用积分法替代梯度计算，提升超声显微成像运动校正效率

Tissue motion correction through image registration is essential for ultrasound localization microscopy (ULM). Parametric image registration is commonly formulated as an optimization problem where motion parameters are iteratively updated to maximize image similarity, and used optimization algorithms typically rely on gradient information, the explicit evaluation of which can become computationally demanding. This work investigates Extremum Seeking Control (ESC) as an alternative to explicit derivative evaluation in image registration. By obtaining descent information via integrating perturbed and demodulated image similarity metric across iterations, ESC avoids differentiation of the image similarity metric with respect to motion parameters in each iteration. The classical ESC, whose optimization behavior approximates that of classical gradient descent (GD), is first compared with GD for affine image registration using simulated ground-truth motions derived from a beating ex vivo porcine heart dataset. The results show that ESC achieves registration accuracy and convergence behavior comparable to GD while reducing per-iteration computational cost by approximately 3.5-fold. ESC is subsequently employed in a two-stage motion correction pipeline, where affine registration compensates for global tissue motion and B-spline registration corrects residual local deformation. The proposed method is applied to ULM imaging of a beating ex vivo porcine heart and achieves a spatial resolution of 219 um, substantially below the half-wavelength diffraction limit of 321 um associated with 2.4 MHz diverging-wave imaging. These results demonstrate that ESC provides an effective alternative to explicit derivative evaluation in ULM image registration, enabling accurate motion correction and high-quality super-resolution imaging.