Purpose: Diffusion kurtosis imaging (DKI) is a recent improvement over diffusion tensor imaging that characterizes tissue by quantifying non-gaussian diffusion using a 3D fourth-order kurtosis tensor. DKI needs to consider three constraints to be physically relevant. Further, it can be improved by considering the Rician signal noise model. A DKI estimation method is proposed that considers all three constraints correctly, accounts for the signal noise and incorporates efficient gradient-based optimization to improve over existing methods. Methods: The ternary quartic parameterization is utilized to elegantly impose the positivity of the kurtosis tensor implicitly. Sequential quadratic programming with analytical gradients is employed to solve nonlinear constrained optimization efficiently. Finally, a maximum likelihood estimator based on Rician distribution is considered to account for signal noise. Results: Extensive experiments conducted on synthetic data verify a MATLAB implementation by showing dramatically improved performance in terms of estimation time and quality. Experiments on in vivo cerebral data confirm that in practice the proposed method can obtain improved results. Conclusion: The proposed ternary quartic-based approach with a gradient-based optimization scheme and maximum likelihood estimator for constrained DKI estimation improves considerably on existing DKI methods.