This paper presents novel unmixing and demosaicing methods for snapshot spectral imaging (SSI) systems utilizing Fabry-Perot filters. Unlike conventional approaches that perform unmixing after image restoration or demosaicing, our proposed methods leverage Fabry-Perot filter deconvolution and extend the "pure pixel" framework to the SSI sensor patch level, enabling improved unmixing accuracy and introducing the concept of localized spectral purity. Through extensive experimentation on synthetically generated data and real images captured by SSI cameras, we demonstrate the superiority of our methods over state-of-the-art techniques. Furthermore, our results showcase the effectiveness of the proposed approach over our recently proposed joint unmixing and demosaicing method based on low-rank matrix completion.