edfas.org ELECTRONIC DEVICE FAILURE ANALYSIS | VOLUME 26 NO. 2 12 Mallat’s original work[13,15] and previous publications.[1–3] The first-order invariants S1 consist of 64 numbers, which are constructed according to the analysis results by a system of wavelets with eight scales and each scale with eight orientations. The invariants are then plotted according to the scale index J and the orientation index L, that is, J = 1, L = 1,..., J = 1, L = 8, J = 2, L = 1,..., J = 8, L = 1,..., J = 8, L = 8. A curve connecting those data points according to this order is called a µSHD curve, and the invariant on a certain scale J and orientation L after taking the logarithm (base 10) is called a µSHD value or simply a µSHD. The µSHD curves in Fig. 1d represent, systematically, quantitatively, and for the first time, what the differences are between the secondary and backscattered images of the same object. In terms of features on different scales, the µSHDs on scales J = 1 and J = 2 and for all orientations on curve b from the backscattered electron image are larger than those on curve a from the secondary electron image. However, the µSHDs on scales from J = 3 to J = 8 on curve a grow increasingly larger with scales and become larger than their counterparts on curve b. The features in different orientations of both curves seem to match very well, that is, both curves show peaks at L = 3 and L = 7 on smaller scales, i.e., J = 1 and J = 2, but consistently at L = 5, that is, the horizontal direction, from J = 3 to J = 8. Reexamining Figs. 1a and b can find that the majority of the information from the aluminum line is indeed horizontal. Note that the µSHD approach differs significantly from the currently available AI approach in MFA. First, the µSHD approach is based on the mathematical principles behind AI and is interpretative and can eliminate the deep learning process intensive with computation. In other words, machine learning and AI-based models can only become more efficient and interpretative if µSHD is used directly as a quantitative descriptor. Furthermore, µSHD treats spectra and images as systems rather than individual signals and can utilize the concept of structural hierarchy to derive deeper insights behind the data. In summary, all the spectra and images in MFA can be systematically quantified within the same framework using the µSHD. With the maturity of data collection and benchmarking for different applications, µSHD enabled interpretative AI framework for MFA can be expected. The following section provides a few examples that highlight such efforts. CASE STUDIES SCANNING ACOUSTIC MICROSCOPY Figures 2a, b, and c show the SAM images of an IC package with an in- crease in measured frequency com- Fig. 2 SAM images of an IC package measured with a frequency and focal length setting of (a) 75 MHz and 10 mm, (b) 120 MHz and 5 mm, and (c) 180 MHz and 5 mm. (d) Redraws of (a) with the wavelets of the analysis superimposed. (e) Plots of the µSHD curves of the SAM images. (a), (b), and (c) are adapted from Hartfield et al.[17] (a) (b) (d) (e) (c)
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