A continuous sampling pattern design algorithm for atomic force microscopy images

Undersampling is an efficient way to increase the imaging rate of atomic force microscopy (AFM). One major challenge in this approach is retaining the quality of the final images while reducing the amount of measurements enough to have a significant improvement in the imaging time. Clearly, the decision of where to acquire data points plays an important role in producing accurate image reconstructions of the sample surface with the best allocation of measurements depending on the specific sample under study. This work focuses on the development of an algorithm to design continuous non-raster scanning patterns for effective undersampled image acquisition in AFM. We assume the frequency structure of the images to be sampled, and in particular the locations of the large frequency coefficients, is partially known. Based on this knowledge, a two-stage algorithm is used to find a continuous scan pattern that minimizes the expected reconstruction error assuming reconstruction is performed using an algorithm known as Simplified Matching Pursuit (SMP). In the first stage, a collection of short, disconnected scan paths is designed to minimize the reconstruction error while in the second, these paths are connected into a continuous trajectory by solving a mixed integer linear program (MILP) that minimizes the total scan length. To demonstrate the performance of the proposed method, we design scanning patterns for four groups of sample surface images from four different materials acquired by AFM. In each case a large sample area is tiled into sixteen sub-images. The first image is raster scanned and the results used to design an optimized scan pattern that is then applied to the remaining 15 tiles. The designed patterns sampled 10-15% of the total pixels (depending on the particular sample) leading to an imaging time that was 15-20% of the time to acquire a full raster of the same area. These results were compared to sub-sampling using spiral and Lissajous scan patterns designed to have an equivalent scanning time and to sample a similar percentage of the pixels. The results show that our approach yields significantly better reconstruction quality than either of these alternative scan patterns. Finally, the method was implemented and used to image a calibration grating by sampling 18% of the total pixels, using four different tip speeds. The results were compared to full raster-scans of the sample with tip speeds selected so as to match the imaging time of the non-raster scans, showing that the proposed method produces higher-quality images than raster scanning at these fast image rates.