Phase retrieval using coherent imaging systems with linear transfer functions
Introduction
At optical and higher frequencies, the extreme rapidity of wavefield oscillations implies that one is only able to directly measure the square modulus of the complex amplitude of such a wavefield, averaged over many cycles [1]. To combat this limitation, many means have been developed for so-called “phase-contrast imaging”, which may be defined as any means of imaging which is sensitive to phase variations across a given electromagnetic or matter wavefield. These methods for phase-contrast imaging are necessarily indirect, for they measure the modulus of a suitably modified wavefield in order to make qualitative and/or quantitative inferences about the phase of the unmodified wavefield.
The most famous method for phase contrast imaging, namely interferometry, is one of many methods for rendering visible the transverse phase shifts in a coherent wavefield, such as might be imprinted upon a plane wave traversing a sample. Other methods include Zernike phase contrast [2], Lorentz microscopy [3], in-line holography [4], differential interference contrast (DIC) microscopy [5], Schlieren imaging [6], focus variation methods [7] and diffraction enhanced imaging (DEI) [8].
After briefly examining the features generic to all such means of phase-contrast imaging (Section 2), we discuss the case of images, obtained using a coherent shift-invariant linear imaging system, for which the transfer function may be taken to be a linear function of spatial frequency (Section 3). This linear approximation to the transfer function (which we term a “linear transfer function”) is applicable when the spread of spatial frequencies, in a given two-dimensional complex wavefield, is sufficiently narrow (with respect to the characteristic length of variation of the transfer function of an imaging system taking such a wavefield as input). Note that the assumption of a linear transfer function, while rather strong at first sight, is applicable to a number of realistic imaging systems, as shall be outlined later in the paper.
We develop several algorithms for phase-amplitude reconstruction of a given wavefield; these algorithms require, as input, one or more images of the disturbance which are taken using a coherent imaging system possessing a linear transfer function (Section 4). The four-image reconstruction algorithm, which requires as input data four different images of the same two-dimensional wavefield which are obtained using four different states of the imaging system, makes no further assumptions regarding the imaging system beyond those listed previously. However, we show that one can perform the phase retrieval using fewer images, if one makes additional assumptions. In particular, one can make do with a single image provided that: (i) one is using plane waves to image an object which is composed of a single-material sample of interest, embedded in a substrate of approximately uniform thickness; (ii) the projection approximation is valid, i.e., the phase and amplitude shifts of the probe radiation, upon passage through the object, are the same as would be obtained if the object's potential distribution were projected onto a plane perpendicular to the optic axis. Within its restricted domain of validity, the application of such a single-image algorithm is simpler than methods which require two or more images as data.
As a detailed case study and experimental implementation of aspects of the general formalism developed in Section 4, Section 5 considers the problem of quantitative diffraction-enhanced imaging. This technique, which has generated much recent interest in the X-ray imaging community, achieves phase contrast by diffracting a given wavefield from a so-called analyzer crystal inclined at an angle to an incident wave-front [8] (see Fig. 1). The physical mechanism for this means of phase contrast, the origins of which date back at least as far as 1959 [9], is the extreme sensitivity of the crystal's reflection coefficient to phase gradients in the incident wave [10]. We use this technique to demonstrate quantitative phase-amplitude reconstruction, using a single DEI image of inclusions in a breast phantom. Significantly, by suitable analysis of the phase-contrast effects in a given DEI image, the method is able to generate the absorption radiographs which would have been obtained if (i) the radiation dose was higher and (ii) all scattered radiation was eliminated. The stability of this algorithm is studied, leading to the conclusion that it is robust in the presence of realistic levels of noise in the data. Since it requires only a single image as input, the single-image algorithm may be applied to real-time image data obtained using DEI imaging (Section 5) and DIC microscopy (Section 6).
Section snippets
General remarks on phase-contrast imaging
The most well-known means of coherent phase-contrast imaging is interferometry, which renders visible the transverse phase variations in a given two-dimensional wavefield by first modifying the wavefield through superposing it with a reference beam, and then measuring the resulting intensity. In the language of Fourier optics, and for the case of a plane reference wave, the action of an interferometer may be viewed as greatly amplifying a single spatial frequency in the two-dimensional Fourier
Coherent image formation using a shift-invariant linear imaging with a linear transfer function
Let us consider the problem of imaging a coherent complex scalar wavefield, the spatial part of which is denoted by ψ(x,y,z), where x,y,z are Cartesian coordinates in three-dimensional space. We suppress the harmonic time dependence exp(−iωt) of the coherent wavefield, where ω denotes angular frequency and t is time. The plane z=z0 is chosen to be the entrance surface of a given imaging system; this may either be a physical surface or a mathematical one, and is defined to be the plane over
Quantitative phase-amplitude reconstruction
Here, we show how one may use Eq. (11) to recover a given input complex wavefield ψIN(x,y), given as data one or more images of this wavefield which are obtained using a coherent linear imaging system whose action is well approximated by Eq. (5). For brevity, this system will henceforth be called the “imaging system”.
Application: diffraction enhanced imaging using hard X-rays
The work of Lang [9] contains the central idea behind the technique of diffraction enhanced imaging. His “projection topograph”, which is formed by imaging the reflection of a well-collimated X-ray beam from an imperfect crystal, is extremely sensitive to dislocations and other imperfections in the crystal. In terms of geometric optics, this sensitivity may be roughly pictured as due to the fact that perturbations introduced by the crystal's defects strongly affect the intensity of a reflected
Application: quantitative differential interference contrast microscopy (DIC)
Here, we show how the ideas of this paper may be used to render quantitative the images which are taken using a differential interference contrast microscope, provided that the (quasi) homogeneous object to be imaged is sufficiently weak, in a sense which shall soon be made precise. An ability to quantitatively analyse visible-light DIC microscope images is of utility in the context of measuring the dry mass of biological objects in vivo [28], and in the quantitative analysis of real-time
Conclusion
We have considered the problem of quantitative phase-amplitude retrieval from images obtained using a coherent shift-invariant linear imaging system with a linear transfer function. Special cases of such systems include those used for differential interference contrast microscopy with visible light or X-rays, and diffraction enhanced imaging with hard X-rays. We gave several algorithms for reconstructing both the phase and amplitude of a given two-dimensional coherent wavefield, given as input
Acknowledgements
David Paganin and Konstantin Pavlov acknowledge support from the Australian Research Council. Marcus Kitchen acknowledges the receipt of an Australian Postgraduate Award. The authors acknowledge stimulating discussions with Michael Morgan, Karen Siu and Andrew Pogany. The experimental data, used in the reconstructions given in this paper, was obtained with the kind assistance of Chris Hall, Alan Hufton, and the superb staff of the SYRMEP beamline at Elettra.
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