Statistical methods for automatic crack detection based on vibrothermography sequence-of-images data' by M. Li, S. D. Holland and W. Q. Meeker: Rejoinder
We would like to thank Professors Volf and Guerin for taking the time to carefully read our paper and prepare the thought-provoking discussions.
Professor Volf raises an important question about robustness of the matched-filter method when the crack signature is not correctly specified. Of course this is important because the crack signature is never specified exactly; different cracks can have different spatial shapes. Although the Gaussian shape provides a good agreement to most of the crack signals that we have seen, some cracks provide a signal with two peaks, one at each of the crack tips. We conducted some simple experiments, especially with respect to the spatial shape, while writing our paper and came to the same conclusion as Professor Volf. We believe that most of the value of the matched filter in this application comes from the strong and consistent temporal pattern shown on the left-hand side of Figure 3 of our paper. This pattern can be expected to be consistent across various crack morphologies. Thus, the performance of the matched filter could be expected to be robust to misspecification of the spatial pattern. It would, however, be interesting to extend Professor Volf's sensitivity study to three dimensions and systematically perturb the crack signature in various ways that are consistent with expected departures in order to get a better quantification of the misspecification effect. We are in the process of developing an improved analysis that takes advantage of a-priori knowledge of the Green's function of thermal conduction. This work tells us that the temporal pattern is more gentile (slower heating and cool-down) at the edge of the indication than the center, but that the shape of the curve is very characteristic. The spatial pattern is controlled by the geometry of the specific crack and is therefore not-suited to mathematical analysis. Since both the crack heating (neglecting noise) and the matched filter are nonnegative, and since the cross-correlation of two nonnegative functions must itself be nonnegative, we can say for certain that misspecification will in the worst case reduce the magnitude of the indication from what would have been detected with the correct matched filter.
For our application, we needed an objective detection criterion that would make an unambiguous detect/no-detect decision. As described in Section 7.2 of our article, we used an SNR criterion that has been used successfully in other NDE applications to make a decision on the basis of the output of the matched filter. Professor Volf asks whether other statistical textural characteristics of the matched-filter output might be used for making detection decisions. He provided plots of such characteristics that were computed by moving a window around the outputs of the matched filter for regions with and without a crack. There is a sharp contrast between the two sets of plots, but these differences raise other questions: How can one objectively discriminate between such pairs of plots (perhaps with a pre-selected set of the most interesting rows) to make the needed objective decision? What are the key characteristics of a real indication and what is the best domain in which to identify them?
We agree with Professor Volf that it would be interesting to compare our procedure with other possible procedures for crack detection based on sequence-of-images data and, as mentioned in