Category: Alumnae/i

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.

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

In physics the phase problem is the name given to the problem of loss of information concerning the phase that can occur when making a physical measurement. The name itself comes from the field of x-ray crystallography, where the phase problem has to be solved for the determination of a structure from diffraction data.

The phase problem is also met in the fields of imaging and signal processing. Various approaches have been developed over the years to solve it.

More specifically, we know that light detectors can only measure the intensity of light. However, light doesn’t only comprise of intensities. The phase information that light carries is also very important for various analytic purposes. For example, these phase information are crucial to calculating the distance between Braggs planes. Thus, methods have been developed to retrieve this information.

1)  Guessing Structures: A structure can be approximated and the phase information of the diffraction pattern of this guessed structure is added to the experimental pattern (without phases). Then, this augmented pattern can then be Fourier transformed to return the real original structure.  We can use this for more basic shapes of C. Elegans.

2) Direct Methods: For a crystal, if we assume it is made up of similarly-shaped atoms that all have positive electron density, there will be statistical relationships between sets of structure factors (resulting diffraction pattern including phase info). These statistical relationships can be used to deduce possible values for the phases. Direct methods exploit such relationships, and can be used to solve small molecule structures.

3) Multiple isomorphous replacement: A change will be made to the original structure so that it will perturb the resulting patterns and, by the way that these patterns are perturbed, some deductions can be made about possible phase values.

4) Phase Oversampling: Oversampling is developed originally to avoid aliasing by taking more samples (more k-space lines with the same overall k-space coverage). More recently, it’s been suggested that sampling the diffraction pattern of a finite specimen more finely than the Nyquist frequency (the inverse of the size of the diffracting specimen) can help retrieve phase information. This is the method that we will most likely use for more complicated shapes of C. Elegans.

Other methods that are less related to sampling objects but more related to crystallography include Anomalous Dispersion, Molecular replacement, Patterson Function and so on.

A few interesting references sources include:

Phase Problem: http://www-structmed.cimr.cam.ac.uk/Course/Basic_phasing/Phasing.html

Pictorial 2-D Transforms: http://www.ysbl.york.ac.uk/~cowtan/fourier/fourier.html

“The link between C. elegans‘ physical and neurological responses to vibrations”

” Impact of environmental toxins on the development and reproduction of the nematode, C. Elegans.”

“Utilizing C. elegans’s brain wiring to run an electronic robot that could one day be a model for a cheap, artificial eel that can locate explosive mines at sea.”

“The crosstalk between thermotaxis and chemotaxis of C. elegans”

“Inducing Caenorhabditis elegans worms to perform activities such as reversing direction, changing speeds, and laying eggs through shining lasers on several of its neurons.”

…

“C. elegans has become a favorite model system for the study of development due to the recent discovery that its genetic makeup differs very little from that of a fly, fish, mouse, or human.”

–Professor Greg Hermann

C. Elegans.

I never actually thought that such small creature could still draw so much attention from a variety of disciplines nowadays. Because of their simple neuron structures and direct corresponding movements in the water, their behaviors are of interest to neuroscience,  biology, and just recently through our studies–physics.

Why is it related to physics all of a sudden?

Here is the reason–lasers are great motion detectors since their travel paths are very accurate and any small alterations in movements can be reflected in pattern changes in laser.

Thus, one important component of this research is to carry out the laser diffraction  for the movements of C. elegans in a curvette and record their patterns while they are allowed to freely swim in three dimensions. The advantage of such observations, compared to using a microscope, is that the C. elegans are allowed to freely move about instead of being restricted to two dimensions in Petri dishes under the microscope.

So far, the diffraction patterns have been analyzed to understand the frequency of the worm’s movement and how its movement is affected by its environment. The experiment then moved to a higher stage which is allowing for the Fourier transform of the original diffraction pattern. Fourier transform allows very scattered and different patterns to be expressed  very uniformly through similar terms. Thus, we now have the worm and the diffraction patterns of its movements.

Where are we looking forward to going with all this?

Of course, we all know the common phase problem with the Fourier transform analysis, where phase information of the original pattern is lost, which is exactly what happened to our Fourier transform results. Thus, by comparing the Fourier transform results and the real image of the worm moving at that precise moment, we can gather more info regarding the operational methods of the transform.

In the end, we hope to retrieve the phase information of the transform, and eventually be able to predict the original shape of the worm movement. Currently, we are able to reconstruct a small section of a worm. The general shape of the worm has been quite successfully outlined, even thought the location of the worm is not exactly correct.

We have much to hope in this line of pursuit. We will go on to try retrieving the phase information of the worm movement, and discover better algorithms to return the diffraction patterns to its original image. Moreover, we’re hoping that in the future, a 3-dimensional analysis of the movement of the worm is possible through laser diffraction. This way, the advantage of the Fourier transformation method can more clearly demonstrate its effectiveness and large potential in the field of biology and related observational sciences.