Edge detection remains challenging under the strong noise and low-contrast conditions typical of scanning electron microscopy (SEM) images. While fractional-order operators improve robustness over classical derivatives, selecting the fractional order is critical, and existing adaptive strategies either rely on noise-sensitive local variance or require heavily tuned noise-suppression pipelines. This work introduces ABC-AED, an adaptive edge detection method based on the Atangana–Baleanu fractional operator in the Caputo sense, which exploits its smoothing and fading-memory properties to build a fractional edge mask whose order is adaptively determined from a bilateral-filter-based local intensity map, without training data or per-image calibration. The method is evaluated on natural images, noise-corrupted natural images, and SEM images of nanoparticles, and compared against a local-variance-based fractional operator (2D-AFCD) and an adaptive anisotropic diffusion scheme combined with Canny (AAPDE-Canny) using Jaccard index, F1-score, Precision, and Recall. ABC-AED achieves accuracy comparable to, and in several SEM cases slightly better than, AAPDE-Canny, while being more than eight times faster than AAPDE-Canny, offering a favorable accuracy–efficiency trade-off for SEM-based nanoparticle characterization.