In diffusion MRI, the reconstructed Ensemble Average Propagator (EAP) from the diffusion signal provides detailed insights on the diffusion process and the underlying tissue microstructure. Recently, the Simple Harmonic Oscillator based Reconstruction and Estimation (SHORE) basis was proposed as a promising method to reconstruct the EAP. However, the fitting of the basis is sensitive to noise. To solve this we propose to use the Laplacian of the SHORE basis as a natural regularization functional. We provide the derivation of the Laplacian functional and compare its effect on EAP reconstruction with that of separated regularization of the radial and angular parts of the SHORE basis. To find optimal regularization weighting we use generalized cross-validation and validate our method quantitatively on synthetic and qualitatively on human data from the Human Connectome Project. We show that Laplacian regularization provides more accurate estimation of the signal and EAP based microstructural measures. Abstract In diffusion MRI, the reconstructed Ensemble Average Propagator (EAP) from the diffusion signal provides detailed insights on the diffusion process and the underlying tissue microstructure. Recently, the Simple Harmonic Oscillator based Reconstruction and Estimation (SHORE) basis was proposed as a promising method 8 to reconstruct the EAP. However, the fitting of the basis is sensitive to noise. To solve 9 this we propose to use the Laplacian of the SHORE basis as a natural regularization functional. We provide the derivation of the Laplacian functional and compare its effect on EAP reconstruction with that of separated regularization of the radial and angular parts of the SHORE basis. To find optimal regularization weighting we use generalized cross-validation and validate our method quantitatively on synthetic and qualitatively on human data from the Human Connectome Project. We show that Laplacian regularization provides more accurate estimation of the signal and EAP 16 based microstructural measures.