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Summary: There exists a strong motivation to increase the spatial resolution of magnetic resonance imaging (MRI) acquisitions so that MRI can be used as a microscopy technique in the study of porous materials. This work introduces a method for identifying novel data sampling patterns to achieve undersampling schemes for compressed sensing MRI (CS‐MRI) acquisitions, enabling 3D spatial resolutions of 17.6 µm to be achieved. A data‐driven learning approach is used to derive k‐space undersampling schemes for 3D MRI acquisitions from 3D X‐ray microcomputed tomography (µCT) datasets acquired at a higher spatial resolution than can be acquired using MRI. The performance of the new sampling approach was compared to other, well‐established sampling strategies using simulated MRI data obtained from high‐resolution µCT images of rock core plugs. These simulations were performed for a range of different k‐space sampling fractions (0.125–0.375) using images of Ketton limestone. The method was then extended to consideration of imaging Estaillades limestone and Fontainebleau sandstone. The results show that the new sampling approach performs as well as or better than conventional variable density sampling and without need for time‐consuming parameter optimisation. Further, a bespoke sampling pattern is produced for each rock type. The novel undersampling strategy was employed to acquire 3D magnetic resonance images of a Ketton limestone rock at spatial resolutions of 35 and 17.6 µm. The ability of the k‐space sampling scheme produced using the new approach in enabling reconstruction of the pore space characteristics of the rock was then demonstrated by benchmarking against the pore space statistics obtained from high‐resolution µCT data. The MRI data acquired at 17.6 µm resolution gave excellent agreement with the pore size distribution obtained from the X‐ray microcomputed tomography dataset, while the pore coordination number distribution obtained from the MRI data was slightly skewed to lower coordination numbers. This approach provides a method of producing a k‐space undersampling pattern for MRI acquisition at a spatial resolution for which a fully sampled acquisition at that spatial resolution would be impractically long. The approach can be easily extended to other CS‐MRI techniques, such as spatially resolved flow and relaxation time mapping. Lay Description: Magnetic resonance imaging (MRI) is widely used to study the microstructure of, and fluid transport phenomena in porous media relevant for engineering applications. A major application is the study of water and hydrocarbon transport in porous sedimentary rocks, which typically have pore sizes smaller than 100 µm. The spatial resolution of routine MRI acquisitions, however, is limited to several hundred µm due to the relatively low sensitivity of the magnetic resonance method. Therefore, there exists a strong motivation to increase the spatial resolution of MRI by one to two orders of magnitude to be able to study these rocks at a pore scale. This work reports the initial step towards achieving this. Three‐dimensional images of rock pore structure are acquired at both 35 and 17.6 µm spatial resolution. In ongoing work, these methods are now being incorporated into magnetic resonance velocity imaging methods, thereby enabling imaging of both pore structure and hydrodynamics at these much higher spatial resolutions than were hitherto possible. Although X‐ray microcomputed tomography (µCT) produces high spatial resolution images, it is far more limited in being able to spatially map transport processes (i.e. flow) in porous media. This work reports a strategy for accelerating the image acquisition time such that sufficient signal‐to‐noise ratio (SNR) is achieved to increase the spatial resolution, that is, the voxel size within which there is sufficient SNR within the resulting image. To achieve this, a technique known as compressed sensing is used which exploits undersampling of the acquired data relative to the standard fully sampled image. In MRI, data are acquired in so‐called k‐space and Fourier transformed to yield the real space image. The challenge, when undersampling, is to optimise the specific points in k‐space that are acquired because these will influence the quality of the resulting image. This work reports a straightforward, robust strategy for identifying the optimal sets of k‐space points to acquire. The method introduced uses simulated MRI images calculated from high‐resolution µCT images of the rocks of interest, from which optimised MRI sampling patterns are obtained. The method does not require any optimisation of parameters for its implementation, which is a significant advantage compared to other strategies. Moreover, we show that the pore space characteristics of the acquired MRI images are in excellent agreement with the same characteristics obtained from a high‐resolution µCT image. [ABSTRACT FROM AUTHOR]