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Recent image guidance developments for preclinical synchrotron microbeam radiotherapy represent a necessary step for future clinical translation of the technique. Image quality can be further improved using x-ray phase contrast, which is readily available at synchrotron facilities. We here describe a methodology for phase contrast image guidance at the Imaging and Medical Beamline at the Australian Synchrotron. Differential phase contrast is measured alongside conventional attenuation and used to improve the image quality. Post-processing based on the inverse Riesz transform is employed on the measured data to obtain noticeably sharper images. The procedure is extremely well suited for applications such as image guidance which require both visual assessment and sample alignment based on semi automatic image registration. Moreover, our approach can be combined with all other differential phase contrast imaging techniques, in all cases where a quantitative evaluation of the refractive index is not required.

We describe the principles of using orientation-independent differential interference contrast (OI-DIC) microscopy for mapping optical path length (OPL). Computation of the scalar two-dimensional OPL map is based on an experimentally received map of the OPL gradient vector field. Two methods of contrast enhancement for the OPL image, which reveal hardly visible structures and organelles, are presented. The results obtained can be used for reconstruction of a volume image. We have confirmed that a standard research grade light microscope equipped with the OI-DIC and 100 × / 1.3 NA objective lens, which was not specially selected for minimum wavefront and polarization aberrations, provides OPL noise level of ? 0.5 ?? nm and lateral resolution if ? 300 ?? nm at a wavelength of 546 nm. The new technology is the next step in the development of the DIC microscopy. It can replace standard DIC prisms on existing commercial microscope systems without modification. This will allow biological researchers that already have microscopy setups to expand the performance of their systems.

For paraxial wave-fields in two dimensions, it is known that the defocused OTF is given by a section through the ambiguity function. This is no longer true for high aperture fields. It is shown that there is a simple relationship, even for high aperture fields, between the two-dimensional (one transverse and one longitudinal) generalized OTF and the spectral correlation function. The connection with the Wigner distribution function is also discussed. The generalization to three-dimensional wavefields is also considered.

X-ray Talbot moire interferometers can now simultaneously generate two differential phase images of a specimen. The conventional approach to integrating differential phase is unstable and often leads to images with loss of visible detail. We propose a new reconstruction method based on the inverse Riesz transform. The Riesz approach is stable and the final image retains visibility of high resolution detail without directional bias. The outline Riesz theory is developed and an experimentally acquired X-ray differential phase data set is presented for qualitative visual appraisal. The inverse Riesz phase image is compared with two alternatives: the integrated (quantitative) phase and the modulus of the gradient of the phase. The inverse Riesz transform has the computational advantages of a unitary linear operator, and is implemented directly as a complex multiplication in the Fourier domain also known as the spiral phase transform.

The significance of the similarity theorem for the fractional Fourier transform is discussed, and the properties of self-similar functions considered. The concept of the fractional Hankel transform is developed for use in the analysis of diffraction and imaging in symmetrical optical systems. The particular case of Fresnel diffraction from a circular aperture is discussed and the effects of the similarity theorem are described.

At present the best methods for rotation of discrete sampled images use a combination of (fast) Fourier interpolation followed by cubic interpolation onto a rotated grid. A method is presented which uses only Fourier interpolation. The new method has a similar computational complexity to the old, and is exactly reversible. The method uses the well-known decomposition of rotation into three pure shears. Each shear is performed using a 2D extension of the 1D Fourier shift theorem. This allows the fast Fourier transform (FFT) to be used. With appropriate data padding (such as zero padding) in both the real and Fourier domains, the procedure gives near perfect results and minimal loss of information in multiple rotation tests.

Summary In an imaging system, it is known that the information capacity, rather than the resolution, is invariant. Thus super-resolution can be achieved by encoding/decoding additional information on to independent parameters of the imaging system. The concept of information capacity is extended to the full three-dimensional case.

For complete spherical concentration of light the maximum theoretically possible total energy density for a given power input can be, in principle, achieved by appropriate choice of polarization and angular amplitude variation. Illumination of a focusing system with a plane-polarized wave creates at the focus equal electric and magnetic energy densities. By appropriate choice of radial variation this energy density can be maximized. For hemispherical concentration the electric energy density can be seven sixteenths of the maximum possible for a given power input, and the total energy density can be seven eighths of the maximum possible. Focusing by optical systems satisfying the sine condition amongst others is also considered. For a system satisfying the sine condition, the total energy density can be 64/75 of the maximum possible.

Observations on white-light interference fringes show that the effects due to the introduction of a variable geometric phase, which is independent of the wavelength, differ significantly from those due to a change in the optical path difference.

We present a method for measuring the optical transfer function (OTF) of a camera lens using a tartan test pattern containing sinusoidal functions with multiple frequencies and orientations. The method is designed to optimize measurement accuracy for an adjustable set of sparse spatial frequencies and be reliable and fast in a wide range of measurement conditions. We describe the pattern design and the algorithm for estimating the OTF accurately from a captured image. Simulations show the tartan method is significantly more accurate than the International Organization for Standardization 12233 standard slanted-edge method. Experimental results from the tartan method were reproducible to 0.01 root mean square and in reasonable agreement with the slanted-edge method.

We present a novel method for accurately measuring the optical transfer function (OTF) of a camera lens by digitally imaging a tartan test pattern containing sinusoidal functions with multiple frequencies and orientations. The tartan pattern can be tuned to optimize the measurement accuracy for an adjustable set of sparse spatial frequencies. The measurement method is designed to be accurate, reliable, and fast in a wide range of measurement conditions, including uncontrolled lighting. We describe the design of the tartan pattern and the algorithm for estimating the OTF accurately from a captured digital image. Simulation results show that the tartan method has significantly better accuracy for measuring the modulus of the OTF (the modulation transfer function, or MTF) than the ISO 12233 standard slanted-edge method, especially at high spatial frequencies. With 1% simulated imaging noise, the root mean square (RMS) error of the tartan method is on average 5 times smaller than the RMS error of the slanted-edge method. Experiments with a printed tartan chart show good agreement (0.05 RMS) with MTFs measured using the slanted-edge method and that, like the slanted-edge method, our method is tolerant to wide variations in illumination conditions.

The effect of numerical aperture on the fringe spacing in interferometry is analyzed by the use of wave optics. The results are compared with published experimental results, and the influence of apodization of the wave front is discussed. The effects of central obscuration and surface tilt are also considered.

Two new two dimensional (2-D) complex operators for estimating the energy and orientation of 2-D oriented patterns are proposed. The starting point for our work is a new 2-D extension of the Teager-Kaiser energy operator incorporating orientation estimation. The first new energy operator is based on partial derivatives and can be considered a local (point-based) estimator. Using a nonlocal (pseudo-differential) operator we derive a second and more general energy operator. A scale invariant nonlocal operator is derived from the recently proposed spiral phase quadrature (or Riesz) transform. The Teager-Kaiser energy operator and the phase congruency local energy are unified in a single equation for both 1-D and 2- D. Robust orientation estimation, important for isotropic demodulation of fringe patterns is demonstrated. Theoretical error analysis of the local operator is greatly simplified by a logarithmic formulation. Experimental results using the operators on noisy images are shown. In the presence of Gaussian additive noise both the local and nonlocal operators give improved performance when compared with a simple gradient based estimator.

Image watermarking is the robust, imperceptible embedding of a small quantity of data into a digital image, and the subsequent recovery of this data, perhaps after the watermarked image has been distorted. We present a new watermarking technique which is robust to many image distortions, in particular arbitrary affine transformations of the image. The method achieves its robustness through the use of one-dimensional chirp functions. An affine-invariant detection method exists for such functions using a Radon transform, yet they are not detected trivially by a malicious attacker. The method also provides a way to determine any affine transformation applied to the watermarked image by using an affine-invariant property of groups of intersecting lines.

We propose a new mathematical model for human fingerprint images. The model can be summarized by the phrase “fingerprints are holograms”. The model unifies the analysis, compression, classification, matching, and re‐synthesis of fingerprints, in a self‐consistent formalism. The parsimony of this model is demonstrated by the reconstruction of fingerprint images with extreme compression ratios (typically >200x). At the heart of the method is a recently proposed method for demodulating two‐dimensional fringe patterns, such as holograms. Demodulation uses a spiral‐phase quadrature transform combined with a two‐dimensional orientation estimator that also uses spiral‐phase Fourier operators. Finally, the fingerprint decomposition itself achieves compactness by splitting the phase modulation into two unique parts, one of which is a pure spiral‐phase function. Spiral‐phase inexorably emerges as a central theme of the work.

In phase shifting interferometry, phase errors due to harmonic components of a fringe signal can be minimized by applying synchronous phase shifting algorithms with more than four samples. However, when the phase shift calibration is inaccurate, these algorithms cannot eliminate the effects of a non-sinusoidal waveform. It is shown that by taking a number of samples beyond one period of the fringe pattern, phase errors due to the harmonic components of the fringe signal can be eliminated, even when there exists a constant error in the phase shift interval. A general procedure for constructing phase shifting algorithms that eliminate these errors is derived. A seven-sample phase shifting algorithm is derived as an example, in which the effect of the second harmonic component can be eliminated in the presence of a constant error in the phase shift interval.

A common goal of many watermarking techniques is to produce a mark that remains detectable after the geometric transformations of Rotation, Scale and Translation; also known as RST invariance. We present a simple approach to achieving RST invariance using pixel-by-pixel addition of oscillating homogeneous patterns known as Logarithmic Radial Harmonic Functions [LRHFs]. LRHFs are the basis functions of the Fourier-Mellin transform and have perfect correlation, orthogonality, and spread-spectrum properties. Once the patterns have been embedded in an image they can be detected directly regardless of RST and with great sensitivity by correlation with the corresponding complex LRHFs. In contrast to conventional methods our approach is distinguished by the utilization of signal phase information and the absence of interpolation artifacts. Data encoding is based on the information in the relative centre positions of multiple spatially overlapping patterns.

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.

Utilizing the asymptotic method of stationary phase, I derive expressions for the Fourier transform of a two-dimensional fringe pattern. The method assumes that both the amplitude and the phase of the fringe pattern are well-behaved differentiable functions. Applying the limits in two distinct ways, I show, first, that the spiral phase (or vortex) transform approaches the ideal quadrature transform asymptotically and, second, that the approximation errors increase with the relative curvature of the fringes. The results confirm the validity of the recently proposed spiral phase transform method for the direct demodulation of closed fringe patterns.

The connection between the Wigner function and the generalized OTF, and between the ambiguity function and the generalized OTF is investigated for non-paraxial scalar wavefields. The treatment is based on two-dimensional (2-D) wavefields for simplicity, but can be extended to the three- dimensional case.© (2001) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

Differential interference contrast (DIC) is frequently used in conventional 2D biological microscopy. Our recent investigations into producing a 3D DIC microscope (in both conventional and confocal modes) have uncovered a fundamental difficulty: namely that the phase gradient images of DIC microscopy cannot be visualized using standard digital image processing and reconstruction techniques, as commonly used elsewhere in microscopy. We discuss two approaches to the problem of preparing gradient images for 3D visualization: integration and the Hilbert transform. After applying the Hilbert transform, the dataset can then be visualized in 3D using standard techniques. We find that the Hilbert transform provides a rapid qualitative pre-processing technique for 3D visualization for a wide range of biological specimens in DIC microscopy, including chromosomes, which we use in this study.

In this paper we investigate the use of a geometric phase- shifting (GPS) technique which allows us to convert conventional transmission or reflection differential interference contrast (DIC) microscopy into a quantitative mode. A phase-shifting algorithm is employed to extract the specimen phase gradient from the mixture of phase and amplitude information which is common in DIC. Fourier techniques are then used to recover the exact phase (i.e. optical path length variations) throughout the biological specimen viewed. In addition to this quantitative 'phase map,' we demonstrate that the GPS process simultaneously yields an 'amplitude-only' representation in which various absorption and transmission properties of the specimen are displayed as intensity variations in the image, similar to brightfield microscopy. These two resulting images can then be analyzed or further processed in a number of ways that are not possible with conventional DIC and which improve the microscopist's ability to correctly identify, interpret and measure features in the specimen.

We have developed a prototype fluorescence microscope which, using tomographic image acquisition and reconstruction techniques, can automatically combine conventional and/or confocal image stacks taken at a number of orientations into a single, very-high-resolution 3D image. We use the term `microtomography' in a broad sense to denote digital image reconstruction from multiple imaging operations which are not necessarily projections. Our system holds a biological specimen inside a thin capillary tube which is rotatable over a 360 degree range beneath an immersion objective. 3D fluorescence image data volumes are acquired by frame-grabbing a through-focus series of 2D images at each angle of rotation. Digital reconstruction of the multi-angle data volumes produces a single very-high-resolution 3D image and involves algorithms which perform rotation, interpolation, alignment and normalization operations in frequency (Fourier) space.

We have developed digital 3D Fourier transform methods for comparing the 3D spatial frequency content and hence the axial and transverse resolution of confocal versus conventional microscope images. In particular, we have utilized these techniques to evaluate the performance of our recently-developed confocal transmission microscope for bright field and Nomarski DIC imaging. We have also found that Fourier methods, such as the Hibert transform, can be successfully employed to overcome the difficulty of visualizing differentially-shaded phase objects, in 3D, that have been acquired using transmission DIC optics.

The unique imaging properties of all confocal microscope systems are based on the fundamental condition that the illumination and collection (detection) optical paths contribute equally to the formation of the final image. However, many of the optical and imaging properties of confocal microscopes can be analyzed effectively by considering the illumination and detection paths to be two distinct entities. In this chapter, we explore one of these optical paths—the illumination path—and describe the ideal function and practical limitations of the total path and the optical components therein. Several other chapters in this volume present additional specific information concerning some of these optical components (e.g., light sources, intermediate optics, objective lenses). Our purpose is to treat the illumination path as an integral system and describe its contribution to the overall performance of the confocal microscope.

We have designed and constructed an experimental confocal specimen-scanning microscope which has the capability of producing high resolution 3D images in a variety of optical modes, many of which are not currently available on commercial confocal microscopes. The transmission Nomarski differential interference contrast mode is particularly interesting because it can be utilized to image small changes in refractive index within complex biological specimens which are transparent in standard brightfield. The three-color reflection configuration can produce a color 3D image, which means that stained or pigmented objects will be similar in appearance to images obtained from conventional white light microscopes which makes them more recognizable.

The aim of this paper is to explore the implications of a multiplicative model of optical imaging.

A new seven-sample algorithm has been developed which is particularly useful for phase evaluation in systems with non-sinusoidal periodic waveforms. The algorithm performs well in systems with significant amounts of second, third, fourth, and sixth harmonic content and in the presence of phase shift errors. The algorithm is characterized by an equi-spaced symmetrical sampling pattern in which the end samples are one full period apart. Principal features of the algorithm in terms of the Fourier theory of phase-shifting are presented. Performance of the seven-sample algorithm is outlined and compared with several conventional algorithms.© (1993) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

The mathematical properties of the arctangent function are investigated, revealing the mechanism of error propagation in digital phase-shifting applications. The analysis gives a systematic approach to the assessment of errors and, as such, provides a valuable tool for understanding the propagation of errors in any phase-shifting algorithm.

An Optical Surface Profiler (OSP130) has been developed for the metrology of master tooling used in the coin stamping process. The OSP130 measure, in a non-contacting manner, the surface relief of tools ranging in diameter from 10 mm to 300 mm. Rapid measurements are performed simultaneously on a large grid of equispaced points across the surface of the tool. From the relief data, many parameters such as the location of high and low features, volume of impression, background curvatures and various diameters can be quickly evaluated. The technique used is phase-shifting moire profilometry. A white light projector illuminates a periodic transmission grating which is then imaged onto the object surface. The light pattern on the object is viewed by a high resolution TV camera connected to a computer. The grating is shifted under computer control to a number of positions and corresponding intensity images of the deformed pattern on the object surface are stored in the computer. From the intensity images a phase map, representing the deformation of the periodic grating by the surface relief, is evaluated and compared with an undeformed pattern. This results in an accurate contour map of the surface relief with an uncertainty less than 1% of the relief excursion on the object. Details of the instrument and its use at the Royal Australian Mint are presented.

We propose a coherent mathematical model for human fingerprint images. Fingerprint structure is represented simply as a hologram - namely a phase modulated fringe pattern. The holographic form unifies analysis, classification, matching, compression, and synthesis of fingerprints in a self-consistent formalism. Hologram phase is at the heart of the method; a phase that uniquely decomposes into two parts via the Helmholtz decomposition theorem. Phase also circumvents the infinite frequency singularities that always occur at minutiae. Reliable analysis is possible using a recently discovered two-dimensional demodulator. The parsimony of this model is demonstrated by the reconstruction of a fingerprint image with an extreme compression factor of 239.

A new method of estimating the phase-shift between interferograms is introduced. The method is based on a recently introduced two-dimensional Fourier-Hilbert demodulation technique. Three or more interferogram frames in an arbitrary sequence are required. The first stage of the algorithm calculates frame differences to remove the fringe pattern offset; allowing increased fringe modulation. The second stage is spatial demodulation to estimate the analytic image for each frame difference. The third stage robustly estimates the inter-frame phase-shifts and then uses the generalised phase-shifting algorithm of Lai and Yatagai to extract the offset, the modulation and the phase exactly. Initial simulations of the method indicate that high accuracy phase estimates are obtainable even in the presence of closed or discontinuous fringe patterns.

The angle-impact Wigner function for highly convergent three-dimensional scalar wave fields is derived directly by use of the three-dimensional generalized optical transfer function rather than from a six-dimensional Wigner function. The angle-impact Wigner function is a real four-dimensional function from which the intensity at any point in space is readily determined.

The focal shift for a lens of finite value of Fresnel number can be defined in terms of the second moment of the intensity distribution in transverse planes. The connection with the optical transfer function is described. The specification of the focused amplitude in terms of the fractional Fourier transform is discussed, and the connections among the fractional Fourier transform, the Wigner distribution, and the ambiguity function are described, leading to a model for effects of Fresnel number in terms of a rotation in phase space. The uncertainty principle is discussed, including the significance of the beam propagation factor M 2 and the width of optical fiber beam modes. Calculation of the moments in terms of the modulus and the phase of the illuminating wave is presented, and the use of the Kaiser‐Teager energy operator is also described. © 2000 Optical Society of America [S0740-3232(00)00404-X] OCIS codes: 060.2270, 070.2580, 070.2590, 140.3300, 260.1960.

The phase-retrieval problem for a physical system with strong support constraints is investigated. Propagation of an optical field in a system with no variation along one transverse axis results in cylindrical wave fronts. Scalar propagation in such systems is a purely two-dimensional process. We show that, given the optical intensity in a plane, the phase of the wave field can be calculated directly if the system has this special symmetry. The procedure relies on a simple geometric relation between the system pupil function (or angular spectrum) and the system optical transfer function in the Debye theory of scalar wave focusing. The inherent autocorrelation operation can be undone, and the phase directly retrieved, with a simple coordinate transformation.

Two complementary methods have now been proposed for deriving phase-shifting algorithms that correct for both nonlinear and spatially nonuniform phase shifts. Some advantages of symmetrical algorithms are outlined, and a simple method for ensuring algorithmic symmetry is presented. Noise susceptibility of the algorithms and the phase-shifter calibration are also discussed briefly.

In phase-shifting interferometry spatial nonuniformity of the phase shift gives a significant error in the evaluated phase when the phase shift is nonlinear. However, current error-compensating algorithms can counteract the spatial nonuniformity only in linear miscalibrations of the phase shift. We describe an error-expansion method to construct phase-shifting algorithms that can compensate for nonlinear and spatially nonuniform phase shifts. The condition for eliminating the effect of nonlinear and spatially nonuniform phase shifts is given as a set of linear equations of the sampling amplitudes. As examples, three new algorithms (six-sample, eight-sample, and nine-sample algorithms) are given to show the method of compensation for a quadratic and spatially nonuniform phase shift.

A simple nonlinear (quadratic) filter is shown to demodulate bandpass sampled AM signals efficiently. The filter is based upon a discrete version of the recently introduced Teager-Kaiser energy operator, but also closely resembles a complex digital sampling demodulator. Such a filter can also be implemented in analogue circuitry.

The article "Image processing: the art of science" by Nick Efford (August pp36–40) was flawed by its complete omission of Mac software. Although one could argue that PCs dominate the commercial market, this is not true to the same extent in the academic and research world.

In phase measurement systems that use phase shifting techniques, phase errors that are due to nonsinusoidal waveforms can be minimized by applying synchronous phase shifting algorithms with more than four samples. However, when the phase shift calibration is inaccurate, these algorithms cannot eliminate the effects of nonsinusoidal characteristics. It is shown that, when a number of samples beyond one period of a waveform such as a fringe pattern are taken, phase errors that are due to the harmonic components of the waveform can be eliminated, even when there exists a constant error in the phase shift interval. A general procedure for constructing phase shifting algorithms that eliminate these errors is derived. It is shown that 2j + 3 samples are necessary for the elimination of the effects of higher harmonic components up to the jth order. As examples, three algorithms are derived, in which the effects of harmonic components of low orders can be eliminated in the presence of a constant error in the phase shift interval.

Conventional phase-shifting algorithms based on a least-squares estimate use N samples over an incomplete period of the sampled waveform. We introduce a class of phase-shifting algorithms having N + 1 samples symmetrically disposed over one full period of the sampled waveform. Fourier analysis techniques are used to derive these algorithms and modify them to improve their performance in the presence of phase-shift errors. The algorithms can be used in phase measurement systems having periodic, but not necessarily sinusoidal, waveforms.

The ambiguity function and the Wigner distribution function have both been applied in the optical area for many years. Later, the fractional Fourier transform has also been used. The connection between the ambiguity function and the defocused optical transfer function has also been described. Here we consider the connections with the generalized optical transfer function, first proposed in 1965, which is a two (three) dimensional optical transfer function for the two (three) dimensional case. The two dimensional form can be used as the basis for phase retrieval algorithms, but is also valid in the non‐paraxial domain.