Riesz transforms in statistical signal processing and their applications to speckle metrology: a review <s> Representation of a 2-D real-valued signal by a vector signal <s> Preface 1. Elements of probability theory 2. Random (or stochastic) processes 3. Some useful mathematical techniques 4. Second-order coherence theory of scalar wavefields 5. Radiation from sources of any state of coherence 6. Second-order coherence theory of vector electromagnetic fields 7. Some applications of second-order coherence theory 8. Higher-order correlations in optical fields 9. Semiclassical theory of photoelectric detection of light 10. Quantization of the free electromagnetic field 11. Coherent states of the electromagnetic field 12. Quantum correlations and photon statistics 13. Radiation from thermal equilibrium sources 14. Quantum theory of photoelectric detection of light 15. Interaction between light and a two-level atom 16. Collective atomic interactions 17. Some general techniques for treating interacting systems 18. The single-mode laser 19. The two-mode ring laser 20. The linnear light amplifier 21. Squeezed states of light 22. Some quantum effects in nonlinear optics References Author index Subject index. <s> BIB001 </s> Riesz transforms in statistical signal processing and their applications to speckle metrology: a review <s> Representation of a 2-D real-valued signal by a vector signal <s> The analytic signal is one of the most capable approaches in one-dimensional signal processing. Two-dimensional signal theory suffers from the absence of an isotropic extension of the analytic signal. Accepting the fact that there is no odd filter with isotropic energy in higher dimensions, one tried to circumvent this drawback using the one-dimensional quadrature Alters with respect to several preference directions. Disadvantages of these methods are an increased complexity, the loss of linearity and a lot of different heuristic approaches. In this paper we present a filter that is isotropic and odd, which means that the whole theory of local phase and amplitude can directly be applied to images. Additionally, a third local property is obtained which is the local orientation. The advantages of our approach are demonstrated by a stable orientation detection algorithm and an adaption of the phase congruency method which yields a superior edge detector with very low complexity. <s> BIB002
In 1-D signal processing, Hilbert transform and its associated analytic signal play critical roles and provide mathematical basis to describe statistical signal. Many interesting properties of analytic signal and its statistical behaviors as 1-D complex random process have been discussed and proposed [2] [3] BIB001 . To extend this important concept, Riesz transform has been introduced as a natural generalization to 2-D signal processing with its resulting 3-D vector representation  BIB002  . Consider a 2-D real-valued function ( , ) f x y , which might represent an intensity fluctuation of an image at point ( , ) xy. By a similar procedure used in monogenic signal  , the vector representation in space domain can be introduced as (2) ( f A x y k x y e k x y e x y e f x y  with their Fourier transforms . The vector representation . As we will see in the next section, Figure 3 shows an example of the vector representation of a speckle as a typical real 2-D signal generated from the Riesz transform, where is modulus of real vector. In this example, the image size of 450 pixels by 450 pixels has been chosen as the truncated speckle pattern with its average speckle width of 45 pixels.