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Spectral unmixing of geological mixtures, such as rocks, is a challenging inversion problem because of nonlinear interactions of light with the intimately mixed minerals at a microscopic scale. The fine-scale mixing of minerals in rocks limits the sensor’s ability to identify pure mineral endmembers and spectrally resolve these constituents within a given spatial resolution. In this study, we attempt to model the spectral unmixing of two rocks, namely, serpentinite and granite, by acquiring their hyperspectral images in a controlled environment, having uniform illumination, using a laboratory-based imaging spectroradiometer. The endmember spectra of each rock were identified by comparing a limited set of pure hyperspectral image pixels with the constituent minerals of the rocks based on their diagnostic spectral features. A series of spectral unmixing paradigms for explaining geological mixtures, including those ranging from simple physics-based light interaction models (linear, bilinear, and polynomial models) to classification-based models (support vector machines (SVMs) and half Siamese network (HSN)), were tested to estimate the fractional abundances of the endmembers at each pixel position of the image. The analysis of the results of the spectral unmixing algorithms using the ground truth abundance maps and actual mineralogical composition of the rock samples (estimated using X-ray diffraction (XRD) analysis) indicate a better performance of the pure pixel-guided HSN model in comparison to the linear, bilinear, polynomial, and SVM-based unmixing approaches. The HSN-based approach yielded reduced errors of abundance estimation, image reconstruction, and mineralogical composition for serpentinite and granite. With its ability to train using limited pure pixels, the half-Siamese network model has a scope for spectrally unmixing rock samples of varying mineralogical composition and grain sizes. Hence, HSN-based approaches effectively address the modelling of nonlinear mixing in geological mixtures.

<p>Geological materials are mixtures of different endmember constituents with most of them having particles smaller in size than the path length of incident light. The obtained spectral response (reflectance) from such mixtures is nonlinear which can be attributed to multiple scattering of light and the receiver sensor’s height from the incident surface. Assuming a sensor’s fixed instantaneous field of view (IFOV), variation in its field of view (FOV) by shifting its height affects the spatial resolution of acquired spectra. We propose to estimate the point spread function (PSF) for which the spectral responses of fine-resolution pixels acquired by a sensor are mixed to produce a coarse-resolution pixel obtained by the same. Our approach is based on the sensor’s unchanged IFOV obtaining spectral information from a smaller ground resolution cell (GRC) at a lower FOV and a larger GRC with an increased sensor’s FOV. The larger GRC producing a coarse resolution pixel can be modeled as a gaussian PSF of its corresponding center and neighboring fine-resolution subpixels with the center exerting the maximum influence. Extensive experiments performed using a point-based sensor and a push broom scanner revealed such variational effects in PSF that are dependent on the sensor’s FOV, the spatial interval of acquisition, and optical properties. The coarse-resolution pixels’ spectra were regressed with their corresponding fine-resolution subpixels to provide estimates of the PSF values that assumed the shape of a two-dimensional Gaussian function. Constraining these values as sum-to-one introduced sparsity and explained variability in the spectral acquisition by different sensors.  The estimated PSFs were further validated through the linear spectral unmixing technique. It was observed that the fractional abundances obtained for the fine-resolution subpixels convolved with our estimated PSF to produce its corresponding coarse-resolution counterpart with minimal error. The obtained PSFs using different sensors also explained spectral mixing at different scales of observation and provided a basis for nonlinear unmixing integrating spatial as well as spectral effects and addressing endmember variability. We performed our experiments with various coarse-grained and fine-grained igneous and sedimentary rocks under laboratory conditions to validate our results which were compared with available literature. </p>