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Spectral imaging enables spatially-resolved identification of materials in remote sensing, biomedicine, and astronomy. However, acquisition times require balancing spectral and spatial resolution with signal-to-noise. Hyperspectral imaging provides superior material specificity, while multispectral images are faster to collect at greater fidelity. We propose an approach for fusing hyperspectral and multispectral images to provide high-quality hyperspectral output. The proposed optimization leverages the least absolute shrinkage and selection operator (LASSO) to perform variable selection and regularization. Computational time is reduced by applying the alternating direction method of multipliers (ADMM), as well as initializing the fusion image by estimating it using maximum a posteriori (MAP) based on Hardie's method. We demonstrate that the proposed sparse fusion and reconstruction provides quantitatively superior results when compared to existing methods on publicly available images. Finally, we show how the proposed method can be practically applied in biomedical infrared spectroscopic microscopy.