Overview of the Iterative Algorithm for Phase Retrieval

In the previous post, the reason that only oversampled patterns can be reconstructed was introduced.

The next question is then–how do we construct these patterns and how can we retrieve the phase quantitatively? Here’s a overview of the iterative algorithm that is popular in the “Phase Retrieval world”, especially for nonperiodic objects.

First, let’s list out a few possible constraints, including the one we introduced in the last post, that we usually apply to the retrieval process to make sure that the phase we get back is what we originally have:

1) Creating known-valued pixels. For example, we could create an object with some non-scattering density (zero-valued pixels) inside it, such as the center of the object. In this method, a concept similar to the oversampling ratio comes about, which is a ratio calculated by “total pixel number/unknown-valued pixel number”. This has to be larger than 2 for the reconstruction, just as the oversampling ratio has to be larger than 2.

2) The previous introduced Oversampling method. Basically, with an oversampling ratio larger than 2, we can create a finite support for the object where the pixels outside this support are all zero, creating zero-pixels constraints again.

3) Apart from the external constraints of 1) and 2), we also have an internal constraint which is the positivity constraint. A complex valued object density can be expressed using complex atomic scattering factor, f1+if2. f1 is the effective number of electrons that diffract the photons in phase, which is usually positive. f2 is the attenuation and is also positive for ordinary matter. So the fact that these two values should usually be positive could serve as positivity constraints for the phase retrieval process.

Now that we’ve learned all the constraints, we should look at how the retrieval method is actually carried out through the iterative algorithm:

1) The measured magnitude of the Fourier transform is obtained through the diffraction pattern. We will combine it with a randomly created phase set and generate a new Fourier transform.

2) This Fourier transform is then inversely fast Fourier Transformed to create a new “image density”.

3) Through the oversampling ratio, a finite support is defined in real space for the separation of the density and no density region. For density outside the support, we enforce it to be 0, and for the density inside, we enforce the positivity constraints. These are usually enforced by the following equations, where f ‘ is the object density before applying the constraints and f is object density after conforming to constraints (which is also the S set).  The second line is to set pixels outside the support gradually to zero, and the “f1” “f2” increase at every iteration until both positive.

After these are constraints enforced, we can obtain a new image density f that belongs to S.

4) With the new image density after the enforced constraints, we obtain a new Fourier transform of the image and adopt its phase set while restoring its central pixels to zero (the center of a diffraction pattern can not be experimentally measured). We have a new phase set which we can combine with the magnitude of the Fourier Transform again.

Usually after a few hundreds to thousands of iterations like this, convergence would be complete and we will be able to reconstruct the original image through retrieved phases.

Dummies Intro to Oversampling Phasing Method

Before introducing the concept of oversampling, let’s first talk about an effect named “aliasing” that is just as important.

Aliasing

An example of aliasing can be seen in old movies, especially when watching wagon wheels on old Western films. You would occasionally see the wheels as if they going in reverse. This phenomenon occurs as the rate of the wagon wheel’s spinning approaches the rate of the sampler (the camera operating at about 30 frames per second).

The same thing happens in data acquisition between the sampler and the signal we are sampling.

Nyquist Theorem

A theorem that states the relationship between the acquired data and sampling frequency (rate of sampler) is stated as the Nyquist Theorem. It states that 2 samples per “cycle” of input signal is needed to define it the input signal. Thus, a signal with frequency f can be accurately measure as long as you are sampling it at greater than 2f.

The following picture is a Frequency versus amplitude plot showing an aliased signal, fa, which occurs due to “aliasing back” from the original signal of 70MHz where

R (sampling rate) = 100MS/s
fs (signal being sampled = 70MHz
fN (the Nyquist frequency) = 50MHz
fa (aliased frequency) = 30MHz

Oversampling a signal

Applying the concept of Nyquist Theorem, we can see that oversampling is sampling at a rate beyond twice the highest frequency component of interest in the signal and is usually desired. Because real-world signals are not perfectly filtered and often contain frequency components greater than the Nyquist frequency, oversampling can be used to increase the foldover frequency (one half the sampling rate) so that these unwanted components of the signal do not alias into the passband.

Oversampling an object through diffraction

In the case of sampling an object, the Nyquist frequency becomes the inverse of the size of the diffracting specimen, and the sampling rate is the laser frequency. As suggested in the last post, reconstruction of an image through its diffraction pattern is a very important subject in our current research, and “the phase problem” that was introduced will be closely tied to the the Nyquist Theorem, where the diffraction pattern of a finite specimen has to be more finely sampled than the Nyquist frequency.

According to the oversampling phase method, the method above corresponds to surrounding the electron density of the specimen with a no-density region. When the no-density region is bigger than the electron-density region, sufficient information is recorded so that the phase information can in principle be retrieved from the oversampled diffraction pattern.

Reference Cites:

yoksis.bilkent.edu.tr/pdf/10.1364-AO.39.005929.pdf

http://zone.ni.com/devzone/cda/tut/p/id/3000

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