Symmetrical non-separable wavelet transforms in remote sensing image fusion

The research presented in this work is concerned with the problem of image fusion by Symmetric Non-Separable wavelet transforms. The main aim was to give a solid foundation for the application of certain types of Symmetrical Non-Separable wavelets considering their mathematical properties and to implement the Fusion of Remote Sensing Images using these wavelets. Two classes of Symmetric Non-Separable wavelets are considered in this thesis: spherically symmetric wavelets and wavelets with square symmetric spectrum. The first type of wavelet was known from the early 1990s while the wavelets with square symmetric spectrum are outlined and studied in this research for the first time. The study of symmetrical non-separable wavelets started with the investigation of corresponding Continuous Wavelet Transforms in the context of different mathematical models of images. Spherically symmetric wavelets were studied in and distributional models and wavelets with square symmetric spectrum were studied in and general models. For each case the conditions guaranteeing pointwise convergence and convergence in mean were formulated and proved as previously unknown theorems using the revealed relations between non-separable wavelet and Fourier integrals. These theorems constituted the theoretical framework for their subsequent application in the fusion of remotely sensed images. Obtained theorems guarantee that the usage of symmetrical wavelets from considered classes is correct and valid. Such theoretical background is not only crucial for further steps in current research but also considerably enhances the methodology of investigation of wavelet based image fusion for future researches. Indeed with proven theorems one can study not only single non-separable wavelets as it was done before but the whole classes of wavelets and obtain general results. Numerical implementation of the fusion of Remote Sensing Images was carried out in MATLAB environment. Nine remote sensing images of different sceneries were chosen as source data for the numerical experiments and spherical Haar and square Shanon wavelets were selected as representatives of non-separable wavelets. The source data were degraded spatially using blurring filter and spectrally using ‘RGB to monochrome’ conversion function. Then the degraded images were fused using non-separable wavelets (spherical Haar and square Shanon) and separable wavelets (Daubechies 2, classical Haar and Symlet). For the fusion with non-separable wavelets we performed numerical calculations of integral transforms while for separable wavelets standard MATLAB functions were applied. The results of fusion based separable and non-separable wavelets were compared using five different quality measures: RMSE, RASE, ADV, ERGAS and SAM. Four metrics representing spatial precision accuracy (RMSE,RASE, ADV and ERGAS) showed that the best fusion outputs were produced by the application of non-separable spherical Haar wavelet and SAM quality measure attributing to spectral precision displayed the supremacy of non-separabe square Shanon wavelet over the others. Thus in all cases non-separable wavelets performed better than their separable counterparts. Finally it was found out that the fusion based spherically symmetric non-separable wavelets showed better spatial precision in the processing of imageries of spherically symmetric objects and patterns rather than the approaches based on separable wavelets and wavelets with square symmetry. On the other hand the best spectrum accuracy was shown by image fusion based wavelets with square symmetric spectrum in the processing of heterogeneous images with sharp edges such as the images of urban areas. Hereby obtained results not only supported known heuristic suggestion that image fusion based non-separable wavelets perform better accuracy but also revealed the correlation between the symmetry of used non-separable wavelets and the structure of processed images.

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