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Hyperspectral unmixing, which decomposes pixel spectra into a collection of constituent spectra, is a preprocessing step for hyperspectral applications like target detection and classification. It can be considered as a blind source separation (BSS) problem. Independent component analysis, which is a widely used method for performing BSS, models a mixed pixel as a linear mixture of its constituent spectra weighted by the correspondent abundance fractions (sources). The sources are assumed to be independent and stationary. However, in many instances, this assumption is not valid. In this paper, a complexity-based BSS algorithm is introduced, which studies the complexity of sources instead of the independence. We extend the 1-D temporal complexity, which is called complexity pursuit that was proposed by Stone, to the 2-D spatial complexity, which is named spatial complexity BSS (SCBSS), to describe the spatial autocorrelation of each abundance fraction. Further, the temporal complexity of spectrum is combined into SCBSS to account for the spectral smoothness, which is termed spectral and spatial complexity BSS. More importantly, a strict theoretic interpretation is given, showing that the complexity-based BSS is very suitable for hyperspectral unmixing. Experimental results on synthetic and real hyperspectral data demonstrate the advantages of the proposed two algorithms with respect to other methods.