Training-free autofocus via turning-point analysis of a frequency-slope metric

Robust autofocus in optical microscopy remains challenging under low-contrast, transparent, and illumination-varying conditions, where conventional focus metrics often exhibit unstable axial responses or significant depth deviations. Although deep learning-based approaches improve robustness, their reliance on labeled datasets, substantial computational overhead, and limited interpretability constrain practical deployment across diverse imaging scenarios. This work presents a deterministic, training-free autofocus framework grounded in the intrinsic geometric structure of the Fourier-based frequency-slope response, f_FS(z). From a physics-informed perspective, the axial evolution of f_FS(z) is governed by defocus-induced transitions in the optical transfer function, leading to highly reproducible turning-point structures that delineate the boundary between structure-dominated and background-dominated regimes. Based on this principle, a rule-based decision strategy is developed using only three descriptors-the two localized turning points (z_L, z_R) and the global maximum (z_max)-to accurately localize the focal plane and classify challenging imaging regions without any trial-and-error parameter tuning. Unlike conventional extremum-dependent metrics, which degrade under high-illumination and overexposed conditions, the proposed framework explicitly incorporates transition geometry, allowing all decision thresholds to remain fixed across entirely distinct specimen types. Extensive experimental validation spanning 5,400 continuous line-scans on reflective metallic and biological specimens demonstrates focus tracking accuracy of up to 98-100%, matching the performance of convolutional neural network-based approaches while eliminating GPU dependence. These results establish an interpretable, reliable, and broadly applicable autofocus paradigm that bridges empirical image processing with predictable microscopic automation.