and then reinforced (Grevera & Udupa, 1996, 1998). These authors showed that linear interpolation of a surface representing the distance map of the pixels from the boundary of the object in the scene proves superior to simple linear interpolation of the pixel intensities. The aforementioned authors have also reported a statistical comparison of their shape-based method, and this was developed within the three-dimensional context against a series of other methods including the classic trivariate linear interpolation.
Recent applications of interpolation procedures devote attention to a large variety of applications and paradigms for the estimation of missing data. Hereto is given a list of works that have been brought just recently to the attention of the scientific community pointing to the diversity of the approaches and the applications in science.
Li and Orchard (2001) devised an improved paradigm of the bivariate linear function through the application of the adaptive covariance based approach with applications to grey scale image resolution enhancement. Varadarajan and Krolik (2003) have employed interpolation for the improvement of radar detection. Angelini et al. (2003) employed interpolation to estimate the factors of the multivariate analysis in econometric projections and in the work of these authors, the interpolation paradigm is linked indirectly to the signal through the factors of the multivariate analysis at the aim to estimate missing data in very large datasets.