Image processing with the radial Hilbert transform: theory and experiments

Real-time image edge enhancement with a spiral phase filter and graphic processing unit

Isotropic image edge enhancement with high contrast can be achieved using a spiral phase filter (SPF) in a 4f optical system. However, real-time application of edge enhancement with SPF has generally been limited due to the requirement of coherent light or complex phase-shifting operation. In this paper, we demonstrate a real-time image edge enhancement method using a SPF and a graphic processing unit (GPU). By implementing the process of virtual spiral phase filtering on GPU, we are able to speed up the whole procedure by more than 8.3× with respect to CPU processing, and ultimately achieve video rate for megapixel images. In particular, our implementation can achieve higher speedup for more multiple images. These developments are increasing the potential for image edge enhancement of moving objects.

Multi-step vortex filtering for phase extraction

A quantized version of a continuous spiral phase filter with unitary topological charge, here denominated multi-step spiral phase filter (MSSPF), is proposed to extract phase from rotated spiral interferograms. Spiral interferograms are usually obtained from phase objects by registering the interference of its vortex filtered complex amplitude with a reference complex amplitude. The structure found in this kind of interferograms, depend on the number of steps used in the MSSPF that usually are assumed with an infinite number of steps for the continuous spiral phase filter. Reducing the number of steps of the MSSPF affects the residual phase error obtained after the phase extraction method. This error is therefore analysed here using a numerical simulation of a Mach-Zender interferometer with a MSSPF and a reduced number of steps. It is shown that, for our proposed method of rotated spiral interferograms, a residual error persists as the number of steps is increased approaching the residual error reported for the phase extraction method of single-shot spiral interferograms. Furthermore, it is shown that this novel technique can be applied without further modifications for phase contrast measurement. Experimental results show similar performance of this phase extraction technique, when compared to the results obtained with a commercial interferometer and with the numerical simulations.

Radial Hilbert transform with the spatially variable half-wave plate

We analyzed the point spread function (PSF) of the typical 4f optical image processing system by use of a spatially variable half-wave plate as the spatial filter and found that the PSF is an elementary vector beam. Theoretical analysis and real experiments show that the optical system can be used for a radially symmetric Hilbert transform that permits two-dimensional edge enhancement as the spiral phase plate. This kind of radial Hilbert transform is useful for image processing because it can enhance the edges of an input image selectively by exerting a polarization analyzer before the output plane. The optical system also can be used for generation of vector beams with arbitrary array and shape in real time conveniently.

STED microscope with spiral phase contrast

Stimulated Emission Depletion (STED) microscopy enables superresolution imaging of fluorescently marked nano-structures in vivo. Biological investigations are often hindered by the difficulty of relating super-resolved structures to other non-labeled features. Here we demonstrate that the similarity in optical design of Spiral Phase Contrast (SPC) and STED microscopes allows straightforward implementation of a phase contrast channel into a STED microscope in widefield and scanning modes. This method allows dual imaging and overlay in two contrast modes in fixed and in living specimens, in which double labeling is especially challenging. Living GFP- and YPF-stained neurons are imaged in one label-free phase contrast and one high-resolution STED channel. Furthermore, we implement SPC in widefield and scanning modes demonstrating that scanning confocal SPC yields the highest optical contrast. The latter configuration can provide contour detection or highlights and shadows reminiscent of differential interference contrast.

High-quality optical vortex-beam generation by using a multilevel vortex-producing lens

In the present work, we propose a method to generate high-quality optical vortices with a reduced number of phase levels by using multilevel vortex-producing lenses (VPLs). The VPL is implemented in a liquid-crystal spatial light modulator with limited capacity to project phase levels. The proposed method significantly improves the quality of the optical vortex obtained by employing spiral phase plates with the same number of phase levels. Simulations and experimental results confirming the effectiveness of the method are presented.

Terahertz bandwidth photonic Hilbert transformers based on synthesized planar Bragg grating fabrication

Terahertz bandwidth photonic Hilbert transformers are proposed and experimentally demonstrated. The integrated device is fabricated via a direct UV grating writing technique in a silica-on-silicon platform. The photonic Hilbert transformer operates at bandwidths of up to 2 THz (~16 nm) in the telecom band, a 10-fold greater bandwidth than any previously reported experimental approaches. Achieving this performance requires detailed knowledge of the system transfer function of the direct UV grating writing technique; this allows improved linearity and yields terahertz bandwidth Bragg gratings with improved spectral quality. By incorporating a flat-top reflector and Hilbert grating with a waveguide coupler, an ultrawideband all-optical single-sideband filter is demonstrated.

Mapping of phase singularities with spiral phase contrast microscopy

In spiral phase contrast (SPC) microscopy the edge-enhancement is typically independent of the helicity of the phase vortex filter. Here we show that for layered specimens containing screw-dislocations, as are e.g. present in mica or some crystallized organic substances, the intensity distribution in the filtered image acquires a dependence on the rotational direction of the filter. This allows one to map the distribution of phase singularities in the topography of the sample, by taking the intensity difference between two images recorded with opposite handedness. For the demonstration of this feature in a microscopy set-up, we encode the vortex filter as a binary off-axis hologram displayed on a spatial light modulator (SLM) placed in a Fourier plane. Using a binary grating, the diffraction efficiencies for the plus and minus first diffraction orders are equal, giving rise to two image waves which travel in different directions and are Fourier filtered with opposite helicity. The corresponding two images can be recorded simultaneously in two separate regions of the camera chip. This enables mapping of dislocations in the sample in a single camera exposure, as was demonstrated for various transparent samples.

A multi-modal stereo microscope based on a spatial light modulator

Spatial Light Modulators (SLMs) can emulate the classic microscopy techniques, including differential interference (DIC) contrast and (spiral) phase contrast. Their programmability entails the benefit of flexibility or the option to multiplex images, for single-shot quantitative imaging or for simultaneous multi-plane imaging (depth-of-field multiplexing). We report the development of a microscope sharing many of the previously demonstrated capabilities, within a holographic implementation of a stereo microscope. Furthermore, we use the SLM to combine stereo microscopy with a refocusing filter and with a darkfield filter. The instrument is built around a custom inverted microscope and equipped with an SLM which gives various imaging modes laterally displaced on the same camera chip. In addition, there is a wide angle camera for visualisation of a larger region of the sample.

Fractional spiral zone plates

In this paper, we generalize the concept of classical spiral zone plates (SZPs) to fractional spiral zone plates (FSZPs). By using an SZP with a fractional topological charge and controlling the starting orientation, we can break down the symmetry of the focusing process to give orientation-selective anisotropic vortex foci. Numerical results show that its binary structure gives additional high-order foci on the optical axis and the intensities in the foci can be controlled by properly choosing the fractional topological charge. Our study reveals the feasibility to control the intensity in the foci by means of FSZPs.

Selective edge enhancement in three-dimensional vortex imaging with incoherent light

We demonstrate a new imaging method enabling a selective edge contrast enhancement of three-dimensional amplitude objects with spatially incoherent light. The imaging process is achieved in a spiral modification of Fresnel incoherent correlation holography and uses a vortex impulse response function. The correlation recordings of the object are acquired in a one-way interferometer with the wavefront division carried out by a spatial light modulator. Two different methods based on applying a helical reference wave in the hologram recording and a digital spiral phase modulation in image reconstruction are proposed for edge enhancement of amplitude objects. Results of both isotropic and anisotropic spiral imaging are demonstrated in experiments using an LED as an incoherent source of light.

Selective edge enhancement using anisotropic vortex filter

In optical image processing, selective edge enhancement is important when it is preferable to emphasize some edges of an object more than others. We propose a new method for selective edge enhancement of amplitude objects using the anisotropic vortex phase mask by introducing anisotropy in a conventional vortex mask with the help of the sine function. The anisotropy is capable of edge enhancement in the selective region and in the required direction by changing the power and offset angle, respectively, of the sine function.

Fraunhofer diffraction of a partially blocked spiral phase plate

The Fraunhofer diffraction pattern from a partially blocked spiral phase plate (SPP) produces a partial vortex output pattern that is rotated by 90 degrees compared with the input. The rotation direction depends on whether the angular phase pattern increases in the clockwise or counterclockwise direction. In this work, we present an explanation of this effect based on careful examination of classical diffraction theory and show new experimental results. This approach is very convenient for easily determining the sign of the vortex charge.

Data transmission by hypergeometric modes through a hyperbolic-index medium

We study the existence of a novel complete family of exact and orthogonal solutions of the paraxial wave equation. The complex amplitude of these beams is proportional to the confluent hypergeometric function, which we name hypergeometric modes of type-II (HyG-II). It is formally demonstrated that hyperbolic-index medium can generate and support the propagation of such a class of beams. Since these modes are eigenfunctions of the photon orbital angular momentum, we conclude that an optical fiber with hyperbolic-index profile could take advantage over other graded-index fibers by the capacity of data transmission.

The feasibility of endorectal MR elastography for prostate cancer localization

The objectives of this study were to evaluate the feasibility of using a rigid radio-frequency receiver endorectal coil for intracavitary prostate magnetic resonance elastography (MRE) and to demonstrate the capability of this technique for generating stiffness maps over a typical prostate volume. An endorectal coil is currently used to help improve the signal-to-noise ratio of images acquired with multiparametric magnetic resonance imaging. We propose that this same coil could also serve to generate shear waves in the prostate gland during imaging, opening up the possibility of incorporating prostate stiffness characterization into multiparametric magnetic resonance imaging. Prostate cancer has been shown to change the elasticity of tissue, suggesting that stiffness imaging (elastography) may provide supplementary diagnostic information. A rigid endorectal coil was mechanically coupled to a piezoceramic actuator and used to investigate full volume (27 slices, 2-mm thick) endorectal MRE in a prostate mimicking phantom. The low-amplitude vibrations (± 8-38 μm displacements) necessary to perform endorectal MRE did not affect the signal-to noise ratio of the coil and endorectal MRE was capable of resolving 0.1 cc (0.6 cm diameter) spherical inclusion volumes. Therefore, the results of this study, in combination with current clinical practice, motivate clinical evaluation of endorectal MRE in patients.

Isotropic edge-enhancement by the Hilbert-transform in optical tomography of phase objects

In optical tomography, isotropic edge-enhancement of phase-object slices under the refractionless limit approximation can be reconstructed using spatial filtering techniques. The optical Hilbert-transform of the transmittance function leaving the object at projection angles ϕ∈(0°,360°), is one of these techniques with some advantages. The corresponding irradiance of the so modified transmittance is considered as projection data, and is proved that they share two properties with the Radon transform: its symmetry property and its zeroth-moment conservation. Accordingly, a modified sinogram able to reconstruct edge-enhanced phase slices is obtained. In this paper, the theoretical model is amply discussed and illustrated both with numerical and experimental results.

Directional edge enhancement in optical tomography of thin phase objects

In this paper, we make a proposal to obtain the Hilbert-transform for each entry of the projection data leaving the slice of a thin phase object. These modified projections are stacked in such a way that they form a modified sinogram called Hilbert-sinogram. We prove that the inverse Radon-transform of this sinogram is the directional Hilbert-transform of the slice function, and the reconstructed image is the directional edge enhancement of the distribution function on the slice. The Hilbert-transform is implemented by a 4f optical Fourier-transform correlator and a spatial filter consisting of a phase step of π radians. One important feature of this proposal is to perform a turn of 180° in the spatial filter at a certain value of the projection angle within the range [0°, 360°]. The desired direction of enhancement can be chosen by the proper selection of such turning angle. We present both the mathematical modeling and numerical results.

Analysis of multilevel spiral phase plates using a Dammann vortex sensing grating

Optical vortices can be easily generated using a multilevel spiral phase plate (SPP). However the quality of the generated vortex beam depends on the number of phase segments. We review the theory for this multilevel SPP using a Fourier expansion of integer topological charged vortices. We then experimentally examine the validity of this expansion using a fabricated Dammann vortex diffraction grating spectrum analyzer. The Dammann vortex diffraction grating is fabricated using SU-8 photoresist on glass substrate and yields uniform diffraction efficiency across a desired number of diffracted orders. Experimental results show the extra angular harmonics of a multilevel SPP and agree with the theory.

Focusing properties of Fresnel zone plates with spiral phase

Focusing properties of Fresnel zone plates with spiral phase with integer and fractional topological charges illuminated by plane wave are studied. Numerical results show that hollow beams can be generated and can also be controlled by the number of the zones and the topological charge, which implies the potential applications of such kind of zone plate in trapping and manipulating particles.

Explicit relations and optimal parameters for sidelobe suppression in optical vortices with a modified Bessel function

This paper establishes explicit relations between the radial modulation and the optimal sidelobe expression effect in the Bessel-like modulation technique. It verifies that both central and outer ring areas of the phase plate result in sidelobes in the diffraction pattern, and the corresponding structural dimensions can be determined quantitatively. This approach gives rise to complete sidelobe suppression by taking optimal modulation parameters.

Diffraction-grating-type phase converters for conversion of Hermite-Laguerre-Gaussian mode into Gaussian mode

Diffraction gratings that can be used to convert the Hermite-Laguerre-Gaussian (HLG) mode into the Gaussian mode were obtained; these modes are space modes of light beams. The HLG mode is intermediate between the Hermite-Gaussian and the Laguerre-Gaussian modes. Generally, gratings produced from interfering two beams as holograms contain information of not only the phase modulation factor but also the amplitude modulation factor. Thus, they cannot be expected to provide high conversion efficiency. To produce high-efficiency gratings, the interference equation is divided into two factors. The gratings produced by considering only the phase factors act as phase converters.

Orientation-selective edge detection and enhancement using the irradiance transport equation

We present a method for orientation-selective edge detection and enhancement based on the irradiance transport equation. The proposed technique distinguishes the sign of the derivative of the intensity pattern along an arbitrarily selected direction. The method is based on the capacity of liquid-crystal displays to generate simultaneously a contrast reverted replica of the image displayed on it. When both images (the original one and its replica) are imagined across a slightly defocused plane, one obtains an image with enhanced first derivatives. Unlike most Fourier methods, the proposed technique works well with a low-coherence light source, and it does not require precise alignment. The proposed method does not involve numerical processing, and thus it could be potentially useful for processing large images in real-time applications. Validation experiments are presented.

Spiral phase filtering and orientation-selective edge detection/enhancement

A spiral phase plate with an azimuthal structure exp[iphi](0phi<2pi) has been used as a filter in a 4f system to achieve edge enhancement. Generally such edge-enhanced effect is isotropic, i.e., each edge of an input pattern is enhanced to the same degree regardless of its orientation. We found that one can achieve anisotropic edge enhancement by breaking down the symmetry of the filtering process. This can be done in two ways: first, by use of a fractional spiral phase filter (SPF) with a fractional topological charge and a controllable orientation of the edge discontinuity, and second, by the lateral shifting of the SPF. We interpret this process as a vortex formation due to the diffraction of the Fourier spectrum of the input pattern by a SPF with an integer and fractional topological charge. Optical experiments using a spatial light modulator were carried out to verify our proposal.

Laguerre-Gaussian radial Hilbert transform for edge-enhancement Fourier transform x-ray microscopy

An efficient technique to achieve isotropic edge enhancement in optics involves applying a radial Hilbert transform on the object spectrum. Here we demonstrate a simple setup for isotropic edge-enhancement in soft-x- ray microscopy, using a single diffractive Laguerre-Gaussian zone plate (LGZP) for radial Hilbert transform. Since the LGZP acts as a beam-splitter, diffraction efficiency problems usually associated with x-ray microscopy optics are not present in this system. As numerically demonstrated, the setup can detect optical path differences as small as lambda/50 with high contrast.

Improved focusing with hypergeometric-gaussian type-II optical modes

We present a novel family of paraxial optical beams having a confluent hypergeometric transverse profile, which we name hypergeometric Gauss modes of type-II (HyGG-II). These modes are eigenmodes of the photon orbital angular momentum and they have the lowest beam divergence in the waist of HyGG-II among all known finite power paraxial modes families. We propose to exploit this feature of HyGG-II modes for generating, after suitable focusing, a "light needle" having record properties in terms of size and aspect ratio, possibly useful for near-field optics applications.

Adjustable refractive power from diffractive moiré elements

We show how suitable combinations of cascaded diffractive optical elements (DOEs) can form a combined "moiré DOE" of adjustable refractive power and high diffraction efficiency. The optical power can be adjusted continuously by a mutual rotation of one DOE with respect to the other. Fresnel lenses and axicons of variable refractive power or spiral phase plates of adjustable helical charge can be realized this way.

Hypergeometric-Gaussian modes

We studied a novel family of paraxial laser beams forming an overcomplete yet nonorthogonal set of modes. These modes have a singular phase profile and are eigenfunctions of the photon orbital angular momentum. The intensity profile is characterized by a single brilliant ring with the singularity at its center, where the field amplitude vanishes. The complex amplitude is proportional to the degenerate (confluent) hypergeometric function, and therefore we term such beams hypergeometric-Gaussian (HyGG) modes. Unlike the recently introduced hypergeometric modes [Opt. Lett. 32, 742 (2007)], the HyGG modes carry a finite power and have been generated in this work with a liquid-crystal spatial light modulator. We briefly consider some subfamilies of the HyGG modes as the modified Bessel Gaussian modes, the modified exponential Gaussian modes, and the modified Laguerre-Gaussian modes.

Diffraction of a finite-radius plane wave and a Gaussian beam by a helical axicon and a spiral phase plate

We derive what we believe to be new analytical relations to describe the Fraunhofer diffraction of the finite-radius plane wave by a helical axicon (HA) and a spiral phase plate (SPP). The solutions are deduced in the form of a series of the Bessel functions for the HA and a finite sum of the Bessel functions for the SPP. The solution for the HA changes to that for the SPP if the axicon parameter is set equal to zero. We also derive what we believe to be new analytical relations to describe the Fresnel and Fraunhofer diffraction of the Gaussian beam by a HA are derived. The solutions are deduced in the form of a series of the hypergeometric functions. We have fabricated by photolithography a binary diffractive optical element (a HA with number n=10) able to produce in the focal plane of a spherical lens an optical vortex, which was then used to perform rotation of several polystyrene beads of diameter 5 microm.

Sidelobe contrast reduction for optical vortex beams using a helical axicon

We present a new approach for generating an optical vortex pattern with reduced sidelobes without increasing the radius of the vortex and without excessive energy loss. Our technique combines the spiral phase plate with a weak axicon to form a helical axicon. Experimental results using a liquid crystal display agree with theory.

Spiral interferogram analysis

Interference microscopy using spatial Fourier filtering with a vortex phase element leads to interference fringes that are spirals rather than closed rings. Depressions and elevations in the optical thickness of the sample can be distinguished immediately by the sense of rotation of the spirals. This property allows an unambiguous reconstruction of the object's phase profile from one single interferogram. We investigate the theoretical background of "spiral interferometry" and suggest various demodulation techniques based on the processing of one single interferogram or multiple interferograms.

Radial Hilbert transform with Laguerre-Gaussian spatial filters

We analyze the point spread function (PSF) of the image processing system for radial Hilbert transform and propose a novel spiral phase filter, called the Laguerre-Gaussian spatial filter (LGSF). Theoretical analysis and real experiments show that the LGSF possesses some advantages in comparison with the conventional spiral phase plate (SPP). For example, the PSF of the imaging system with a LGSF presents smaller suboscillations than that with the conventional SPP, which allows us to realize a radial Hilbert transform for achieving a high contrast edge enhancement with high resolution.

Generation of helical Ince-Gaussian beams with a liquid-crystal display

We generate helical Ince-Gaussian (HIG) beams by using complex amplitude and phase masks encoded onto a liquid-crystal display (LCD). These beams display an intensity pattern consisting of elliptic rings, whose number and ellipticity can be controlled, and a phase exhibiting a number of in-line vortices, each with a unitary topological charge. We show experimental results that display the properties of these elliptic dark hollow beams. We introduce a novel interference technique for generating the object and reference beams by using a single LCD and show the vortex interference patterns. We expect that these HIG beams will be useful in optical trapping applications.

Optical vortex coronagraph

We describe a method to observe dim exoplanets that eliminates light from the parent star across the entire exit pupil without sacrificing light from the planet by use of a vortex mask of topological charge m = 2.

Azimuthal prism effect with partially blocked vortex-producing lenses

We report a new effect in which the diffraction from a partially blocked angular phase pattern occupying half of the input plane produces a partial vortex output pattern that is rotated by 90 degrees compared with the input. The energy is sent into a different quadrant of the output plane from the input plane. The rotation direction depends on whether the angular phase pattern is clockwise or counterclockwise. When we combine clockwise and counterclockwise angular phase patterns on separate horizontal halves of the input plane, we create an interference effect in one half of the output plane, while the other half remains dark.

Spiral interferometry

We present a surprising modification of optical interferometry. A so-called spiral phase element in the beam path of a standard microscope results in an interferogram of phase samples, for which the interference fringes have the shape of spirals instead of closed contour lines as in traditional interferograms. This configuration overrides the basic problem of interferometry, i.e., that elevations and depressions cannot be distinguished. Therefore a complete sample profile can be reconstructed from a single exposure, promising, e.g., high-speed metrology with a single laser pulse. The method is easy to implement, it does not require a spatially separated reference beam, and it is optimally stable against environmental noise.

Shadow effects in spiral phase contrast microscopy

Recently it has been demonstrated that spatial filtering of images in microscopy with a spiral phase element in a Fourier plane of the optical path results in a strong edge enhancement of object structures. In principle the operation is isotropic, i.e., all phase edges of a sample object are highlighted simultaneously, independent of their local direction. However, here we demonstrate that the symmetry can be broken intentionally by controlling the phase of the central area of a spiral phase hologram, which is displayed at a computer controlled spatial light modulator. This produces an apparent shadow effect which can be rotated at video rate. The resulting relieflike impression of the sample topography with a longitudinal resolution in the subwavelength regime is demonstrated by imaging a standard low contrast test sample consisting of a human cheek cell.

Spiral phase contrast imaging in microscopy

We demonstrate an optical method for edge contrast enhancement in light microscopy. The method is based on holographic Fourier plane filtering of the microscopic image with a spiral phase element (also called vortex phase or helical phase filter) displayed as an off-axis hologram at a computer controlled high resolution spatial light modulator (SLM) in the optical imaging pathway. The phase hologram imprints a helical phase term of the form exp(i phi) on the diffracted light field in its Fourier plane. In the image plane, this results in a strong and isotropic edge contrast enhancement for both amplitude and phase objects.

Programmable birefringent lenses with a liquid-crystal display

We study the properties of a novel birefringent lens constructed by combination of a regular glass lens and a programmable diffractive lens addressed to a liquid-crystal display (LCD). The LCD affects only the vertical polarization state. Consequently the birefringent lens produces two images of an input object with different locations and magnifications for the two orthogonal polarization states. Using a properly oriented analyzer polarizer produces interference fringes. We then show how the imaging system acts as a common-path polarization interferometer for wave-front analysis of objects in the input plane. Finally, we subtract the two images to produce an edge-enhanced version of the input image. All these effects can be controlled because we can program lenses with different focal lengths onto the LCD.

Optical processing with vortex-producing lenses

We discuss two types of optical processing using vortex-producing angular phase plates. In the most common spatial-filtering operation, an input object is Fourier transformed (either by Fraunhofer diffraction or with a lens system). The Fourier transform is then multiplied by an angular phase pattern, and the product is again Fourier transformed. The output is a space-invariant, edge-enhanced version of the input object. Alternatively we can directly image the object using a lens multiplied by the angular phase. The space-variant image is severely distorted along the optical axis of the system. We encode the phase plates onto a liquid-crystal display and present experimental results on both systems.

Selective edge enhancement of images with an acousto-optic light modulator

We show experimental results for image processing using an acousto-optic light modulator (AOLM) where the image can be edge enhanced with respect to the input object. We can select which edges are enhanced and the degree to which they are enhanced by changing the amplitude of the acoustic wave of the AOLM. We relate this technique to the fractional Hilbert transform and the fractional derivative image-processing operations and discuss its application to phase-only input images.

Fractional derivatives-analysis and experimental implementation

The fractional derivative spatial-filtering operator is useful for image-processing applications, particularly for examination of phase objects. Experimental implementation is difficult because the mask function combines both amplitude and phase. We present a simple one-dimensional analysis of the fractional derivative operation and note similarities with the fractional Hilbert transform. We demonstrate how to encode these amplitude and phase masks using a phase-only liquid-crystal spatial light modulator and present experimental results. Finally, we introduce a radially symmetric extension of this operation that is more useful for objects having an arbitrary shape.

Natural demodulation of two-dimensional fringe patterns. I. General background of the spiral phase quadrature transform

It is widely believed, in the areas of optics, image analysis, and visual perception, that the Hilbert transform does not extend naturally and isotropically beyond one dimension. In some areas of image analysis, this belief has restricted the application of the analytic signal concept to multiple dimensions. We show that, contrary to this view, there is a natural, isotropic, and elegant extension. We develop a novel two-dimensional transform in terms of two multiplicative operators: a spiral phase spectral (Fourier) operator and an orientational phase spatial operator. Combining the two operators results in a meaningful two-dimensional quadrature (or Hilbert) transform. The new transform is applied to the problem of closed fringe pattern demodulation in two dimensions, resulting in a direct solution. The new transform has connections with the Riesz transform of classical harmonic analysis. We consider these connections, as well as others such as the propagation of optical phase singularities and the reconstruction of geomagnetic fields.