edfas.org 11 ELECTRONIC DEVICE FAILURE ANALYSIS | VOLUME 26 NO. 2 of magnitude involved in MFA. A systematic approach must be developed to overcome this challenge on multiple scales. C.S. Smith first proposed the concept of structural hierarchy to describe the multiscale nature of material structures.[11] He pointed out that what matters is the interlocking of the smaller scales that generate the larger overarching structures.[12] Unfortunately, no quantitative descriptor for the hierarchy of structures has been developed. S.M. Mallat studied the mathematics behind AI for image recognition,[13] that is, DCNNs, and found that the features learned from DCNNs resemble wavelet systems, which inherently maintain a structural hierarchy. Wavelet-like physical structures exist in our ears and brains, as physiology research has identified.[14] Based on these connections, the Mallat scattering transform (MST)[15] was proposed and used to dramatically reduce computing costs but maintain AI power and accuracy for image recognition. In short, the invariants from the MST are good descriptors, like the ones learned from the DCNNs. Therefore, one can naturally connect this MST-invariant system with a quantitative descriptor of the structural hierarchy, that is, µSHD, which is generally applicable to digital signals such as spectra and images. Figures 1a and b illustrate the secondary and backscattered electron images of an aluminum line with a tungsten cap layer after electromigration, and Figs. 1c and d show the analysis Morlet wavelets and the µSHD curves. Note that all input images for µSHD analysis in this work are trimmed to the same size of 512 × 512 pixels2. The first eight columns in Fig. 1c show the real part, while the rest show the imaginary part of the analysis wavelets. Each column from top to bottom corresponds to an increasing scale from J = 1 to J = 8 with the same orientation, L. Similarly, each row from left to right corresponds to an increasing orientation from L = 1 to L = 8 on the same scale J. It is noted that multiresolution wavelet analysis in the images is the first step toward the µSHD curve. To make the descriptor more robust, e.g., invariant under small image distortion or rotation, a series of invariants must be obtained following the MST. Mathematics is heavily involved in this step, and details can be referred to Fig. 1 SEM images and corresponding µSHD curves of an aluminum line with a tungsten cap layer after electromigration. (a) Secondary electron image, (b) backscattered electron image, (c) real and imaginary parts of the Morlet wavelet system for analysis, and (d) µSHD curves. (a) and (b) are adapted from Vanderlinde.[16] (a) (b) (c) (d)
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