Quantitative phase retrieval using coherent imaging systems with linear propagators
David Paganin*, Timur E. Gureyev , Konstantin M. Pavlov*, Robert A. Lewis* and Marcus Kitchen* *School of Physics and Materials Engineering, Monash University, # Victoria 3800, Australia Commonwealth Scientific and Industrial Research Organisation, Manufacturing and Infrastructure Technology, Private Bag 33, Clayton South, Victoria 3169, Australia
Abstract We consider the problem of quantitative phase retrieval from images obtained using a coherent shift-invariant linear imaging system whose associated propagator (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 propagator is applicable when the spread of spatial frequencies, in a given input twodimensional complex wavefield, is sufficiently narrow when compared to the propagator of the 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, then the phase-amplitude reconstruction can be performed using a single image.
1. Two quite different imaging techniques yield similar images: why? Below are a pair of images obtained using quite different imaging techniques. The image on the left is a visible-light differential interference contrast (DIC) micrograph, while the image on the right is an x-ray diffraction-enhanced image (DEI). The similarity between these images suggests that the associated phase-retrieval problem can be attacked from a common theoretical viewpoint. (a) DIC image of neuroblastoma cells* (b) DEI image of methane bubbles in intestines of one-day old mouse
* Source: http://www.cellnucleus.com/Structure%20Database /Nucleolus/Nls.htm
2. Coherent imaging systems with a linear propagator Coherent linear imaging systems can be described by a propagator formalism. One takes the Fourier transform of an input two-dimensional complex wavefield, and then multiplies by the “propagator”, to give the Fourier transform of the output wavefield. Suppose that the Fourier transform of the input wavefield is non-negligible over the region of Fourier space indicated by the red dot. Suppose the propagator, denoted by the contour lines, is well approximated by a linear function over the “red patch” region. We call this a “linear propagator”. This is a good approximation for DEI
3. The forward problem: From input complex wavefield to output image If a wavefield with intensity IIN(X,Y) and phase IN(X,Y) is input into a coherent linear imaging system with a linear propagator, the output intensity IOUT(X,Y) is:
2 * * IN (X, Y) ImST loge I IN (X, Y) S 2ReST X X (1) I OUT (X, Y) I IN 2 2 1 2 T IN (X, Y) loge I IN (X, Y) . X 4 X
Here, S and T are complex numbers that completely specify the state of an imaging system which has a linear propagator. Note from (1) that the output of the imaging system is sensitive to the input intensity IIN(X,Y), together with the gradient of both the input intensity and the input phase. Since the output depends on the phase IN(X,Y) of the input, IOUT(X,Y) is a form of phase-contrast image.
4. The inverse problem: From output image(s) to input wavefield Eq (1) can be solved for IIN(X,Y) and IN(X,Y), given images taken using four different states of the imaging system. If we neglect the non-linear term in the braces of (1), phase-amplitude reconstruction can be done with three images. Stronger approximations give a twoimage algorithm. If we neglect the non-linear term in (1), and assume the input intensity to be constant (=I0), the reconstruction requires only a single image:
2 I OUT (X, Y) S S S (2) IN (X, Y) dX Re sign Re Im . 2 T I0 T T T
If we again neglect the non-linear term, and assume that the sample comprises a single material of projected thickness T(X,Y) and absorption coefficient , the reconstruction of T(X,Y) requires a single
(3) T(X, Y) μ loge F | S | iξ FI OUT ( X, Y),
1 1 2 1
τ γReST Im(ST ) ,
where is reciprocal to X, F = Fourier transform, and = constant.
5. Application: Quantitative reconstruction using a single DEI image DEI makes use of the extreme angular sensitivity of x-ray reflectivity from a crystal, in order to render visible the slight phase gradients imposed on hard x-rays traversing a sufficiently weakly scattering sample. DEI is a special case of our formalism. Neglect the non-linear term in eq. (1), and introduce the rocking curve R(), to give:
R( ξ 0 ) R' (ξ ξ 0 ) IN (X, Y) X (4) I DEI (X, Y) I IN (X, Y) α log e I IN (X, Y) . X
Here, IDEI(X,Y) is the DEI image, 0 is a point on the rocking curve depending on the experimental geometry, R’(-0) is the rocking curve slope, and is a constant. The last term of (4) is often neglected.
We demonstrate the single-image algorithm (3), which exactly solves (4) to give the projected thickness T(X,Y) of a single-material sample.
(a) DEI image of four nylon fibres embedded in breast phantom. (b) DEI image of three plastic discs embedded in breast phantom.
b (c) Single-image reconstruction of projected thickness, obtained from a. (d) Single-image reconstruction of projected thickness, obtained from b.