We present a ternary quartic based approach with an improved gradient based optimization scheme for diffusion kurtosis imaging to estimate constrained and physically realistic diffusion and kurtosis tensors. We account for the signal noise by considering a maximum likelihood estimation based on the Rician noise model. Diffusion kurtosis imaging (DKI) is a recent important improvement over the diffusion tensor imaging (DTI) model that quantifies the degree of non-Gaussian diffusion in a tissue using a 3D 4th order tensor. However, DKI estimation needs to consider three constraints to be physically relevant. By adopting the implicit ternary quartic parameterization which allows to elegantly impose a positivity constraint on the kurtosis tensor and by employing gradient based optimization schemes, we show dramatically improved performance in terms of estimation time and quality. We derive the mathematical framework and show results on extensive synthetic data experiments.