**2. Materials and Methods**

The SPOT 6/7 multispectral and panchromatic dataset over Pretoria, South Africa was used for the study. SPOT 6 and SPOT 7 are identical sun-synchronous optical satellites launched on 12 September 2012 and 30 June 2014, respectively that co-orbit in the constellation at an altitude of 694 km and are phased at 180 degrees (Airbus, Toulouse, France, 2018). The spectral configuration of the satellites consists of blue (450–520 nm), green (530–590 nm), red (625–695 nm) and near-infrared (760–890 nm) multispectral bands with a spatial resolution of 6 m and a panchromatic (450–745 nm) band with a spatial resolution of 1.5 m and dynamic range of 12 bits per pixel. SPOT 6/7 are capable of contiguous image segments of more than 120 km × 120 km or 60 km × 180 km from a single pass along one orbit.

To meet the operational needs of generating a national wall-to-wall mosaic of South Africa, we selected established pansharpening methods for quantitative quality assessment. The Bayesian (BAY), Brovey transform (BRO), color normalized spectral (CNS) sharpening, Ehlers fusion technique (EHLERS), Gram–Schmidt (GRS), local mean and variance matching (LMVM), modified intensity hue saturation (MIHS), Pansharp algorithm (PANSHARP), principal component analysis (PCA), ratio component substitution (RCS), and wavelet resolution merge (WAVELET) techniques were evaluated in the study.

The PANSHARP algorithm available in the PCI Geomatica software is a statistics-based fusion technique aimed at maximizing spatial detail while minimizing color distortions [26]. It attempts to preserve the spectral characteristics of the data. Developed by Zhang [27], the algorithm uses the least-squares method to approximate the grey value relationship between the original multispectral, panchromatic, and fused images to achieve the best color representation. The modified intensity hue saturation (MIHS) fusion technique merges high-resolution panchromatic data with lower resolution multispectral data to produce a pansharpened image that retains sharp spatial detail and a realistic resemblance of the original multispectral scene colors. This approach assesses the spectral overlap between each multispectral band and the high-resolution panchromatic band and weighs the merge based on these relative wavelengths. The MIHS method was developed to address a shortcoming of the intensity-hue-saturation (IHS) transformation where color distortions occurred due to discrepancies in spectral characteristics between panchromatic and multispectral bands. The IHS fusion transforms the RGB (red, green, and blue) space into the IHS color space and subsequently replaces the intensity band with a high-resolution pan image in the fusion before performing a reverse IHS transformation. The Ehlers (EHLERS) fusion technique uses an IHS transform coupled with Fourier domain filtering and aims to maintain the spectral characteristics of the fused image [22]. This is achieved by using the high-resolution panchromatic image to sharpen the multispectral image while avoiding adding new grey level information to its spectral components by first separating the color and spatial information. The spatial information content is then embedded as an adaptive enhancement to the images using a combination of color and Fourier transforms [22]. The Brovey transform (BRO) algorithm applies a ratio algorithm to combine the images. This is done by first multiplying each multispectral band by a high-resolution pan band and subsequently dividing each product by the sum of the multispectral bands. It is known to preserve the relative spectral contributions of each pixel but substitutes scene brightness with the high-resolution panchromatic (PAN) image [28]. The principal component analysis (PCA) transform converts intercorrelated Multispectral (MS) bands into a new set of uncorrelated components. The first component that resembles a high-frequency band is replaced by a high-resolution panchromatic band for the fusion. The panchromatic band is fused into low-resolution multispectral channels by performing a reverse PCA transform. A high-resolution fused image is generated after the reverse PCA transformation [29]. The color normalized spectral sharpening (CNS) algorithm implemented in Environment for Visualizing Images (ENVI) software is employed to simultaneously sharpen any defined number of bands and retain the characteristics of the original bands in terms of data type and dynamic range. In this case, the higher resolution bands are used to sharpen the lower resolution bands and in the ENVI implementation, the lower resolution multispectral bands are expected to fall in the same spectral range with the high-resolution panchromatic channel [30,31]. The multispectral bands are clustered into spectral segments defined by the spectral range of the high-resolution panchromatic sharpening band. The pansharpened image is generated by multiplying the high-resolution panchromatic with each lower resolution multispectral band before normalizing the computation by dividing the sum of the input spectral channels in each segment.

The wavelet resolution merge (WAVELET) fusion approach sharpens low-resolution multispectral bands using a matching high-resolution panchromatic band by first decomposing the high-resolution panchromatic band into a set of low-resolution multispectral bands with corresponding wavelet coefficients (spatial details) for each level. This is done by infusing the high-resolution spatial into each of the multispectral bands by performing a reverse wavelet transform on each MS band together with the corresponding wavelet coefficients. In a sense, wavelet-based processing is akin to Fourier transform analysis, except fast Fourier transform analysis uses long continuous (sine and cosine) waves, whereas wavelet transform analysis applies short and discrete wavelets [32–35]. The Gram–Schmidt (GRS) pansharpening algorithm available in the ENVI fuses the high-resolution panchromatic band to the lower resolution multispectral bands by simulating the panchromatic band from the multispectral band by averaging the multispectral bands. A Gram–Schmidt transformation is computed from the simulated panchromatic band and the multispectral band, whereby the simulated panchromatic band is used as the first band. Further, the high spatial resolution panchromatic band is substituted with the first Gram–Schmidt band before applying an inverse Gram–Schmidt transformation to generate the pansharpened multispectral bands [36,37]. The ratio component substitution (RCS) pansharpening algorithm implemented in Orfeo ToolBox [38] fuses orthorectified panchromatic (PAN) and multispectral (XS) images using a low pass sharpening filter as shown in the computation below (OTB, 2019).

$$\frac{\text{\\$S}}{\text{Filtered (PAN)}} \text{PAN E} \tag{1}$$

where E is a vector of random errors that is considered to be stochastically independent of Z.

The Bayesian fusion (BAY) applies elementary calculus in the fusion of the panchromatic and multispectral images to generate a pansharpened image [38]. This fusion approach uses the statistical relationships amongst the spectral bands and the panchromatic band. Bayesian pansharpening techniques use three images that include a panchromatic band and a multispectral image resampled to the same spatial resolution as the panchromatic band. The panchromatic band is weighted in comparison to the multispectral bands. A thorough mathematical description of the Bayesian pansharpening algorithm implemented in Orfeo ToolBox is provided by [39]. This pansharpening technique is dependent on the notion that the variables of interest, expressed as vector Z, are not directly observable and related to observable variable Y through an error-like equation.

$$\mathbf{Y} = \mathbf{g}(\mathbf{Z}) + \mathbf{E} \tag{2}$$

where g(Z) is considered a set of functionals.

The LMVM pansharpening algorithm implemented in OTB software uses an LMVM filter that applies a normalization function at a local scale within the images to equate the local mean and variance values of the high spatial resolution panchromatic band with those of the lower resolution multispectral image [38,40]. The resulting small residual differences are then considered to arise from the high-resolution panchromatic band [40]. Rubiey [40] further notes that this form of filtering improves the correlation between the pansharpened image and the original multispectral image. The LMVM algorithm is highlighted below.

$$\mathbf{F}\_{\mathbf{i},\mathbf{j}} = \frac{\left(\mathbf{H}\_{\mathbf{i},\mathbf{j}} - \overline{\mathbf{H}}\_{\mathbf{i},\mathbf{j}}\right) \cdot \mathbf{s}\left(\mathbf{L}\right)\_{\mathbf{i},\mathbf{j}\left(\mathbf{w},\mathbf{h}\right)}}{\mathbf{s}\left(\mathbf{H}\right)\_{\mathbf{i},\mathbf{j}\left(\mathbf{w},\mathbf{h}\right)}} \to \tag{3}$$

where Fi,j refers to the fused image, Hi,j and Li,j denote high and low spatial resolution images respectively at pixel coordinates i,j. (H)i,j(w,h) and (L)i,j(w,h) are local means calculated inside the window of size (w, h). s denotes the local standard deviation.
