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A quantitative measure of relevance is proposed for the task of constructing visual feature sets which are at the same time relevant and compact. A feature's relevance is given by the amount of information that it contains about the problem, while compactness is achieved by preventing the replication of information between features in the set. To achieve these goals, we use mutual information both for assessing relevance and measuring the redundancy between features. Our application is speechreading, that is, speech recognition performed on the video of the speaker. This is justified by the fact that the performance of audio speech recognition can be improved by augmenting the audio features with visual ones, especially when there is noise in the audio channel. We report significant improvements compared to the most commonly used method of dimensionality reduction for speechreading, linear discriminant analysis.