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We revisit Gaussian blurring mean-shift (GBMS), a procedure that iteratively sharpens a dataset by moving each data point according to the Gaussian mean-shift algorithm (GMS). (1) We give a criterion to stop the procedure as soon as clustering structure has arisen and show that this reliably produces image segmentations as good as those of GMS but much faster. (2) We prove that GBMS has convergence of cubic order with Gaussian clusters (much faster than GMS's, which is of linear order) and that the local principal component converges last, which explains the powerful clustering and denoising properties of GBMS. (3) We show a connection with spectral clustering that suggests GBMS is much faster. (4) We further accelerate GBMS by interleaving connected-components and blurring steps, achieving 2x--4x speedups without introducing an approximation error. In summary, our accelerated GBMS is a simple, fast, nonparametric algorithm that achieves segmentations of state-of-the-art quality.