Among all enhancement techniques being developed over the past two decades, anisotropic diffusion has received a lot of attention and has experienced significant developments, with promising results and applications in several specific domains. The elegant property of the technique is that it can enhance images by reducing undesirable intensity variability within the objects in image, while improving signal-to-noise ratio (SNR) and enhancing the contrast of the edges in scalar and, more recently, in vector-valued images, such as color, multispectral and hyperspectral imagery. In this paper, we firstly analyze two complementary schemes-variational methods and nonlinear diffusion partial differential equations (PDEs), in terms of edge enhancement. Based on these analyses, a general flexible class of hyperspectral forward-and-backward (FAB) diffusion process will be proposed, which can achieve the main requirements for edgepreserving regularization with image enhancement. In addition, we use additive operator splitting (AOS) scheme to speedup the numerical evolution of the nonlinear diffusion equation with respect to traditional explicit schemes. The performance of the vector-valued FAB diffusion PDE is studied using some hyperspectral remote sensing images. Experimental results on these images are shown the validity and effectiveness of the proposed method.