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Wavelet multiresolution adaptive methods are known to provide efficient schemes for detecting and processing edges, and reducing noise in images. However, they introduce oscillations around edges. Methods based on diffusion equations have been used to enhance the edges in images and to reduce oscillations around the edges but are likely to introduce noise around the edges. In this paper, we provide a scheme that combines the edge detection properties of wavelets and the edge enhancing properties of diffusion equations there by reducing the noise and the oscillation around the edges. Our results indicate that the scheme using curvelets out performs the one with two-dimensional tensor product of Daubechies wavelets, and that using just the diffusion equations when these schemes are applied to a noisy image.