functions. Thus, the phasor representation of the measured phase distribution can be mathematically manipulated without concern for the discontinuities associated with phase wrapping. Even with the most careful ESPI measurements, speckle noise can be substantial, so some data filtering is needed in addition to the use of phasor notation. The continuous character of the phasor notation facilitates such filtering because it avoids any concerns for the discontinuities associated with wrapped phase data. Even so, the gauge length associated with the strain evaluation must be limited to encompass a phase difference less than 2π, else aliasing errors will occur. Data filtering must be done sparingly to avoid data smearing and consequent loss of spatial resolution. In addition, care must be taken when using phasor notation because filtering can move the phasors away from their required unit magnitude. To combat both effects, repetitive filtering [11] is used, where a modest filter is repeatedly applied to the data, with phasor renormalization done after each filter application. Repetitive filtering is highly effective, with very smooth displacement maps obtained even with small filtering kernels. However, it must be done with moderation, else errors can propagate and distort large areas. An ESPI strain calculation algorithm based on phasor format data and repetitive phasor mean filtering is described here. Particular focus is given to maintaining performance in areas of high strain concentrations. 2.2 Proposed Strain Calculation Approach ESPI measurements indicate the phase change at each pixel that is caused by deformation of the imaged surface. Given knowledge of the ESPI interferometer geometry, this phase change enables direct evaluation of the surface deformation. The most common way of making ESPI measurements uses the phase stepping technique. A sequence of speckle pattern images is taken with a specified phase step added for each image using a piezo actuator mounted within one beam of the ESPI interferometer [4]. A popular choice is to take four images with a π/2 phase step between them. For this procedure, the local phase angle φ at a given pixel can be determined from the light intensities Ii measured at that pixel within the 1 i 4 measured images: φ ¼ atan I4 I2 I1 I3 ¼ atan n d ð 2:1Þ where n and d respectively are the numerator and denominator of the fraction within the atan function. The details of the contents of these quantities will vary according to the particular phase stepping technique used, but almost all will produce a atan(n/d) type of result. The phase angle φ can be determined within the range π <φ π using the two-argument arctangent function. Phase angles outside this range are “wrapped” with modulo 2π, thus an angle 1.1 π appears as 0.9 π. Many mathematical techniques have been developed to unwrap the phase angle through comparisons with the phase of adjacent pixels [8]. However, such unwrapping can be difficult to do accurately in the presence of significant measurement noise and substantial phase gradients. Thus, it would be attractive to do the strain evaluations of interest here without the need for initial phase unwrapping. ESPI measurement noise also seriously impedes attempts to determine the phase gradients to yield the desired surface strain information. This occurs because the needed differentiation is a numerically sensitive operation that enlarges the noise present in the measured data. Thus, it is necessary to do some initial smoothing of the measured phase data to reduce highfrequency noise and allow the strain evaluation to proceed reasonably. However, the amount of smoothing must be kept modest so that it does not significantly smear the underlying data and impair spatial resolution. Initial smoothing to facilitate unwrapping is impeded by the presence of the 2πjumps present in the raw (wrapped) phase data. Conversely, initial unwrapping to facilitate smoothing is impeded by the presence of noise. A way forward is to choose a smoothing method that does not require prior phase unwrapping. The approach chosen here is to express the phase angle φ in Eq. (2.1) in phasor format [10]: Φ¼cosφþi sinφ¼ d r þ i n r ð 2:2Þ where r ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffifffifi d2 þn2 p and i ¼ ffiffiffiffiffiffi 1 p ð2:3Þ 26 J. Heikkinen and G.S. Schajer
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