Currently used imaging methods in Atomic Force Microscopy (AFM) including the use of a Lock-In Amplifier or a Phase-Locked Loop, are suboptimal. In this report, the image resolution in AFM is improved by detecting the tip-sample interaction using complete measurements of the input of the cantilever and its measured deflection. Two methods are studied while assuming that the tip-sample interaction is sparse, namely a model-based approach and a data-driven approach. Real-life experiments have shown that the model-based approach improves the image resolution with a factor of 7.5 to 0.555 nm compared to the conventional imaging method, according to a metric using Fourier Ring Correlation in which a reference image is unnecessary. The data-driven approach can be used in the model-based approach to further improve the resolution. In addition to improved resolutions, a Linear Time-Invariant model of the mechanically driven AFM-cantilever immersed in liquid – from piezo input to cantilever deflection – has been obtained through subspace identification with a Variance Accounted For of 79.2%. Recommendations for future research include applying the latter model in detecting the tip-sample interaction, improving the data-driven approach, reducing the computational effort of the model-based approach and implementing algorithms for detecting the tip-sample interaction online.