The hybrid input-output (HIO) algorithm for phase retrieval is a modification of the
error reduction algorithm for retrieving the phases in
coherent diffraction imaging
Coherent diffractive imaging (CDI) a computational microscopy method that reconstructs images from coherent diffraction patterns without the use of lenses. It was first experimentally demonstrated in 1999 by Miao and collaborators using synchro ...
. Determining the phases of a diffraction pattern is crucial since the diffraction pattern of an object is its
Fourier transform
In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...
and in order to properly invert transform the diffraction pattern the phases must be known. Only the amplitude however, can be measured from the intensity of the diffraction pattern and can thus be known experimentally. This fact together with some kind of
support constraint can be used in order to iteratively calculate the phases. The HIO algorithm uses negative feedback in Fourier space in order to progressively force the solution to conform to the Fourier domain constraints (support). Unlike the error reduction algorithm which alternately applies Fourier and object constraints the HIO "skips" the object domain step and replaces it with negative feedback acting upon the previous solution.
Although it has been shown that the method of error reduction converges to a limit (but usually not to the correct or optimal solution)
there is no limit to how long this process can take. Moreover, the error reduction algorithm will almost certainly find a local minimum instead of the global solution. The HIO differs from error reduction only in one step but this is enough to reduce this problem significantly. Whereas the error reduction approach iteratively improves solutions over time the HIO remodels the previous solution in Fourier space applying negative feedback. By minimizing the
mean square error
In statistics, the mean squared error (MSE) or mean squared deviation (MSD) of an estimator (of a procedure for estimating an unobserved quantity) measures the average of the squares of the errors—that is, the average squared difference between ...
in Fourier space from the previous solution, the HIO provides a better candidate solution for inverse transforming. Although it is both faster and more powerful than error reduction, the HIO algorithm does have a uniqueness problem.
Depending on how strong the negative feedback is there can often be more than one solution for any set of diffraction data. Although a problem, it has been shown that many of these possible solutions stem from the fact that HIO allows for mirror images taken in any plane to arise as solutions. In
crystallography
Crystallography is the branch of science devoted to the study of molecular and crystalline structure and properties. The word ''crystallography'' is derived from the Ancient Greek word (; "clear ice, rock-crystal"), and (; "to write"). In J ...
, the scientist is seldom interested in the atomic coordinates relative to any other reference than the molecule itself and is therefore more than happy with a solution that is upside-down of flipped from the actual image. A downside is that HIO has a tendency to escape both global and local maxima. This problem also depends on the strength of the feedback parameter, and a good solution to this problem is to switch algorithm when the error reaches its minimum. Other methods of phasing a coherent diffraction pattern include
difference map algorithm
Difference commonly refers to:
* Difference (philosophy), the set of properties by which items are distinguished
* Difference (mathematics), the result of a subtraction
Difference, The Difference, Differences or Differently may also refer to:
Mu ...
and "relaxed averaged alternating reflections" or RAAR.
See Also
*
Phase retrieval
Phase retrieval is the process of algorithmically finding solutions to the phase problem. Given a complex spectrum F(k), of amplitude , F (k), , and phase \psi(k):
::F(k) = , F(k), e^ =\int_^ f(x)\ e^\,dx
where ''x'' is an ''M''-dimensional spat ...
*
Gerchberg-Saxton algorithm
References
{{DEFAULTSORT:Hybrid Input Output (Hio) Algorithm For Phase Retrieval
Diffraction