edfas.org 13 ELECTRONIC DEVICE FAILURE ANALYSIS | VOLUME 26 NO. 2 bined with a decrease in focal length.[17] In contrast to Fig. 2a measured at 75 MHz and 10 mm focal length, Hartfield et al. claimed that better resolution and more details were obtained with an increase in frequency from 120 to 180 MHz and a focal length of 5 mm, as shown in Figs. 2b and c, respectively. Figure 2d redraws Fig. 2a with the wavelets of analysis superimposed. The real parts of the wavelets on the first to fifth scales are plotted with labels from J = 1 to J = 5, and wavelets with different orientations, labeled L = 1 to L = 7, are also shown on the scale J = 5. The feeling of better resolution and more details becomes concrete and quantified by referring to the µSHD curves of the SAM images shown in Fig. 2e. It is obvious that the µSHDs of the best image, that is, Fig. 2c, are the largest on smaller scales from J = 1 to J = 3, but become almost the smallest from J = 5 to J = 8, with the µSHDs of the three images being almost the same at J = 4. Features on larger scales, for example, from J = 5 to J = 8, sketch the outline of the image, which are already available in the images captured with lower frequencies, i.e., Figs. 2a and b. However, the features on smaller scales enrich the fine details of the images. Carefully examining the µSHD curves, one can find that the µSHDs on smaller scales in Figs. 2a and b are very similar, with levels obviously below 2c. Therefore, information on smaller scales provides better resolution and more detail in SAM images. Furthermore, the peaks on the µSHD curves appear to coincide very well in general, regardless of the measurement frequency and focal length settings. This means that the directional sensitivity under different measurement settings is similar, but fine details may differ. For example, it is the value of µSHD in L = 1, that is, the vertical direction, the maximum under the 75 MHz and 10 mm settings, but L = 5, that is, the horizontal direction, under the other two settings, on scale J = 8. With observation and understanding of the features presented in the µSHD curves of SAM images, we can establish, based on sufficient data, quantitative relationships between the µSHDs of SAM images and image quality in one direction and the measurement frequency and focal length settings in another direction. The supervised learning approach, which requires a labeled image quality score, by training a shallow neural network, can lead to a quantitative relationship between µSHDs, 64 values with a system of J = 8 and L = 8, and the image score. Similarly, a quantitative relationship between the µSHDs of an image or the low-level statistics of the µSHD curve, such as the mean, standard deviation on smaller and larger scales, and the measurement parameters, can also be obtained. Using directly the 64 µSHD values as a quantitative descriptor, unsupervised learning, such as self-organizing maps (SOM), can be used to automatically classify SAM images into different clusters. SOM results can also reveal how the measuring parameters, µSHD features, and image quality are mutually related. It should be noted that the number of features on a µSHD curve can be adjusted by varying J and L, or by using the second-order invariant S2, which generates substantially more features. After data insights are obtained, one can, for example, from the µSHDs of the SAM image, derive the measurement frequency and focal length setting, or judge whether or not the image quality is good enough under the current measurement parameters. MAGNETIC FIELD IMAGING Using the MFI technique, Hechtl[18] obtained current distribution maps in a flip-chip packaged die with top and bottom interposer metallization as shown in Figs. 3a to d, in which the current path detected by the superconducting quantum interference device (SQUID) sensor at a constant current of 1 mA for the same die with a thickness of 200, 75, 40, and 5 µm, respectively. The current image is blurred more for the case where the die is the thickest, that is, Fig. 3a. With the die becoming thinner to 75 µm and below, the image resolution improves, and the current paths on the die or even the interposer, located below the die, can be clearly identified. Being objects of interest for MFI, the current paths are better characterized by sharp lines, which correspond to µSHD curves with low energy (or low information). Therefore, it is reasonable to find that the image with the best resolution, that is, Fig. 3d, corresponds to the µSHD curve with the lowest energy, that is, the curve d in Fig. 3i. Note that the thickness of the die gradually decreases from 200 µm to 5 µm for Figs. 3a to d, but the µSHD curves do not drop accordingly. Although the µSHDs of Figs. 3a, b, and c vary little on scales from J = 1 to J = 4, the µSHDs of Fig. 3c are located between Fig. 3a and b on scale J = 5 and onward. The reason why the µSHDs on larger scales of Fig. 3c do not drop from the level of Fig. 3b lies in that larger scale features exist on top of Fig. 3c, which indeed appear to be larger than the counterpart in Fig. 3b. Hechtl further studied the effect of current, ranging from 1 mA to 25 µA with the die thickness fixed at 5 µm, on image quality, as shown in Figs. 3e to h. It is obvious that the information captured by the sensor decreases with decreasing current, although the electrical fault can still be located around the brightest area of the image with the lowest current of 25 µA. Among the µSHD curves shown in Fig. 3j, the best image, that is, Fig. 3e, maintains the lowest energy, while other images exhibit higher energy in the µSHD curves as their resolution power on current paths weakens. An obvious decrease in energy
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