Preface

Linear canonical transforms (LCTs) are a three-parameter family of linear integral transformations, which have a quadratic-phase kernel. For this reason, they have also been called quadratic-phase transforms or quadratic-phase systems (as well as other names). They are unitary transforms that correspond to linear, area-preserving distortions in phase space, a fact which underlies certain invariance properties. Combinations of LCTs are again LCTs. The family includes important operations or transforms such as chirp multiplication, chirp convolution (Fresnel transforms), fractional Fourier transforms, and of course the ordinary Fourier transform, as special cases. Arbitrary LCTs can be written as combinations of these simpler transforms. This leads to fast algorithms for approximately calculating LCTs, much as the ordinary Fourier transform can be calculated with fast algorithms. LCTs have been rediscovered many times in different contexts, a fact we consider evidence of their ubiquity. Their significance in optics was recognized at least as early as the 1970s. Later, interest in the fractional Fourier transform during the 1990s led to renewed interest in LCTs from new perspectives. This book deals with LCTs primarily from the perspective of signal and image processing, and optical information processing. Part I presents the mathematical theory of LCTs in the style of signal theory and analysis, as well as the foundations of how LCTs are related to optical systems. Part II deals with issues of degrees of freedom, sampling, numerical implementation, and fast algorithms. Part III is a survey of various applications. No attempt is made here to discuss canonical transformations as they appear in classical Hamiltonian mechanics and symplectomorphisms. These are well-established subjects in physics. However, we note that it is quite possible that a crossover of concepts and techniques between the different approaches to these transforms may be quite fruitful, and we hope this book may contribute to that end, in addition to being useful for its primary audience in the areas of signal processing and optics.