Signal Resilient to Interpolation: The Mathematics: The Mathematics of the Signal Resilient to Interpolation

The calculation of the intensity-curvature terms is related to the discrete sample of signal values and is also related to the analog/continuous transformation necessary to the function to become an interpolator. Such calculation is undertaken within the entire spatial extent of the sampling location, and as shown in Ciulla (2009), leads to the measurement of the energy level change determined through the interpolation function which is called: the Intensity-Curvature Functional. The idea is therefore to equate the intensity-curvature term calculated with the signal (image) through the given mathematical function in two conditions: (i) the given signal as it has been sampled and (ii) the signal calculated at locations where is unknown because of the limitations of the sampling instrument. Through mathematical developments that illustrate concepts of algebra and calculus, the equation of the two intensity-curvature terms furnish the instrument apt for deriving a new signal. This new signal is dependent on the given model function and is called Signal Resilient to Interpolation (SRI). The book includes illustration of the math processes, logical reasoning, which show how to generate the Signal Resilient to Interpolation. Within the book, the Signal Resilient to Interpolation is derived on the basis of quadratic and cubic polynomials in one, two and three dimensions, embedding and not embedding the pixel (voxel in three dimensions) to be re-sampled. Once the signal is modeled through an interpolator, it is possible to calculate the second order derivatives, and thus the curvature of the model interpolator which is nonetheless the modeled representation of the curvature of the signal. Geometrically, the curvature of the signal is the tangent to the first order derivative curve of the signal. There are two types of curvature treated in this book and they are: (i) classic-curvature and (ii) resilient curvature.

CARLO CIULLA had been undergraduate and graduate student at the University of Palermo, Italy, RUTGERS University, U.S.A. and the New Jersey Institute of Technology, U.S.A. from the year 1987 to the year 2002. He has earned the following graduate degrees: Laurea in Management Engineering (Italy); an M.S. in Information Systems and a Ph.D. in Computer and Information Science (U.S.A.). Carlo was pre-doctoral student at the National Institute of Bioscience and Human Technology (NIBH) in Tsukuba, Japan (1995-1997) and he worked with Magnetoencephalography (MEG) studying the spontaneous alpha rhythm of the human brain. Following the completion of the Doctoral degree, Carlo's former academic appointments were: Research Associate at Yale University (2002-2003); Postdoctoral Scholar at the University of Iowa (2004-2005); Postdoctoral Scholar at Wayne State University (2005-2007); Assistant Professor of Computer Science at Lane College (2007-2009). During the years 2009-2012 Carlo is a self employed scholar whom devoted his time to his research interest related to the development of innovative methods of signal interpolation and also to the development of educational software for students. During the course of his career the research interests remain in the domain of mathematics in computational engineering: Artificial Neural Networks, Image Registration in fMRI, Signal-Image Interpolation, and MEG Alpha Rhythm. He has authored and co-authored numerous papers in journals and conference proceedings, and is the author of the books: (i) Improved Signal and Image Interpolation in Biomedical Applications: The Case of Magnetic Resonance Imaging (MRI); (ii) AUTOALIGN: Methodology and Technology for the Alignment of Functional Magnetic Resonance Imaging Time Series: Image Registration: The Case of Functional MRI; (iii) SIGNAL RESILIENT TO INTERPOLATION: An Exploration on the Approximation Properties of the Mathematical Functions; and (iv) Computer Science Signal Processing Applications in Higher Learning. In September 2012, Carlo had been hired at the University of Information Science and Technology in Ohrid, Republic of Macedonia, where he is assistant professor.