Texture analysis is an image processing task that can be conducted using the mathematical framework of multifractal analysis to study the regularity fluctuations of image intensity and the practical tools for their assessment, such as (wavelet) leaders. A recently introduced statistical model for leaders enables the Bayesian estimation of multifractal parameters. It significantly improves performance over standard (linear regression based) estimation. However, the computational cost induced by the associated nonstandard posterior distributions limits its application. The present work proposes an alternative Bayesian model for multifractal analysis that leads to more efficient algorithms. It relies on three original contributions: A novel generative model for the Fourier coefficients of log-leaders; an appropriate reparametrization for handling its inherent constraints; a data-augmented Bayesian model yielding standard conditional posterior distributions that can be sampled exactly. Numerical simulations using synthetic multifractal images demonstrate the excellent performance of the proposed algorithm, both in terms of estimation quality and computational cost.