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In this paper, we address the problem of fully automated decomposition of hyperspectral images for transmission light microscopy. The hyperspectral images are decomposed into spectrally homogeneous compounds. The resulting compounds are described by their spectral characteristics and optical density. We present the multiplicative physical model of image formation in transmission light microscopy, justify reduction of a hyperspectral image decomposition problem to a blind source separation problem, and provide method for hyperspectral restoration of separated compounds. In our approach, dimensionality reduction using principal component analysis (PCA) is followed by a blind source separation (BSS) algorithm. The BSS method is based on sparsifying transformation of observed images and relative Newton optimization procedure. The presented method was verified on hyperspectral images of biological tissues. The method was compared to the existing approach based on nonnegative matrix factorization. Experiments showed that the presented method is faster and better separates the biological compounds from imaging artifacts. The results obtained in this work may be used for improving automatic microscope hardware calibration and computer-aided diagnostics.