Phase retrieval using coherent imaging systems with linear transfer functions

D.P. would like to dedicate this paper to the memory of Armando Paganin
https://doi.org/10.1016/j.optcom.2004.02.015Get rights and content

Abstract

We consider the problem of quantitative phase retrieval from images obtained using a coherent shift-invariant linear imaging system whose associated transfer function (i.e., the Fourier transform of the complex point-spread function) is well approximated by a linear function of spatial frequency. This linear approximation to the transfer function is applicable when the spread of spatial frequencies, in a two-dimensional complex wavefield, is sufficiently narrow when compared to the characteristic length of variation of the transfer function for an imaging system taking such a wavefield as input. We give several algorithms for reconstructing both the phase and amplitude of a given two-dimensional coherent wavefield, given as input data one or more images of such a wavefield which may be formed by different states of the imaging system. When an object to be imaged consists of a single material, or of a single material embedded in a substrate of constant thickness, the phase-amplitude reconstruction can be performed using a single image. As a first application of these ideas, we write down an algorithm for using a single diffraction-enhanced image (DEI) to obtain a quantitative reconstruction of the projected thickness of a single-material sample which is embedded within a substrate of approximately constant thickness. This algorithm is used to quantitatively map inclusions in a breast phantom, from a single synchrotron DEI image of the same. In particular, the reconstructed images quantitatively represent the projected thickness in the bulk of the sample, in contrast to raw DEI images which greatly emphasise sharp edges (high spatial frequencies). Lastly, we point out that the methods presented here are also applicable to the quantitative analysis of differential interference contrast (DIC) images, obtained using both visible-light and X-ray microscopy.

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.

References (31)

  • F. Zernike

    Physica

    (1942)
  • M. De Graef

    Magnetic Imaging and its Application to Materials: Experimental methods in the Physical Sciences

    (2001)
  • R.A. Lewis et al.

    Medical Imaging 2002: Physics of Medical Imaging, Proc. SPIE

    (2002)
  • M. Born et al.

    Principles of Optics

    (1993)
  • D. Gabor

    Nature

    (1948)
  • G. Nomarski et al.

    Rev.Metall.

    (1955)
  • J.R. Meyer-Arendt (ed.), Selected papers on Schlieren optics, SPIE Milestone series vol. MS61, The International...
  • W. Coene et al.

    Phys. Rev. Lett.

    (1992)
  • D. Chapman et al.

    Phys. Med. Biol.

    (1997)
  • A.R. Lang

    Acta Crystallogr.

    (1959)
  • T.E. Gureyev et al.

    Il Nuovo Cimento

    (1997)
  • M. Nieto-Vesperinas

    Scattering and Diffraction in Physical Optics

    (1991)
  • A. Papoulis

    Systems and Transforms with Applications in Optics

    (1968)
  • L. Pismen

    Vortices in Nonlinear Fields: From Liquid Crystals to Superfluids, from Non-equilibrium Patterns to Cosmic Strings

    (1999)
  • L. Yaroslavsky et al.

    Fundamentals of digital optics: Digital signal processing in optics and holography

    (1996)
  • Cited by (78)

    • X-ray phase-contrast imaging: A broad overview of some fundamentals

      2021, Advances in Imaging and Electron Physics
      Citation Excerpt :

      To gain an overview of the plethora of other approaches that have been developed to address the same question, see e.g. the review article by Wilkins et al. (2014), together with the suite of relevant chapters from the collection edited by Russo (2018). A variant, of the method in Eq. (121), has been developed for analyzer based phase-contrast imaging and other phase-contrast imaging systems that yield first-derivative phase contrast (Paganin, Gureyev, Pavlov, et al., 2004). Another variant has been developed for phase-contrast imaging systems that simultaneously yield both first-derivative and second-derivative phase contrast (Pavlov et al., 2005, 2004).

    • Characterization of mouse spinal cord vascular network by means of synchrotron radiation X-ray phase contrast tomography

      2016, Physica Medica
      Citation Excerpt :

      The tomography data was acquired in half acquisition mode with 4000 projections covering a total angle range of 360 and an acquisition time of 1 s per projection. Phase retrieval was performed using a single distance method developed by Paganin [29]. Image analysis were carried out by means of ImageJ software and 3D rendering representations were generated using VG Studio Max 2.0 (Volume Graphics, Heidelberg, Germany).

    • Principles of Electron Optics, Volume 4: Advanced Wave Optics

      2022, Principles of Electron Optics, Volume 4: Advanced Wave Optics
    View all citing articles on Scopus
    View full text