Atmospheric corrections for pansharpening

Among remote sensing image fusion applications, panchromatic (Pan) sharpening, or pansharpening, of a multispectral (MS) image has received considerable attention over the last quarter of century. Pansharpening techniques take advantage of the complementary characteristics of spatial and spectral resolutions of MS and Pan data, in order to synthesize a unique product that exhibits as many spectral bands as the original MS image, each with same spatial resolution as the Pan image. After the MS bands have been interpolated and co-registered to the Pan image, spatial details are extracted from Pan and added to the MS bands according to a certain injection model. The detail extraction step may follow the spectral approach, originally known as component substitution (CS) or the spatial approach, which may rely on multiresolution analysis (MRA). The Pan image is preliminarily histogram-matched, that is, radiometrically transformed by constant gain and offset in such a way that its low-pass version (PanL) exhibits mean and variance same as the component that shall be replaced, for CS methods, or the MS band that shall be sharpened for MRA methods. In some cases, histogram matching is implicitly performed by the injection model. The latter rules the combination of the low-pass MS image with the spatial detail extracted from Pan. Such a model is stated between each of the re-sampled MS bands and a low-pass version of the Pan image having same spatial frequency content as the MS bands. Although several models have been introduced, the most popular are: i) the projection model, which may be derived from the Gram-Schmidt (GS) orthogonalization procedure, representing the basis of the GS spectral sharpening and of the context-based decision (CBD); ii) the multiplicative or modulation model, which is the basis of such techniques as high-pass modulation (HPM), Brovey transform (BT), synthetic variable ratio (SVR), Zhang's UNB pansharp, smoothing filter-based intensity modulation (SFIM) and additive wavelet luminance proportional (AWLP). Unlike the projection model, which may be either global, as for GS, or local, as for CBD, the multiplicative model is inherently local, because the injection gain changes at each pixel. The pioneering paper that introduced SFIM firstly gave an interpretation of the multiplicative injection model in terms of the radiative transfer model ruling the acquisition of an MS image from a real-world scene. Accordingly, the kth MS band interpolated at the pixel scale of Pan represents a low spatial resolution spectral radiance, that is, a radiance with a spectral diversity. Pan represents the high spatial resolution solar irradiance, which bears no spectral information, but only spatial and radiometric information. Thus, its lowpass-filtered version, PanL, having the same spatial frequency content as the interpolated MS bands, is a low-resolution irradiance and conveys radiometric information at the spatial resolution of MS. Ultimately, a high resolution MS (HRMS) is synthesized at each pixel as low resolution MS (LRMS) divided by PanL and multiplied by Pan. Since the ratio of spectral radiance to solar irradiance is an estimate of spectral reflectance, when the ratio between LRMS and PanL is multiplied by Pan, a low-resolution reflectance is multiplied by a high resolution irradiance. So far, very few authors have ever considered the path radiance of the kth band, which is an energy scattered by the atmosphere that reaches the aperture of the instrument without being reflected by the Earth's surface. Thus, the path radiance of each band, which appears as a haze in a color composite display of the MS bands, should be estimated and removed from the band before the modulation is accomplished and re-inserted after the sharpening, to restore unbiased (spectral) radiance values in the sharpened image. In this paper, several methods for estimating the path radiance compared in terms of benefits on the quality of fusion. The first method is empirical: if the scene is large enough, the path radiance of the blue (B) band roughly corresponds to the minimum value of (spectral) reflectance over the image. Furthermore, if the presence of water in the scene is detected, also the path radiance of the near infra-red (NIR) band can be approximated by its minimum. The path radiances of green (G) and red (R) bands are generally not related to the corresponding minima and are assumed to be zero. The second method is statistical and allows the path radiances of G and R to be estimated through bivariate linear regressions: in G-to-(B-Bmin) the intercept on the G axis yields the path radiance of G; in R-to-(B-Bmin) the intercept on the R axis yields its path radiance. Unfortunately, the NIR band is little correlated with any of the visible bands; hence, the regression does not provide an acceptable value for the path radiance of NIR. Finally, a third method is based on modelling the atmosphere according to the model proposed by Yu Gu of UCLA, which requires acquisition year, month, day, hour (UTM), longitude, latitude, and possibly type of landscape (urban or rural) for setting aerosols. Such a model directly yields values of path radiance in predefined bands, roughly corresponding to those of MS scanners, like Landsat 7 TM and Landsat 8 OLI. Experiments on IKONOS and GeoEye images are carried out at spatial scales degraded by fours, in order to allow objective quality assessments (SAM, ERGAS, Q4) to be accomplished. Since the values of path radiance are assumed to be constant over the scene, the results found at degraded scale are consistent with those would be found at full scale. The advantages of removing the estimated path radiances are more consistent in terms of spectral quality (colors) and are especially significant on vegetated areas. A combination of empirical and statistical estimation attains the performances of the model, whose uncertainty in the composition of atmosphere and slight mismatch of instrument bands and modelled bands constitute a source of errors. Also a greedy search, unfeasible in practical cases, confirms that the estimates of path radiance are correct and that haze removal is always beneficial for multiplicative pansharpening methods, especially in terms of spectral angle error.