Cost Function Ridge Extaction Algorithm
Ridge detection has an important role to play in the process of using wavelet transforms for fringe pattern analysis, and more specifically, it is the fundamental tool in the phase extraction stage. By definition, the ridge corresponds to the maximum of the CWT modulus. However, it must be born in mind that the modulus will have a maximum value when the daughter wavelet frequency is very close to the frequency of the original signal.
The direct maximum method is the most popular technique that is used in the ridge extraction procedure. This is probably due to the simplicity with which it may be implemented. This technique can be used when the signal to noise ratio of the fringe pattern is high. In the case of a very noisy fringe pattern being processed using the 1D-CWT method, it is very probable that the magnitude of the CWT contribution from the noise will exceed that which is contributed by the signal itself. Consequently, in this scenario the direct maximum method would compute the phase of the noise and erroneously consider it to be the required fringe phase information.
The cost function method is capable of dealing with higher levels of noise than the direct maximum method. This is due to the inherent assumption in this algorithm that both the unwrapped phase and its derivative are continuous. This assumption is valid because the WTP method only has the capability to process fringe patterns that do not contain phase discontinuities. Dynamic programming is used to search for the optimum ridge path through the local maxima in the modulus of the CWT.
The use of a cost function was introduced, in conjunction with the 1D-CWT as a tool for ridge determination and is given by:
where sc(b) represents any value of the scaling parameter s, the term b is the shifting parameter in the x-axis, and S[sc(b), b] is the modulus value at both sc(b) and b. The parameters Co and C1 are weighting coefficients of the modulus S[sc(b), b] and the gradient of sc(b) respectively.
Equation (1) calculates the cost of the step variation. As this algorithm searches through many local maxima, then many different potential paths may occur. However, an optimal path should be selected and considered to be the true ridge of the transform.
This algorithm selects the subset of several local maximum points from each column of the modulus, instead of a single global maximum value. These local maxima values will be considered as the potential candidate points for the ridge. A matrix consisting of all the candidate ridge points for all columns will be built up, and many different potential ridge paths can be found. These will be evaluated using the cost function. The ridge with a minimum final cost will be extracted, and this is the path that will be defined as representing the true ridge of the CWT.
For a detailed explanation of the cost function ridge extraction algorithm, please see AbdulBasit PhD thesis. A Matlab program that demonstrates the operation of this algorithm can be downloaded here.
