Riesz transforms in statistical signal processing and their applications to speckle metrology: a review <s> Optical vortex metrology <s> The singularities of complex scalar waves are their zeros; these are dislocation lines in space, or points in the plane. For waves in space, and waves in the plane (propagating in two dimensions, or sections of waves propagating in three), we calculate some statistics associated with dislocations for isotropically random Gaussian ensembles, that is, superpositions of plane waves equidistributed in direction but with random phases. The statistics are: mean length of dislocation line per unit volume, and the associated mean density of dislocation points in the plane; eccentricity of the ellipse describing the anisotropic squeezing of phase lines close to dislocation cores; distribution of curvature of dislocation lines in space; distribution of transverse speeds of moving dislocations; and position correlations of pairs of dislocations in the plane, with and without their strength (topological charge) ±1. The statistics depend on the frequency spectrum of the waves. We derive results for general spectra, and specialize to monochromatic waves in space and the plane, and black–body radiation. <s> BIB001 </s> Riesz transforms in statistical signal processing and their applications to speckle metrology: a review <s> Optical vortex metrology <s> We present the first direct experimental evidence of the local properties of optical vortices in a random laser speckle field. We have observed the Berry anisotropy ellipse describing the anisotropic squeezing of phase lines close to vortex cores and quantitatively verified the Dennis angular momentum rule for its phase. Some statistics associated with vortices, such as density, anisotropy ellipse eccentricity, and its relation to zero crossings of real and imaginary parts of the random field, are also investigated by experiments. <s> BIB002 </s> Riesz transforms in statistical signal processing and their applications to speckle metrology: a review <s> Optical vortex metrology <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> BIB003 </s> Riesz transforms in statistical signal processing and their applications to speckle metrology: a review <s> Optical vortex metrology <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> BIB004 </s> Riesz transforms in statistical signal processing and their applications to speckle metrology: a review <s> Optical vortex metrology <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> BIB005
Just as a random speckle intensity pattern imprints marks on a coherently illuminated object surface, randomly distributed singularities in the map of azimuthal angle  of the speckle-like pattern imprint unique marks on the object surface. We make use of the information about the locations of these singularities before and after the displacement. To do this, we need to identify the corresponding singularities between the pre-and post-displacement azimuthal angle maps. If the displacement is known to be small a priori, we can restrict our search only to the closest neighbor singularities of the same topological charge. However, when the displacement is large and/or non-uniform and no information is given a priori, we cannot uniquely identify the corresponding singularities. To solve this problem, we make use of additional information about the core structure of the singularities in the azimuthal angle map. Similarly to optical vortices in random laser speckle fields BIB001 BIB002 , the changes of the azimuthal angle around the singularities are nonuniform, and the typical core structure around the singularities are strongly anisotropic. Fig. 4(a) , are invariant to the in-plane rigid-body motion of the object involving translation and rotation, and we use these two geometric parameters to describe the local properties of the phase singularities. In addition, each singularity has its own topological charge and vorticity defined by , which we also assume to be invariant to the in-plane rigid-body displacement involving translation and rotation. Just as no fingers have exactly the same fingerprint patterns, no singularities have exactly the same local properties with identical eccentricity e, zero-crossing angle θ RI , topological charge q, and vorticity  . It is this uniqueness of the core structure that enables the correct identification and the tracking of the complicated movements of singularities. The matching between the pre-and post-displacement singularities are done on the basis of a merit function representing the degree of similarity between these parameters expressed by a distance in their parameter space. To measure lateral displacements in the micrometer range and rotational micro-displacements, we introduced controllable lateral and rotational micro-displacements with the precision mechanical stage of the microscope. From the recorded random texture patterns on the surface of the precision stage, we generated a vector signal representation by Riesz transform and retrieved the singularities information from the azimuthal angle map. In the experiment, we adjusted the average speckle size and controlled the density of singularities carefully by applying an additional low pass fileter with a proper bandwidth in the Fourier frequency domain, so that a single speckle includes about 40 pixels along a traversing line. After identifying the corresponding singularities for the object before and after displacement, we measured the given displacement by the proposed optical vortex metrology. Figure 5 (a) shows the parallel movement of singularities in the azimuthal angle map caused by the lateral displacement introduced by the translation stage. The displacements in the x and y directions were found to be ) . Figure 5 (b) shows the rotational movement of singularities caused by the rotational displacement introduced by the rotation stage. The rotation angle was found to be −518milli-radians with the standard deviation 2milli-radians. We have also demonstrated optical vortex metrology for displacement measurement with nanometric resolution BIB003 BIB004 , and for random flow measurements In addition to the information about the anisotropic core structure for an individual singularity in azimuthal angle map as its unique fingerprint for reliable identification, we detected a group of optical vortices (singularity clusters) with specific mutual spatial structure to further strengthen an unambiguous tracking over the entire field of view BIB005 . To demonstrate the performance of the proposed technique, a swimming fugu fish with a speckle-like intensity pattern on its body surface, as seen in Fig.6 . From the recorded images, we generated a vector signal by Riesz transform, and retrieved the azimuthal angle information. Similarly, we adjusted the average speckle size by choosing a proper bandwidth of the low pass filter when Riesz transforms were calculated through Fourier transform. In the experiment, we have controlled the density of the singularities carefully so that the singularities within the fugu body are clearly separated with a large average separation distance. Then, the constellation of singularities in azimuthal angle of the generated vector signal was identified and tracked during the specimen's motion across the entire field of view of the camera. After identifying corresponding singularities for each pair of consecutive images making use of their core structures as fingerprints, we traced the movement of the swimming fugu through its trajectory as shown in Fig. 6 (c) and (d) . From the coordinate information for each singularity in the constellation, we conducted the in vivo measurement and obtain instant information for this swimming fugu about its translation, rotation, and scaling 14 . . AS . w tg: l. w * _i .. We conducted experiments to demonstrate the validity of the proposed technique. We generate the laser speckle pattern by illuminating an object with rough surface by He-Ne laser. The generated speckle pattern was imaged with a lens onto a CCD image sensor. By taking two images of the object before and after displacement, we measured the microdisplacement by the proposed pseudo Stokes vector correlation technique. To make the comparison on equal grounds, we calculated the correlation with a sub-window size of 100x100 pixels, and the shapes of the correlation functions for intensity correlation and PSVC are illustrated in Fig. 7(a) . For both techniques, a main correlation peak can be observed at the coordinate 13   x pixels, which indicates the displacement of the object. Near the main peak, we can also observe other small peaks, which can be a source of discrimination error in displacement measurement. The ratio of the main peak to its highest side-lobe can serve as a reliability measurement of the technique used. In this measure, the ratios of heights are 6.48 for IC and 7.02 for PSVC, respectively. By decreasing the size for the correlation window, we tested the performance of these two correlation techniques as shown in Fig. 7(b) . As expected, the side lobes of the normalized correlation function increase with decreasing the correlation area. Unlike a small ratio equal to 1.62 given by the conventional IC technique, the proposed PSVC gives a very large ratio with 4.74, which indicates a strong robustness against decorrelation. In this case, a small sub-window of 30x30 pixels has been chosen for both methods. Figure 1 demonstrates the validity and advantage of the proposed pseudo Stokes vector correlation technique as compared with conventional intensity correlation. (a) (b) Figure 7 . Distributions of correlation functions with different sub-window size (a) 100x100 pixels (b) 30x30 pixels