Phase retrieval

Phase retrieval is the process of algorithmically finding solutions to the phase problem. Given a complex signal $F(k)$, of amplitude $|F(k)|$, and phase $\phi (k)$:

$F(k)=|F(k)|e^{i\phi (k)}=\int _{-\infty }^{\infty }f(x)\ e^{-2\pi iu\cdot x}\,dx$

where x is an M-dimensional spatial coordinate and k is an M-dimensional spatial frequency coordinate, phase retrieval consists in finding the phase that for a measured amplitude satisfies a set of constraints. Important applications of phase retrieval include X-ray crystallography, transmission electron microscopy and coherent diffractive imaging, for which $M=2$. (Fineup 1982:2759)

Methods

Error reduction algorithm

Schematic view of the error reduction algorithm for phase retrieval

The error reduction or Gerchberg-Saxton algorithm solves the signal equation for $f(x)$ by the iteration of a four-step process. In the first step, an estimate of the object $f(x)$, $g_{k}(x)$ undergoes Fourier transformation:

$G_{k}(u)=|G_{k}(u)|e^{i\phi _{k}(u)}={\mathcal {F}}(g_{k}(x))$

The experimental value of $|F(u)|$, calculated from the diffraction pattern via the signal equation, is then substituted for $|G_{k}(u)|$, giving an estimate of the Fourier transformation:

$G'_{k}(u)=|F(u)|e^{i\phi _{k}(u)}$

Next, the estimate of the Fourier transformation $G'_{k}(u)$ is inverse Fourier transformed:

$g'_{k}(u)=|g'_{k}(x)|e^{i\theta '_{k}(x)}={\mathcal {F}}^{-1}(G'_{k}(u))$

$g'_{k}(x)$ then must be changed so that the new estimate of the object, $g'_{k+1}(x)$ satisfies the object constraints. $g'_{k+1}(x)$ is therefore defined piecewise as:

$g'_{k+1}(x)\equiv {\begin{cases}g'_{k}(u)&x\notin \gamma \\0&x\in \gamma \end{cases}}$

where $\gamma$ is the domain in which $g'_{k}(x)$ does not satisfy the object constraints. As the calculated image, $g'_{k}(x)$, is complex valued, $|g'_{k}(x)|$ is replaced by the experimental value of $|f(x)|$, to give the new estimate of the object. (Fienup and Wackerman, 1986:1899)

$g_{k+1}(x)=|f(x)|e^{i\theta '_{k+1}(x)}$

This process is continued until both the Fourier constraint and object constraint are satisfied. Theoretically, the process will always lead to a convergence (Fineup 1982:2761), but the large number of iterations needed to produce a satisfactory image (generally >2000) results in the error-reduction algorithm being unsuitably inefficient for sole use in practical applications.

Hybrid input-output algorithm

The hybrid input-output algorithm is a modification of the error-reduction algorithm - the first three stages are identical. However, $g_{k}(x)$ no longer acts as an estimate of $f(x)$, but the input function corresponding to the output function $g'_{k}(x)$, which is an estimate of $f(x)$ (Fienup 1982:2762). In the fourth step, when the function $g'_{k}(x)$ violates the object constraints, the value of $g_{k+1}(x)$ is forced towards zero, but optimally not to zero. The chief advantage of the hybrid input-output algorithm is that the function $g_{k}(x)$ contains feedback information concerning previous iterations, reducing the probability of stagnation.

$g'_{k+1}(x)\equiv {\begin{cases}g'_{k}(u)&x\notin \gamma \\g_{k}(x)-\beta {g'_{k}(x)}&x\in \gamma \end{cases}}$

Here $\beta$ is a feedback parameter which can take a value between 0 and 1. For most applications, $\beta \approx 0.9$ gives optimal results.

Shrinkwrap

For a two dimensional phase retrieval problem, there is a degeneracy of solutions as $f(x)$ and its conjugate $f^{*}(-x)$ have the same Fourier modulus. This leads to "image twinning" in which the phase retrieval algorithm stagnates producing an image with features of both the object and its conjugate. In the shrinkwrap technique the object support constraint is repeatedly convolved with a Gaussian and thresholded, leading to a reduction in the image ambiguity. (Fineup and Wackerman, 1986:1900)

References

• Fienup, J.R. (1 August 1982). "Phase retrieval algorithms: a comparison" (PDF). Applied Optics. 21 (15): 2758–2769.{{cite journal}}: CS1 maint: date and year (link)
• Fienup, J. R. and Wackerman, C.C. (3 July 1986). "Phase-retrieval stagnation problems and solutions" (PDF). Journal of the Optical Society of America. 3 (11): 1897–1907.{{cite journal}}: CS1 maint: multiple names: authors list (link)

See also

• Phase problem
• Crystallography
• X-ray crystallography