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Diffusion kurtosis imaging (DKI) is a recent approach in medical engineering that has potential value for both neurological diseases and basic neuroscience research. In this letter, we develop a robust method based on adaptive spherical integral that can compute kurtosis based quantities more precisely and efficiently. Our method integrates spherical trigonometry with a recursive computational scheme to make numerical estimations in kurtosis imaging convergent. Our algorithm improves the efficiency of computing integral invariants based on reconstructed diffusion kurtosis tensors and makes DKI better prepared for further clinical applications.