In this paper, we propose a new endmember estimation method for snapshot spectral imaging (SSI) systems using Fabry-Perot filters. Indeed, such systems only provide a part of the spectral content of a classical multi- or hyperspectral camera and restoring the full hyperspectral datacube from an SSI matrix is named "demosaicing". However, it was recently shown that a joint unmixing and demosaicing method allowed a much better unmixing performance than a two-stage approach consisting of a demosaicing step followed by an unmixing one. In this paper, we propose a new approach to estimate endmembers from the SSI image without requiring a demosaicing step. It inverts the Fabry-Perot filters and extends the "pure pixel" framework to the SSI sensor patch level, that we name the ``pure patch'' assumption. Our experiments show that our proposed scheme significantly outperforms state-of-the-art methods in terms of endmember estimation accuracy.