Numerical Analysis of Interference Patterns

This page is currently a work in progress. According to Kieran G. Larkin, interferogram analysis finds its roots in the 1960's from Carre Rowley and Hamon. Analysis methods can be sorted into two main categories: "temporal (phase shifting) methods and spatial methods." The methods of Carre, Rowley and Harmon lie in the range of temporal methods. This form of analysis was problematic in that the phase shift needed to be precisely calculated (i.e. a larger degree of experimental control must be present). This ushered in the wave of spatial analysis methods which took advantage of the lack of need of multiple fringe patterns to analyze. With experimental control increasing and computing power also increasing methods of the present usually are computationally involved or have a mixture of the control of the lab and the computer. The list below of analysis methods only represents a fraction of the variations of interferogram analysis (i.e. this list should not be considered exhaustive, but rather an overview of the main classes of analysis methods).

Phase Shifting Technique

As noted in the introduction, the first phase-shifting algorithms were designed in the 1960's by Carre, Rowley and Harmon. The essence of the phase shifting algorithm lies in the ability to gather the necessary phase information from (theoretically) three to (more realistically) five interference patterns, called frames, so that a nearly complete image may be rendered of the surface in question.

In recent years combinations of spatial and temporal methods have been combined and thus form a subcategory of the phase-shifting method, known as "self calibrating [sic] spatio-temporal algroithms." Among these methods are the Fourier Method, the Lissajou ellipse fitting method, the interferogram correlation method and others. Each of these have their high points and low points, mostly stemming from either the type of surface they best work for or the the number of frames required.

• requires three phase shifted fringe patterns
• the phase shift must be known
• carefully controlled conditions must be maintained

Fourier Analysis Method

A "Mexican Hat" function. (courtesy of [1])

The Fourier Analysis method of interferograms was created in 1982 by M. Takeda, H. Ina and S. Kobayashi and originally intended as an alternative to Moire Topography and the phase-shifting technique [1,2]. However, this method was ineffective at analyzing closed fringe patterns. A revision to the method, solved this problem by utilizing a Cartesian to polar coordinate transform [3]. The result of which could then be analyzed using the original method proposed.

The revised Fourier Analysis method does have several limitations. The first requirement is that the "measurement wave front be a monotonic function in the direction of the carrier frequency" [3]. For instance, if the surface to be analyzed resemble the image to the right were analyzed by the above method, it would look no different than a surface that decreased or increased from top to bottom. In order to analyze such a fringe pattern generated by such a surface, an additional fringe pattern giving the carrier frequency must be used.

[1] takeda et al 1982

[2] cuevas et al 2002

[3] ge et al 2001

Regularization Algorithms

The regularization method was created for the specific purpose of automatically demodulating "noisy" fringe patterns without any further unwrapping of the phase. Regularization algorithms involve evaluating the estimated phase field with a cost function against the actual pattern and then imposing the smoothness criterion. This method is repeated for each pixel on the phase field, until a global minimum is reached in the cost function.

Variations of the regularization method involve demodulating certain points in the low frequency region of the fringe pattern. This point can then be used to seed the estimated phase field. The algorithm then follows a somewhat analogously to crystal growing.

A drawback to this method arises from the fact "that a low-pass filtering and a binary threshold operation are required."

Cuevas, Servin

Phase-Locked Loop Algorithm

• computer simulated oscillator (VCO) needed
• phase error between the fringe pattern and the VCO vanishes

Artificial Neural Network Method

• requires carrier phase
• non-algorithmic (i.e. must have learning phase)
• types of learning include: supervised, unsupervised and reinforcement
• multi-layer: input, output, hidden neurons present

The artificial neural network approach utilizes the ability of automatons programmed with simple rules to be trained to solve complex problems when connected together in a flexible network.

Genetic Algorithm

• compares generated surfaces to goal via a cost function
• mutations alter generated solutions which are then evaluated
• this process loops until cost function conditions (set by the user) are met

Simulated Annealing

explanation goes here

ParSA

Here [2] is the link the the ParSA documentation.

The ParSA (Parallel Simulated Annealing) library is a set of classes written in C++ that can be used to solve optimization problems via a process know as simulated annealing.

The ParSA library contains many different types of

The Equation for convergence speed is:

$P\left(X_{n}\notin Cost_{min}\right)\approx \left({\frac {K}{n}}\right)^{\alpha }$

(1)

Where K and $\alpha$ are problem specific constants and $X_{n}$ is a solution of length n. Using equation (1) and test runs on smaller problems of lower order, K and $\alpha$ can be determined. Along with some suggestions provided in the ParSA documentation, progress can be made towards finding higher quality solutions at a much faster rate.

$\ln P=\alpha \left(\ln K-\ln n\right)$

(2)

The equation for warming temperature in the Aarts scheduler:

$T={\bar {\Delta C^{(|)}}}\left(\ln {\frac {m_{2}}{m_{2}\chi _{0}-(1-\chi _{0})m_{1}}}\right)^{-1}$

(3)

Table of Proposed Runs

Clustering

SA Clustering Solver

Clustering Scheduler

• SA Aarts
• SA Easy Scheduler

Multiple Independent Runs (M.I.R.)

MIR_Solver

MIR_Scheduler

Use Combinations of the Different Solver/Scheduler Classes