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In this letter, we propose a novel algorithm for reconstructing unreliable spectrographic data, a method applicable to missing feature-based automatic speech recognition (ASR). We provide quantitative analysis illustrating the high compressibility of spectrographic speech data. The existence of sparse representations for spectrographic data motivates the spectral reconstruction solution to be posed as an optimization problem minimizing the lscr1-norm. When applied to the Aurora-2 database, the proposed missing feature estimation algorithm is shown to provide significant improvements in recognition accuracy relative to the baseline MFCC system. Even without an oracle mask, performance approaches that of the ETSI advanced front end (AFE) , with less complexity.