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Standard approaches to high-speed AFM rely on moving the tip faster along the raster-scan pattern. Recent work has developed a non-raster scheme for string-like samples such as biopolymers that reduces imaging time by reducing the total number of measurements. This local raster-scan algorithm steers the tip in a sinusoidal path along the sample, acquiring only data near the sample of interest. It has been shown that for a particular class of samples, an order-of-magnitude reduction in imaging time is possible. In this paper we build upon that approach but consider time-lapse imaging. In this scenario, the prior knowledge from previous images can be used to setup a time-optimal control problem between successive crossing points along the string-like sample. Using recent results in time-optimal control, we solve this problem and illustrate that it can, in principle at least, yield at least an additional order of magnitude improvement in the imaging rate.