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Diffusion MRI data are generally acquired using hyperpolarized gases during patient breath-hold, which yields a compromise between achievable image resolution, lung coverage, and number of b-values. In this paper, we propose a novel method that accelerates the acquisition of diffusion MRI data by undersampling in both the spatial and b-value dimensions and incorporating knowledge about signal decay into the reconstruction (SIDER). SIDER is compared with total variation (TV) reconstruction by assessing its effect on both the recovery of ventilation images and the estimated mean alveolar dimensions (MADs). Both methods are assessed by retrospectively undersampling diffusion data sets (n=8) of healthy volunteers and patients with Chronic Obstructive Pulmonary Disease (COPD) for acceleration factors between x2 and x10. TV led to large errors and artifacts for acceleration factors equal to or larger than x5. SIDER improved TV, with a lower solution error and MAD histograms closer to those obtained from fully sampled data for acceleration factors up to x10. SIDER preserved image quality at all acceleration factors, although images were slightly smoothed and some details were lost at x10. In conclusion, we developed and validated a novel compressed sensing method for lung MRI imaging and achieved high acceleration factors, which can be used to increase the amount of data acquired during breath-hold. This methodology is expected to improve the accuracy of estimated lung microstructure dimensions and provide more options in the study of lung diseases with MRI.