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Two new extensions of latent Dirichlet allocation (LDA), denoted topic-supervised LDA (ts-LDA) and class-specific-simplex LDA (css-LDA), are proposed for image classification. An analysis of the supervised LDA models currently used for this task shows that the impact of class information on the topics discovered by these models is very weak in general. This implies that the discovered topics are driven by general image regularities, rather than the semantic regularities of interest for classification. To address this, ts--LDA models are introduced which replace the automated topic discovery of LDA with specified topics, identical to the classes of interest for classification. While this results in improvements in classification accuracy over existing LDA models, it compromises the ability of LDA to discover unanticipated structure of interest. This limitation is addressed by the introduction of css-LDA, an LDA model with class supervision at the level of image features. In css-LDA topics are discovered per class, i.e., a single set of topics shared across classes is replaced by multiple class-specific topic sets. The css-LDA model is shown to combine the labeling strength of topic-supervision with the flexibility of topic-discovery. Its effectiveness is demonstrated through an extensive experimental evaluation, involving multiple benchmark datasets, where it is shown to outperform existing LDA-based image classification approaches.