Generic algorithms

This algorithms use known values of the phase shift   , which means that only three unknown of the interference equation remain to be determined:  a, b , Δφ. Therefore, only three interferograms are needed in order to solve the problem. A generic formulation of an algorithm with N interferograms can be obtained considering the least square criteria. Greivenkamp [7] used this approach in 1984. The interferogram can be written:

The unknown are now: a0=a, a1=b cos( Δφ) and a2=b sin(Δφ). In the least square sense, the interferograms must minimise the following criterion:

is minimum when the partial derivatives with respect to the three parameters are equal to zero:

This leads to a linear system of three equations with three unknowns. We have:

which can also be written as:

with

Inverting this equation gives:

Therefore, by inverting , we obtain a0, a1 and a2 and the optical phase by:

The amplitude of modulation at each point of the interferogram is

If we judiciously choose the phase shift, the matrix   becomes diagonal. Indeed, if with , the expression giving Δφ becomes simple:

With and , we recover the 4-images-algorithm previously described:

and with and , we recover the 3-images-algorithm [6,8] :

However, the algorithms built according to this scheme do not always have an optimal behavior in the presence of error sources. This is why other approaches have been developed. Phase shifting algorithms robust to random errors can be conceived by using the maximum likelihood theory, or, for systematic errors, by using a combination of pre-existing algorithms.

For example, in 1983 J. Schwider [9] showed that an average value of the optical phase can be computed using a classical algorithm. The strategy relies on evaluating the phase with the N-1 first interferograms, and with the N-1 last interferograms, and to calculate the mean of the two results. If we consider and , we have one one hand

and on the other hand

and the average value is given by:

This type of algorithm is less sensitive to calibration errors of the element producing the phase shift.