Riesz transforms in statistical signal processing and their applications to speckle metrology: a review <s> INTRODUCTION <s> The paper extends to three dimensions (3-D) the two‐dimensional (2-D) Hilbert transform relations between potential field components. For the 3-D case, it is shown that the Hilbert transform is composed of two parts, with one part acting on the X component and one part on the Y component. As for the previously developed 2-D case, it is shown that in 3-D the vertical and horizontal derivatives are the Hilbert transforms of each other. The 2-D Cauchy‐Riemann relations between a potential function and its Hilbert transform are generalized for the 3-D case. Finally, the previously developed concept of analytic signal in 2-D can be extended to 3-D as a first step toward the development of an automatic interpretation technique for potential field data. <s> BIB001 </s> Riesz transforms in statistical signal processing and their applications to speckle metrology: a review <s> INTRODUCTION <s> An extension of the notion of the analytical signal to multidimensional signals is presented. The real and imaginary parts of this signal are a linear combination of the original signal and of its complete and partial Hilbert transforms. Its Fourier image exists only in one orthant of the multidimensional frequency space. The orthant is a half-axis in one dimension, a quadrant in two dimensions, an octant in three dimensions, etc. A multidimensional complex signal makes it possible to introduce the definitions of instantaneous amplitude, instantaneous phase, and partial instantaneous complex frequencies (partial derivatives of the instantaneous phase) and to formulate a modulation theory of a multidimensional harmonic carrier. The 2-D equivalent of 1-D single-sideband modulation is defined and called single quadrant modulation. It is shown that the multidimensional complex signal with a signal orthant spectrum may be defined as a convolution of the real signal with the multidimensional complex delta distribution. > <s> BIB002 </s> Riesz transforms in statistical signal processing and their applications to speckle metrology: a review <s> INTRODUCTION <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> BIB003 </s> Riesz transforms in statistical signal processing and their applications to speckle metrology: a review <s> INTRODUCTION <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> BIB004 </s> Riesz transforms in statistical signal processing and their applications to speckle metrology: a review <s> INTRODUCTION <s> As an alternative to correlation-based techniques widely used in conventional speckle metrology, we propose a new technique that makes use of phase singularities in the complex analytic signal of a speckle pattern as indicators of local speckle displacements. The complex analytic signal is generated by vortex filtering the speckle pattern. Experimental results are presented that demonstrate the validity and the performance of the proposed optical vortex metrology with nano-scale resolution. <s> BIB005 </s> Riesz transforms in statistical signal processing and their applications to speckle metrology: a review <s> INTRODUCTION <s> A new technique for displacement measurement is proposed that makes use of phase singularities in the complex signal generated by a Laguerre-Gauss filter operation applied to a speckle pattern. The core structures of phase singularities are used as unique fingerprints attached to the object surface, and the displacement is determined by tracing the movement of registered phase singularities with their correspondence being identified by the fingerprints. Experimental results for translational and rotational displacement measurements are presented that demonstrate large dynamic range and high spatial resolution of the proposed optical vortex metrology. <s> BIB006 </s> Riesz transforms in statistical signal processing and their applications to speckle metrology: a review <s> INTRODUCTION <s> A new technique for fluid mechanics measurement is proposed that makes use of the elliptic anisotropy of phase singularities in the complex signal representation of a speckle-like pattern. Based on the formal analogy between the polarization field of a vector wave and the gradient field of the complex signal, the Poincare sphere representation has been used to characterize the phase singularities that serve as unique fingerprints attached to the seeding particles moving with the flow. Experimental results for flow velocity and acceleration measurement are presented that demonstrate the validity of the proposed optical vortex metrology for fluid mechanics measurement. <s> BIB007 </s> Riesz transforms in statistical signal processing and their applications to speckle metrology: a review <s> INTRODUCTION <s> A novel technique for biological kinematic analysis is proposed that makes use of the pseudophase singularities in a complex signal generated from a speckle-like pattern. In addition to the information about the locations and the anisotropic core structures of the pseudophase singularities, we also detect the spatial structures of a cluster of phase singularities, which serves as a unique constellation characterizing the mutual position relation between the individual pseudophase singularities. Experimental results of in vivo measurements for a swimming fish along with its kinematic analysis are presented, which demonstrate the validity of the proposed technique. <s> BIB008 </s> Riesz transforms in statistical signal processing and their applications to speckle metrology: a review <s> INTRODUCTION <s> : As an improvement of the intensity correlation used widely in conventional electronic speckle photography, we propose a new technique for displacement measurement based on correlating Stokes-like parameters derivatives for transformed speckle patterns. The method is based on a Riesz transform of the intensity speckle pattern, which converts the original real-valued signal into a complex signal. In closest analogy to the polarisation of a vector wave, the Stokes-like vector constructed from the spatial derivative of the generated complex signal has been applied for correlation. Experimental results are presented that demonstrate the validity and advantage of the proposed pseudo-Stokes vector correlation technique over conventional intensity correlation technique. <s> BIB009 </s> Riesz transforms in statistical signal processing and their applications to speckle metrology: a review <s> INTRODUCTION <s> In one-dimensional (1D) signal analysis, the complex analytic signal built from a real-valued signal and its Hilbert transform is an important tool providing a mathematical foundation for 1D statistical analysis. For a natural extension beyond 1D signal, Riesz transform has been applied to high-dimensional signal processing as a generalized Hilbert transform to construct a vector signal representation and therefore, to enlarge the traditional analytic signal concept. In this paper, we introduce the vector correlations as new mathematical tools for vector calculus for statistical speckle analysis. Based on vector correlations of a real-valued speckle pattern, we present the associated correlation properties, which can be regarded as mathematical foundation for the vector analysis in speckle metrology. <s> BIB010
In physics and engineering, it's common practice and usual treatment to represent a real-valued signal by its related complex-valued signal. The use of the associated complex signal simplifies calculations, especially when averages of quadratic quantities in the field variable have been involved. Meanwhile, the introduced instantaneous frequency, envelop and phase in this complex signal also provide more convenient means for analyzing, processing and understanding the available information from the original real-valued signal. Since the introduction of the analytic signal built from the Hilbert transform for a real one-dimensional (1-D) signal by Gabor in 1946 1 , the complex signal representation has become an important tool in 1-D signal processing. It is well-known that an analytic signal consists of two parts: the real part is the original signal and the imaginary part is the Hilbert transform of the real part [2] [3] BIB003 . Although the concept of analytic signal underpins many aspects of modern 1-D signal processing, the Hilbert transform has restricted its application of the analytic signal concept to multiple dimensions due to its inherent preferred directions  BIB002 BIB004  . In the areas of optics, image analysis and visual perception, many attempts have been made to extend the Hilbert transform beyond one dimension naturally without implying a preferred direction  BIB002 BIB004  BIB001 . Among all these attempts with proposed filters and generated signals, the signal obtained from Riesz transform has become the most promising one as the attempt for the natural extension of concept of 1-D analytic signal    . Although Riesz transforms have found successes in many applications such as fringe analysis  , pattern recognition  BIB004 and optical metrology BIB005 BIB006 BIB007 BIB008 BIB009 , little work has been done to apply such transforms to statistically analyze random signal like speckle patterns. No knowledge has been obtained about the statistical properties of the associated signals when the Riesz transform is applied. In this paper, we will review the Riesz transforms in statistical signal processing and some applications to speckle metrology. Through the application of Riesz Transform to a 2-D speckle pattern as a typical random signal, the corresponding vector representation has been established in Cartesian and spherical coordinate systems as the natural generalization of 1-D analytic signal. Some correlation properties associated with the vector representation of a real-valued 2-D speckle pattern have been presented, which share the formal analogy to the statistics of the 1-D analytic signal and may serve as a mathematical foundation for the statistical analysis of 2-D signal BIB010 . Under the framework of the vector signal representation generated by Riesz transforms, the principle and applications of several techniques, such as Optical Vortex Metrology BIB005 BIB006 BIB007 BIB008 and Pseduo-Stokes vector correlation