February_EDFA_Digital

edfas.org ELECTRONIC DEV ICE FA I LURE ANALYSIS | VOLUME 24 NO . 1 12 traditional typewriter used to work or an older cathode ray tube television). After the flyback, the beam typically has a longer dwell time at the left-edge to allow for any hysteresis in the scan to dampen out and the left-edge of the scan to be aligned at the same location for each row. The beamsize is typically the same regardless of themag- nification of the image, which can be as small as ~0.1 nm for an aberration corrected (Cs-corrected) STEM. [7,8] In a low magnification image, the area of the scan is large and the pixel size is therefore much larger than the size of the beam. For example, to continue the discussion for a Cs-corrected STEM, in a 1000 x 1000-pixel scan covering 1 x 1 mm of sample, the pixel size is 1 µm, i.e., 1000x the size of the beam. To achieve atomic resolution, themagni- fication of themicroscope is increased to the point where the pixel size approaches the atomic separation, i.e., ~0.1-0.5 nm. For many of the most impressive atomic resolution STEM images that have been obtained from beam stable samples, the magnification is turned up to a level where the pixel size is much smaller than the probe size, leading to an oversampling of the image. When a SEM/STEM is running at low-magnification, beam damage is typically not an issue that any experi- mentalist must face, as the distance between the beam locations is very large, and the likelihood that the scan hits exactly the same location in successive sweeps is very small; damage still occurs, but it is below the scale of the intended image resolution. It is only when the beam and pixel size start to overlap that the damage becomes serious, and this is of course the condition that the microscope aims to achieve for the highest spatial resolution (Fig. 1). Thinking about the problem from the perspective of overlappingbeampositions and their effect on themeasurable damage, then it is clear that increasing the spacing of the beam positions at high magnification allows for avoiding/reducing the beam damage problem that plagues high resolution SEM/STEM (Fig. 1). Of course, the issue with this “sparse sampling” approach is there needs to be a means to “reconstruct the full image” from this sub-sampled acquisition. As the quality of the image then would obviously depend on how much sampling was included, the best or optimal sampling would be the one where the physics of the beam damage process was minimized and the ability to reconstruct the image was maximized. SUB-SAMPLING AND IMAGE RECONSTRUCTION As seen from Fig. 1, it is possible to obtain a scanned imagebyusingbotha set of “random”beampositions, and a “random walk” or “line-hop” scan. Practically, the line- hop approach is easier to implement on electron micro- scopes as it avoidsmuch of the hysteresis issue present in conventional scanning systems, permitting the system to run at the fastest possible speed. [5] Implementation of the line-hop on any electron microscope can be achieved by plugging into theexternal scanport on themicroscopeand implementing a signal generator to create the scans and record the images. [5] In viewof its simplicity to implement, the remainder of this article will discuss the implementa- tion and application of this line-hopmethod. Please note that it is possible to design a hardware solution to reduce hysteresis and permit true random scanning, but such a solution is not retrofittable to existing systems. The key challenge for all sub-sampling methods is to reconstruct the sub-sampled image. Compressive sensing and/or inpainting is a method of efficient signal acquisition and reconstruction via the solving of a set of underdetermined linear equations. [9] Like traditional image compression techniques, it relies upon the fact that given an appropriate coordinate system (or “dictionary”), complex high dimensional signals such as an image can be expressed within a margin of error by a (potentially) much smaller set of parameters, describing a linear com- bination of signal patterns with their respective scalar coefficients. The goal for any image reconstruction is to form a complete signal (with the smallest error) from as few measurements as possible. As an example of this process, consider the case of a simple one-dimensional (1D) signal, such as a wave shown in Fig. 2. Here, a series of dictionary elements (in this case Fourier components) can be used to recon- struct a true signal (Fig. 2b). But now what happens if the complete signal is not measured? Figures 2c-f show the effect of sub-sampling the true signal. For relatively high levels of sub-sampling (missing only a minimal amount of information), the dictionary elements can be fit to the observation, effectively “inpainting” the missing level of sampling in the experiment. As the level of sampling is reduced, the ability to “fit” to the data with a minimal error is reduced, until when it gets to only 2.5%of the data, THE GOAL FOR ANY IMAGE RECONSTRUCTION IS TO FORM A COMPLETE SIGNAL FROM AS FEW MEASUREMENTS AS POSSIBLE.