Changes of pressure and temperature in the atmosphere cause changes in its density and refractive index (Goodman 1985). These, in turn, if they are inhomogeneous, result in the distortion of the wavefront passing through the atmosphere from any object point. Such changes are always present in the natural atmosphere. They vary randomly and are called turbulence. As a consequence of turbulence, an image formed through the atmosphere will, in general exhibit scintillation, motion, and blurring effects. They pose an ultimate limit on the image quality attainable through the atmosphere. In astronomical observation this effect is called seeing and represents the major limitation on the resolution and geometrical accuracy. The effect of atmospheric turbulence is a spatial filtering of the image, which is smeared by the loss of spatial coherence over the pupil plane (Fried 1966). A full interpretation of images is then possible if the system response (point spread function or PSF) is known. Iterative deconvolution techniques have been developed (maximum entropy, clean and Lucy, Hanish & White 1993) which aim to estimate the ideal image that would be observed if no image degradation were present. From this point of view, full restoration of the spatial details acquired by means of a push-broom imaging spectrometer is not possible, or at least, very difficult. The reason is due to the fact that the lines forming an image are acquired at different times. The system response along the slit can be known by taking the image line of a point source. However, due to the optical aberrations, it is a function of position along the slit. The across slit response is unknown, due to the random nature of the atmospheric turbulence. The method we are presenting here is an image sharpening procedure which improves the spatial content of an image by utilizing spectral correlations among adjacent pixels. From the previous discussion we have seen how the atmosphere causes the blurring of an image: photons coming from the instantaneous field of view pertaining a certain pixel, are spread over a larger number of picture elements. A planetary surface is never composed of a single uniform material, and therefore the spatial blurring also causes mixing of mineralogically different regions of the surface. This situation is illustrated in Fig. 1, where two spectrally distinct units are indicated by letters A and B (Fig. 1a). At the A-B border, blurring causes mixing of A and B spectra and generates a third zone (C) having a spectrum intermediate between the A and B spectra (Fig. 1b). Image sharpening can then be accomplished by means of spectral unmixing procedures. Note that spectral unmixing is particularly effective with imaging spectroscopy data, because the spectrum of each pixel is defined with high accuracy, due to the great number of bands. In planetary multiband imaging (generally less than 10 spectral bands), the pixel spectrum is less detailed and spectral unmixing is then less effective. If the scale of mixing is large (as in planetary remote sensing), mixing generally occurs in a linear fashion while for microscopic or intimate mixtures, the mixing is generally not linear (Singer & McCord 1979). Spectral unmixing is usually done by using known endmembers, seeking to derive the apparent fractional abundance of each endmember material in each pixel, given a set of known or assumed spectral endmembers (Adams et al. 1986). These known endmembers can be drawn from the data (averages of regions picked using previous knowledge), drawn from a library of pure materials by interactively browsing through the imaging spectrometer data to determine what pure materials exist in the image, or determined using expert systems to identify materials. The mixing endmembers matrix is made up of spectra from the image or a reference library. The problem can be cast in terms of a linear least squares problem. The mixing matrix is inverted and multiplied by the observed spectra to get least-squares estimates of the unknown endmember abundance fractions. Constrains can be placed on the solutions to give positive fractions that sum to unity. Shade and shadow are included either implicitly (fraction sum to 1 or less) or explicitly as an endmember (fractions sum to 1). A possible drawback of this method is that the image endmembers may themselves represent mixtures rather than true compositional endmembers. For example, on Moon this happens at proximity of craters where the materials resulting from an impact event are mixed with preexisting soils. A simple measure of pixel spectral homogeneity is to reduce the least squares error to a significant level through a trial-and-error method, by continually refine the endmember selection. However, this procedure can be very time consuming, specially when the spectra to process are composed of many bands. For this reason, we adopted a solution based on the use of another linear technique, the Principal Component Analysis (henceforth PCA). This technique is very sensitive to the spectral content of the data, and can then be used for spectral unmixing.
|Figure 1: Sketch illustrating the sharpening method. a) Two distinct spectral units without spatial blurring. b) The effect of blurring is to produce a third spectral unit (C) and to blur the edges|
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