Texture analysis is central in many image processing problems. It can be conducted by studying the local regularity fluctuations of image amplitudes, and multifractal analysis provides a theoretical and practical framework for such a characterization. Yet, due to the non Gaussian nature and intricate dependence structure of multifractal models, accurate parameter estimation is challenging: standard estimators yield modest performance, and alternative (semi-)parametric estimators exhibit prohibitive computational cost for large images. This present contribution addresses these difficulties and proposes a Bayesian procedure for the estimation of the multifractality parameter c2 for images. It relies on a recently proposed semi-parametric model for the multivariate statistics of log-wavelet leaders and on a Whittle approximation that enables its numerical evaluation. The key result is a closed-form expression for the Whittle likelihood. Numerical simulations indicate the excellent performance of the method, significantly improving estimation performance over standard estimators and computational efficiency over previously proposed Bayesian estimators.