Theory of quantitative phase-contrast computed tomography

Phase-contrast x-ray computed tomography (CT) is an emerging imaging technique that can be implemented at third-generation synchrotron radiation sources or by using a microfocus x-ray source. Promising results have recently been obtained in materials science and medicine. At the same time, the lack of a mathematical theory comparable with that of conventional CT limits the progress in this field. Such a theory is now suggested, establishing a fundamental relation between the three-dimensional Radon transform of the object function and the two-dimensional Radon transform of the phase-contrast projection. A reconstruction algorithm is derived in the form of a filtered backprojection. The filter function is given in the space and spatial-frequency domains. The theory suggested enables one to quantitatively determine the refractive index of a weakly absorbing medium from x-ray intensity data measured in the near-field region. The results of computer simulations are discussed.     

Zeroth-order phase-contrast technique

What we believe to be a new phase-contrast technique is proposed to recover intensity distributions from phase distributions modulated by spatial light modulators (SLMs) and binary diffractive optical elements (DOEs). The phase distribution is directly transformed into intensity distributions using a 4f optical correlator and an iris centered in the frequency plane as a spatial filter. No phase-changing plates or phase dielectric dots are used as a filter. This method allows the use of twisted nematic liquid-crystal televisions (LCTVs) operating in the real-time phase-mostly regime mode between 0 and p to generate high-intensity multiple beams for optical trap applications. It is also possible to use these LCTVs as input SLMs for optical correlators to obtain high-intensity Fourier transform distributions of input amplitude objects.     

Differential phase-contrast BioCD biosensor

Common-path differential phase-contrast interferometry measures the spatial gradient of surface dipole density on a bio-optical compact disk (BioCD) and is sensitive to small changes in dipole density following molecular binding of target molecules out of solution. The recognition molecules are antibody IgG proteins that are deposited in periodic patterns on the BioCD using soft lithography or photolithography on the silanized silica surfaces of dielectric mirrors. Spatial carrier-wave sideband demodulation extracts the slowly varying protein envelope that modulates the protein carrier frequency. The experimental interferometric profilometry has surface height sensitivity down to 20 pm averaged over a lateral scale of 70 microm with a corresponding scaling mass sensitivity limit of 1.5 pg/mm. Under the conditions of an IgG immunoassay with background changes caused during incubation, the scaling mass sensitivity is approximately 7 pg/mm. A saturated reverse immunoassay performed with IgG at 100 ng/ml showed false positive and false negative rates of 0.2%.     

Phase-space formulation for phase-contrast x-ray imaging

Phase-space formulation based on the Wigner distribution has been presented for analyzing phase-contrast image formation. Based on the statistical nature and affine canonical covariance of Wigner distributions in the phase space, we show that the partial coherence effects of incident x-ray wave field on image intensity are simply accounted for by a multiplication factor, which is the reduced complex degree of coherence of the incident x-ray wave field. We show especially that with the undulator sources one cannot obtain the phase-contrast intensity by summing over the contributions from all electron positions, since the van Cittert-Zernike theorem fails in general for undulators. We derive a comprehensive formula that quantifies the effects of partial spatial coherence, polychromatic spectrum, body attenuation, imaging-detector resolution, and radiation dose on phase-contrast visibility in clinical imaging. The results of our computer modeling and simulations show how the formula can provide design guidelines and optimal parameters for clinical x-ray phase-contrast imaging systems.     

Vectorial, high-numerical-aperture study of phase-contrast microscopes

We describe, using a high-numerical-aperture vectorial model, the image formation of phase-contrast microscopes. In particular, imaging of a weak phase object is considered. We show that, partly owing to the fact that phase-contrast microscopes are interference microscopes, their image formation is fundamentally different from that of conventional transmission optical microscopes. Our detailed analysis reveals a number of yet undocumented properties of these microscopes, including that depending on the given configuration, they can exhibit an improved lateral resolution when larger detectors are used in comparison with that obtained for a small detector size. We present numerical examples to explain this phenomenon and discuss our analysis in detail.     

Effects of incident illumination on in-line phase-contrast imaging

Effects of incident illumination on phase-contrast images obtained by means of free-space propagation are investigated under the "transport-of-intensity" approximation. Analytical expressions for image intensity distribution are derived in the cases of coherent quasi-plane and quasi-spherical incident waves, as well as for spatially incoherent and quasi-homogeneous sources and some other types of sources. Practical methods for measuring the relevant parameters of the incident radiation are discussed together with formulas allowing one to calculate the effect of these parameters on the image intensity distribution. The results are expected to be useful in quantitative in-line imaging, phase retrieval, and tomography with polychromatic and spatially partially coherent radiation. As an application we present a method for simultaneous "automatic" phase retrieval and spatial deconvolution in in-line imaging of homogeneous objects using extended polychromatic x-ray sources.     

Noninterferometric phase-contrast images obtained with incoherent x-ray sources

We report on what are believed to be the first full-scale images obtained with the coded aperture concept, which uses conventional x-ray sources without the need to collimate/aperture their output. We discuss the differences in the underpinning physical principles with respect to other methods, and explain why these might lead to a more efficient use of the source. In particular, we discuss how the evaluation of the first imaging system provided promising indications on the method's potential to detect details invisible to conventional absorption methods, use an increased average x-ray energy, and reduce exposure times-all important aspects with regards to real-world implementations.     

X-ray phase-contrast imaging: transmission functions separable in Cartesian coordinates

In-line, x-ray phase-contrast imaging is responsive to both phase changes and absorption as the x radiation traverses a body. Expressions are derived for phase-contrast imaging of objects having transmission functions separable in Cartesian coordinates. Starting from the Fresnel-Kirchhoff integral formula for image formation, an expression is found for the phase-contrast image produced by an x-ray source with nonvanishing dimensions. This expression is evaluated in limiting cases where the source-to-object distance is large, where the source acts as a point source, and where the weak phase approximation is valid. The integral expression for the image is evaluated for objects with simple geometrical shapes, showing the influence of the source dimensions on the visibility of phase-contrast features. The expressions derived here are evaluated for cases where the magnification is substantially greater than one as would be employed in biological imaging. Experiments are reported using the in-line phase-contrast imaging method with a microfocus x-ray source and a CCD camera.     

Wavelet-SVM classification and automatic recognition of unstained viable cells in phase-contrast microscopy

Irradiation of individual cultured mammalian cells with a pre-selected number of ions down to one ion per single cell is a useful experimental approach to investigating the low-dose ionising radiation exposure effects and thus contributing to a more realistic human cancer risk assessment. One of the crucial tasks of all the microbeam apparatuses is the visualisation, recognition and positioning of every individual cell of the cell culture to be irradiated. Before irradiations, mammalian cells (specifically, Chinese hamster V79 cells) are seeded and grown as a monolayer on a mylar surface used as the bottom of a specially designed holder. Manual recognition of unstained cells in a bright-field microscope is a time-consuming procedure; therefore, a parallel algorithm has been conceived and developed in order to speed up this irradiation protocol step. Many technical problems have been faced to overcome the complexity of the images to be analysed: cell discrimination in an inhomogeneous background, among many disturbing bodies mainly due to the mylar surface roughness and culture medium bodies; cell shapes, depending on how they attach to the surface, which phase of the cell cycle they are in and on cell density. Preliminary results of the recognition and classification based on a method of wavelet kernels for the support vector machine classifier will be presented.     

Adaptive optics with advanced phase-contrast techniques. I. High-resolution wave-front sensing

High-resolution phase-contrast wave-front sensors based on phase spatial light modulators and micromirror/ liquid-crystal arrays are introduced. Wave-front sensor performance is analyzed for atmospheric-turbulence-induced phase distortions described by the Kolmogorov and the Andrews models. A high-resolution phase-contrast wave-front sensor (nonlinear Zernike filter) based on an optically controlled liquid-crystal phase spatial light modulator is experimentally demonstrated. The results demonstrate high-resolution visualization of dynamically changing phase distortions within the sensor time response of approximately 10 ms.     

Adaptive optics with advanced phase-contrast techniques. II. High-resolution wave-front control

A wave-front control paradigm based on gradient-flow optimization is analyzed. In adaptive systems with gradient-flow dynamics, the output of the wave-front sensor is used to directly control high-resolution wavefront correctors without the need for wave-front phase reconstruction (direct-control systems). Here, adaptive direct-control systems with advanced phase-contrast wave-front sensors are analyzed theoretically, through numerical simulations, and experimentally. Adaptive system performance is studied for atmospheric-turbulence-induced phase distortions in the presence of input field intensity scintillations. The results demonstrate the effectiveness of this approach for high-resolution adaptive optics.     

Analysis of ideal observer signal detectability in phase-contrast imaging employing linear shift-invariant optical systems

Phase-contrast imaging methods exploit variations in an object's refractive index distribution to permit the visualization of subtle features that may have very similar optical absorption properties. Although phase-contrast is often viewed as being desirable in many biomedical applications, its relative influence on signal detectability when both absorption- and phase-contrast are present remains relatively unexplored. In this work, we investigate the ideal Bayesian observer signal-to-noise ratio in phase-contrast imaging for a signal-known-exactly/background-known exactly detection task involving a weak signal. We demonstrate that this signal detectability measure can be decomposed into three contributions that have distinct interpretations associated with the imaging physics.     

Automated phase sensing and control of an external Talbot cavity laser with phase-contrast imaging

We describe a novel phase-sensing and control system, based on phase-contrast imaging, operating within a linear external cavity laser consisting of 18 GaAlAs edge-emitting gain stripes. The system is used to achieve single-spatial-mode operation and diffraction-limited output from the linear cavity, which uses diffractive coupling at a Talbot plane to achieve coherent operation.     

Relation between computed tomography and diffraction tomography

The relationship between computed tomography (CAT) and diffraction tomography (DT) is investigated. A simple condition with a clear physical meaning is derived for the applicability of CAT. Corrections due to scattering are incorporated into CAT, and it is shown that the effect of scattering may be characterized by a two-dimensional fractional Fourier transform. The implications of these results for the three-dimensional imaging of weakly scattering objects are also discussed.     

Differential phase-contrast microscope with a split detector for the readout system of a multilayered optical memory

A transmission differential phase-contrast microscope with a split detector is used as a readout system for a multilayered three-dimensional optical memory. The system is applicable to data recorded as refractive-index changes. The system is compact and easy to use. The three-dimensional optical transfer function for the system is derived. This shows that the spatial bandwidth of the system is the same as that of a conventional microscope with incoherent illumination but with much improved contrast. Six layers of information are recorded in a photopolymer and are read out with little cross talk and high contrast.     

Diffuse optical tomography guided quantitative fluorescence molecular tomography

We describe a method that combines fluorescence molecular tomography (FMT) with diffuse optical tomography (DOT), which allows us to study the impact of heterogeneous optical property distribution on FMT, an issue that has not been systemically studied. Both numerical simulations and phantom experiments were performed based on our finite-element reconstruction algorithms. The experiments were conducted using a noncontact optical fiber free, multiangle transmission system. In both the simulations and experiments, a fluorescent target was embedded in an optically heterogeneous background medium. The simulation results clearly suggest the necessity of considering the absorption coefficient (mu(a)) and reduced scattering coefficient (mu'(s)) distributions for quantitatively accurate FMT, especially in terms of the accuracy of reconstructed fluorophore absorption coefficient (mu(a(x-->m))). Subsequent phantom experiments with an indocyanine green (ICG)-containing target confirm the simulation findings. In addition, we performed a series of phantom experiments with low ICG concentration (0.1, 0.2, 0.4, 0.6 and 1.0 microM) in the target to systematically evaluate the quantitative accuracy of our FMT approach. The results indicate that, with the knowledge of optical property distribution, the accuracy of the recovered fluorophore concentration is improved significantly over that without such a priori information. In particular absolute value of mu(a(x-->m) ) from our DOT guided FMT are quantitatively consistent with that obtained using spectroscopic methods.     

Compensation of the inherent wave front curvature in digital holographic coherent microscopy for quantitative phase-contrast imaging

An approach is proposed for removing the wavefront curvature introduced by the microscope imaging objective in digital holography, which otherwise hinders the phase contrast imaging at reconstruction planes. The unwanted curvature is compensated by evaluating a correcting wave front at the hologram plane with no need for knowledge of the optial parameters, focal length of the imaging lens, or distances in the setup. Most importantly it is shown that a correction effect can be obtained at all reconstruction planes. Three different methods have been applied to evaluate the correction wave front and the methods are discussed in detail. The proposed approach is demonstrated by applying digital holography as a method of coherent microscopy for imaging amplitude and phase contrast of microstructures.     

Theory of total-internal-reflection tomography

A method is presented to reconstruct three-dimensional tomographic images of weakly scattering objects with subwavelength resolution. The method may be applied to data available in phase-sensitive, total-internal-reflection microscopy. The results follow from an analysis of the near-field inverse scattering problem with evanescent waves.