The forthcoming text of this chapter is intended to raise the debate about the content of the book. To fulfill this purpose, is presented in first instance a section that summarizes on how the literature is reviewed throughout the book and also the MRI database employed to validate the unifying theory. Following this presentation, the chapter introduces to the reader the concept behind the basic issue relating to signal-image interpolation, which is the preservation of the signal (image) energy after processing with interpolation. This concept is strictly related to the approximation properties of the interpolation functions and consequently to the interpolation error. This concept is widely accepted in literature and is viewed in this works with a different and innovative perspective. The discussion undertaken in this chapter then introduces the signal-image processing techniques that were developed under the umbrella of the unifying theory. These are: (i) the SRE-based interpolation functions, (ii) the spectral power evolutions, (iii) a three layered artificial neural network with the added capability to generate its internal architecture during the learning process such to adjust to the given pattern classification task.
TopIntroduction
Since its inception (Lauterbur, 1973; Mansfield & Grannell, 1973) Magnetic Resonance Imaging (MRI) has produced a beneficial and revolutionary trend in several biomedical imaging related applications. Diagnostic imaging, among others, is one of the most relevant. Following the discovery of MRI and parallel to the widespread use of computer technology, an immense research literature has been produced on signal-image processing techniques devoted to Magnetic Resonance Imaging. Several text books exist in literature that explain the concepts behind the MRI physics and the concepts that drive the need and the types of signal-image processing techniques that were and are being developed (Cho et al., 1993; Haacke et al., 1999; Jezzard et al., 2001; Liang & Lauterbur, 2000).
Interpolation is indeed one of the most relevant signal processing applications related to MRI and also to any scientific discipline that requires the estimation of the signal at time-space locations where the signal is not known because of the limitations to sampling as imposed by the recording (imaging) equipment. This fact should provide with an immediate answer as to what is the relevance of the contents of this manuscript. The main objective of this book is to present a unifying theory, supported through empirical validation, for the improvement of the approximation properties of mathematical functions employed in signal-image interpolation to estimate the value of the signal at those space-time locations that are not sampled. Given the relevance of MRI, this book presents the validation of the unifying theory employing Magnetic Resonance Imaging signals (images).
The forthcoming text of this chapter is intended to raise the debate about the content of the book. To fulfill this purpose, is presented in first instance a section that summarizes on how the literature is reviewed throughout the book and also the MRI database employed to validate the unifying theory. Following this presentation, the chapter introduces to the reader the concept behind the basic issue relating to signal-image interpolation, which is the preservation of the signal (image) energy after processing with interpolation. This concept is strictly related to the approximation properties of the interpolation functions and consequently to the interpolation error. This concept is widely accepted in literature and is viewed in this works with a different and innovative perspective. The discussion undertaken in this chapter then introduces the signal-image processing techniques that were developed under the umbrella of the unifying theory. These are: (i) the SRE-based interpolation functions, (ii) the spectral power evolutions, (iii) a three layered artificial neural network with the added capability to generate its internal architecture during the learning process such to adjust to the given pattern classification task.
The chapter also introduces the very basic concept of the unifying theory which is the distinctive feature of the SRE-based interpolation functions: the local curvature of the function. The main implication of the theory is introduced as it relates to the possibility to calculate the re-sampling locations through a computational approach based on the combined information content of signal intensities and curvature (second order derivative) of the interpolation function. Finally the chapter gives a general overview on the implications of the unifying theory in signal-image processing and particularly in motion correction in functional MRI. The discussion highlights that the creation of signal-image artifacts, which are detrimental to signal reconstruction, can derive directly from the approximate nature of interpolation, but can be reduced through SRE-based interpolation.