Analysis of speckle photographs by subtracting phase functions of digital Fourier transforms Part 2

Comparison to Image Correlation

The data set that generated the data presented in Table 5 was alternatively analyzed by image correlation via the equation,

Where Cab(m,n) is the correlation of the two specklegram images, a and b, F-1 is the inverse Fourier transform operator, and  Fa and Fb are the Fourier transforms of the two specklegrams where the asterisk indicates the complex conjugate.

This image correlation results in a peak whose displacement from zero equals the displacement of one image relative to the other. Several issues arise with this method of measurement. First, zero displacement for these operations normally lies in the upper left corner (for the DADisP program), so the quadrants of the correlation array must be re-arranged to center the zero-displacement point, and care must be taken in doing this to keep track of which point actually corresponds to zero displacement.

Next, in order to measure the image displacement to subpixel accuracy, the discrete values calculated by Eq. (9) must be modeled by a continuous function and its maximum located between the discrete values. As pointed out by Sjodahl and Benckert10, the function used to do this modeling will influence the results obtained. The model chosen here is a Gaussian function. An array of 10 by 10 values surrounding the peak was selected and the natural logarithms calculated of those values. Ideally, these values should be fitted to a bi-quadratic function, to fit the data to a Gaussian function, but a cubic spline function was used instead, because that was available in the DADisP program, and interpolation was performed to 100th of  the spacing between the correlation values. For this data, the magnification was 12.56 so that the pixel spacing on the object was 92.94 ^m, and interpolation to 100th of that spacing resulted in a resolution of 0.93

Table 7. Comparison of displacement measurement obtained via the FFT shift theorem and via image correlation using the same data as for table 5.

Discussion and Conclusions

Based on these results, we may state that the process presented here, using incoherent speckles and 8-bit digitization, can measure displacements to within a few micrometers and strains to approximately 30 microstrain. In this setup, the gage length for the strain measurement is approximately 15 mm owing to the magnification of the lens system. With one-to-one imaging, the gage length could be reduced to about 1.2 mm, and the accuracy should remain the same. The process is considerably less accurate using laser speckles, due probably to speckle decorrelation. The laser speckle decorrelation results, most likely, from the relatively small aperture of the camera lens and the small fill factor of the detector array. The distinction between laser and incoherent speckles observed here should be observable with image correlation as well and should be investigated.

The displacement results obtained for image correlation are shown here to be of similar accuracy to the method of transform phase subtraction. Each method has advantages and problems, however, due to the digital nature of the data. With the phase subtraction method, there is a clear connection between the phase difference of Fourier transforms and displacement via the shift theorem, and it is natural to fit the digital values obtained to a linear function of transform coordinates. This method requires applying phase unwrapping to the wrapped phase difference of two transforms; however, the unwrapping method used here, via calculated phase unwrap regions, is extremely fast and robust. Although this whole process has not been integrated into a single program, there is no reason to believe that such an integrated program would not be as rapid and easy to implement as existing correlation programs. The correlation method is straightforward in that it measures the translation needed to get the best correlation between the two images. Subpixel resolution can only be obtained, however, by interpolating between the discrete values obtained, and this is dependent upon the nature of the model of the continuous function used to interpolate between pixels. A Fourier series expansion was shown to work well in Ref. 9 and a cubic spline fit to the natural logarithm of the correlation values was shown to work well here; however, there really is no compelling argument for any model over another beyond its demonstrated accuracy. In any case, the two measurement methods are distinctly different and equivalent only to the extent that they give similar results when analyzing the same data. It is not the intent of this communication to present the FFT phase subtraction method as better than the correlation method but simply to present it as an alternative that, with future investigation, may possibly be shown to have advantages.